Real Algebraic Geometry: Proceedings of the Conference held in Rennes, France, June 24-28, 1991 (Lecture Notes in Mathematics) (English and French Edition)

Lecture Notes in Mathematics Editors: A. Dold, Heidelberg B. Eckmann, Ztirich E Takens, Groningen

1524

M. Coste

L. Mah6

M.-F. Roy (Eds.)

Real Algebraic Geometry Proceedings of the Conference held in Rennes, France, June 24-28, 1991

Springer-Verlag Berlin Heidelberg NewYork London Paris Tokyo Hong Kong Barcelona Budapest

Editors Michel Coste Louis Mah6 Marie-Fran~oise Roy IRMAR (CNRS, U.R.A. 305) Universit6 de Rennes I Campus Beaulieu F-35402 Rennes Cedex, France

Mathematics Subject Classification (1991 ): Primary: 14P Secondary: 12D15, 32B, 32C, 58A35, 68Q40

ISBN 3-540-55992-2 Springer-Verlag Berlin Heidelberg New York ISBN 0-387-55992-2 Springer-Verlag New York Berlin Heidelberg This work is subject to copyright. All rights are reserved, whether the whole or part of the material is concerned, specifically the rights of translation, reprinting, re-use of illustrations, recitation, broadcasting, reproduction on microfilms or in any other way, and storage in data banks. Duplication of this publication or parts thereof is permitted only under the provisions of the German Copyright Law of September 9, 1965, in its current version, and permission for use must always be obtained from Springer-Verlag. Violations are liable for prosecution under the German Copyright Law. © Springer-Verlag Berlin Heidelberg 1992 Printed in Germany Typesetting: Camera-ready by author/editor 46/3140-543210 - Printed on acid-free paper

PREFACE

The meeting on "Real Algebraic Geometry" was held in La Turballe, on the seashore not far from Rennes, from June 24 to 28, 1991. It took place ten years after the first meeting on "G6omttrie Algtbrique Rtelle et Formes Quadratiques" (*). These Proceedings contain survey papers on some of the developements of real algebraic geometry in the last ten years, and also contributions by the participants. Every paper has been submitted to a referee, and we want to thank all of them for their collaboration. The meeting, and the collaboration between the european teams which made it possible, received support from the Universit6 de Rennes 1, the GDR MathtmatiquesInformatique (CNRS), and the programs Rtseau Europ6en de Laboratoires, Acces, Alliance, Actions Inttgrtes Franco-Espagnoles. We would like also to thank SpringerVerlag for publishing this volume, and to express our gratitude to Ms. Yvette Brunel, for her precious help for the secretary of the meeting. Michel Coste Louis Mah6 Marie-Fran~oise Roy

(*) Lecture Notes in Mathematics 959, Springer (1982)

TABLE OF CONTENTS

Survey papers Knebusch M.: Semialgebraic topology in the last ten years

. . . . . . .

Parimala R.: Algebraic and topological invariants of real adgebraic varieties Polotovskii G.M.: On the classification of decomposing" plaae algebradc curves

1

37 52

Schcidcrcr C.: Read adgebra and its applications to geometry in the last ten years: some m a j o r developments and results

. . . . . . . . . .

Shustin E.I.: Topology of real plane algebraic curves

75

. . . . . . . .

97

Silhol R.: Moduli problems in real dgebraic geometry . . . . . . . .

110

Other papers Akbulut S., King H.: Constructing strange read Mgebraic sets

. . . . .

120

Andradas C., Ruiz J.: More on basic semialgebraic sets . . . . . . . .

128

Borobia, A.: Mirror property for nonsingular m i x e d configurations o f one line and k points in R 3 . . . . . . . . . . . . . . . . . . . .

Br6cker L.: Families of semialgebraic sets and limits

. . . . . . . .

140

145

Brumfiel G.W.: A H o p f flxed point theorem for semi-algebraic m a p s .

163

Brumfiel G.W.: On regular open semi-algebraic sets . . . . . . . . .

170

Castilla, A.: S u m s of 2n - th-powers o f meromorphic functions with compact zero set . . . . . . . . . . . . . . . . . . . . . . . .

174

Charzynski Z., Skibinski P. : T h e pseudoorthogonality o f the coordinates o f a holomorphic m a p p i n g in two variables with constant dacobian

178

Costc M., Reguiat M.: Trivialit4s en famille . . . . . . . . . . . .

193

Degtyamv A.: Complex structure on a r e d dgebraic variety . . . . . .

205

Dcnkowska Z.: Subanaly~icity and Hilbert's 16th problem

221

. . . . . .

Franqois¢ J-P., Ronga F.: T h e decidability of real dgebraic sets by the index formula . . . . . . . . . . . . . . . . . . . . . . . .

Gamboa J.M., Ueno C.: Proper polynomial maps: the real case . . . . .

235 240

Gondard-Cozette D.: Sur les ordres de nivean 2" et sur une extension du 17~me probl~me de Hilbert . . . . . . . . . . . . . . . . .

257

Itenbcrg I.V.: Curves o f degree 6 with one non-degenerate double point and groups o f m o n o d r o m y of non-singular curves

. . . . . . . . . .

267

VIII Jaworski E: The 17th ttilbert problem for noncompact reaJ analytic manifolds

289

Korchagin A.: Construction of new M-curves of 9-th degree . . . . . .

296

Krasinski T., Spodzieja S.: On linear differentiaJ operators related to the ndimensional jacobian conjecture . . . . . . . . . . . . . . .

308

Kurdyka K.: On a subanMytic stratification satisfying a W h i t n e y property with exponent 1 . . . . . . . . . . . . . . . . . . . . .

316

Lombardi H.: Une borne sur les degr~s pour le th~orbme des z~ros r~els effectif 323 Marshall M., Walter L.: Minimal generation of basic sem_./aJgebra/c sets over an arbitrary ordered field . . . . . . . . . . . . . . . . . 346 Mazurovskii V.E: Configurations of at most 6 lines in R P 3

. . . . . .

354

Mikhalldn G.: Extensions of Rokhlin congruence for curves on surfaces .

372

Mostowski T., Rannou E. : Complexit~ de la construction des strafes A multiplicit~ constante d'un ensemble alg~brique de C n . . . . . . 378 Shustin E.I.: Real plane Mgebraic curves with many singularities

. . . .

389

Vorobjov N.: Effective stratification of regular real Mgebra/c varieties

402

List o f participants

416

. . . . . . . . . . . . . . . . . . . .

Semialgebraic

topology

in the last ten years

Manfred Knebusch

Contents §1 Brumfiels program §2 The two approaches §3 The state of art in 1981 §4 Sheaves and homology §5 Locally semiMgebraic spaces §6 Abstract semialgebraic functions and real closed spaces §7 Cohomology with supports §8 Borel-Moore homology §9 Base field extension and comparison theorems §10 Homotopy sets §11 Weakly semialgebraic spaces §12 Generalized homology §13 Novel features of semialgebraic topology: three examples §14 An outlook References

1

2 5 8

10 11 14 16

18 20 22 24 27 31

34

§1 Brumfiel's p r o g r a m Before discussing the subject named in the title it seems appropriate to outline the situation in semialgebraic topology in 1981, at the time of the first Rennes conference on real algebraic geometry. Already in the seventies, in the long introduction to his book "Partially ordered rings and semialgebraic geometry" [B], G.W. Brumfiel had laid down a program for what we now call "semialgebraic topology". Here Brumfiel advocated a new way of handling topological problems which is closer to the spirit of algebraic geometry than traditional topology. Let me just quote the following passage: "It thus seems to me that a true understanding of the relations between algebraic geometry and topology must stem from a deeper understanding of real algebraic geometry, or, actually, semi-algebraic geometry. Moreover, real algebraic geometry should not be studied by attempting to extend classical algebraic geometry to nonalgebraically closed ground fields, nor by regarding the real field as a field with an added structure of a topology. Instead, the abstract algebraic treatment of inequalities originated by Artin and Schreier should be extended from fields to (partially ordered)

algebras, with real closed fields replacing the algebraically closed fields as ground fields" [B, p.2]. In the main body of the book [B] Brumfiel develops a "real algebra" by studying partially ordered commutative rings and various sorts of convex ideals, with the perspective that this real algebra should perform a similar role in semialgebraic geometry as commutative algebra does in present day algebraic geometry. But the book does not go very far into semialgebraic topology. §2 T h e two a p p r o a c h e s Even today not much semialgebraic topology has been done using Brumfiel's rather intricate real algebra from the seventies. Around 1979 two other approaches to semialgebraic topology emerged independently which turned out to be successful. These are the "abstract" approach by M. Coste and M.F. Roy, and the "geometric" approach by H. Delfs and M. Knebusch. Before we get into this let me remind you of what are perhaps the two most serious difficulties which one encounters if one works over a real closed base field R different from R. a) R '~ is totally disconnected in the strong topology (i.e. the topology coming from the ordering of R). b) R '~ has very few reasonable (i.e. geometrically relevant) compact subsets. In particular, the closed unit ball in R n is not compact. Let M be a semialgebraic subset of some R". In the abstract approach one adds to M "ideal points" which turn M into an honest (albeit not Hausdorff) topological space. More precisely, one passes from M to the corresponding constructible subset .£/of the real spectrum Sper R[T1,..., T,~] of the polynomial ring R[T1,..., T,,I (cf. [BCR, Chap. 7]). -~/turns out to have only finitely many connected components, a n d / ~ / i s quasicompact. Thus in some sense the difficulties described above are overcome. The subspace topology of M in M is the strong topology we started with. One could also pass from M to the subspace/17/m~x of closed points of ~/, which still contains M as a dense subset and is a compact Hausdorff space with only finitely many connected components. But although this compactification .~:/m~x of M has its merits (cf. [B1]), the more interesting and more useful space is /~/ itself. The main reason for this is that .~/is a spectral space, as defined by Hochster [Ho], and that the constructible subsets Y of M correspond bijectively with the semialgebraic subsets N of M via the relation Y = N. A very nice consequence of this is that the "semialgebraic structure" of M is encoded to a large extent in the topology of 37/, since the lattice .q(/~/) of constructible subsets of M is by definition the boolean lattice generated by the lattice J~(~¢) of quasicompact open subsets of M, and thus

J~(37/) is completely determined by the topology of 37/ (cf. [Ho]). We call the space the abstraction of the semialgebraic set M. The wisdom of passing back and forth between the semialgebraic sets and their abstractions has been displayed well in the book [BCR] by Bochnak, Coste and Roy. Curiously another very important and fascinating aspect of the abstract approach is scarcely touched on in that book: One can study the constructible subsets of the real spectrum Sper A of any commutative ring A. Thus the abstract approach opens the door for an "abstract" semialgebraic topology where no base field (real closed or not) needs to be present. Coste and Roy were certainly well aware of this aspect at an early stage (cf. for example Roy's paper on abstract Nash functions [R]) but chose not to give much space to this in their book with Bochnak. The geometric approach (cf. [DK]) relies on the following two ideas, the first one being very simple. 1° Don't consider any subset of a semialgebraic set M C R n which is not semialgebraic or any map f: M --~ N between semialgebraic sets which is not semialgebraic! In this context a map f is called semialgebraic if the graph of f is a semialgebraic set and f is continuous with respect to the strong topologies of M and N. 2 ° Install on M a Grothendieck topology such that the semialgebraic functions, i.e., semialgebraic maps to R on the open semialgebraic subsets U of M (open with respect to the strong topology) form a sheaf OM of R-algebras! Instead of studying M as a semialgebraic subset of R n study the ringed space (M, OM)! Let me give some comments and explanations on these ideas. A d 1°: The reason that this idea makes sense is Tarski's principle. It guarantees that many of the usual constructions of new sets and maps from given ones give us semialgebraic sets and maps if we start with such sets and maps. In particular, if f: M --* N is a semialgebraic map between semialgebraic sets then the image f(A) of a semialgebraic subset A of M is semialgebraic and the preimage f - l ( B ) of a semialgebraic subset B of N is semialgebraic. Continuity of f is not necessary for this but is appropriate since we want to do "topology". A d 2°: The Grothendieck topology on M is defined as follows. The underlying category is the category ~ ( M ) of open semialgebraic subsets of M (i.e. semialgebraic subsets which are open in the strong topology), the morphisms being the inclusion mappings. An admissible open covering (Ui[i E I) of some U E ~ ( M ) is a family (Uili E I) in ~ ( M ) with U = U Ui, such that there exists a finite subset J of I with

iEl U = U Ui. {Thus a property similar to quasicompactness is forced to hold.} Then iEJ

4 the semialgebraic functions on the sets U E ~ ( M ) indeed form a sheaf OM. It turns out that a morphism from (M, OM) to (N, ON) is determined by the underlying map f from M to N, and that these maps f are just the semialgebraic maps from M to N as introduced above (of. [DK, §7], by definition the morphism has to respect the R-algebra structures of the structure sheaves). Replacing a semialgebraic set M C R n by the ringed space (M, OM) allows us to forget the embedding M ~ R n. We call any ringed space of R-algebras which is isomorphic to such a space (M, OM) an affine semialgebraic space over R. By abuse of notations we do not distinguish between a semialgebraic set M and the corresponding ringed space (M, OM).

A semialgebraic path in M is a semialgebraic map from the unit interval [0,1] (which is a semialgebraic subset of R 1) to M. Having this notion of paths at hand one defines the path components of M in the obvious way. It turns out, that M has only finitely many path components M 1 , . . . , M r and that these are semialgebraic in M and closed, hence also open in the strong topology, cf. [DK]. Every Mi is "semialgebraically connected", i.e. Mi is not the union of two disjoint non empty open semialgebraic subsets, since this holds for [0,1], as is easily seen. Thus we have dealt with the first difficulty mentioned above, exploiting only idea N ° 1. By the way, the abstractions 2~/1,..., ~/r are the connected components of the topological space In order to cope with the second difficulty one also needs idea N ° 2. The category of affine semialgebraic spaces over R has fiber products. Thus we can define proper morphisms as in algebraic geometry. We call a semialgebraic map f: M --* N closed, if the image f(A) of a closed semialgebraic subset A of M is again closed. We call f proper if f is universally closed, i.e. for any semialgebraic map g: N ~ --* N the cartesian square

M ×N N ~

/ ~

¢1 M

N~

lg f

.~N

gives us a closed semialgebraic map f~. We call an affine semialgebraic space M complete if the map from M to the one-point space is proper. Even more than in algebraic geometry over an algebraically closed field, it is true for many purposes, that complete spaces are the right substitute for compact spaces in topology. For example, a semialgebraic function on a complete space attains its maximum and minimum.

It turns out that there exist in abundance relevant complete affine semialgebraic spaces. Namely, the following analogue of the Heine-Borel theorem holds: A semialgebraic subset M of R n is a complete space iff M is closed and bounded in R n. §3 T h e s t a t e of a r t in 1981 I give a rough sketch of the technical progress up till 1981. This is just to give an impression of the state of art at the first Rennes conference. It is not meant, of course, as a complete account of everything done up to that time. In the geometric theory we have the following list. 1) 2) 3) 4) 5) 6)

Connected components Complete affine semialgebraic spaces and the semialgebraic Heine-Borel theorem Dimension theory Existence of triangulations Hardt's theorem Semialgebraic homology

Here are s~me comments on these. N ° 1 and N ° 2 have been described above. One may add to N ° 2 that in 1981 we also had a good insight into the nature of proper maps between affine semialgebraic spaces [DK, §9 and §12]. A d 3: The dimension d i m M of a semialgebraie set can be defined as the maximal integer d such that M contains a subspace N which is isomorphic to the unit ball in R a {[DK, §8], there a different but equivalent definition had been given}. This notion of dimension behaves very well, better than in classical topology. For example, if a partition of M into finitely many semialgebraic subsets A 1 , . . . , Ar is given, then dim M is the maximum of the numbers dim AI,. •., dim A t . A d 4: If M is an affine semialgebraic space and A 1 , . . . , Ar are finitely many semialgebraic subsets of M then there exists a finite simplicial complex X over/it and an isomorphism of spaces (p: X ~ , M such that, for every i E { 1 , . . . , r}, the set ¢p-l(Ai) is a subcomplex of X [DK, §2]. Here the word "simplicial complex" is used in a non classical meaning: X is the union of finitely many open simplices a l , . . . , ~rt in some R N such that the intersection ~i fl ~j of the closures of any two simplices ¢ri, aj is either a common face of them or empty. Thus the closure )~" of X is a classical finite simpIicial complex (,~ finite polyhedron), and X is obtained from )( by omitting some open faces. Also "subcomplex" means just the union of some of the sets ( r l , . . . , ~rt. Clearly X = 3~ iff M is complete. In the case R = R the triangulation theorem has been well known since the sixties, even for semianalytic sets [L, Gi].

A d 5: Hardt's theorem states that for every semialgebraic map f: M --~ N there exists a partition of N into finitely many semialgebraic subsets N 1 , . . . , Nr such that f is trivial over each Nj, i.e. f-l(Nj) is isomorphic over Nj to a direct (= cartesian) product Nj × Fj, cf [DK1, §6]. The theorem had been proved for R = R by R. Hardt around 1978 [Ha]. A d 6: In his thesis [D] Delfs constructed homology and cohomology groups with arbitrary constant coefficients for affine semialgebraic spaces over any real closed field R. In the case R = R these groups coincide with the singular groups known from classical topology. Certainly Dells' homology theory was the most profound achievement in semialgebraic topology up till 1981. But the proofs of the triangulation theorem and of Hardt's theorem also needed new ideas beyond the known proofs for R = I~. The triangulation theorem is the main technical tool in developing semialgebraic homology (and also semialgebraic homotopy theory, cf. §10 below). Hardt's theorem is very useful if one wants to profit from semialgebraic homology. For a good example, cf. [DK1, §7]. I will say more about semialgebraic homology in the next section §4. R e m a r k . Only recently (1989) I learned from Cert-Martin Greuel about the unpublished dissertation of Helmut Brakhage [Bra] (Heidelberg 1954, thesis advisor F.K. Schmidt). Here Brakhage studies semialgebraic topology over an arbitrary real closed field. He exploits idea N ° 1 of the geometric theory (cf. §2) to an enormous extent and obtains many of the results we had found up to 1981, in particular the triangulation theorem. The introduction to Brakhage's thesis reads very much like the talks Dells and I used to give around 1980. He would have saved us a lot of work if we only would have known about his thesis. Brakhage is now a professor at Kaiserslautern, working mostly in applied mathematics. It is difficult to give a good picture of the state of art in semialgebraic topology in 1981 on the abstract side, since in the abstract theory the main bias was towards algebraic problems. Topology seems to have been studied mainly as an aid for solving algebraic problems of current interest. I give the following list. 1) 2) 3) 4) 5) 6)

Connected components Compactness of constructible sets Specialization theory Dimension theory Abstract Nash functions Separation of connected components by global quadratic forms

Here only N ° 1 - 4 truly belong to semialgebraic topology, but N ° 5 and 6 use topology in an essential way, and have also turned out to be stimulating for semialgebraic topology since 1981.

N ° 1 has been discussed above, N ° 2 alludes to the easily accessible but extremely important fact, that the real spectrum Sper A of any commutative ring A is compact in the constructible topology. This means that, if X is a constructible subset and (Y~[i E I) is a family of constructible subsets of SperA with X C [,.J Y~, then there iEl exists a finite subset J of I with X C U Y~" The quasicompactness of J~/stated iEJ above is a rather special consequence of this. A d 3: If x and y are points of a topological space X then we say that y is a specialization of x (and x is generalization of y) if y lies in the closure of the set {x}. We write x ~- y for this. N ° 3 alludes to some - again simple but important facts about specializations in a real spectrum SperA, cf. [CR2], [BCR, 7.1], [KS, III §3 and §7]. In particular, the specializations of a given point x in Sper A form a chain, i.e. if x ~- y and x ~- z then y :,-- z or z ~- y. Moreover if neither x ~ y nor y ~- x then there exist disjoint open subsets U, V in Sper A with x E U and y E V.

-

A d 4: The dimension dim X of a constructible subset X of Sper A is defined as the supremum of the lengths of the specialization chains in X. {Up till now it has been adequate to put dim X = oo if the lengths do not have a finite bound.} The main result is that, if M is a semialgebraic set over some real closed field, then this "combinatorial" dimension dim J~/ of the abstraction /~¢ coincides with the semialgebraic dimension dim M from above, cf. [CR2], [BCR]. A d 5 a n d 6: One of the most important achievements in the early work of Coste and Roy is the construction of a sheaf of "abstract Nash functions" 91A on the real spectrum of an arbitrary commutative ring A JR], which generalizes the sheaf of classical Nash functions for algebraic manifolds over R. Indeed, right from the beginning they had the idea of constructing the real spectrum as a ringed space (Sper A, 92A) [CR], [CR1], thus bringing semialgebraic geometry close to the spirit of abstract algebraic geometry in the sense of Grothendieck. The sheaf 91A is more algebraic in nature than the sheaf of semialgebraic functions discussed in §2. It does not belong to semialgebraic topology, but nevertheless relies on the topological fact that every etale morphism A ~' ~ B induces a local homeomorphism Sper qo: Sper B Sper A. Building on this, Mah6 was able to solve one of the main open problems of quadratic form theory from the seventies [K1, Problem 16] affirmatively, namely the separation by global quadratic forms of the connected components of the set V(FI) of real points of an affine algebraic variety V, and later, together with Houdebine, also of a projective algebraic variety V over R [M], [HM]. In fact, they prove such a theorem over any real closed field R, and also for the real spectrum of any commutative ring. Mah~'s theorem in [M] is probably the first result which signaled to the outside world

that something new in principle had happened in real algebraic geometry around 1980. §4 S h e a v e s a n d h o m o l o g y After 1981 semialgebraic topology has been dominated by two major new trends: A strong interaction between the geometric and the abstract theory, and the employment of new spaces. An important instance of the first trend is sheaf theory. Let M be a semialgebraic set over some real closed field R. Then a (set valued) sheaf over M is essentially the same object as a sheaf over the abstraction _~/. Indeed, as was already known before 1981 [CR2], [D], [De], a semialgebraic subset U of the affine semialgebraic space M is open iff the abstraction/) is open in/~/. Moreover, a family (Ui[i E I) of open semialgebraic subsets of M is an admissible open covering of U iff (/)i[i E I) is an open covering o f / ) . The reason for this is the definition of the Grothendieck topology on M on the one hand, and the quasicompactness of U on the other. Since the quasicompact open subsets of/~/form a basis of the topology of/iT/, all of this gives us a canonic£1 isomorphism Y H .9~ from the category of sheaves on i to the category of sheaves on Jl~/, via the rule ~-(U) = 5g'(U). Henceforth we only consider sheaves of abelian groups. Recall that M is dense in/~/. For x E M the stalks ~'z and fi'z are equal. It may well happen that all stalks .T'x, x E M, are zero, but .T is not zero. {An example is given in [92, 1.1.7].} This is by no means astonishing: Of course, Y # 0 iff fi" # 0. Then, since ~/ is an honest topological space, there exists some a E M with ~a # 0. But it may happen that none of these points a lies in M. This discussion makes it clear that most often sheaf theoretic techniques work better in the abstract settir~g than the geometric one. Only there one can argue "stalk by stalk" without further justification. Now is a good moment to say something about the semialgebraic homology theory of Hans Dells, sinc e he has been able to simplify his theory greatly by using sheaves and the interplay back and forth between semialgebraic sets and their abstractions [D1]. I first describe the main problem in defining homology groups Hq(M,G) for a semialgebraic set M over some real closed field R and some abelian group of coefficients G. Let us assume for simplicity that M is complete. We choose a triangulation 90: X ~ * M. Here X is a finite simplicial complex in the classical sense but over R; X may be regarded as the realization [K[/~ over R of an abstract finite simplicial complex K, a purely combinatorial object (cf. [Spa, 3.1]; the realization is defined exactly as in the case R = R). It is intuitively clear that Hq(M,G) should coincide, up to isomorphism, with the combinatorial homology group Hq(K,G) from classical topology. To make an honest

definition out of this, one has to verify that (up to natural isomorphism) the group Hq(K, G) does not depend on the choice of the triangulation. The now traditional way to prove this is to define a complex C.(M, G) of singular chains and to verify the seven Eilenberg-Steenrod axioms for the homology groups [ES, I §3]. Then one obtains, in a well known manner, that tIq(C.(M, G)) ~- Hq(I(, G) for the triangulation ~ above. {One also has to consider noncomplete spaces M and the relative chain complex C.(M, A; G) for A a semialgebraic subset of M. I omit these technicalities.} We can indeed define singular chain groups Cq(M, G) along classical lines, decreeing that a singular simplex is a semialgebraic map from the q-dimensional standard simplex Aq to M. Six of the seven Eilenberg-Steenrod axioms can bc proved as in the classical theory, always using semialgebraic maps instead of continuous maps. But the excision axiom is difficult. The classical way to prove it is to make a given singular cycle Z "small" with respect to a given covering of M by two open (semialgebraic) sets U1, U2, by iterated barycentric subdivision of Z. This means that every singular simplex occuring in the subdivided cycle has its image either in U1 or U2. But if the base field R is not archimedian then this procedure completely breaks down, since then usually a given bounded semialgebraic set cannot be covered by finitely many semialgebraic sets all whose diameters are smaller than a given c > 0. In his thesis [D] Delfs found the following way out of this difficulty. He defined cohomology groups Hq(M, G) as the sheaf cohomoIogy groups of the constant sheaf GM on M, and similarly relative cohomology groups Hq(M, A; G) as the sheaf cohomology groups of a suitable sheaf GM,A on M. {Recall that M is equipped with a Grothendieck topology.} For these groups Hq(M, A; G) Delfs succeeded in verifying the Eilenberg-Steenrod axioms. Then he knew that Hq(M,G) is isomorphic to the combinatorial group Hq(K, G). Thus Hq(K, G) is independent of the choice of the triangulation, up to natural isomorphism. From this Delfs concluded that also Hq(K, G) is independent of the choice of triangulation [D]. The verification of six of the seven Eilenberg-Steenrod axioms for the groups Hq(M,A,G) is straightforward, but this time the homotopy axiom causes difficulties. Delfs surmounted these difficulties in [D] by a complicated geometric procedure. Later Delfs found an easier way [D1]. He realized that the homotopy axiom follows from the statement that, for any sheaf ~" on M, the adjunction homomorphism 5r --~ rr.r*~', with 7r the projection from M × [0, 1] to M, is an isomorphism and Rqr,(rr*.T") = 0 for q > 1. [D1, Prop. 4.2 and 4.4]. This then could be deduced via a stalk by stalk argument from the fact that Hq([0, 1], G) = 0 for q > 1 and any abelian group G, which in turn can be verified in an easy geometric way. The crucial point is that one needs the fact Hq([0, 1], G) = 0 not just over R but ovcr the residue class fields k(x) of all points x E M. Roughly one can summarize that Delfs reduced the verification of the homotopy axiom to an easy special case using sheaf theory, at the expense of enlarging the real dosed base field in many ways.

10 §5 L o c a l l y s e m i a l g e b r a i c spaces Dells and I had already introduced "scmialgcbraic spaces" over a real closed field R before 1981 by gluing together finitely many affine semialgebraic spaces over R along open subspaces [DK, §7]. What then was still missing was a handy criterion for a semialgebraic space M = (M, (-gM) to be again affine. Such a criterion would allow the building of semialgebraic spaces M from semialgebraic sets in an "abstract" manner, i.e. without explicitly looking at polynomials, such that M eventually turns out to be an affine space, in other words, a semialgebraic set. In 1982 R. Robson proved his imbedding theorem [Ro] which gives such a criterion. The theorem says that a semialgebraic space M over R is affine iff M is regular, i.e. a point x and a closed semialgebraic subset A of M with x ~/A can be separated by open semialgebraic neighbourhoods. {A subset A of M is called closed semialgebraic if the complement M - A is an open semialgebraic, i.e., an admissible open subset of

M.} Robson's theorem reMly paved the way for the trend of employing new spaces in the geometric theory. Before I go into details about this I should say some words about covering maps. Having semialgebraic paths at hand we may define the fundamental group 7rl(M, x0) for M a semialgebraic space over R and x0 E M, as in the classical theory, by considering homotopy classes of semialgebraic loops with base point x0. Of course, homotopies also have to be defined in the semialgebraic sense, starting from the unit interval [0,1] in R, cf. §10 below. It turns out that for affine M the group 7rl (M, x0) is very respectable. It is finitely presented and coincides with the topological fundamental group in the case R = R. {These are consequences of the two comparison theorems on homotopy sets [DK2, III §3 and §5], to be discussed in §10 below.} Assume since now that M is affine and path connected. The question arises whether the subgroups of 7rl(M, x0) classify "semialgebraic covering spaces" of M, as one might expect from classical topology. It seems clear what a semialgebraic covering map 7r: N ~ M has to be: N should be a semialgebraic space and 7r a semialgebraic map. Further there should exist an admissible open covering (Uili E [) of M such that ~ is trivial over each Ui with discrete fibers, i.e. rr-](Ui) ~ Ui × Fi over Ui for a discrete semialgebraic space Fi. But what does it mean for a semialgebraic space F to be discrete? Reasonable answers, one can think of, are: dim F = 0; the path components of F are one-point sets; the one-point sets in M are open in F; the one-point sets in M form an admissible open covering of M. - All of these properties mean the same thing, namely that the space F consists of finitely many points. We conclude that every semialgebraic covering map r: N ~ M has finite degree.

]1 Working with path lifting techniques one verifies that the semialgebraic coverings ~r:N --~ M of M are indeed classified by the conjugacy classcs of subgroups of ~rl(M, x0) of finite index [K3]. Using Robson's embedding theorem one also sees that N is again affine. Having verified this in 1982 [DK3], Dells and I realized that the category of semialgebraic spaces is not broad enough. There should exist some sort of covering space N of M corresponding to any given subgroup H of ~rl (M, x0), in particular a "universal covering", corresponding to H = {1}. This led us to introduce locally semialgebraic spaces. A locally semialgebraic space M over R is obtained by gluing together (maybe infinitely many) affinc semialgebraic spaces over R along open semialgebraic subspaces. Of course, the gluing is meant in the sense of ringed spaces with Grothendieck topologies, cf. [DK2, I §1]. The nice locally semiatgebraic spaces are those which are regular (defined in the same way as above) and paracompact, as defined in [DK2, I §4]. The category LSA(R) of regular paracompact locally semialgebraic spaces over R contains the category of affine semialgebraic spaces over R as a full subcategory. In LSA(R) we have a fully satisfactory theory of covering spaces. In particular ever), space M E LSA(R) has a universal covering (cf. [DK3, §5]; a full treatment of this topic still awaits publication [K3]). In LSA(R) there exist fibre products. There is also a good notion of subspaces. Namely, if M is a locally semialgehraic space and (Mill • I) is an admissible open covering of M, such that every M / i s an affine semialgebraic space, then a subset A of M is cMled a subspace if A N M/ is semialgebraic in Mi for every i • I. Indeed, collecting the affine semialgebraic spaces A N M / w e obtain on A the structure of a locally semialgebraic space over R, which is independent of the choice of the covering (Mi[i • I). This space A is regular and paracompact if M has these properties [DK2, I, §3 and §4]. Up to now LSA(R) has proved to be the appropriate basic category for all geometric studies over R, as long as one does not pass to abstract spaces. In particular the triangulation theorem for semialgebraic sets (eft §3 above) extends to a triangulation theorem of equal strength for these spaces (simultaneous triangulation of M and a locally finite family of subspaces of M, cf. [DK2, Chap.II]). Also the homology theory of Dells discussed above extends to these spaces [DK2, Chap.III]. And we have a fairly good homotopy theory in LSA(R) at hand, to be discussed below. §6 A b s t r a c t s e m i a l g e b r a i c f u n c t i o n s a n d real closed spaces We come back to the relationships between a semiatgebraic set M over R and its abstraction/~/. Recall from §4 that the sheaves on the affine semialgebralc space M correspond uniquely with the sheaves on 37/. In particular we have a sheaf of rings

12 (-')M on 5-I which corresponds to the sheM 0 M of semialgebraic functions on M. The question arises whether (~M generalizes in a natural way to a sheaf of rings Ox on any constructible subset X of any real spectrum Sper A, which then can be regarded as a sheaf of "abstract" semialgebraic functions on X. This is indeed the case. Around 1983 G. Brumfiel [B4] and N. Schwartz [S] gave two solutions of this problem. A (slightly "corrected", cf. [D, 1.7], [S, Example 58]) version of Brumfiel's definition runs as follows. Let p: Sper A[T] --* Sper A be the natural map from the real spectrum of the polynomial ring A[T] in one variable over A to Sper A, induced by the inclusion A ~ A[T]. For any quasicompact open subset U of the space X the elements of Ox(U) are the continuous sections s of plp-l(u): p-I (U) --~ U such that s(U) is a closed constructible subset of p-1 (U). What does this mean? For any x E A we may identify p-1(x) with the real spectrum Sper k(x)[T], where k(x) denotes the residue class field of Sper A at x, a real closed field. This real spectrum is the abstraction of the real affine line over k(x). Thus k(x) injects into p - l ( x ) as a dense subset (cf. §2). For a section s ms above, s(z) lies in this subset and hence corresponds to an element f(x) of k(z), which should be regarded as the value of the abstract semialgebraic function f given by s. The section s is completely determined by the values f(x) and should be regarded as the graph of f. N. Schwartz defined an abstract semialgebraic function f on U directly as a family (f(x)lx E U) E l-Leu k(x) with compatibility relations between the values f(x) coming from canonical valuations Ax,y: k(x) --~ n(x, y) U c~. For any pair (x, y) with x E U and y a specialization of x in U, ~(x, y) is an overfield of k(y), and A~,v has to map f(x) to f(y) E k(y) C t~(x, y). The definition of Schwartz has the advantage that here it is immediately clear that Ox(U) is a ring, while in Brumfiel's definition one has to work for this. Then Dells proved that the definitions of Brumfiel and Schwartz give the same sheaf Ox [D1, §1]. The stalks of Ox are local rings. In the geometric case, i.e., if A = R[T1,...,Tn] and X = /17/ with M C R n a semialgebraic set, we indeed have Ox = (gM. From now on we call the ringed space (_~/, OM) - instead of just the topological space .ltT/- the abstraction of the affine semialgebraic space (M, OM). In the paper [B4] cited above Brumfiel introduced abstract semialgebraic functions as a tool to prove a vast generalization of Mah6's theorem on the separation of connected components by global quadratic forms. For every commutative ring A there is a natural homomorphism from the Witt ring W(A) to the orthogonal Kgroup KOo(Sper A) of the real spectrum of A. Brumfiel proves that both the kernel and the cokernel of this homomorphism are 2-primary torsion groups. Thus, from our viewpoint, the localization 2 - ~ W ( A ) of W(A) at the prime 2 is a purely topological object.

]3 Brumfiel's paper is a bold step into the realm of abstract semialgcbraic topology. A full understanding of it is a challenge even today, since some arguments are only sketched. For a discussion cf. [K, §6], and for a treatment in the geometric case cfi [BCR, 15.3]. N. Schwartz studied in [S] the spaces (X, Ox), with X a constructible subset of some real spectrum Sper A, for their own sake. The ring O(X) of global sections of Ox is a sort of "real closure" of the ring A. Schwartz describes how to obtain O(X) from the ring A in a constructive way. He further makes the important discovery that the natural map from X to Sper O(X) is an embedding which makes X a dense subspace of Sper O(X). Even more is true: the closed points and also the minimal points of Sper O(X) all lie in X. In the special case that X is convex in Sper A with respect to specialization, it turns out that the ringed spaces (X, Ox) and Spec O(X) are equal. In the geometric case X = ~ / t h i s happens iff the semialgebraic set M C R n is locally closed in R n. Later Schwartz realized that all we have said above about (X, Ox) remains true if X is a proconstructible subset of Sper A, i.e., the intersection of an arbitrary family of constructible subsets of X [$1], [$2]. He called any ringed space isomorphic to such a space (X, Ox) an aO~ne real closed space. He then introduced the category 7"~ of real closed spaces as a full subcategory of the category of all locMly ringed spaces. The definition of a real closed space is simple: a ringed space (X, Ox) - always with X a genuine topological space, no Grothendieck topology - is called real closed if every point x E X has an open neighbourhood U such that (U, Ox[U) is an affine real closed space. The books [$1], [$2] are both versions of Schwartz's Habilitationsschrift [S]. For the insiders they constitute a sort of bible of abstract semialgebraic topology - an incomplete bible, I should add, since more can and should be written down with the methods developed there. The shorter version [S1] is easier to read, while [$2] is closer to the original Habilitationsschrift and contains much more material. In IS] Schwartz defined real closed spaces using as building blocks only constructible subsets of real spectra, instead of proconstructible ones. I will call these more special ringed spaces here "abstract locally semialgebraic spaces" and denote their category by 7~0. The analogy with locally semialgebraic spaces over a real closed field R is striking. But there is more than analogy. One can attach to any locally semialgebraic space (M, OM) over R an abstract locally semialgebraic space (/~, 0 M ) in a rather obvious way, starting from the abstractions of affine semiMgebraic spaces discussed above. Schwartz proves that this gives an embedding of LSA(R) into the category ~0, making LSA(R) a full subcategory of the category of abstract locally semialgebraic spaces over Sper R [S], [$1], [$2]. A good thing about ~ is that here more constructions - in particular more quotients - are possible than in LSA(R).

14 In view of what has been said above about affine real closed spaces, it is clear that real closed spaces are close to schemes and many of them are in fact schemes. This constitutes a rather thorough algebraization of semialgebraic topology, since the notion of schemes originates from polynomial equalities and non-equalities ( f = 0, f 0) instead of inequalities ( f > 0, f > 0). The books [$1], [$2] give ample evidence that scheme theoretic notions and techniques work well in the category 7~. The books are close in spirit to the foundations of Grothendieck's abstract algebraic geometry [EGA I], [EGA I*]. In particular, the transition from a locally semialgebraic space to its abstraction is fully analogous to the transition from an algebraic variety over an algebraically closed field to the associated scheme. Of course, in many respects [$1] and [$2] are simpler than Grothendieck's theory, since here only "topological" phenomena have to be captured. This is already reflected by the fact that no nilpotent elements occur in the structure sheaves. It also pays well to pass from the category 7-4o to 7~. For example, a finite subset of a real spectrum Sper A is always proconstructible but only rarely constructible. This implies that for a real closed space (X, Ox) every finite subset S of X gives us a "subspace" (S, Os) of (X, Oz) which is again real closed. Especially useful are the two-point spaces S = {~, x} coming from the real spectrum Sper 0 of a convex subring 0 of a real closed field K with ~ the minimal point ("general point") and x the closed point ("special point") of Sper 0. {Recall that 0 is a valuation ring.} These two-point spaces occur in valuative criteria for various properties of morphism between real closed spaces, cf. ~13 A below. By the way, Sper 0 = Spec 0 as a ringed space. My student Michael Prechtel has given an interesting classification of all real closed spaces which contain only finitely many points [P].

§7 Cohomology w i t h s u p p o r t s Starting with this section, a geometric space means a regular paracompact locally semialgebraic space over some real closed field R, and an abstract space means a regular paracompact abstract locally semialgebraic space. A word of explanation for these terms: A real closed space X is called regular if the specializations of a point x E X form a chain. X is called paracompact if X has a locally finite covering (Xi]i E I) by affine open subsets Xi. There are several reasons why this terminology makes sense. On the one hand the abstraction M of a locally semialgcbraic space M over some real closed field R is regular if M is regular and is paracompact if M is paracompact. Thus the abstraction of a geometric space is an abstract space. Oil the other hand, if X is an abstract space then the subspace X m~x of closed points of X is a paracompact topological space [D2, Chap. I].

15 Abstract spaces are very amenable to sheaf cohomology with supports, as defined in classical topology (cf. e.g. [Bre D. This has been amply demonstrated by Dells in Chapter II of his tIabilitationsschrift [D2]. Recall that if • is a family of supports on X (= antifilter of closed subsets), and ~- is a sheaf on X (as always, with values in abetian groups), then F~(X, 7 ) denotes the group of sections s E .T(X) with support supps E ~, and .T" ~ H{(X,F) is the q-th derived functor of the left exact functor

There is a close connection with the sheaf cohomology on the subspace X max, provided X max is dense in X and every point of X has a specialization in X max (which holds in many applications). Then we have a canonical continuous retraction r: X ~ X raax. The direct image functor r, from the category of sheaves on X to the category of sheaves on X max is easily seen to be equal to the inverse image functor i* for i the inclusion X max ~-~ X. For every sheaf ~" on X this gives us canonical isomorphisms tl~(X,~) ~= ~ u qv "~ v m ~ ,z. , ~. . j , and for every sheaf ~ on X, canonical isomorphisms H{(X ma*,~) ~ H{(X, i,G), with • denoting the set of intersections {A N xm~*Ia E ~}. In the case that X is affine (or, more generally, a "normal spectral space") this important observation goes back to Carral and Coste [CC]. If M is a geometric space then we have a canonical isomorphism fi" H ~" from the category of sheaves on M to the category of sheaves over/~'/, as explained above in the special case that M is affine (§4). A family of supports ¢b on M is by definition an antifilter of closed subspaces of M. We can define cohomology groups H~(M,U) in much the same way as in the classical theory. {Fo(M, U) is the group of all s E .T(M) such that sIM - A = 0 for some A E ¢.} Let ~ denote the antifilter of closed subsets of/~/generated by the abstractions A of all A C ¢. Then it is evident that

U (M, 7) = H (i4, for all q _> 0. There are many useful families of supports ~, even more than in classical topology. In particular we can choose for ~ the family of all complete sub@aces of M, "complete" being defined as in §2. We denote this family by c, suppressing its dependence on M in the notation. In the abstract setting we also have the notion of complete spaces. Here, for X an abstract space, we denote by c the antifilter of closed subsets generated by the complete subspaces of X. The groups IIq~(M,F) resp. H~(X, J:) are the analogues of the cohomology groups with compact support in classical topology. For M a geometric space we have H~(M, J:) = Hqc(~l,~). In Chapter II of [D4] Dells develops the formal theory of sheaf cohomology with supports for geometric and abstract spaces virtually to the same extent as the classical theory (e.g. [BreD . Some important topics are omitted, in particular cup products,

16 but it is evident that these topics can be dealt with by the same methods. Dells has often talked to me about such things in very clear terms. As a sample of his theory I will now say something about the cohomology of fibers [D2, II §8]. Let f : M ~ N be a morphism between geometric spaces and Y" a sheaf on M. Let f:/~/ --* N denote the abstraction of f . For every c~ C fir the fibre f-l(c~) is the abstraction of a geometric space over k(~). In particular, if y C N, then f - 1 (y) = I - 1 (y)~. Fix some q C No. For every c¢ E fir we consider the cohomology group which we denote more briefly by H ~ ( f - l ( c ~ ) , ~ ) . The question arises as to how these groups are related for varying ~. A first answer is that there exists a sheaf (5 on N, namely ~5 = Rqf!.T", such that, for every o~ C /V, H ~ ( f - l ( c ~ ) , ~ ) is the stalk of ~ at c~,

H~(f-~(oO, )elf-~(~))

(*)

(R f!.T) c} for some c > 0. This is an ultrafilter which gives us a closed point a E Frin (M) fixed under f . Brumfiel's fixed point theorem is important for his "real spectrum compactification" Mg of the Teichmfiller space/kIg of compact Riemann surfaces of genus g [B1], since it implies that every element of the Teichmfiller modular group has a fixed point in Mg.

§14 A n outlook What can be said about the situation in semialgebraic topology now, in the year 1991, and about prospects for the future, without too much speculation? Certainly abstract spaces contain a rich potential for further research. They should be "superior" to geometric spaces. But up to now nearly all the deeper results in homology and homotopy theory rely heavily on arguments in the geometric catcgory,

32 using triangulations, simplicial approximations, the homotopy extension property and the like. It would be highly desirable to have a homotopy theory for real closed spaces. There can be no doubt that we have found the "right" homotopy groups in the geometric setting. But all this does not tell us how to proceed in the abstract setting. It is only clear that an abstract homotopy theory should give us the geometric theory if we apply it to the abstractions of geometric spaces. For abstract cohomology the analogous problem seems to be easier. There exists an approach by N. Schwartz using inverse real closed spaces which looks promising. Schwartz talked about this at Luminy in October 1989, and then in the Ragsquad seminar at Berkeley in October 1990. In abstract homotopy one could proceed along the lines given in chapter II of Baues' book [Ba], where the hornotopy theory of a "cofibration category" has been developed. Or one could try to associate with a real closed space X (perhaps fulfilling some conditions like regularity or "tautness" [D2, I §3]) a simplicial set or, perhaps better, something like a simplicial space V X (in the classical sense, with Hausdorff topology), such that the homotopy type of V X encodes the honaotopy information about X. In his talk at Oberwolfach in June 1990 C. Scheiderer proposed a simplicial space V X (together with a "quasiaugmentation" V X --~ X ) which looks promising. The definition of V X employs chains of real valuations of commutative rings. Sche~derer was able to verify that, for any sheaf .T on X, the cohomology groups Hq(X,.T) can be computed from V X . Also, for X the abstraction of a geometric space M, V X represents the hornotopy type of M. These are hints that V X gives the right homotopy type. In the last years the interest in abstract spaces also has been enhanced by the discovery of the Spanish school (Andradas, Ruiz, ...), that questions in semianalytic geometry can be treated by considering real spectra. I just mention two basic observations: If A is a local ring of real analytic functions then the constructible subsets of Sper A are in natural one-to-one correspondence with the germs of semianalytic sets [Rz]. If A is the ring of global analytic functions on a compact real analytic manifold M then the constructible subsets of Sper A correspond bijectively with the globally semianalytic subsets of M [Rzl]. The homotopy or homology of such a constructible set (which is a real closed space) should have a close relation to the homotopy resp. homology of the corresponding semianalytic object. Concerning the two trends in the last ten years mentioned above (§4) we can safely say that the first one has run its course. Interaction between the geometric and the abstract theory is no longer a "trend" but a well established and widely used technique. However the first trend may continue in a wider context if we allow other meanings for the word "geometric".

33 Geometric spaces might be built, for instance, from semianalytic sets, or perhaps even from subanalytic sets. J. Denef and L. van den Dries have found a new approach to subanalytic sets [DvdD] which gives much hope that subanalytic sets are amenable to techniques similar to the ones we now have in the geometric semialgebraic topology. Subanalytic sets (more precisely, subsets of R" which are subanalytic in Pn(R)) and semialgebraic sets can both be subsumed under o-minimal structures, a notion stemming from model theory, cf. [PiS]. In his talk in the Ragsquad seminar at Berkeley, April 1991, van den Dries has outlined a "tame topology of o-minimal structures" with many of the features we are used to for semialgebraic sets, in particular N ° 1 - N ° 5 of the list in §3 for the geometric theory (cf. his forthcoming book [vdD], as soon as it appears). Thus o-minimal structures seem to provide a good framework for highly interesting lmw geometric spaces in the years to come. In 1990 R. Huber introduced "semirigid functions" [Hull as an offspring of his abstract approach to rigid analytic geometry [Hu]. They give us semirigid sets, which are vaguely analoguous to semianalytic sets in real analytic geometry. To understand these sets well, one definitely needs abstract spaces which are derivates of the real valuation spectra of commutative rings [Hul]. This, and the frequent occurrence of valuations in the theory of real closed spaces, are hints that valuation spectra will play a rote in a further development of semialgebraic topology, for which the word "semiatgebraic" is probably no longer appropriate. An introduction to valuation spectra has been given in [Hu, Chap. I] and [HuK]. From all of this it is pretty clear that tile second trend mentioned in §4, i.e., the employment of new spaces - both geometric and abstract, will persist. A c k n o w l e d g e m e n t s . I thank Roland Huber, Claus Scheiderer, Colm Mulcahy, and Victoria Powers for helpful criticism, both in mathematics (R.H., C.S.) and in language (C.M., V.P.). I further thank Michel Coste, Louis Mah~, and MarieFrancoise Roy for organizing the conference and thus providing - among many other things - a very pleasant opportunity to give a talk, of which this article is an expanded version.

34 References [Ba] t t . J . Baues, "Algebraic homotopy". Cambridge studies advanced math. 15, Cambridge Univ. Press 1988. [BCR] J. B o c h n a k , M. Coste, M.-F. Roy, "GSom~trie alg6brique r~elle". Ergebnisse Math. Grenzgeb. 3. Folge Band 12, Springer 1987. [BH] A. Borel~ A. Haefliger, La classe d'homologie fondamentale d'un espace analytique, Bull. Soc. Math. France 89,461 - 513 (1961). IBM] A. Borel, J . C . M o o r e , Homology for locally compact spaces. Mich. Math. J. 7, 137- 159 (1960). It. B r a k h a g e , "Topologische Eigenschaften algebraischer Gebilde fiber einem [Bra] beliebigen reell-abgeschlossenen GrundkSrpcr". Dissertation Heidelberg 1954. [Bre] G. B r e d o n , "Sheaf theory". McGraw Hill 1967. [Br] L. BrScker, Real spectra and distribution of signatures. In: "G~om6trie alg6brique rSelle et formes quadratiques" (Ed. J.-L. Colliot-Th~l~ne, M. Coste, L. Mah6, M.F. Roy), pp. 249 - 272. Lecture Notes Math. 959, Springer 1982. [B] G . W . B r u m f i e l , "Partially ordcred rings and semialgebraic geometry". London Math. Soc. Lecture Notes 37, Cambridge Univ. Press 1979. [B1] - - , The real spectrum compactification of Teichmfiller spaces. In: "Geometry of group representations" (Ed. W.M. Goldman, A.R. Magid), Contemporary Math. 74, A.M.S. 1988, pp 51 - 75. [B2] - - , A semi-algebraic Brouwer fixed point theorem for real affine spaces, ibid., pp 77 - 82. , A Hopf fixed point theorem for semialgebraic maps. Preprint Stanford [B3] University. [24] - - , Wittrings and K-theory. Rocky Mountain J. Math. 14, 733 - 765 (1984). [Bs] - - , Quotient spaces for semialgebraic equivalence relations. Math. Z. 195, 69 - 78 (1987). [CC] M. C a r r a l , M. Coste, Normal spaces and their dimensions. J. Pure Appl. Algebra 30, 227 - 235 (1983). [CR] M. Coste, M.-F. C o s t e - R o y , Le spectre rdel d'un anneau est spatial. C.R.A.S. Paris 290, 91 - 94 (1980). , Le topos ~tale r~el d'un anneau. In: 3 e colloque sur les categories [CR1] • , dedier a Charles Ehresmann, Cahiers de topologie et g~om. diff. 22-1, 19 - 24 (1981). , , La topologie du spectre r~el. In: "Ordered fields and real algebraic [CRy] geometry" (Ed. D.W. Dubois, T. Recio), Contemp. Math. 8, 27 - 59 (1982). [D] H. Delfs, Kohomologie affiner semialgebraischer R£ume. Dissertation Regensburg 1980. [D,] - - , The homotopy axiom in semialgebraic cohomology. J. reine u. angew. Math. 355, 108 - 128 (1985). - - , Homology of locally semialgebraic spaces. Lecture Notes in Math. 1484. Springer, to appear ( ~ Habilitationsschrift Regensburg 1984).

35

[D3] - - ,

Semialgebraic Borel-Moore homology, Rocky Mountain J. Math. 14, 987 990 (1984). [DK] H. Delfs, M. K n e b u s c h , Semialgebralc topology over a real closed field Ih Basic theory of semialgebraic spaces. Math. Z. 178, 175 - 213 (1981). [DK,] - - , , On the homology of algebraic varieties over real closed fields. J. reine u. angew. Math. 335, 122 - 163 (1982). [DK2] , --, "Locally semialgebraic spaces". Lecture Notes Math. 1173 (1985). [DKa] , , An introduction to locally semialgebraic spaces. Rocky Mountain J. Math. 14, 945 - 963 (1984). [De] C.N. Delzell, A finiteness theorem for open semialgebraic sets, with applications to Hilbert's 17th problem. In: "Ordered fields and real algebraic geometry" (Ed. D.W. Dubois, T. Recio), Contemporary Math. 8, 79 - 97 (1982). [DvdD] J. D e n e f , L. van den Dries, p-adic and real subanalytic sets. Ann. Math. 128, 79 - 138 (1988). [ES] S. E i l e n b e r g , N. S t e e n r o d , "Foundations of algebraic topology". Princeton Univ. Press 1952. [ci] B. Giesecke, Simpliziale Zerlegung abzghlbarer analytischer Rgume. Math. Z. 83, 177- 213 (1964). [EGA II A. G r o t h e n d i e c k , J. D i e u d o n n 6 , "l~lements de g6om~trie alg6brique h Le language des sch~ma.s. Publ. Math. N ° 4, Inst. Hautes Etudes Sci. 1960. [EGA I*] , , "Elements de gdom~trie alg~brique I". Grundlehren math. Wiss. 166, Springer 1971. [Hal R. H a r d t , Semialgebraic local triviality in semi-algebraic mappings. Amer. J. Math. 102, 291 - 302 (1980). [Ho] M. H o c h s t e r , Prime ideal structure in commutative rings. Trans. Amer. Math. Soc. 142, 43 - 60 (1969). [HM] J. H o u d e b i n e , L. Mah6, S~paration des composantes connexes r~elles dans le cas des vari~t6s projectives. In: "Gdometrie alg4brique r6elle et formes quadratiques (Ed. J.-L. Colliot-Thdl~ne, M. Coste, L. Mahd, M.F. Roy), pp 358 - 370. Lecture Notes Math. 959, Springer 1982. [Hu] R. H u b e r , "Bewertungsspektrum und rigide Geometrie". Habilitationsschrift, Regensburg 1990. [Hul] - - , Semirigide Funktionen. Preprint Regensburg 1990. [HuK] R. H u b e r , M. K n e b u s c h , On valuation spectra. To appear in the proceedings of the RAGSQUAD-seminar Berkeley 1990/91, Amer. Math. Soc. [HuS] R. H u b e r , C. S c h e i d e r e r , A relative notion of local completeness in semialgebraic geometry. Arch. Math. 53, 571 - 584 (1989). [El M. K n e b u s c h , An invitation to real spectra. In: "Quadratic and hermitian forms" (Ed. C.R. Riehm, I. Hambleton), Canadian Math. Soc. Conference Proceedings 4, 51 - 105 (1984). [KI] - - , Some open problems. In: "Conference on quadratic forms - 1976" (Ed. G. Orzech). Queen's Papers Pure Appl. Math. 46, 361 - 370 (1977). [K21 - - , "Weakly semialgebralc spaces". Lecture Notes Math. 1367, Springer 1989.

36 [K3] - - , "Semialgebraic fibrations and covering maps". In preparation. [KS] M. K n e b u s c h , C. S c h e i d e r e r , "Einffihrung in die reelle Algebra", Vieweg,

[L] [M] [May]

[Mi] [P] [PiS] [Rol [RI

[ 11 [Sch]

iS] [Si]

IS3] [S4] [Spa] [Sw]

Braunschweig-Wiesbaden 1989. S. Lojasiewicz, Triangulation of semi-analytic sets. Ann. Scuola Norm. Sup. Pisa (3) 18, 449- 474 (1964). L. Mah~, Signatures et composantes connexes. Math. Ann. 260, 191 - 210 (1982). J.P. M a y , "Simplicial objects in algebraic topology". Van Nostrand Math. Studies 11, Van Nostrand 1967. J. M i l n o r , The geometric realization of a semi-simplicial complex. Annals of Math. 65, 357 - 362 (1957). M. P r e c h t e l , "Endliche semialgebraische R~.ume". Diplomarbeit Regensburg 1988. A. Pillay, C. S t e i n h o r n , Definable sets in ordered structures, I. Trans. Amer. Math. Soc. 295, 565 - 592 (1986). R. Robson, Embedding semialgebraic spaces. Math. Z. 183, 365 - 370 (1983). M . - F . Roy, Faisceau structural sur le spectre r~el et fonctions de Nash. In: "G~om~trie alg~brique r~elle et formes quadratiques", pp. 406 - 432. Lecture Notes Math. 959, Springer 1982. J . M . Ruiz, Basic properties of real analytic and semianatytic germs, Publ. Inst. Recherche Math. Rennes 4, 29 - 51 (1986). -- , On the real spectrum of a ring of global analytic functions, ibid., 84 - 95. C. S c h e l d e r e r , Quotients of semi-algebraic spaces. Math. Z. 201, 249 - 271 (1989). N. Schwartz, "Real closed spaces". Habilitationsschrift Ludwig-MaximiliansUniversit£t Mfinchen 1984. -- , "The basic theory of real closed spaces". Memoirs Amer. Math. Soc. 77, N ° 397 (1989). -- , "The basic theory of real closed spaces". Regensburger Mathematische Schriften 15, Fakult~t ffir Mathematik der Universit~t Regensburg 1987. , Inverse real closed spaces. Illinois J. Math., to appear soon. , Open morphisms of real closed spaces. Rocky Mountain J. Math. 19, 913 - 939 (1989). E . H . Spanier, "Algebraic topology". McGraw-Hill t966. R . M . Switzer~ "Algebraic topology - Homotopy and homology". Grundlehren math. Wiss. 212, Springer 1975.

A d d r e s s : Fakult£t f/Jr Mathematik, Universit£t Regensburg, Universit£tsstr. 31, D-8400 Regensburg

Algebraic Geometric Methods in Real Algebraic Geometry R. P a r i m a l a

This article is based on the talk given at the conference. It is a brief overview of certain recent developments in the study of the topology of a real algebraic variety through a combination of techniques from algebraic geometry and the theory of quadratic forms. Our underlying theme is to illustrate how the higher dimensional variations of some basic theorems of Witt ([W]) for real curves have encompassed a wide range of mathematical activities in the last decade. Thus, this article is in no sense a comprehensive survey of the results linking algebraic geometry and real algebraic geometry. For instance, an area left out is the construction of real moduli spaces. Let X be a real variety and X(JT~) its set of/R-rational points, equipped with the strong (Euclidean) topology. We consider principally two classes of maps which relate geometric invariants of X to topological invariaats of X(~). I. Suppose X is smooth and X ( ~ ) compact. We discuss the cycle map from the Chow group of X to the homology groups of X ( ~ ) with 2~/2 coefficients, its kernel and image. II. We discuss two kinds of fibre maps: 1) The signature map from the Witt group of X to the group of 2~-valued continuous functions on X ( ~ ) . 2) The "mod-2" signature map from the unramified 6tale cohomology of X to the cohomology of X(L~). The study of these maps for real curves is contained in the above mentioned paper of Witt.

38 We fix the following notation for the rest of this article. By a variety X over a field k, we mean a reduced irreducible scheme over k and by a subvariety a closed subscheme which is reduced and irreducible.

1

T h e cycle m a p

Let X be a nonsingular variety o v e r / R such that X ( ~ ) is compact. Let Zk(X) denote the group of )~-cycles on X; namely the free abelian group on k-dimensional sub-varieties of X. Let Pk(X) denote the subgroup generated by cycles which are rationally equivalent to zero. Let z*kh(x) denote the subgroup of Zk(X) generated by "thin cycles", i.e. z~h(x) is generated by k-dimensional subvarieties Y of X with Y(/R) not Zariski dense in Y. The cycle map clk: Zk(X) ~ H~,(X(J~),2g/2) is defined as follows: Let Y be a k-dimensional subvariety of X. If Y is thin, then we define clk(Y) = O. Otherwise, let cry E Hk(Y(J~£),~/2) be the fundamental class of Y ( ~ ) . Then c4(Y) = i(ay), where i : Hk(Y(JT~), ~ / 2 ) --, Hk(X(~), 2g/2) is induced by the inclusion Y(/R) ---, X ( ~ ) (cf. [Bor-H] 5.12). The image of the cycle map is called the group of algebraic homology cla88e~, denoted H."tg(X(~), 2~/2). If p: H*( X (1R), 2~/2 ) ~ H.( X (1R), 2g/2 ) denotes the Poincar@ duality isomorphism, then p-l(H~.tg(X(~),~,/2)) = H*la(X(~),2g,/2 ) is called the

group of algebraic cohomology classe~. We first look at the ken,tel of the cycle map. This map respects rational equivalence ([Sor-H] 5.13) so that clk(Pk(X)) = 0. Further, by definition, c4(Z~h(x)) = 0. The question whether the kernel of the cycle map is precisely the group Pk(X) + z~h(x) for smooth projective varieties X over remained open until recently. We first indicate how the results of Witt imply that for a smooth projective real curve X, kernel of clo is equal to Po(Z) + z~h(x). Let X(/R) = UCi, {Ci} denoting the set of connected components of X ( ~ ) . Each Ci is topologically a circle. Suppose we axe given an even number of points {P1, P 2 , . . . , P2,~} on some Ci. Witt proves that there is a rational function f E K~(X), regular on X(/R) which changes sign on Ci exactly at the points Pi. Knebusch ([K]x 4.5 a, p. 61) refines this into the following theorem: There exists a rational function f E / R ( X ) ,

39

regular on X(K/) such that divf=

Pi+D,

~ l d + 1, where X is a smooth quasiprojective variety of dimension d over a real closed field R. In particular, for i > O,n >__d + 1, H i ( X , ~ n) ~ H~(X,T.2~,/2). The beautiful remark of Scheiderer (cf [S]1) is, that for any variety X over a real closed field R, even possibly singular, RqT.(2~/2) = 0, so that Hi(X, ~.2~/2) ~_ H i (Spec~X, Z~/2) = H~,(X i (R), 2~/2) (cf [D-K]). Specializing R to the field of real numbers, we obtain an isomorphism h,~: H'(X,7-/'~) ~ H'(X(~),2~/2), for n k d + 1, i _> 0. In particular, if X is a smooth surface over ~ with X(/R) compact, and s is the number of connected components of Z ( ~ ) , the degeneracy of the Bloch-Ogus spectral sequence ([C-P]) gives, for i _> 4,

Hi(X) ~_ H°(X, 7-14)o g 1 ( x , ~ 3 ) o H2(X, 7-t2) ,.,

(cf [C-P] 3.2.1 and [S]1).

e

47

The following observation of Colliot-Th61~ne [C] allows us to dispense with the smoothness assumption on X in the above theorems. Cox in [Co] proves that, for any variety X over/R, there is a natural isomorphism

H:t(X, 2g/2) ~m

G

O__2 d + 1. This yields an isomorphism 7-/r~ "~ T.2~/2, for m >__2 d + 1, in view of the above mentioned remarks of Scheiderer. Since for an affine scheme X, and for m >__d + 1 , H ~ ( X ¢ , 2 ~ / 2 ) = 0, the Kummer sequence yields a long exact sequence in etale cohomology:

H'~(X~T,z~/2) ..-, Hm(X, 2~/2) uS.-;' Hm+l(X,z~/2)---, H'~+I(X(T,2~/2), which, for m > d + 1, yields an isomorphism 7"/m -% 7-/m+] of sheaves and which commutes with h, so that one has hm : 7-/"* & ~.2~/2, for m > d + 1. In [S]2, Scheiderer tries to interpret the results of Cox in the general setting of the real spectrum of any scheme. This leads to a natural isomorprism lJ.m H'~(X, 2~) '2_@ Hi(Spe%Z, 2~/2), -'-+n i>0

for any affine scheme X in which 2 is invertible. This places in an abstract setting, the results of Arason-Elman-Jacob for fields ([A-E-J], 2.4). One of the interesting consequences of this analysis is that the connected components of the real spectrum of any commutative ring are separated by the @tale cohomology classes. A c k n o w l e d g e m e n t s . I thank J.-L. Colliot-Th6l~ne for his help in the preparation of this talk. I also thank M.A. Knus and J. Barge for critically reading the manuscript.

References [A-K] S. Akbulut, H. King, A relative Nash theorem, Trans. Amer. Math. Soc. 267 (2), 465-481 (1981). [Ar-E-J] J.Kr.Arason, R.Elman, B.Jacob, The graded Witt ring and Galois cohomology, CMS.Conf. Proc. Vol.4, Providence: AMS, 1984, 17-50.

48

lay]

G. Ayoub, Le groupe de Witt d'une surface r6elle. Comment. Math. Helv. 62, 74-105 (1987).

[B]

H. Bass, Unitary algebraic K-theory, Hermitian K-theory and Geometric Applications, Battelle Institute Conference 1972. Algebraic K-theory III, SLN 34:3, 1973, pp. 57-265.

[Ba-O] J. Barge, M. Ojanguren. Fibr6s a/g6briques sur une surface r6elle. Comment. Math. Helv. 62, 616-629 (1987). [Be-D l R, Benedetti, M. Dedo, Counterexamples to representing homology classes by real algebraic subvarities up to homeomorphism. Compositio Math. 53, 143-151 (1984). [Be-T]1 R. Benedetti, A. Tog:noli, Sur les fibr~s vectoriels alg~briques rfiels. C.R. Acad. Sc. Paris, t. 287, S~rie A-831-833 (1978). [Be-T]2 R. Benedetti, A. Tognoli, Remarks and counterexamples in the theory of real algebraic vector bundles and cycles, Gfiom~trie alg~brique r~elle et formes quadratiques, Rennes 1981, SLN 959, pp. 198-211.

[Bo]

J. Bochnak, Topology of real analytic sets - Some open problems. Rennes 1981, SLN 959, pp. 212-217.

[Bo-B-K] J. Bochnak, M. Buchner and W. Kucharz, Vector bundles on real algebraic varieties, K.-theory, 3, 271-298 (1990). [Bo-C-R] J. Bochnak, M. Coste, M-F.Roy, G6om6trie alg6brique r6elle, Ergeb. der Math. 3. Folge, Bd. 12, Springer 1987. [Bo-K]I J. Bochnak, W. Kucharz, Representation of homotopy classes by algebraic mappings, J. Reine. Angew. Math. 377, 159-169 (1987). [Bo-K]2 J. Bochnak, W. Kucharz, On real algebraic morphisms into even dimensional spheres, Ann. of Math. 128, 415-433 (1988). [Bo-K]3 J. Bochnak, W. Kurcharz, K-theory of real algebraic surfaces and threefolds, Math. Proc. Camb. Phil. Soc. 106, 471-480 (1989).

49

[Bo-K]4 J. Bochnak, W. Kucharz, Algebraic models of smooth manifolds, Invent. Math. 588-611 (1989).

[Bor-H]

A. Borel, A. Haefliger, La classe d' homologie fondamentale d'un espa~e analytique. Bull. Soc. Math. France, 89, 461-513 (1961).

[Br]l L. BrScker, Reelle Divisoren, Arch. Math. 35, 140-143 (1980). [Br]2 L. BrScker, Minimale Erzeugung von Positivbereich. Geom. Dedication 16. 335-350 (1984).

[Br]

L. BrScker, Spaces of orderings and semialgebraic sets, Quadratic and hermitian forms, CMS Conf. Proc. Vol. 4, Providence: AMS, 1984, pp. 231-248.

[Bru] G.W. Brumfiel, Witt rings and K-theory. Math. 14, 733-765 (1984).

[c]

Rocky Mountain J. of

J.-L. Colliot-Th61~ne, Abstract of a talk at OberwoKach, 1990.

[c-i]

J.-L. Coltiot-Th@l~ne, Ischebeck, L' @quivalence rationnelle sur les cycles de dimension z@ro des vari@t@s alg@briques r@etles. C.R. Acad. Sci. Paris, Serie I, 292, 723-725 (1981).

[c-P]

J.-L. Colliot-Th@l~ne, R. Parimala, Real components of algebraic varieties and @tale cohomology, Invent. Math. 101, 81-99 (1990).

[c-s]

J.-L. Colliot-Th@l&ne, J.-J. Sansuc, Fibr@s quadratiques et composantes connexes r@elles, Math. Ann. 244, 105-134 (1979).

[Co]

D.A. Cox, The @tale homotopy type of varieties over ~ , Proc. Amer. Math. Soc. 76, 17-22 (1979).

[D-K] H. Delfs, M. Knebusch, Semialgebraic topology over a real closed field II: Basic theory of semialgebraic spaces. Math. Z. 178, 175-213

(1981).

IF]

R. Fossum, Vector bundles over spheres are algebraic. Invent. Math. 8, 222-225 (1969).

50

[G-al

A.V. Geramita, L.G. Roberts, Algebraic vector bundles on projective space. Invent. Math. 10, 298-304 (1970).

[Ge]l

W.D. Geyer, Ein algebraischer Beweis des Satzes yon Weichold fiber reelle algebraische FunktionenkSrper. In : Algebraische Zahlentheorie, Oberwolfach, 1964.

[Ge]

W.D. Geyer, Dualit/~t bei abelschen Varietiiten fiber reelle abgeschlossen K5rpern, J. reine, angew. Math. 293-294, 62-66 (1977).

[H-MI

J. Houdebine, L. Mah6, S6paration des composantes connexs r6elles dans le cas des variet6s projectives, G6om6trie alg6brique r6elle et formes quadratiques, Rennes 1981, SLN 959, p. 358-370.

[I-Sc]

F. Ischebeck, H-W. Schfilting, Rational and homological equivalence for real cycles, Invent. Math. 94, 307-316 (1988).

[K]I

M. Knebusch, On algebraic curves over real closed fields I. Math. Z.

150, 49-70 (1976). M. Knebusch, On algebraic curves over real closed fields I1. Math. Z.

151, 189-205 (1976).

[KI3

M. Knebuseh, Some open problems, Queen's Papers in Pure and Appl. Math. 46, 361-370 (1977).

[M]I

L. Mah6, Signatures et componentes connexes. Math. 191-210 (1982).

[MI2

L. Mah6, Th6or&me de Pfister pour les vari6t6s et anneaux de Witt r6duits, Invent. Math. 85, 53-72 (1986).

Ann. 260,

[O-P-S] M. Ojanguren, R. Parimala and R. Sridharan, Symplectic bundles over affine surfaces, Comment. Math. Helv. 61,491-500 (1986).

[PI

W. Pardon, A relation between Witt groups and zero cycles in a regular ring, Algebraic K-theory, Geometry and Analysis, SLN 1046, 1984, pp. 261-328.

51

R. Parimala, Witt groups of affine threefolds, Duke Math. J. 57, 947-954 (1988).

[Sll

C. Scheiderer, A remark on the paper, Real components of algebraic varieties and 6tale cohomology by J.-L. Colliot-Th616ne, R. Parimala, Preprint, 1990.

[S]2 C. Scheiderer, Real and 6tale cohomology, Preprint, 1991. [Sc] H.W. Schfilting, Algebraische und topologische reelle Zykeln unter birationalen Transformationen, Math. Ann. 272, 441-448 (1985).

[si] R.

Silhol, Cohomologie de Galois et cohomotogie des vari6t6s alg6briques r6elles; Applications aux surfaces rationnelles, Bull. Soc. Math. France 115, 107-125 (1987).

[Su] R. Sujatha, Witt group of real projective surfaces, Math. Ann. 288, 89-101 (1990). [Sw]l R. Swan, Vector bundles and projective modules. Math. Soc. 105, 264-277 (1962).

Trans.

Amer.

[Sw]2 R. Swan, Topological examples of projective modules, Trans. Amer. Math. Soc. 230, 201-234 (1977).

[Sw]3

R. Swan, K-theory of quadric hypersurfaces, Arm. of Math. 122, 113-154 (1985).

IT]

A. Tognoli, Su una congettura di Nash. Ann. Scuola Norm. Sup. Pisa 27, 167-185 (1973).

[w]

E. Witt, Zerlegung reeller algebraischer Funktionen in Quadrate, SchiefkSrper fiber reelle Funktionenk6rper. J. Reine Angew. Math. 171, 4-11 (1934).

School of Mathematics Tara Institute of Fundamental Research Bombay- 400005 India. E-mall : [email protected]

ON THE CLASSIFICATION OF DECOMPOSING PLANE ALGEBRAIC CURVES G.M. Polotovskii (Nizhni~ Novgorod, USSR} This r e p o r t represents a survey of results on the arrangement of curves which are decomposed into product of curves "in general position"

in

real

projective

plane

P2 ~,

The

problem

consideration is in close connection with the classical

under

question

of the Hilbert 18-th problem on the topology and

arrangement

of

non-singular algebraic varieties. Note

some

we

that

receive, in fact, the classification of the

in

cases

non-singular

affine

plane algebraic curves (see sections I I I - VI below}. Let's denote the non-singular plane real curve of degree m by symbol C m, and the set of points of this curve in P0~a by the same symbol. Let's denote the union of transversal intersecting curves C m and Cn ( ~ n ) are

by C muCn *>

equivalent

if

Two of such curves C m u C n

topological

triples

( R ~ , C ~ u C ~ , C '} are homeomorphic,

or

m=n

and

C'm u c'n

(R'J~a,OmUCn,Cm} and

C=C"

and

and

6 =C

(i.e. curves differ only by the sequence of writing the factors}. A

De/~n~t~on. The c u r v e Cmu C n is called the

curue if C a is (M-9}-curve, Cn

is

(M-~}-decompos£n~

(M-~)-curve,

the

number

of

A

common real points of C m and CniS equal to mn-r and p+@+r=~. The similar definitions are used for the case C m U C n U C k ; particular,

these

curves

have

only

non-degenerate

in

two-fold

singular points. The problem of classification of decomposing curves is (as in the case of non-singular curves} most interesting and complicated ~> The notations of the factors are distinguished m-n by "the hat",

in

the

case

53 in

"maximalities

cases".

We

shall

M-,(M-I)-

discuss

(M-£)-cases mainly.

and

A

I. C m u C nin cases m+nt.

It

is

examples

of

(s, t)

in

it

consider

evident that s+~ is even and 2 1, to be a sum of 2n-th powers is not an elementary property in the coefficients of f , contrary to the case n = 1! Another way to put this is that in representations fro(x) = x

+ mx

+ 1 =

\ hr. /

'

i=1

for positive integers m, one cannot keep the Nm and the degrees of the hm bounded at the same time for ra -+ ~ , although this is possible for the Nm alone. The substitution criterion was later refined in [Prestel 87], where the degrees of the Lanrent polynomials were bounded by computable functions. At this point I want to digress for a moment, to mention recent work on R(X), the field of rational functions in one variable. Becker had already shown P4 (R(X)) < 36 [Beeker 82b] where P4 is the Pythagoras (or Waring) number for sums of fourth powers. This was improved to 24 in [Schmid 88]. But now in {Choi Lain Prestel Rezniek 91}, P4(R(X)) _< 6 is shown! Also for higher exponents the upper bounds from [Becker 82b] are slightly improved. This uses a new characterization of ~R(X) 2n. In addition to the obvious necessary conditions for a polynomial f (initial coefficient positive, multiplicities of real roots and degree divisible by 2n) - - which by Beeker's criterion suffice if R is archimedean --, a new condition is needed. If one partitions the upper half plane of R(v/'L'i) into equivalence classes, by defining two points to be equivalent if their hyperbolic cross-ratio and its inverse are bounded in absolute value by some integer, the condition says that the number of nonreal roots of f lying in each equivalence class must be a multiple of n. Obviously, this condition is implied by the others if R is archimedean. If one keeps these cross-ratios in each class bounded (and fixes the degree of the polynomial), then also the complexity of representations as sums of 2n-th powers can be bounded. A related result had been proved before over IR in [Prestel Bradley 89].

86 I conclude this part with a series of beautiful results about ideals in real holomorphy rings. Here once more geometrical methods are used (coming from algebraic geometry and topology) to obtain purely algebraic theorems. There was a long-standing question by Gilmer, who had asked if any finitely generated ideal in a Priifer domain has actually two generators. Sch/ilting disproved this by taking the real holomorphy ring H(F) of F = R(X, Y) and showing that the fractional ideal (1, X, Y) is not generated by two elements [Sch/ilting 79]. Let #(H) be the minimal number of generators sufficient for all finitely generated ideals of H. Since H(F) has Krull dimension d, for F/]R a ddimensional real function field, a general result of Heitmann gives #(H(F)) < d + 1. After SchNting had given his counterexample in 1979, people began to wonder whether equality might hold in general. In the end, this turned out to be true, but the general result was established only quite recently: After R. Swan had shown equality for F = R ( X 1 , . . . , Xd) [Swan 84] and Schiilting had proved # (H(F)) = 3 for all function fields of real surfaces {Schiilting 86b}, W. Kucharz settled the general case in [Kucharz 89a], showing # (H(F)) = d + 1 for F / N real of dimension d. As Becker had observed much 2n earlier, this implies that there are f l , . . - , f d + l C F such that f~" + . . . + f~+l is not a sum of d 2n-th powers for any n _> 1. If F / R is purely transcendental one can take 1 , X 1 , . . . , X d (this came out already from Swan's proof), and thereby obtain a generalization of a classical theorem of Cassels! Kucharz's proof uses the description of H(F) as those rational functions which are defined on the real points of some smooth projective model. Since his techniques build upon the ideas of Schiilting and Swan, I will skip over the details of their work. Kucharz proceeds roughly like this: A fractional ideal of H(F) corresponds to a line bundle on some smooth projective model X of FIR. Blowing up X in a real point x0 one gets ~r : X I --, X. For the class a of the exceptional fibre E = 7r-l(x0) one shows that a d 7~ 0 in H d (X'(R), Z/2). If the fractional ideal corresponding to E were generated by d elements, a calculation with Stiefel-Whitney classes shows that there would be a multi-blow-up a : X " --* X' for which a d maps to zero in H d ( x " ( R ) , Z / 2 ) . But one can show that the map in cohomology induced by such a a is always injective. Actually, Kucharz's paper provides much more information than just it(H) = d + 1. For example, Sch/ilting had studied the binary class group BE of F, i.e. the quotient of the class group of H(F) by the classes of those ideals which admit two generators. This group plays also a role in Becker-Rosenberg's theory of reduced Witt rings of higher level {Becker Rosenberg 85b}. Sch/ilting had found that BE ~ (Z/2) s if d = 2, and IBFI = oo for d > 3, where s is the number of connected components of a smooth projective model {Schiilting 86b}. These results are widely generalized by Kucharz to other quotients of the class group. The result it(H) = d + 1 was later generalized to relative real holomorphy rings H = H ( F / R ) over arbitrary real closed fields R {Kucharz 89c}. [Buchner Kucharz 89] also studied relative real holomorphy rings H(F/V), where V is an affine irreducible smooth R-variety and F its function field. Here one has it(H(F/V)) < dim V if and only if V(R) has no compact connected components. Also this is a generalization of the above main result of Kucharz.

87 3. S e l e c t e d a p p l i c a t i o n s o f real a l g e b r a to g e o m e t r y I want to start with a few words on the work of J.M. Ruiz on global semianalytic geometry. He not only made available real spectrum methods and applied them with great success; rather, on his way to this goal, he established remarkable results of algebraic nature which can certainly claim independent interest. The key result, which makes the real spectrum a highly useful tool in semianalytic geometry, is an Artin-Lang property for global analytic functions on a compact analytic set X. It means that the globally semianalytic subsets of X correspond bijectively, via an operator "tilda" which preserves set-theoretic operations, to the constructible sets in the real spectrum of O(X). The Artin-Lang property in turn rests on the study of the real spectrum formal fibres of local analytic rings. The basic result, whose use pervades the whole theory, is the following extension theorem of Ruiz. Suppose A is an excellent local domain and a is an ordering of its field of fractions. If ol makes the maximal ideal of A convex, then it extends to an ordering of the total ring of quotients of A, the completion of A. Other forms of this result are a formal curve selection lemma and a real dimension theorem, both for excellent rings. See [Ruiz 85b], [Ruiz 86a+b]. As an immediate application, his theorem allowed Ruiz to solve the 17th Hilbert problem for meromorphic functions in the amrmative, on a compact irreducible analytic set [toc.cit.]. Before, this had been known only in very special cases. Another application is the real Nullstellensatz for global analytic functions. It must be pointed out that the same results for compact analytic sets (ArtinLa~g property, Hilbert 17, real Nullstellensatz) have been proved independently by P. Jaworski. In his approach, part of the algebraic reasoning of Ruiz is replaced by the use of Hironaka's theorems [Jaworski 86]. But more effort was needed to establish basic properties of well-behavedness for the real spectra of (local or global) rings of analytic functions. Ruiz showed that the class of excellent rings is a suitable environment: It contains the rings arising in analytic geometry, and on the other hand has "excellent" properties also for the real spectrum. Apart from the above mentioned theorem (which is central for many arguments), this includes dimension theory, the going-down property for regular homomorphisms, constructibility of closures of constructible sets and (under suitable conditions) of connected components. See [Ruiz 86b], [Ruiz 89a+b], [Alonso Andradas 87], [Andradas Brhcker Ruiz 88]. By Artin-Lang, this gives corresponding geometric theorems: If X is a compact analytic set and Z is a globally semianalytic subset of X, then closure and connected components of Z are again globally semianalytic; and there is a "Finiteness Theorem" saying that the operator tilda preserves closures [Ruiz 86b], [Ruiz 88]. (The compactness hypothesis on X is weakened in [Ruiz 90a].) Another application will be mentioned later in this talk. A recent very nice result is the characterization of sums of 2n-th powers of meromorphic functions, on a compact analytic set X, by their values on curve germs [Ruiz 90b] (in the case of analytic surfaces this was also proved in [Kucharz 89b]): A necessary and sufficient condition for f to lie in Efl4(X) 2n is that for every analytic curve germ a : ]-e, e[ ~ X one has f(a(t)) = at2"k+ higher powers, with a > 0, as long as a ~ Sing X and f o a is neither undefined nor identically zero. This is a remarkable result, and is of course very much reminiscent of Prestel's substitution

88 criterion mentioned before. Tim similarity is "explained" and embedded into a more general context in {Ruiz 91}. In the remainder of this talk I want to record some progress in questions on the complexity of systems of inequalities. A common feature of the following is that it rests on t h e - apparently completely abstract - - theory of preorderings and fans in fields, and on its even more abstract generalization, the spaces of orderings. I will not go here into the purely algebraic aspects of these matters; it seems that the majority of important results in this theory had already been obtained by the beginning of the 80s, and that since then the applications have been more striking. A noteworthy exception are the important papers [Schwartz 83] and [Marshall 84] on saturation and local stability. The problem is how to describe semialgebraic sets as economically (or "cheaply") as possible, i.e. how to minimize the number of functions in a description, regardless of their complexity. The first major step was done by L. Brhcker [Brhcker 84a]. He considered affine algebraic varieties V over a real closed field R, and what was later called basic open (semialgebraic) sets in V(R); that is, sets which can be written S = {x E V ( R ) : fl(x) > 0 , . . . , f r ( x ) > 0} with polynomials fi C R[V]. The question is, how many inequalities are really needed to present S in this form? Brhcker gave the surprising answer that there is a common upper bound for all basic open sets in V(R), which depends only on d = dim V. In fact, writing s(d) for the least upper bound valid for all d-dimensional varieties, he showed that d(d-2)(d-4)...2 deven, d < s(d) 1 inequalities in each residue field of A, then S itself has a presentation by r inequalities. (The real spectrum of A can even be replaced by any intersection of basic subsets.) Marshall essentially follows Mahd's proof; but where Mah~ argues by induction over the dimension to get a weak representation, Marshall uses a more abstract tool, namely" a local global principle (due originally to [Brbcker 82]) for modules over preorderings in rings. So after all, the former "problem of the stability index" has been settled in a truely elementary and satisfying way, and in the greatest generality, with the help of the machinery of abstract real algebra.

References

In the text, references to articles are by name(s) of author(s) and year. For the difference between curly braces and square brackets see footnote 1 in the text. [GARFQ] Gdomdtrie Algdbrique Rdelle et Formes Quadratiques, Proc. Conf. Rennes 1981, ed. J.-L. Colliot-Thdl~ne, M. Coste, L. Mahd, M.-F. Roy, Lect. Notes Math. 959, Springer, 1982. [OFRAG] Proc. Special Session on Ordered Fields and Real Algebraic Geometry, ed. D.W. Dubois, T. Recio, Contemp. Math. 8, 1982. [RAAG] Real Analytic and Algebraic Geometry. Proc. Conf. Trento 1988, ed. M. Galbiati, A. Tognoll, Lect. Notes Math. 1420, Springer, 1990. M.E. Alonso, C. Andradas: [87] Real spectra of complete local rings. Manuscripta math. 58, 155-177 (1987). C. Andradas: [85] Real places in function fields. Comm. Algebra 13, 1151-1169 (1985). [89] Specialization chains of real valuation rings. J. Algebra 124, 437-446 (1989). C. Andradas, L. Br6cker, J.M. Ruiz: [88] Minimal generation of basic open semianalytic sets. Invent. math. 92,409-430 (1988). J.Kr. Arason, A. Pfister: [82] Quadratische Formen fiber affinen Algebren und ein Mgebraischer Beweis des Satzes yon Borsuk-Ulam. J. reine angew. Math. 331,181-184 (1982). S.M. Barton: [88] The real spectrum of higher level of a commutative ring. Ph.D. thesis, Cornell Univ., Ithaca, 1988. E. Becker: [82a] Valuations and real places in the theory of formally real fields. In [GARFQ], pp. 1-40.

92 [82b] [84]

The real holomorphy ring and sums of 2n-th powers. In [GARFQ], pp. 139-181. Extended Artin-Schreier theory of fields. Rocky Mountain J. Math. 14, 881-897 (1984). E. Becker, R. Bert, F. Delon, D. Gondard: {90} Hilbert's 17-th problem for 8urns of 2n-th powers. Preprint 1990. E. Becker, D. Gondard: [89] On rings admitting orderings and 2-primary chains of orderings of higher level. Manuscripta math. 65, 63-82 (1989). E. Becket, J. Harman, A. Rosenberg: [82] Signatures of fields and extension theory. J. reine angew. Math. 330, 53-75 (1982). E. Becket, B. Jacob: [85] Rational points on algebraic varieties over a generalized real closed field: A model theoretic approach. J. reine angew. Math. 357, 77-95 (1985). E. Becker, A. Rosenberg: [85a] Reduced forms and reduced Witt rings of higher level. J. Algebra 92,477-503 (1985). {85b} On the structure of reduced Witt rings of higher level. Preprint 1985 (unpublished). E. Becket, N. Schwartz: {89} Reduced Witt rings of infinite level. Preprint 1989. R. Berr: Reelle algebraische Geometrie hSherer Stufe. Dissertation, Ludwig-Maximilians-Uni[89] versit~it Mfinchen, 1989. {90} Sums of mixed powers in fields and orderings of prescribed level. Preprint 1990. {91} The p-real spectrum of a commutative ring. Preprint 1991. L. BrScker: [74] Zur Theorie der quadratischen Formen fiber formal reellen KSrpern. Math. Ann. 210, 233-256 (1974). [82] Positivbereiche in kommutativen Ringen. Abh. Math. Sem. Univ. Hamburg 52, 170178 (1982). [84a] Minimale Erzeugung yon Positivbereichen. Geom. Dedicata 16, 335-350 (1984). [84b] Spaces of orderings and semialgebraic sets. In: Quadratic and Hermitian Forms, Conf. Hamilton 1983, ed. I. Hambleton, C.R. Riehm, Canad. Math. Soc. Conf. Proc. 4, 1984, pp. 231-248. {85} Description of semialgebraic sets by few polynomials. Lecture CIMPA, Nice 1985. {87} Characterization of basic semialgebraic sets. Preprint 1987. [88a] On the separation of basic 8emialgebraic sets by polynomials. Manuscripta math. 60, 497-508 (1988). {88b} On basic semi-algebraic sets. Preprint 1988 (to appear Expos. Math.). [90] On the stability index of noetherian rings. In [RAAG], pp. 72-80. L. Br5cker, H.-W. Schiilting: [86] Valuations of function fields from the geometrical point of view. J. reine angew. Math. 365, 12-32 (1986). L. BrScker, G. Stengle: [90] On the Mostowski number. Math. Z. 203, 629-633 (1990). R. Brown: [87a] Real closures of fields at orderings of higher level. Pacific J. Math. 127, 261-279 (1987). [87b] The behavior of chains of orderings under field extensions and places. Pacific J. Math. 127, 281-297 (1987). G. Brumfiel: [82] Some open problems. In [OFRAG], pp. 19-25.

93 M.A. Buchner, W. Kucharz: [89] On relative real holomorphy rings. Manuscripta math. 63, 303-316 (1989). A. Campillo, J.M. Ruiz: [90] Some remarks on pythagorean real curve germs. J. Algebra 128, 271-275 (1990). M.D. Choi, Z.D. Dai, T.Y. Lam, B. Reznick: [82] The Pythagoras number of some affine algebras and local algebras. J. reine angew. Math. 336, 45-82 (1982). M.D. Choi, T.Y. Lam, A. Prestel, B. Reznick: {91} Sums of 2m-th powers of rational functions in one variable over real closed fields. Preprint 1991. M.D. Choi, T.Y. Lam, B. Reznick: [87] Even symmetric sextics. Math. Z. 195,559-580 (1987). [91] Positive sextics and Schur's inequalities. J. Algebra 141, 36-77 (1991). J.-L. Cotliot-Th~l~ne: [86] Appendix to: K. Kato, A Hasse principle for two dimensional global fields. J. reine angew. Math. 366, 181-183 (1986). J.-L. Colliot-Th~l~ne, U. Jannsen: [91] Sommes de carr~s dans les corps de fonctions. C. R. Acad. Sci. Paris 312, sdr. I, 759-762 (1991). Z.D. Dai, T.Y. Lam: [84] Levels in algebra and topology. Comment. Math. Helvetici 59, 376-424 (1984). Z.D. Dai, T.Y. Lam, R.J. Milgram: [81] Applications of topology to problems on sums of squares. Ens. Math. (2) 27, 277-283 (1981). Z.D. Dai, T.Y. Lain, C.K. Peng: [80] Levels in algebra and topology. Bull. Am. Math. Soc. (N.S.) 3, 845-848 (1980). F. Delon: [86] Corps et anneaux de Rolle. Proc. Am. Math. Soc. 97, 315-319 (1986). C.N. Delzell: [823] A finiteness theorem for open semi-algebraic sets, with applications to Hilbert's 17 th problem. In [OFRAG], pp. 79-97. [82b] Case distinctions are necessary for representing polynomials as sums of squares. In: Proc. Herbrand Syrup., Logic Coll. 1981, ed. J. Stern, North Holland, 1982, pp. 87103. [84] A continuous, constructive solution to Hilbert's 17th problem. Invent. math. 76, 365-384 (1984). M. Dickmann: [88] The model theory of chain closed fields. J. Symb. Logic 53, 73-82 (1988). E. Fortuna, M. Galbiati: [89] S~paration de semi-alg~briques. Geom. Dedicata 32, 211-227 (1989). D. Gondard: [87] Th~orie du premier ordre des corps chMnables et des corps chaine-clos. C. R. Acad. Sci. Paris 304, s~r. I, 463-465 (1987). [90] Chainable fields and real algebraic geometry. In [RAAG], pp. 128-148. J. Harman: [82] Chains of higher level orderings. In [OFRAG], pp. 141-174. B. Jacob: [81] The model theory of generalized real closed fields. J. reine angew. Math. 323,213-220 (1981).

94 P. Jaworski: [86] Extension of orderings on fields of quotients of rings of real analytic functions. Math. Nachr. 125, 329-339 (1986). J.L. Kleinstein, A. Rosenberg: [78] Signatures and semisignatures of abstract Witt rings and Witt rings of semilocal rings. Canad. J. Math. 30, 872-895 (1978). [80] Succinct and representational Witt rings. Pacific J. Math. 86, 99-137 (1980). M. Knebusch: [77] Some open problems. In: Conf. Quadratic Forms, Kingston 1976, ed. G. Orzech, Queen's Papers Pure Appl. Math. 46, 1977, pp. 361-370. [81] On the local theory of signatures and reduced quadratic forms. Abh. Math. Sem. Univ. Hamburg 51,149-195 (1981). [82] An algebraic proof of the Borsuk-Ulam theorem for polynomial mappings. Proc. Am. Math. Soc. 84, 29-32 (1982). W. Kucharz: [89a] Invertible ideals in real holomorphy rings. J. reine angew. Math. 395,171-185 (1989). [89b] Sums of 2n-th powers of real meromorphic functions. Monatsh. Math. 107, 131-136 (1989). {89c} Generating ideals in real holomorphy rings. Preprint 1989. F.-V. Kuhlmann, A. Presteh [84] On places of algebraic function fields. J. reine angew. Math. 353, 181-195 (1984). K.Y. Lam: [84] Topological methods for studying the composition of quadratic forms. In: Quadratic and Hermitian Forms, Conf. Hamilton 1983, ed. I. Hambleton, C.R. Riehm, Canad. Math. Soc. Conf. Proc. 4, 1984, pp. 173-192. [85] Some new results on composition of quadratic forms. Invent. math. 79, 467-474 (1985). K.Y. Lain, P.Y.H. Yiu: [89] Geometry of normed bilinear maps and the 16-square problem. Math. Ann. 284, 437-447 (1989). T.Y. Lain, T.L. Smith: {90} On Yuzvinsky's monomiM pairings. Preprint, Berkeley 1990. L. Mah6: [86] Th@or~me de Pfister pour les varidt@s et anneaux de Witt r~duits. Invent. math. 85, 53-72 (1986). [89a] Une d~monstration 61@mentaire du th~or&me de BrScker-Scheiderer. C. R. Acad. Sci. Paris 309, s@r. I, 613-616 (1989). {89b} On the geometrical stability index of a ring. Lecture, Quadratic Forms Conf., Oberwolfach 1989. [90] Level and Pythagoras number of some geometric rings. Math. Z. 204, 615-629 (1990). M. Marshall: [84] Spaces of orderings: Systems of quadratic forms, local structure, and saturation. Comm. Algebra 12, 723-743 (1984). {89} Minimal generation of constructible sets in the real spectrum of a Noetherian ring. Preprint 1989. {91} Minimal generation of basic sets in the real spectrum of a commutative ring. Preprint 1991. M. Marshall, C. Mulcahy: [90] The Witt ring of a space of signatures. J. Pure Appl. Algebra 67, 179-188 (1990).

95 C. Mulcahy: [88a] An abstract approach to higher level forms and rigidity. Comm. Algebra 16, 577-612 (1988). [88b] A representation theorem for higher level reduced Witt rings. J. Algebra 119, 105-122 (1988). J.J. Ortega: [91] On the Pythagoras number of a real irreducible algebroid curve. Math. Ann. 289, 111-123 (1991). A. Pfister, S. Stolz: [87] On the level of projective spaces. Comment. Math. Helvetici 62, 286-291 (1987). V. Powers: [87] Characterizing reduced Witt rings of higher level. Pacific J. Math. 128, 333-347 (1987). [89] Finite spaces of signatures. Canad. J. Math. 41,808-829 (1989). A. Prestel: [84] Model theory of fields: An application to positive semidefinite polynomials. Mdm. Soc. Math. France (N.S.) 16, 53-65 (1984). [87] Model theory applied to some questions about polynomials. In: Contributions to General Algebra 5, Proc. Salzburg Conf. 1986, Wien 1987, pp. 31-43. A. Prestel, M. Bradley: [89] Representations of a real polynomial f(X) as a sum of 2m-th powers of rational functions. In: Ordered Algebraic Structures, ed. J. Martinez, Kluwer 1989, pp. 197207. B. Reznick: [87] A quantitative version of Hurwitz' theorem on the arithmetic-geometric inequality. J. reine angew. Math. 377, 108-112 (1987). [89] Forms derived from the arithmetic-geometric inequality. Math. Ann. 283, 431-464 (1989). J.M. Ruiz: [84] A note on a separation problem. Arch. Math. 43, 422-426 (1984). [85a] Pythagorean real curve germs. J. Algebra 94, 126-144 (1985). [85b] On Hilbert's 17th problem and real Nullstellensatz for global analytic functions. Math. Z. 190, 447-454 (1985). [86a] C6nes locaux et compl6tions. C. R. Acad. Sci. Paris 302, s6r. I, 67-69 (1986). [86b] On the real spectrum of a ring of global analytic functions. Publ. Inst. Rech. Math. Rennes 4, 84-95 (1986). [88] On the connected components of a global semianalytic set. J. reine angew. Math. 392, 137-144 (1988). [89a] A dimension theorem for real spectra. J. Algebra 124, 271-277 (1989). [89b] A going-down theorem for real spectra. J. Algebra 124, 278-283 (1989). [90a] On the topology of global semianalytic sets. In [RAAG], pp. 237-246. [90b] A characterization of sums of 2nth powers of global meromorphic functions. Proc. Am. Math. Soc. 109, 915-923 (1990). {91} The substitution test for sums of even powers. Preprint 1991. C. Scheiderer: [89] Stability index of real varieties. Invent. math. 97, 467-483 (1989). J. Schmid: [88] Eine Bemerkung zu den hSheren Pythagoraszahten reeller KSrper. Manuscripta math. 61, 195-202 (1988).

96 H.-W. Schfilting: [79] Uber die Erzeugendenanzahl invertierbarer Ideale in Prfiferringen. Comm. Algebra 7, 1331-1349 (1979). [82a1 On real places of a field and their holomorphy ring. Comm. Algebra 10, 1239-1284 (1982). [82b1 Real holomorphy rings in real algebraic geometry. In [GARFQ], pp. 433-442. [86a] Prime divisors on real varieties and valuation theory. J. Algebra 98, 499-514 (1986). {86b} The binary class group of the real holomorphy ring. Preprint 1986. N. Schwartz: [83] Local stability and saturation in spaces of orderings. Canad. J. Math. 35, 454-477 (1983). [84] Chain signatures and real closures. J. reine angew. Math. 347, 1-20 (1984). [87] Chain signatures and higher level Witt rings. J. Algebra 110, 74-107 (1987). D.B. Shapiro: [84] Products of sums of squares. Expos. Math. 2,235-261 (1984). 0. Simon: [89] Aspects quantitatifs de Nullstellensgtze et de Positivstellens£tze. Nombres de Pythagore. Comm. Algebra 17, 637-667 (1989). G. Stengle: [84] A lower bound for the complexity of separating functions. Rocky Mountain J. Math. 14, 925-927 (1984). [89] A measure for semialgebraic sets related to boolean complexity. Math. Ann. 283, 203-209 (1989). S. Stolz: [89] The level of real projective spaces. Comment. Math. Helvetici 64, 661-674 (1989). R.G. Swan: [84] n-generator ideals in Prfifer domains. Pacific J. Math. 111,433-446 (1984). P.Y.H. Yiu: [86] Quadratic forms between spheres and the non-existence of sums of squares formulae. Math. Proc. Camb. Phil. Soc. 100, 493-504 (1986). [87] Sums of squares formulae with integer coefficients. Canad. Math. Bull. 30, 318-324 (1987). [90] On the product of two sums of 16 squares as a sum of squares of integral bilinear forms. Quart. J. Math. Oxford (2) 41,463-500 (1990). S. Yuzvinsky: [81] Orthogonal pairings of euclidean spaces. Michigan Math. J. 28, 131-145 (1981). [84] A series of monomial pairings. Linear Multilin. Algebra 15, 109-119 (1984). G. Zeng: A characterization of preordered fields with the weak Hilbert property. Proc. Am. [881 Math. Soc. 104, 335-342 (1988). [91] On preordered fields related to Hilbert's 17th problem. Math. Z. 206, 145-151 (1991).

TOPOLOGY OF REAL PLANE ALGEBP~IC CURVES E.i. Shustin (Math. Dept., Samara State University, ul. Acad. Pavlova 1, 443011 S~nara, USSR) In this article we'll consider some ideas, methods, results found over the last ten years in topology of real plane algebraic curves. We'll also speak of some unsolved problems. Topology of real curves is expounded in many known surveys ~ , 4, 9, 18, 26, 28, 29~. We'll consider matter not included in them. The text consists of 4 parts: in the first part we speak of methods of getting restrictions, in the second part we speak of real curves constructions, in the third part we speak of connection between algebraic curves and singularities smoothings, in the fourth part we formulate some conjectures. Used below (without refering) classical results in real curves topology are described in ~I, 4, 9, 18, 26, 28, 29J quite completely. Introduction By plane real algebraic curve we mean a homogeneous real polynomial

F C~o~cI~

~m)

(being considered modulo a constant

factor) and the set

Connected non-singulaz' two-sided components of ~ are called ovals. The classification problem of real curves of a given degree can be considered from three points of view: (i) The real cl+~ssification - up to continuous isotopies in ~ p z (i.e. the ovals arrangements classification) (ii) The equivariant one - up to equivariant isotopies in ~p~ connectin~ the given cu~.ves complexifications

(iii) The rigid one - up

to

isotopies consisting of algebra-

ic curves of a given degree. We'll deal with the first problem. I. Restrictions All the kno~m restrictions to an ovals arrangement arise from investigation of various structures on objects connected with a real curve. Such objects are

(i) the pairs (~p2 2 ~ F ) ~

(~P~j C F ) 3

g8

(ii) the pair ('5 ~

S ~= t P ~

4 ; IRF~={eeRP21EF(e)>--D3,

and also the double covering of

~2

branched a t

E=_+, ~F

or at

(~F/~H~) U ~

, if a curve degree is even. The technical means of investigation are algebraic-topological (Smith theory, complex orientations, quadratic form of intersection index in 2-dimensional homology of double covering etc.) or algebraic-geometrical (Bezout's theorem, a position of a curve with respect to special straigh~ lines or conics etc.). Here we'll give the detailed description of one method of getting restrictions (see [15, 24, 28, sec 4.15]), which is based on some topological and geometrical properties of curves. Namely, let F be a real non-singular curve of the even de&Tee p/~ , let p (resp. kW ) denote the number of ovals lying inside even (odd) nmnber of other ovals. Put j---~#- 1 - p -- ~ ; where $ is a curve genre. Let ~ ~ ~ --~ ~ p ~ be the double covering with branching at ~ F The complex conjugation C~H~'~ ~"p~--p #~' p a is covered by tv¢o antihoiomorphic involutions ~ H ~ + 2 ~ ' on ~ . Assume that ~ £ X ~_) is non-orientable. It is well-knovaa [2, 11] that ~ a ( ~ is the unimodalar lattice (without torsion) with respect to intersection index, and it contains the maximal sublattice H ÷ ~ H _ ~ H where

~,

H= { d, e H= ( ~.) l r..o~4j±;.~:-,~ _.

•

"

"~ H= 1', "~ H+ = (~2_ 3 ~ ) / 2

~'

.

~'

--

.4"

+ p -in,

z4H_ = (~"-sm)/Z * 2 -p+~;

6"*/H)= -/, 6--(H)=O: ~-+(~)=0~ 0,..., f , ( x ) > O} where s < dim(X). On the other hand the problem of characterizing basic open sets in geometric terms seems to be wide open. First of all, there is an immediate necessary condition for a semialgebraic open set S to be basic; namely, that S does not meet the Zariski closure of its usual boundary. If this is the case, a theorem of BrScker says that S is basic if and only if its intersection S N Y with any irreducible subvariety Y of X is generically basic, which means that S n Y \ Z is basic for some proper subvariety Z C Y. This brings the problem to the birational setting, or in other words to the fields K:(Y) of rational functions of the Y's. Then using the so-called tilda operation, which relates semialgebraic subsets of Y and constructible subsets of the space of orderings of the field K:(Y), the theory of fans can be used to decide whether S N Y is or not generically basic. For the details of all of this we refer to [Br3] or the forthcoming book [AnBrRz2]. The purpose of this paper is to improve the theorem of BrScker quoted above in the following way:

* PartiM1y supported by a grant Del Amo, Unlversidad Complutense de Madrid, 1990, and DGICYT, PB 89-0379-C02-02

129

1. Let S C X be an open semialgebraic set which does not meet the Zariski closure of its boundary. Then S is basic if and only if S fq Y is generically basic for any irreducible algebradc surface Y C X . Theorem

This theorem shows that the obstructions to basicness appear in the smallest possible dimension, because as is well known everything is basic in dimension 1. T h e result will follow f r o m a more precise statement characterizing generically basic sets t h r o u g h their intersections with irreducible surfaces. Of course the main tool in the proofs is the theory of fans mentioned above, and more specifically, the Theorem 1.2 below: an approximation theorem which shows t h a t for the m a t t e r we are discussing it is enough to consider fans built up from real prime divisors. Since we have not even said w h a t a fan is, let us at least describe a fan built up from a real prime divisor. First we start with a prime divisor V of the field K:(X), t h a t is, V is a discrete rank one valuation ring o f / C ( X ) whose residue field k is a finitely generated extension of I t of transcendence degree d - 1 where d = d i m ( X ) . Let t be a uniformizer of V. T h e n every element f E V can be written in the form f = ut n, where n is the value of f and u is a unit of V, so that its residue class ~ in k is not zero. Now, we pick two orderings ~'1 and r2 in k. (In particular, we are assuming that k is formally real, and that is why V is called a real prime divisor.) It is easy to llft rl and r2 to get four orderings al,a3 and a2,a4 in V, and so in the field/C(X). Namely, for an f = utn as above, put a l ( f ) = 7"1(~),

an(f) = (-1)nrl(~)

a2(f) = r2(~),

a4(f) = (-1)nv2(~)

(for an ordering c~ and an element x we let a ( x ) denote the sign of x in a). In other words, the units of V have the sign of their residue classes, and the uniformizer is declared either positive or negative. It follows from the construction t h a t a4 = al • a2 • a3, which is in fact the definition of a 4-element fan F = {or1 , a2, an, a4 } (orderings are multiplied as signatures). Now we can formulate rigorously our vague assertion that the tilda operation makes the connection between our initial problem and fans. To start with, given a semialgebraic set S C X we consider the set S C Specr(IC(X)) defined by any formula t h a t also defines S. T h e n we will prove: 2. Let X C I t " be an irreducible algebraic variety, and S C X a semiMgebraic set. Then S is generically basic if and only if # ( S fl F ) 7~ 3 for every 4-element fan F built up from a t e a / p r i m e divisor.

Theorem

This improves the following theorem of BrScker:

130

T h e o r e m 3. Let X C It" be an irreducible algebraic variety, and S C X a semialgebraic

set. Then S is generically basic if and only if # ( S f3 F) ~ 3 for every 4-element fan F. W h a t is new in our statement is the very special type of fans which are enough to check generic basicness. To give some hint of how this helps in our special problem, suppose S is not generically basic. Then, by the if part of Theorem 2, there are a real prime divisor V of ]C(X), two orderings rl and r2 in its residue field k, and four Iiftings a l , a 3 and a2,cr4, such that a l , a a , a 2 E S and a4 ¢ S.Here is the clue for the proof of Theorem 1: by a suitable aplication of Bertini's theorem we can decrease the transcendence degree of the residue field k, till in the end we get transcendence degree 1. Hence k is the function field of a real algebraic curve, and we get a homomorphism Y --* k[[t]] (where t is a uniformizer of V). After some work with M. Artin's approximation theorem we may suppose the homomorphism is actually into the ring of algebraic power series Y --~ k[[t]]~g. Furthermore, we can suppose that V contains the ring T~(X) of polynomial functions on X , and so get a homomorphism :P(X) --* k[[t]]~ag. This already gives the surface we were looking for: it is the zero set of the kernel of the homomorphism above. Finally, the four orderings ai we had in V extend easily to k[[t]], then restrict to n(p). Thus we obtain a 4-element fan in n(p) = / C ( Y ) which shows that the intersection S f3 Y is not generically basic (only if part of Theorem 2). The paper is organized as follows. Section 1 contains all results on fans in function fields needed in Section 2 to prove Theorems 1 and 2. We finish in Section 3 with a counterexample showing why our result fails in both the Nash and the analytic categories. This work started while the authors where participating in the Special Year 1990-91 on Quadratic Forms and Real Algebraic Geometry organized by Profs. T.Y. Lain and R. Robson in the University of California, Berkeley. We are glad to thank the RAGSQUADers, who listened to our first presentation of these ideas. We also want to thank Profs. J. Bochnak, M. Coste and P. Milman, who helped us to understand the Nash and analytic cases; in particular, we completed the construction of the counterexample which ends the paper during the conference in La Turballe, using a surface found by E. Bierstone and P. Milman.

131

1. T h e a p p r o x i m a t i o n

theorem

for fans

Let K be a field and V a valuation ring of K with residue field k. Consider an ordering a of K . We say t h a t a and V are compatible if the maximal ideal m y of V is convex with respect to a. T h e n a induces in the obvious way an ordering r in k, and we write a --* r. Now consider a fan F = {al, a2, a3, a4 } of K , that is, a set of four orderings of K such that the p r o d u c t of three of t h e m gives always the fourth. We say t h a t F and V are compatible if every al is compatible with V; we say that F trivializes along V if the ai's induce in k no more t h a n two distinct orderings. T h e n the situation looks like ffl

~3

\/

if2

~4

\/

T h e key result concerning the relation between fans and valuations is the following ([Brl]):

Theorem

1.1. Let F be a fan of the t~eld K. Then F trivializes along a valuation ring

of K . Henceforth we will think of a 4-element fan as a tuple (al, a2, a3,0"4) rather t h a n a set {al, a2, a3, a4 }. This way fans become points in the p r o d u c t E of four copies of the space of orderings Specr(K). Since ~ carries naturally the p r o d u c t topology of the Harrison topology, we can make statements in terms of that topology. Formally we should identify points in ~ up to permutations, but such a formalism is not needed for our purposes. From now on we suppose that K is a function field over R , and denote by n its

dimension, t h a t is, its transcendence degree over R. We recall t h a t V is eLprime divisor if it is a discrete rank one valuation ring of K and its residue field k is a function field over 1~ of dimension n - 1. W i t h this terminology, we have:

Theorem

1.2. The set of fans of K which trivialize along prime divisors is dense in the

set of all fans. Proof: Let X be a compact model of K , that is, a compact irreducible algebraic variety X C R n whose field of rational functions is K . T h e n every polynomial is b o u n d e d on X , from which it follows that every real valuation ring of K contains the ring P ( X ) of polynomial functions of X . Let F = (al,a2,a3,a4) E E b e a f a n o f K and U = U I × U2 × U 3 x U 4 an open neighborhood of F , with Ui = {fil > 0 , . . . , fir, > 0}, fij E 7v(Z); after shrinking the Ui's, we m a y assume they are pairwise disjoint. By T h e o r e m 1.1 F trivializes along a valuation ring V of K : the ai's are compatible with V and induce two orderings in the

132

residue field k of V according to the picture 0.1

0.3

0.2

%/

0.4

%/

(possibly vl = v:). Now we apply resolution of singularities so that after finitely many blowings-up X is non-singular and all the fij's are normal crossings. Let p C ~ ( X ) be the center of V in 7~(X). Then A = 7~(X)p is a local regular ring of dimension say d, and there exists a regular system of parameters x l , . . . , xd such that for all i,j

fij

=

~ux7 ' i ' ' ' "

~,i,,

where the uij are units of A and the a~jk are non-negative integers. In this situation the residue field n(p) of A is a subfield of the residue field k of V, and rl, r2 restrict to two orderings in n(p) that we still denote T1, v2. Then notice that the signs of the elements flj in the ordering ap are completely determined by the signs of the parameters xt in ap and the signs of the units (or more properly of their residue classes) in vq, where p = q mod2. Reordering the parameters we may suppose that al(xt) = aa(xt) for I < r and el (zt) # aa(xt) for l > r. Note that r < d since otherwise al and aa would coincide over the f l j ' s , which is imposible because these elements define a neighborhood of 0.1 which does not contain aa. On the other hand, it might very well happen that r = 0. Recalling that 0.1 "0.2 "0.3 "0"4 ~- 1, we see that also 0.2(xt) = 0.4(xt) for all l < r and a2(xt) ~ 0.4(x/) for all / > r. Consider the diagram A = T ' ( X ) p --~ B = A [ x ~ , . . . , x , - , x , - + l / x d , . . . ,Xd--1/Xd, Xd](xl . . . . . . . . Xd) -'~ W = B(x,~ ) C K

l

l

i

Bt(xd)

c

kw

1 /¢(~)

~

~' = K , ( p ) [ X r + I l X d , . . .

,Xd--1/Xd]

Let us analyse the ingredients of this figure. First, the local ring B is regular of dimension r + 1 with regular system of parameters { x l , . . . , x r , xd}, and dominates A. Furthermore, its residue field is generated over the residue field of A, which is g(p), by the residue classes x r + l / x a , . . . , Xd-1/xd, which are algebraically independent over to(p). Second, the ring W is a discrete rank one valuation ring of K , and it is in fact a prime divisor. Indeed, its residue field k w is by construction a finitely generated extension of R. Moreover, k w is the quotient field of B/(xd), which is a regular local ring of dimension r and whose residue field k t is that of B, and has transcendence degree d - 1 - r over x(p). As the latter has transcendence degree e = d i m ( T ' ( X ) / p ) over R., we see that the transcendence degree of k w over R is > r + ( d l-r) +e = d+e1. But

133

d + e = ht(p) + d i m ( P ( X ) / p ) = d i m ( P ( X ) ) = n, and we find that k w is a function field over l:t of dimension n - 1. Third, we have the following coincidences of signs:

O'I(Xl) ~---o'3(Xl),l < r;

0"l(.TI/Xd) = o'3(xt/xd),r < l < d;

O'l(Xd) # 6r3(Xd)

o'~(zdxd) = o,,(:~dx,d,," < t < d;

o'~(Xd) # ",,(=d)

and since al • a2 " cr3 • a4 = 1,

o'~(zt) = o'4(xt), t _< ,-;

Finally, taking all of this into account, we can extend and lift orderings through the preceding diagram, starting from t~(p) with ri, i = 1, 2, as follows: • Extend ri to an ordering r~ in k' such that r ' ( x t / x a ) = ai(xt/Xd) for r < l < d. This extension is possible since k' is a purely transcendental extension of ~(p) with { x r + l / x r t , . . . , x a - 1 / x a } as a transcendence basis. • Lift r ' to an ordering 7i in k w such that 71(xt) = a i ( x t ) for 1 < l < r. This lifting follows easily after remarking that the completion of B / ( x a ) is a ring of formal power series k'[[g:l,..., g:r]], where { £ 1 , . . . , xr} is the regular system of parameters of B / ( X d ) induced by { x l , . . . , x ~ } . • Lift 7i to two orderings al' and ~'/+2 of I¢ such that o)(=d) = ~A~d). Trds can be done as before using the completion kw[[xd]] of W = B(=d) , or as was described in the introduction for a discrete rank one valuation ring like W. We claim that the four orderings of K just constructed form a 4-element fan F ' which trivializes along the prime divisor W, and that F ' is in the neighborhood U of F fixed at the beginning. Indeed, they form a fan because they are built up from the real prime divisor W, as explained in the introduction, which implies also that F ' trivializes along W. Furthermore, ~ ( u / ) = ~p(ui), ~(=t) = ~p(~) for all p, i, I, and so

for all i , j . Hence a~ E Up for allp, and F' = (a~,a~,a~,a~) E U.

II

134

2. C h a r a c t e r i z a t i o n s

o f basic s e t s

We start by deducing Theorem 2 of the introduction as an immediate consequence of Theorem 1.2: P r o o f of Theorem 2: It is immediate that in case S is generically basic, # ( S N F ) ~ 3 for any fan F, built up or not from a real prime divisor. Conversely, by Theorem 3, if S is not generically basic, there is a fan F such that #(;~ A F ) = 3, and the problem is to replace F by another fan F ' , built up from a real prime divisor. To do it, let hj be the polynomial functions involved in a fixed description of S, and a l , a2, a3, 0"4 the orderings of F. Then consider the neighborhood Ui = {fil > 0 , . . . , fir > 0} of ai where flj = hj if hj(gi) > O, fij = - h i if hj(ai) < 0. Then U = U1 x U2 x (]3 × U4 is a neighborhood of F in the sense of Theorem 1.2, and there is a fan F ' in U which trivializes along a prime divisor, which t t means that F ' = ( a l , a 2 , a3, ' a4) ' is built up from that real prime divisor. By construction the signs of a~ and ai at hj coincide, and so a~ E S if and only if ai E 2~. Consequently, # ( S N F ' ) = # ( S I"1F ) = 3, and we are done. I

Now we are ready to state and prove the characterization of generically basic sets through surfaces:

T h e o r e m 2.1. Let X C R " be an irreducible algebraic variety and S C X a semialgebraic set. Let Z C X be an algebraic set ~ X containing both the singular locus of X and the boundary of S, OS = S \ S °. Then S is generically basic if and only if S M Y is generically basic for any irreducible surface Y C X not contained in Z.

Proof: Assume first that S is generically basic, and let Y C X be an irreducible surface not contained in Z; denote by p the ideal of Y. Since Y is not contained in the singular locus of X the localization P ( X ) p is a regular local ring, and there is a discrete valuation ring V of the quotient field L:(X) of 7~(X) dominating P ( X ) p whose residue field is k = q f ( P ( X ) / p ) = 1C(Y). Now suppose that S N Y is not generically basic. Then there is a fan (rl,v2,r3,v4) of k such that rl,r2,v3 E S and r4 ~ S. Using V one easily lifts this fan to another (al,a2,aa,a4) of K:(X), with a~ -~ rl, and we claim that al,o'2,a3 E S and o'4 ~ S. First, since Z does not contain Y and Z contains OS, ~'i E S \ Z C S ° and consequently al E S, for i = 1,2,3. Second, since r4 E Y \ Z U S , we see that v4 ~ S and a4 ~ S. Whence, S is not generically basic, as claimed. Note that for this implication we do not need fans trivializing along prime divisors. Now assume S is not generically basic. By Theorem 1.2 there is a 4-element fan F of the field K = K:(X) with # ( S N F ) = 3, and F trivializes along a prime divisor V of K; say F m {ffl,ff2, O'3,O'4}, O*1,O"3 ---+ T 1 and a2,a4 --* r2, where r~ and v2 are two orderings in the residue field k of V. Substituting X by its one-point compactification we can assume that

135

X is compact, and so V contains the ring P ( X ) of polynomial functions of X. The residue field k of V is a function field over R: there is an irreducible algebraic variety W C R m whose field of rational functions KJ(W) is k, or in other words, k is the quotient field of the ring 79(W) of polynomial functions on W. Now let H stand for a generic irreducible hyperplane section of W, p for the ideal of H in 79(W) and k ~ for the field of rational functions on H; note that k' is the residue field of p, that is, the quotient field of 79(W)/p. With alI of this we have the following diagram p(x)

c v

~ ,f," =

Jc(w)I[t]] u

T'(W),[[t]] -~/C(H)[[t]] where t is a uniformizer of V and the homomorphism qa is the obvious extension of the canonical mapping 79(W)p -~ KJ(H). Since the ring P ( X ) is an algebra finitely generated over R we can pick finitely many generators fi E 79(X) and add to them the equations involved in a description of the semialgebraic set S and an equation of Z. All these functions fi are in V, and so they have, in its completion r~" =/C(W)[[t]], power expansions fi = fi(t) = ~ t ( g i t / h i l ) t t, where gil, hit E 7:~(W). As our hyperplane section H is generic, we can suppose that no gil, hit vanishes on H (although there are infinitely many gig hit's, their number is countable, and working over the reals we can use Baire's theorem). In particular hit ¢ P implies that the fi(t)'s are well defined elements of 7~(W)p[[t]]. Finally, since the f i ' s generate 79(X) we get ¢(79(X)) C 79(W)p[[t]l and consequently we have the m a p ¢ =

p(x) -. P(H)[[G

Moreover git ~ P implies that the coefficients of the fi(t)'s are units in 7)(W)p and so ¢ ( f i ) = ~,(fi(t)) = ~ t ( g i t / h l t ) t t is a non-zero element of/C(H)[[t]] (here - - stands for residue class mod p). Now we complete the choice of the generic hyperplane section H . We have the two orderings a l , a a in V =/C(W)[[t]], liftings of the same ordering rl in/C(W). It is easy to find an open neighborhood G1 of rl such that for any ordering v~ E G1 its two liftings a 1', rr3' have at the fi's the same signs as a l , a3, where the liftings are chosen so that a~(t) = ai(t). Analogously, we find a neigborhood G2 of r2. This implies that for any two orderings r~ ~ G1 and r~ E G2 the fan F ' built up from them and the prime divisor V verifies # ( S fl F ' ) = 3. Now, G1, G2 will also denote two open semialgebraic subsets of W corresponding to the neigborhoods just constructed. These semialgebraic sets are Zariski dense in W, which guarantees that we can choose the generic hyperplane section H to meet both of them. This implies that there are r~ E G1 and r~ E G2 which make the ideal p of H convex. In other words, r~ and v~ induce two orderings 7t and 72 in the residue field of p, which is KJ(H). Then we lift 71 and 72 to four orderings gl,g3 and g2,6~ which form a fan F " of KJ(H)[[t]] with the conditions gi(t) = ai(t).

136

After this preparation, we have the following diagram P(X)

/ ~(w)p[p]] ~

..L

p(w)p

--~

\ ~(p)[p]]

l

~(p )

~

o~ ~

\/

T~

~

\/

~

, ~ ,

~3 ~

\/

~1

~4

\/

~2

Now consider the kernel q of the homomorphism ¢. Its zero set is an algebraic set Y C X with "P(Y) = T ' ( X ) / q , and dim(Y) = dim(X) - ht(q). Furthermore, the fan F " = {~1, ~2,/~3, ~4} induces a fan F* in/C(Y) such that # ( S A F*) - 3, because by construction the signs of the ~i's at the ~b(f/)'s coincide with those of the cr~'s. Consequently, the semialgebraic set S N Y is not generically basic. Furthermore, since among the f~'s there is an equation of Z, and no fi is in p, Y is not contained in Z. Hence it only remains to show that we can impose the further condition dim(Y) < dim(X) and from that the proof will end by induction. Now consider any other homomorphism ¢ ' : P ( X ) ~ ~(q)[[t]] which approximates ¢ to an order ~,, that is, such that ¢ ' ( f ) _--- ~b(f) m o d t v for every f e P(X). If we look at the requirements that !b : "P(X) --* n(q)[[t]] fulfils, it is clear that for u large enough ¢~ will fulfil them too. Then consider the ideal n = ¢ - I ( t ) and the localization A = P ( X ) , . The homomorphism ¢ extends to the henselization A h, which is a quotient of a ring n ( n ) [ [ x l , . . . , Xr]]atg of algebraic power series. Using this remark and M. Artin's approximation theorem we can arbitrarily approximate ¢ by ¢ ' : A h --~ n(q)[[t]]~g (see [Tg] III §5). Hence, substituting ¢ by ¢ ' we may merely suppose ¢ ( T ' ( X ) ) C t~(q)[[t]]~ag. It follows that ¢ induces an embedding T'(Y) ¢--* ~(q)[[t]]aig, which extends to the quotient fields K:(Y) ~ n(q)((t))~g. Counting transcendence degrees over R we find dim(Y) --- tr.deg.[/C(Y): l:t] < tr.deg.[n(q)((t))~lg : R] ---1 q- tr.deg.[~(q) : R] = 1 + d i m ( H ) < 1 + dim(W) = dim(X) as wanted.

|

We finish this section with the following:

Proof of Theorem 1: It is clear that if S is basic open any intersection S n Y with an irreducible subvariety Y C X is also basic, and so generically basic. Conversely, suppose S is not basic open. Then by Br6cker's theorem there is an irreducible subvariety X t C X such that S fl X ~ is not generically basic. Then, by Theorem 2.1 there is an irreducible surface Y C X t such that S n Y is not generically basic, and we are done. |

137

3. T h e N a s h a n d t h e a n a l y t i c c a s e s The approach followed here to study semialgebraic sets works the same in the Nash and the analytic categories (with some compactness assumption in the latter) [Rz],[AnBrRzl]. Even the approximation theorem for fans proved in Section 1 could be proved in both the Nash and the analytic cases. However any attempt to prove Theorem 2.1 or Theorem 1 would fail. The obstruction is that no Bertini type theorem is available in those cases. An easy counterexample is given by the standard torus T in R3: a lot of hyperplane sections of T consist of two circles, and so cannot be irreducible in either the Nash or the analytic category. More difficult is to produce a counterexample to Theorem 1 itself. Here we present one based on a surface used by Bierstone and Milman in dealing with arc-analytic functions, IBM] Ex.l.2(3). E x a m p l e . Let RP3 be the reM projective 3-space with homogeneous coordinates (x0 : xl : x2 : x3), and Y0 C RPa the surface XoX 1 2 2 = x~ + x~. This surface has two singular points, namely a = (1 : 0 : 0 : 0) and b = (0 : 1 : 0 : 0), and Y0 \ {a,b} consists of two connected components whose closures Y1 and ]I2 meet exactly at a and b. Moreover, Y0 is irreducible from the analytic viewpoint, but has the property that any analytic arc through one singular point has to be completely contained in either Y1 or Y2- This implies that any irreducible analytic curve Z C Yo has to be completely contained in either Y1 or

Y2. After these preliminaries consider the semiaigebraic set S C RPa defined as follows. First S will be a subset of the a ~ n e space R a = {x0 ~ 0} C RP3, in which we take coordinates x = x l / x o , y = x 2 / x o , z = x3/xo. Then Y0 N R a is the surface of equation x2 = y4 + z ~, and we have the sets: •

B1 = bail centered at (1, O, 1) E II1 with a very small radius.

•

B 1+ = B 1 [7 {322 ~__ V4 -1- z 4 ) ,

•

B2 = ball centered at ( - 1 , 0, 1) E Y2 with a very small radius.

• B+ = B ~ n { z ~ > v ' + z ' } ,

B 1 = B 1 [7 {X 2 ~_~ V4 "q- Z4) •

B K = B 2 n { z ~_O, does there exist some mirror configuration on M(h,k)? A satisfactory answer is given when h=l: fcM(1,k) is mirror if and only if kO, M(h,k) will denote the topological space whose points are the nonsingular mixed configurations of h lines and k points in R3. The topology induced by ~3 in M(h,k) is the obvious one. We will call each connected component of M(h,k) a camera. An isotopy will be a continuous function y:[0,1]----~M(h,k), that is, y is a path joining two points that lie in tile same camera. Therefore, if f, g~M(h,k) lie in tile same camera we will say that f and g are isotopic.

f~M(h,k) is said to he mirror or to have the mirror proper O, if f is isotopic to its mirror image in any plane. Otherwise f is said to be nonmirror. Letf~M(h,k) be a mirror configuration. Given any reflection o in a plane of ~3 consider an isotopy T:[0,1]--~M(h,k) such that T(0)=o(]) and T(1)=f. The composition of o with T acts over f permuting its h lines and ils k points. Any permutation of the lines and the points of f obtained in this way will be called a mirror permutation o f f Given f6M(h,k), if "f'~M(h',k') is composed of h'M(I,k) be any isotopy with ~/(0)=f. Then Pl and pj are adjacent in f if and only if their images are adjacents in y(1). Suppose the opposite. Then, as T is a continuous transformation, for some t~[0,1] T(t) would be a configuration that is not in general position. In the same way, if o" is any reflection in a plane of ~3 then p and On the other hand, let y:[0,1]

q are adjacents in f if and only if o(p) and o(q) are adjacents in a(j).

,0,~

YIr','l,vi

vs 3

3.

J~qttce /-t

In what follows we will suppose that f=ll;p,,..,pkleM(1,k ) with k>8 is mirror and q~ is the mirror permutation of f obtained from the composition of one reflection cr with one isotopy T:[O,l]---oM(h,k) such that T(0)=o'(]) and T(l)=f. From tile preceding paragraph we deduce that q) transforms adjacent points into adjacent points. Then q~ induces (by means of ~) the action over Pk of some ~:~D2k. where D2k denote the dihedral group of 2k elements. Let C k denote the cyclic group of k elements, then Lemma 3.- xc CkCD2k. Proof: Suppose xc D2k~Ck. Any non-orieutable element o f D2k acts on Pk as a reflection; then x acts on the set of vertices of Pk as a permutation with 0 or 2 vertices fixed if k is

even, or 1 vertex fixed if k is odd. The remaining vertices of P are divided into disjoint transpositions. As k>8 we have more than 2 transpositions. Let f" be composed of / and 4 points of f whose images by ~ are the vertices corresponding to two transpositions. Then q)(f')=f' and f'~M(l,4) would be mirror. But this is not possible by lemma 2 Corollary 4.- If the lhw I is endou'ed with some orientation then (p reverses it. Proof:

Construct one loop q3 passing through all the points of f and such that any

halfplane of S l intersects ~ in one single point. We can extend the action of cr and ~' over lt.~ in such a way that this properties will be preserved in the images of the loop during the isotopy, and that if we denote by r I to the composition of o" with T then rl(d?)=qb. Consider l and ~ endowed with some orientations. As 11 is composed of one reflection and 1 and ~ are linked, rl must reverse the orientation on l or on ~. By lemma 3, rl doesn't reverse the orientation of ~ • Now, we can prove tile following Lemma 5.- f~ M(l,k) is nonmirror if k>_8.

144

Proof: Suppose f=ll;pcp2,..,pk}cM(1,k ) with k>_8 is mirror. Let to be a mirror permutation o f f . Then, from lemma 3 we deduce that to permutes the set of points o f f in such a way that [pCp2,..,pk I will be divided in disjoint cycles, all of the same length t. We will see that no value for t is possible. (i) t=l, 2 or 3: consider some f'~M(l,6) composed of the line I and the points of 6, 3 or 2 cycles respectively. Then q~(f')=f" and f ' would be mirror. But this is not possible by lemma 2. For t_>4 we will consider some f'~ M(1,0 composed of tile line I and the t points of one single cycle. (ii) t=4, 5 or 6: to(f')=f' and therefore f'cM(1,t) would be mirror, but this is not possible by lemma 2. (iii) t->7 and odd: we will show that M(l,t) doesn't contain mirror configurations. Let g~M(l,i). If i>7 define s(g) to be tile sum of tile linking coefficients of tile i triples that consist of 3 lines determined by three consecutive pairs of adjacent points in g. Then s is preserved under isotopies and antisymmetric in M(l,i). If i is odd, then s(g) is odd and g is nonmirror. (iv) t->8 and even: let p be a point of f', then f'=[/;p,q0(p),..,q]l(p)} and q~(p)=p. We will denote by 1' to tile line determined by p and tot~(p) endowed with some orientation, and by ~ ( f ) to the line determined by toi(p) and toi(totn_(p))=~÷t,/2-)(p) endowed with the orientation induced by ~ and /'. Then, r and to~a(l') denote the same line with opposite orientations. We know that to is composed of one reflection, therefore if the line t of f is endowed with some orientation we will have Ik(l',/) ~ -lk(tp(/'),to(l)) Co~. 4 Ik(tp(/'),/). Hence, applying t/2 times to we obtain lk(r,i)=lk(tp~P--(r),l) which is a contradiction By lemmas 1, 2 and 5 we have tile following: Theorem 6.-f~M(l,k) is mirror if and only if k 0. It remains to show that d(S~. NBp(O),goClBp(O)) < e f o r sufficiently large n where d is the Hausdorff distance and So = limT--,~o ST. Let

U = U~(SonB~(O)) be the e-neighbourhood of SoNBp(O) in N ~. Since Bp(O) is bounded, it is clear t h a t S~,, cl B,~(0) C U for n > no. Now assume t h a t there

153

are infinitely m a n y m • N such that (So A Sp(0)) C U(e, S~., ClB,(0)). In particular, there is a point zm • SoABp(O) such that z,,~ ~ U(e, S~,~NBp(O)). Since So fq Bp(0) is compact, we m a y assume that {z,~} --* Zo. Let U = Ue/v.(zo) in ll~'~ x N. By construction, U n Sy.~ = 0 for all m with :,~ - Zo < e/2. So now T1 := {y • T [ S ~ n U = 0} is infinite and yo • Bd(T~).

On the other hand,

T1 is semialgebraic. So we m a y assume Tx = T. But then z0 ~ limT-~0 ST. Contradiction.

[]

Let us consider further properties of a family S. P r o p o s i t i o n a n d D e f i n i t i o n 3.2 For a semialgebraic family S C R '~ × R k

the following properties are equivalent: a) For any sequence {(xn, y~)} .in S with {(x,~, y~)} -~ (Xo, y0) and y,~ # Yo

one has (Xo, Yo)• Cl(S o) (The vertical closure of S is closed.) b) For any semialgebraic set T C ]~k, Yo • CI(T), one has limT-~o ST C Cl( Svo ).

b') Same as b), but with the further condition that dim(T) = 1. c) lim~:k.~o S~> C CI(S~o). d) For all a • Sper(R[y]), a ~ yo one has li,r~..vo S, C Cl(S~o). d') Same as d), but with the further condition that dim(supp(a)) = 1. e) lin~.}_.y 0 Sv. C Cl(S~o) whenever the left hand side exists.

If these conditions hold, the family S is called semicontinuous. Proof: Obviously, a) ~ b) ---, c) ~ b), b) ---* b') and d) ~ d'). Moreover, by 2.6 : b) ~ d), by 3.1 and the curve selection lemma [B-C-R, 2.5]: b') ~ d') and e) ~ b'). Finally, by 2.7: d) ~ e). So it remains to show b') ~ a): Let {(x,~,y~)} (xo,yo) and assume that (xo, yo) ~ Svo. By the curve selection lemma (loc. cit) we find a semialgebraic arc-~, : [0, e [ ~ S with 0'(0) = (xo, yo) and O'(t) 7r-~(yo) for t > 0. Let T := 7r o 7- Then dim(T) = 1 and by construction (x0, y0) E lim ST , T --~ y o

which contradicts b').

[]

154

R e m a r k 3.3 In fact~ the semicontinuity is a condition at the points yo e R k. "Natural" families will be semicontinuous, but by brute force one can replace

fibres by bad ones at certain points to make it non-semicontinuous. From property c) above we learn, that for an arbitrary semialgebraic family S the subset T C N k of points at which S is not semicontinuous, is semialgebraic. If S is semicontinuous, then ~r(S) is closed. It should be clear, what we do now and how we do it. P r o p o s i t i o n a n d D e f i n i t i o n 3.4 For a semialgebraic family S C R'* x N ~ the following properties are equivalent:

a) For any semialgebraic set T C ~r(S), Yo e Cl(T) one has limT-~o ST = C l( S~o).

a') Same as a), but with the further condition that dim(T) = 1. b) For all a e Sper(R[Y]), a ---+ yo one has lim~_~o s~ = Cl(S~o).

b') Same as b), but with the further condition that dim(supp(a)) = 1. c) Fo~ any sequence {y,} with y~ ~ ~ ( S ) a~d {y.) -~ yo ~ s one has {s~.} -,

Cl( S~o) (in

the Hausdorff topology).

If these conditions hold, the family S is called continuous. Proof: Obviously a) ~ a') and b) --* b'). Moreover, by 2.6: a) --* b) and by 3.1 and the curve selection lemma (loc. cit): a') --* b'). b') --+ c): Let {y,~} -* Yo. Assume also that y~ # yo. Suppose {S~.} does not tend to Cl(Svo ). Then there exists e, p > 0 and infinitely many m e N such that d(Sv.. N Bp(O), Svo N Bp(O)) >_ ~. If (Sv= N Ba(O)) (~ U,(Svo N Ba(O)) for infinitely many m, we find a curve 3' : [0, p[---~ R '~ × R k, 7(]0, p[) C S, 7(0) = ze~r-l(yo) \ (U,(Svo) A Bp(O)) and 7(t) ~ lr-l(yo) for t > 0. Then ~r(3') defines an element a ~ Sper(N[y]) such that a --+ yo. For this, by 3.1, we get z e lim~.~ ° S~, which contradicts b'). So assume that (S~o n Bp(0)) U~(S~., N Bp(0)) for infinitely many m. As in the proof of 3.1 we find Zo e Cl(S~o ) n Bp(O) and a ball U around Zo in ]R~ × IRk such that U N S~.~ = 0 for infinitely many m. Therefore we find a curve 8 : [0, p[---* ~r(S), 8(0) = yo and fi(t) # yo for t > 0, such that S~(t)NU = 0 for all t e]0, p[. Let a be the element in Sper(IR[y]) defined by 5. Then a --* y0 but, by 3.1, z0 ~ lir~-~0 S~,

155

which again contradicts b'). c) --+ a): Let T C ~r(S) be semialgebraic with yo E e l ( T ) .

Then we find

a sequence {y~} --+ Yo, Y,~e T \ {yo}. Clearly Cl(Suo ) = lin~,_.uo S,. C limT-~vo ST. The converse inclusion follows from 3.2. []

R e m a r k 3.5 Again, continuity is defined at points. One might ask for the

size of the set D C re(S) of all points, where S is not continuous. A first information is given by the following partition into continuous parts. There is a decomposition re(S) -- T1,... ,T~ into disjoint semialgebraic sets such that the following hold: a) T~ is open in its Zariski closure. b) SIT~ "is continuous. This is an immediate consequence of Hardt's theorem [Ha]. In particular, D is nowhere dense in re(S). May be that D is also semialgebraic, but I didn't see anything which makes it evident.

4

L i m i t s of c h a i n s

Let R be a real closed field. chains 4.1 A semialgebraic q-chain S in R ~ is a bounded closed semialgebraic set S C R '~ of dimension q such that S = S*, that means, also the local dimension dim,(S) = q for all x E S. The sum of two q-chains S and T is defined to be 0 if S = T or C I ( S A T ) (symmetric difference of S and T) if S ¢ T. The boundary OS of a q-chain S is defined via any triangulation of S but it does not depend on the particular triangulation. For semialgebraic sets A C X C R n the semialgebraic q-chains give rise to a definition of homology groups Hq(X/A, Z/2Z). For R = R these coincide with the usual ones and in general they coincide with those defined in [D-Kn] or

[B-C-R, 11.7] (see [Br, § 4]).

E x a m p l e 4.2 Let S C R '~ be algebraic of dimension q such that S* is

bounded and closed. called algebraic.

Then S* is a semialgebraic q-cycle. Such a cycle is

156 Reduction

o f c h a i n s 4.3

A s s u m e t h a t the real closed field R a d m i t s a real valuation v with residue field R and valuation ring 1/. As before, for a semiMgebrMc set S C V '~ let S := { x ' e R ' ~ ] 3 x e S

: x' = 5}.

If S is a q-chain, we d e f i n e r ( S ) , the

reduction of S, as follows: If d i m ( S ) < q then r(S) := 0. If d i m ( S ) = q, there exist semiMgebraic q-chains T 1 , . . . , T,,~ C ~

such t h a t S* = T1 + . . . + T,n

and the multiplicity #s(a) is constant = ri for all a e ~bi such t h a t d i m ( a ) = q

(see § 1 or [Br, § 4]). N o w r(S) := riT1 +... + r~T,, T h e n r COlrlnlutes with 0 [Br, Th. 4.4] and thus for semiMgebraic sets A C X C V '~, the reduction r defines a b o m o m o r p h i s m

r: H,(X/A, Z / 2 7 / , ) ~ H , ( X / A , Z / 2 Z ) We want to u n d e r s t a n d these multiplicities in the context of limits in a purely geometric way, at least modulo 2. For this we need another tool. Intersection

numbers

4.4

Let r/ be a p - - c h a i n and ~ a q-chain in N '~ w i t h p + q

= n. T h e n r/ and

are called admissible if lr]t cl 10~1 = 0 and [~] N IOr/] = O. In this case, an intersection n u m b e r 7/o ¢"e Z is defined [Do, V I I § 4], [S-T, § 731. We denote by 77 02 T the intersection n u m b e r rood.2. If R is replaced by any real closed field, S 02 T is still defined for admissible semialgebraic p-chains S and qchains T [Br, § 5]. For an isolated point x e S N T we denote by i2(x, S, T) the local intersection n u m b e r mod. 2 at x. If x ~ S N T we set i2(x, S,T) = O. Now let us r e t u r n to families S C N '~ x IRI' of semialgebraic sets.

Such a

family S is called a family of q-chains if S~ is a q-chain for all y e ~r(S). It is easily seen, using H a r d t ' s t h e o r e m [Ha], [B-C-R, 9.3] or similar a r g u m e n t s , that

OS :=

U

OS~

v e ~(s)

is semiMgebraic, so it is a subfamily of S. Now assume t h a t S C ~'~ × ]~k is a family of p-chains, T C IR'~ × N k a family of q-chains, p + q = n and ~r(S) = 7r(T). Let y~ e ~r(S) = ~r(T) for n c N and {y,~} --~ yo. We assume f u r t h e r m o r e , t h a t all Hausdorff limits lira S w =: S0, lim OSw =:(OS)o {v-}-~o {v.}-~yo lira % . = : T o , lira 0 % , = : ( 0 T ) 0 {~.}-~uo {u.}-~uo

157

exist and are bounded. By 2.7 there exists an element a•(~-(S)) ~ with a ~ Yo such that (with respect to the natural valuation v of the specialization a ~ y0), So = lim S~ = S~, ~'-~YO

(OS)o =

lira

(OS)~ = (OS):

cx --~ y 0

To = lim T~ = T~,

(OT)o = lira (OT),, = (OT)~ . ~ "* YO

In this situation we have P r o p o s i t i o n 4.5 Assume that So N (OT)o = 0 and To (3 (OS)o = O. Then a) S~ and T~ are admissible. b) r ( S , ) and r(T~) are admissible. c) There exists no • N such that Sv. and Tv. are admissible and Sy. o2 T~. = S~ o~. T~ for n >_ no.

d) So o: T . = r(So) o: Proof: a): Obvious. b): This follows from the fact that 0 commutes with v [Br, Th. 4.4]. c): Obviously, Su. and Tu. are admissible for sufficiently large n, and from the general theory of intersection numbers [Do, VII § 4] it is also clear t h a t S~. o2 T~ becomes constant. This follows also from b) and the construction below. First of all, by the compactness of (~r(S)) ~ with respect to the constructible topology, there is a semialgebraic set D C r ( S ) = ~-(T) such that a e / ) and S#, T# are admissible for all fl E/). Now for y • D the statement, t h a t S~ 02 Ty takes a certain value, is elementary. Therefore, there is a semialgebralc partition: D = T1 U . . . U T,~ such that Sv o2 Tu is constant for y • T~. But a • T~ for exactly one i • { 1 , . . . , m}, say a • ~bx. By construction, y~ • T1 for infinitely m a n y n and for these we have S,j o~ Ty, = S~ o2 T~

d): This is just stated by the intersection - reduction formula [Br, Th. 5.3]. []

As a consequence, we get a geometric description of multiplicities mod. 2. for this, let the family S of p-chains be given as before, and also the sequence

158

{Y,~} -~ Y0, the limits So and (OS)o and the valuation a which leads to the reduction of chains:

so

r(So) a

C o r o l l a r y 4.6 Let x •-S~ and let T C ~'~ be a q-chain such that O T N S , = ~, T n (-:d-S)~ = ~ and T N ~ = {x}. Then T and S~, are admissible for almost all n and T o2 S~, is stationary, Moreover a) I f x ~ r(S~), then T o: S~. = 0 for almost all n. b) I f x • r(S~), then T 02 S~, = T 02 S*~ for almost all n.

We conclude this work by a result on the non approximability of proper semialgebraic sets by algebraic ones. We consider a relative situation. So let us fix an open semialgebraic set U C N n. D e f i n i t i o n 4.7 A semialgebraic set B C. U is essentially algebraic in U, if there is an algebraic subset W C N '~ such that B* = ( W n U)*. If S is a p-chain and • 7~ S n U is essentially algebraic in U, then OS n U = (k Any p-dimensional semialgebraic set B C U admits a unique maximal essentiatiy algebraic subset of pure dimension p. This subset will be denoted by B ~ (but note that B" does not only depend on B but also on U). T h e o r e m 4.8 Let U C R '~ bc semialgebraic and open. Let S C R'* x R k be a semialgebraic family for which the following conditions hold: -

or(S) is bounded.

- 3 9 is a p-chain for all y e or(S). - Sv N U is essentially algebraic in U for all y • or(S). Let B C U be semialgebraic such that - B is closed in U.

- dim(B) _< p. Then there exists e > 0 such. that - for a l l 5 w i t h O < 5 < e , - for all q-chains T C U \ B ~ where p + q = n and OT N Us(B) = 0.

159

-

for all y ~ ~r(S) with (S,

r3

U)

C

U~(B)

one has T o: S~ = O. This looks a little complicated. So let us first consider an example. Let n = 2 and Q = Q= the unit cube:

Q = {xcR2i[x, I 0}) Here B a = U n {X2 = 0}. Now, of course, for an arbitrarily smM1 6 > 0 we find y e ~r(S) such t h a t (S~ n U) C U~(B), but t h e n the T h e o r e m says, t h a t for any 1-chain T in U with T N B ~ = 0 and OTA U~(B) = 0 one has

T o 2 S ~ = O.

Us(B)

~---.e

It m i g h t be very difficult to find the smallest possible ¢ for which the T h e o r e m holds, even for the above example.

proof of the theorem: 1) Let U ' : = U \ B

a. We set for ~ > 0 H~(6) := H~(U'/(U' \ U~(B)),Z/2Z) and

H~(O) :=

H~(U'I(U' \ B), Z/2Z)

Then

~I~(o) = lira H~(6) ,S>O

160 We need a little more: First of all, Ha(O ) and all Hq(d~) are finite Z / 2 Z vectorspaces. Moreover, by Hardt's theorem [Ha], [Br-C-R, 9.34] there exists e > 0 such that ~r : U' D (U~(B) n U') H ~ is a trivial fibration for 6e]0, e]. In fact, for 0 < 81 < 6~ < e the inclusion

(u', u' \ U6 (B)) c (u', u' \ U,,(B)) is a retraction, hence the canonical maps

Hq(,)

Ha(65) Ha(*1) nq(0)

are isomorphisms. 2) Assume t h a t the statement of the Theorem is wrong. Then there exists a convergent sequence {y,~} --+ Yo, Y,~e lr(S) and a sequence {T,~} of semialgebraic q-chains with the following properties: i)

(S~. n U) C U~(B).

ii)

T~cU'=U\B

iii)

S ~ 02 T~ = 1.

~and0T~nU¼(B)=0.

We choose e according to part 1) of the proof. Since Sv, o2T~ depends only on the class of T~ in Hq (¼) we may assume that n > ~ and all T~ are restrictions of a single semialgebraic q-chain T C U' with OT Q U,(B) = 0. Now pick a e ~r(S) such that each element of the ultrafilter associated to a contains infinitely m a n y y~ (compare the proof of 2.7). Then S,,QU C B and S~ o5 T = 1. On the other hand, by 4.9 below, S, Cl (U x Nk)~ is essentially algebraic in (U x Rk),. Hence by [Br,Cor. 3.8] we get r ( S , ) n U C B ". Since T N B ~ = ~ and T = r T we get r T o5 rS~ = O. But this is a contradiction to the intersection-reduction formula [Br, Th. 5.3] saying S~ohT = rS~,o2rT. []

For the L e m m a below we use the following notation: Let S C R ~ be semialgebraic and let x l , . . . , x,~ e R ~. Then S is called essentially algebraic a~ x l , . . . ,x~, if there is an open set U containing z l , . . . ,z~ such t h a t S is essentially algebraic in U. L e m m a 4.9 Let U C N" be semialgebraic and open and let B C U be semi-

algebraic. Then the statement "B is essentially algebraic in U" is elementary. Proof:

This is an immediate consequence of the following facts.

161

a) B is essentially algebraic in U if and only if B is essentially algebraic at any two points xl, x2 e U. b) Let dim(B) < n - 1. Then B is essentially algebraic at x l , . . . ,x,~ if and only if for all y e S '~-I and all ¢ > 0 there exists z e U~(y) such that vz(B) is essentially algebraic at r ( x l ) , . . . , ~r(x,~), where r , is the orthogonal projection: R ~ -~ (z) ±. c) The statement is true if dim(B) = n - 1. Here a) is clear and b) holds, since the image of an algebraic set under an injective polynomial map is essentially algebraic. For c) first one finds among the polynomials, which describe B, equations for an algebraic set W D B with dim(W) = dim(B). Since ]~[X], X = X~,... ,A~ is factoriM, the degree of equations for the Zariski closure of B can be bounded. []

References [B-C-R]

Bochnak, J., Coste, M., Roy, M.F.: G6om~trie Alg~brique R6elle. Springer, Berlin Heidelberg New York, 1987

[Brt

Brhcker, L.: On the reduction of semialgebraic sets by reM valuations. To appear.

[Ch-Di]

Cherlin, G., Dickmann, M.: Real closed rings II. Model theory. AnnMs of pure and applied Logic 25, 213-231 (1983)

[D-Kn]

Dells, H., Knebusch, M.: On the homology of algebraic varieties over real closed fields. J. reine angew. Math. 335, 122-163 (1982)

[Do]

Dold, A.: Lectures on Algebraic Topology. Springer, Berlin Heidelberg New York 1972

[Ha]

Hardt, R.: Semi-algebraic local triviality in semi-algebraic mappings. Amer. J. Math. 102, 291-302 (1980)

[Kn-Schei] Knebusch, M., Scheiderer, C.: Einffihrung in die reelle Algebra. Vieweg, Braunschweig Wiesbaden 1989

t62

[v.d.D.]

van den Dries, L.: Tarskis Problem and Pfafilan functions. Logic Colloquium 1984, ed. I.B. Paris, A.J. Wilkie, G.M. Wilmers, North Holland 59-90 (1986)

Ludwig BrScker Mathematisches Institut Einsteinstr. 62 4400 Mfinster Germany

A H o p f F i x e d P o i n t T h e o r e m for S e m i - A l g e b r a l c M a p s G. W. BRUMFIEL*

Introduction. This note is a sequel to [2], in which we proved the Brouwer fixed point theorem for the real spectrum compactification of R n, that is, for PV' = S p e c a ( R [ x l . . . xn]), the real spectrum of the polynomial ring in n variables over the real numbers. Here, we fix a real closed field K and a semi-algebraic subset X C K'*. Let ) ( C Pi~ = Speca(K[xl . . . x n ] ) be the constructible subset corresponding to X C K n. Any continuous semi-algebraic map ~0 : X -* X extends to a continuous map Q : X ---* X. Define the Lefschetz number of ~ to be t r ( ~ , ) = ~(--1) I trace(~p, : H i ( X , Q ) --* H i ( X , Q ) ) . Homology of semi-algebraic sets X c K n is defined in [3J. The Hopf fixed point theorem basically states that if tr(q0.) ~ 0 then q3 : .~" -* X has a fixed point. Before giving a precise statement, we make a few remarks. It seems reasonable to investigate topological properties of Fix(Q), the fixed point set of Q. For example, we can show that Fix(Q) C_ X is always closed. I am indebted to Claus Scheiderer for the proof of this fact given below. His proof replaces a more cumbersome argument in an earlier version of this note, which only worked under the added hypothesis that X C K '~ was locally closed. Note that Fix(q3) is obviously closed in the Tychonoff, or constructible topology on X , because the Tychonoff topology is compact and Hausdorff and q3 is Tychonoff continuous. A Tychonoff closed subset is closed in the real spectrum topology, precisely when it is closed under specialization a ~ fl [1, Proposition 2.11]. We will show below that if a, fl E )~, a --* fl, and Q(a) = a, then q3(fl) = ft. Thus we obtain P r o p o s i t i o n 1. I f X C K'* is any semi-algebraic set and T: X continuous semi-algebraic map, then Fix(q3) is closed in f(.

--* X is any

The classical Hopf fixed point theorem concerns maps ~ : Y --~ Y, where Y C R n is a finite closed simplicial complex. If t r ( ~ . ) ¢ 0 then ~o has a fixed point in Y. A key step in our general Hopf fixed point theorem is to extend this result to arbitrary real closed ground fields K. P r o p o s i t i o n 2. Suppose Y C K n is a complete semi-Mgebraic set, that is, Y is dosed and bounded in K n. Suppose ~ : Y --. Y is a continuous semi-Mgebraic map with tr(qo.) 7~ O. Then ~ has a t~xed point in Y . * Research supported by NSF grant DMS 85-06816 at Stanford University

164

The a s s u m p t i o n that Y is complete is equivalent to assuming t h a t Y is a closed finite simplicial complex over K. In fact, we use triangulations in a crucial way in our proofs. Specifically, we exploit the fact that any semi-algebraic set X __ K " can be triangulated and a d m i ~ strong deformation retractions onto complete subspaces Y C X . Define F r i n g ~ X ) C X to be all those points which do not belong to any constructible C C X , where C C X is a complete subspace. Here is a precise s t a t e m e n t of our main theorem. T h e o r e m . i f ~ : X -+ X is any continuous s e m L a l g e b r a l c m a p with t r ( ~ . ) ~ 0 then e i t h e r ~ : X ~ X has a fixed point or ~ : X --+ X has a fixed point in

inge(

). Moreover,

:

always has a closed fixed point,

nd, if either

X C_ K n is locally dosed or if ~he ground field K is R, then ~ : f ( --~ X always has a dosed fixed point in Fringe(.~) U X . The proof of the Brouwer fixed point theorem for A m in [2] was s o m e w h a t unsatisfactory because at one stage a transcendental a r g u m e n t was used to construct certain retractions from R '~ onto closed disks in R". In fact, the details of that transcendental argument, which a m o u n t e d to constructing a suitable gradient vector field and integrating, were not included. The argument in this p a p e r is purely algebraic. Of course, if X C K " is any contractible space, then tr(~p.) = 1 for any : X ~ X , so the H o p f fixed point theorem certainly implies the Brouwer fixed point theorem. Finally, we point out that it would be desirable to have an even m o r e general Hopf-Lefschetz fixed point formula for ~3 : X -~ )(, which expresses tr(~p.) in terms of local d a t a near Fix(~) C ~7. However, this looks rather messy. For example, if K contains an infinitesimal e relative to Q c K , consider the m a p ~ : I ~ I given b y ~ ( z ) = z + ex(1 - x), where I = [0, 1] C K. T h e n ~ : _T~ .T has a very complicated fixed point set. Somehow, only the two b o u n d a r y fixed points 0 and 1 should be relevant, b u t for each r E K , 0 < r < 1, the Dedekind cuts of I C K defined by the sequences r + ne or r - he, n = 1, 2, 3 . . . . , determine points of I which are fixed by ~. T h e r e are also m a n y other fixed points. E v e n when K = R, if X C R" is not compact, the point-set nature of the fixed points of ~ : )~ -+ -~ in Fringe(A ~) will often rcsemble the b a d fixed point set of the m a p ~ : _T--~ T above. Finding a good generalization of the Hopf-Lefschetz fixed point formula looks challenging. P r o o f o f t h e t h e o r e m . We first point out that if ~ : )~" --* .~ has any fixed point then it has a closed fixed point. Namely, if ~ ( a ) = a and a ~ fl with fl closed in ~7. T h e n ~ ( a ) = a --* ~(fl). Since the specializations of a form a chain, either fl ~ ~(fl) or ~(fl) ~ ft. But, since fl is closed, fl ~ qb(fl) implies fl = ~(fl), and, since dim(~(fl)) _< dim(fl), ~(fl) --* fl also implies fl = ~(fl). Next, consider the fringe of )~, defined as Fringe()~) = )~ - -

U cc_x

C complete

C"

165

Since each constructible C is Tychonoff open, Fringe(,Y) is Tychonoff closed, hence compact. If a E Fringe(.~) and ~ ( a ) # a then 9~(a) 7/* a since dim(93(a)) ~ dim(a). Thus, there exists a polynomial f E K[xl,...xn] with

f(a) < 0 and f(~(a)) >_O. T h e set tS'a = .{fl E )( [f ( f l ) , < 0 and f(c~(fl))>_ 0} is a constructible set containing a, with the property that fl E V a implies qb(fl) ~ VaIl qb : ..Y ~ ]~" has no fixed points in Fr.ingeO~'), we can find finitely m a n y such constructible sets ~ with Fringe(,Y) C U ~ . Let ~ ---- l)i A X , a semi-algebraic set in X. Choose a semi-algebraic triangulation of X such that all the Vi are subcomplexes. Recall that a semi-algebraic triangulation is simply a semi-algebraic h o m e o m o r p h i s m between X and some union of open simplices of a closed finite simplicial complex Z C K " . Boundary faces of simplexes of X need not belong to X and also, of course, the I'~ are not necessarily closed subcomplexes of X. Next, we take two barycentric subdivisions Z" of Z. As in [3], there is now a canonical strong deformation retraction, r : X ---* Y, of X onto a closed subcomplex Y of Z ' . T h e picture below illustrates this construction when X is a single closed 2-simplex with one vertex and two edges removed. X c Z = A 2/\

Y c X C Z ~ x

T h e general formulation can be found in [3]. For our purposes, the crucial property of r : X ~ Y is that if a is any open simplex of the original triangulation of X then r(a) C a. Thus, if a C Vi then r(a) C I,~ so r(Vi) C 17/. Also, if r b # Id then a m u s t contain fringe points of )( so cr C Vd, some j. Thus X - Y C UVi. Consider the composition r~0 : X --* X ~ Y. By Proposition 2, r~qy : Y --* Y has a fixed point y E Y since t r ( ( r ~ I y ) . ) = t r ( ~ , ) # 0. If ~(y) ~ Y, say 9~(y) e ~ . T h e n r~(y) E ~ , and it is impossible that y = rqo(y) because y E ~ implies 9~(y) ~ Vi. Therefore _~(y) E Y and we have thus proved that either 9 : X ~ X has a fixed point or ~ : X --* X has a fixed point in Fringe(X). If X C K " is locally closed then Fringe()~) is closed in )~. This is most easily seen by identifying X with a closed semi-algebraic set X ~ C K n+l. T h e n Fringe(~) corresponds to the set of points of ~ t at infinity relative to K, which is closed in X ~. Therefore, in our theorem, we can conclude that for locally closed X, 9~ : ~ --* )( either has a fixed point in X or a closed fixed point in Fringe(.~).

166

We can also make this same conclusion for arbitrary X C W'. Namely, suppose a e Fringe(~r), ~ ( a ) = a, and a ~ fl with/3 closed in )(. We know 9~(/3) =/3. If fl 6 Fringe(X), then ~ C C for some complete C C X. But when the groundfield is R, the only closed points in a complete C are actually points of C, hence fl E C and ~o(fl) = fl is a fixed point of q0 in X. It seems plausible that this stronger conclusion, that ~5 : ~7 -* X either has a fixed point in X or a closed fixed point in Fringe(~7), holds for arbitrary X C K ' . However, if X C K " is not locally closed and K ~ R, there will often exist a E Fringe(X) with a -* fl, /3 closed, and/3 6 Fringe(X) U Z . Conceivably, the only closed fixed points/3 of ~ : )~ ~ )( could be of this type. P r o o f o f P r o p o s i t i o n 1. We introduce the ring C(X) of continuous K-valued semi-algebraic functions on X C K " . If a E X corresponds to a homomorphism ev(a): K[xl... x,] ---* k(a), where k(a) is a real closed field, algebraic over the image of K[xl... xn], then every f E C(X) can also be evaluated at a. T h a t is, ev(a) extends to ev(a): C(Z) ~ k(a). In this way, we obtain a continuous inclusion X C SpecR(C(X)), which, in fact, is a homeomorphism if X is locally closed in K", [4]. But, in any case, the map ~o:X --* X induces a ring homomorphism ~o*: C(X) --+ C(X), which, in turn, induces ~o.: SpecR(C(X)) --+ SpecR(C(X)) such that the diagram below commutes.

2

C SpecR(C(X))

ot X"

C

SpecR(C(X))

It thus suffices to prove that if ~ . ( a ) = a and a ---* fl in SpecR(C(X)), then = 8.

The assertion ~ . ( a ) = a is equivalent to the existence of an order preserving K-endomorphism i: k(a) --. k(a) such ~hat the following diagram commutes ,~(~) c(x)

,

c(x)

,

Since k(a) is real closed and has finite transcendence degre over K, the inclusion i is an isomorphism. Thus, i induces a permutation of the convex valuation rings in k(a) which contain K, and since these valuation rings form a finite chain under inclusion, each such valuation ring maps isomorphicaily onto itself under i. On the other hand, the specializations of a in SpecR(C(X)) correspond, by the real place existence theorem, to the prime ideals of C(X) which are the inverse images under ev(a) of the maximal ideals of those convex valuation rings in k(a) which contain

167

the image of C ( X ) . From the commutative diagram above, we see that for each such prime ideal p, ( ~ . ) - l ( p ) = p, hence a , : SpecR(C(X)) -+ SpecR(C(X)) is the identity map on the specializations of a, as desired. P r o o f o f P r o p o s i t i o n 2. We have a continuous semi-aigebraic map ~o : Y -+ Y, where Y C K n is a closed finite simplicial complex. The graph P(~o) C Y × Y C K'* × K n is defined in terms of finitely m a n y polynomial inequalities with coefficients in K. We know that if K C R and t r ( ~ . ) ~ 0 then T has a fixed point in Y, because the classical proof of the Hopf fixed point theorem works for any Archimedean ground field K. We will deduce the proposition for arbitrary K by transfer from the real algebraic nttmbers, using Tarski's principle. We may assume that Y itself is defined over Z, since we may as well assume the vertices of Y are vertices of some standard simplex A N C K N+I. We regard the coefficients of the polynomials used to define F(~) C Y × Y as parameters and we regard the statement that F(~p) is the graph of a continuous function ~0 : Y --+ Y as an elementary statement about these parameters. Also, the desired conclusion that T has a fixed point is an elementary statement about these parazneters. For example, a fixed point of ~ is just a point of F(~) ¢3 A(Y), where A ( y ) C Y x Y is the diagonal. The crucial point is that the hypothesis tr(~p.) ~ 0 is also an elementary statement about the parameters defining ~. Assuming this, the truth of the proposition is clear, because if the propositon were false, there would exist in K some values of the coefficient parameters for which (a) F(~o) C Y x Y is the graph of a continuous semi-algebraic function, (b) t r ( ~ . ) ~ 0, and (c) F(~0) n A ( y ) = 0. By Tarski's principle, there would also exist in the real algebraic numbers some values of the parameters for which (a), (b), (c) hold, which is a contradiction. The claim that the statement tr(~0.) ~ 0 is an elementary statement amounts to understanding how the homology maps T, : H i ( Y , Q) --+ Hi(Y, Q) can be computed in terms of the polynomial inequalities which define F(~) C Y x Y. We have the cover of Y by (open) semi-algebraic simplices, hi. Since the ~ - 1 ( ~ i ) C Y are also semi-algebraic, we can retriangulate Y so that all the sets ~ - l ( a i ) , as well as the original hi, are unions of (new) simplices. The important point is that this retriangutation can be done constructively. We now have a second closed, finite, simplicial complex Y', semi-algebraically identified with Y. We claim that the original map T: Y' -+ Y is homotopic to a simplicial map ~': Y ' ---+ Y . Specifically, for each vertex v~ E Y', ~(v~) is in some unique open simplex a i of Y. Choose ~'(v~) to be any vertex of this aj. Suppose r ' is an open simplex of Y' with vertices {v~,..., v'm} and suppose a is the simplex of Y with r ' C T - l ( a ) . Then ~o(~') C ~. Thus, the {~'(v})} are vertices of a, hence span a face of a, which shows that our vertex assignment extends to a simplicial map ~o': Y' --+ Y. From the construction it is clear that ~o' is canonically homotopic to ~, since for any y' E Y', both ~0(y') and T'(y') belong to the same closed simplex of Y.

168

The vertex choices made above axe constructive because of the way Y' was obtained from qo: Y --, Y. Thus, qo' induces a constructively obtained chain map t.: C , ( Y ' ) ~ C . ( Y ) . There is also an algebraic subdivision m a p of chain complexes s. : C . ( Y ) -* C.(Y'), since each simplex of Y is a union of simplices of Y'. The chain m a p s, induces a homology isomorphism. Identifying H , ( Y , Q) = H . ( Y ' , Q) via s , , we have tr(w.) = t r ( 8 , t . ) and the Lefschetz number of , , t , : c . ( r ' ) -* C , ( Y ) ~ C . ( Y ' ) can certainly be constructively computed. The above proof of Proposition 2 might seem to just be an adaptation of the classical proof of the Hopf fixed point theorem, using the existence of simultaneous triangulations of a finite collection of semi-algebraic ~ t s , rather than baxycentric subdivisions. If we just wanted to prove that ~ : Y ~ Y had a fixed point whenever tr(w,) # 0 then we could adapt the classicM argument in this way, and it would not be necessary to appeal to the Tarski principle. (In fact, this is essentially what we did in the proof of the main theorem.) However, the example of ~0 : I --~ I discussed in the last paragraph of the introduction shows that there is not such a close connection between fixed points of ~ : Y --* Y and fixed points of W : Y --~ Y. Thus, something additional is needed to derive the strong conclusion of Proposition 2 that ~ : Y ~ Y has a fixed point. In the chain map , . t . : C . ( Y ' ) -~ C . ( Y ) ~ C . ( Y ' ) which we constructed in the course of the proof, we do not claim that all cells of y i are "moved fax away from themselves" if ~ : Y ~ Y has no fixed point. (This would obviously imply t r ( s . t . ) = 0.) Instead, we simply exploit the facts that t r ( s . t . ) = t r ( ~ . ) and that computation of t r ( s , t . ) is constructive to validate the use of Taxski's principle. There is something a little mysterious about this. The classical Brouwer fixed point theorem over K is much easier to prove than the Hopf theorem. T h a t is, if B " C K n+l is the standard unit disk and ~ : B n --~ B n is any continuous semi-algebraic map, then ~ has a fixed point in B " . Namely, if not, one constructs a continuous semi-algebraic retraction r : B " --~ S n-x = OB" in the usual way, by following rays from ~(x) E B n through x C B n until the rays hit OB n. Thus, r I

I Sn-1

= Id, and the existence of the homology functor is enough

to get a contra ction, since

# 0 and

= 0, [3].

Curiously, it isn't clear whether this proof of the Brouwer theorem for B n is adequate to deduce the Hopf theorem for K n, or, more precisely, for/~l~.. The proof of the Hopf theorem depends on a strong deformation retraction K n ~ Y for some finite simplicial complex Y C K n, which is, of course, contractible. W h a t is not clear is whether Y can always be assumed to be semi-algebraically a closed disk. This seems to be closely 1:elated to the problem of semi-algebraic uniqueness of cone neighborhoods of points in semi-algebraic sets. But, in any case, Proposition 2 applies to Y.

169

References

1. E. Becker, "On the reM spectrum of a ring and its application to semi-MgebrMc geometry", Bull. Amer. Math. Soc. 15 (1986)19-60. 2. G.W. Brumfiel, "A semi-algebrMc Brouwer fixed point theorem for reM affine space", Contemporary Mathematic3 74 (1988), Geometry of Group Representations, 77-82. 3. H. Dells arid M. Knebusch, "Homology of algebraic varieties over real closed fields", Journal f~r die reine und angewandte Mathematik, 335 (1982) 122-163. 4. N. Schwartz, "Real closed spaces", Habilitation~schrift, Miinchen, 1984.

On regular open semi-algebraic sets G. W . Brumflel* I n t r o d u c t i o n . Let k be an ordered field with real closure K. If f 6 k[xl"'x,], let U(f) = {~ 6 K " I f ( ~ ) > 0} C K " Given a finite collection of polynomials {fi} C k[zl.., x,], let U{fi} = N U(fi) C It'". The Finiteness Theorem asserts that any semii algebraic subset of K " which is open (in the local sense that it contains a ball around each of its points) is, in fact, a lqnite union of sets of the form U{fl}. [2], [5], [6], [7], [10]. A regular open set V in a topological space is an open set which satisfies V = V*, where bar indicates closure and circle indicates interior. For any open set V, V* is regular. Also, finite intersections of regular open sets are regular. In the semi-algebraic context of the first paragraph above, let V(f) = ~ C K" and V{fi} = N V(fi) C If". By the Tarski principle, the V(fi) are semi-algebraic sets.

i From the second paragraph above, they are also regular open sets. The goal of this note is to prove a regular version of the Finiteness Theorem. Specifically, we show that any open semi-algebraic set U c K " is a finite union of sets of the form V{fi}. In general, U(f) need not be regular. For example, U = U(~x~) = It'" - (0) is not regular, since V ' = K " . Of course,

= Utr(+x,)

and the V ( + x l ) =

are

regular. It is conceivable that some algorithmic proof of the standard Finiteness Theorem always produces regular U(fi) in the process of decomposing a given open U. The argument we give in this note, producing V(fi), is an existence argument only and does not clarify this fine point of whether one can get by with regular U(fi). We pose this as an open question. Actually, we prove a relative form of the regular Finiteness Theorem. T h a t is, we fix a semi-algebraic subset X C K " . The sets U(f) and V(f) above, as well as closures, interiors, open sets, and regular open sets, now refer to the subspace topology on X. It is not true that a regular open set in K " necessarily intersects X in a regular open set in X. For example, i f X = K 1 C K 2 then U = U(x 2 _y2) is regular in K 2 but Uf'IX = K 1 -(0) is not regular in X . The relative regular Finiteness Theorem requires one assumption on X . For example, if X = [0, oo) C K 1 and if V is any set containing an open interval (0, e) C X , then 0 6 V ° (relative to X). Thus, U = (0, co) C X is not a finite union of regular open sets. We will say that X has boundary of codimension greater than 1 if no point of X has a neighborhood semi-algebraically homeomorphic to a neighborhood of a boundary point in a half-space (of any dimension). Note that algebraic sets have this property, for example, by Sullivan's even local Euler characteristic condition, [11]. Also, if X has boundary of codimension greater than 1 and if X ' is obtained from X by deleting a semi-algebraic subset W with the property that for all p 6 W - W, dim(W,p) < dim(X,p), then X ' = X - W * Research supported by NSF grant DMS85-06816 at Stanford University

171

has boundary of codimension greater than 1. Here dim(W,p) and dim(X,p) mean the local dimensions of W and X at p. It is rather easy to see what the boundary condition above says if one makes use of the existence of "triangulations" of semi-algebraic sets X, [8]. This means that X is semialgebraically homeomorphie to some finite union, Y, of open simpliees (of any dimensions) of some closed, linear, finite simplicial complex in affine space over K. Boundary faces of a simplex of Y need not belong to Y. The condition "boundary codimension greater than 1" on X just says that the picture below cannot occur as an open subset of Y. The simplex a must adhere to at least one other simplex/t ~ r with dim(/~) > dim(a).

~//~/,/..

dim(r) = dim(a) + 1

s

Triangulations Y of X can be chosen so that any preassigned finite fasnily of semialgebraic subsets of X correspond to subcomplexes of Y. We will use triangulations in the proof of the rational, relative, regular Finiteness Theorem, which is stated as follows. T h e o r e m : If the semi-algebraic set X C K n has boundary of codimension greater than 1, then every open semi-algebraic subset of X is a finite union of regular open sets

of the form V{gi}, g~E k[=l--. =.].

R e m a r k : This result was "conjectured", at least in a special case, in [3, page 233], and also at the end of [4]. I would like to thank J. M. Ruiz for some correspondence concerning the proof which follows. P r o o f : The proof will use properties of the real spectrum, SpecR(K[xl • • • x,]), which can be found in [1] and [9]. As far as the "rationality" aspect of the theorem is concerned, that is, finding polynomials with coefficients in k rather than K , the point is that SpecR(K[xl-.. x,]) is homeomorphic, in both the constructible and Harrison topologies, to the subset of SpecR(k[xl ... x,]) which extends the given order on k. Let U C X be open and let W -%X - U._.By the Artin-Lang Theorem, X, U, and W correspond to constructible sets X, U, and W C SpecR(K[=I "'" x,]). By the Finiteness Theorem, f f is open in X. Now, U and W are compact in the constructibleN topology. Fix a E U. We will first construct a certain constructible cover of W and extract a finite subcover. This will lead to a regular open neighborhood V{gi} of a, disjoint from W, that is, contained in U. Since these neighborhoods cover U, which is compact, the Theorem follows. Here, if g E k[xl...x,], we abuse notation somewhat and also write U(g) and V(g) for the constructible subsets of SpecR(g[xl... x,]) corresponding to U(g) and V(g) C Kn. This is so that bars, circles, and tildes don't accumulate too much. The point is, the Artin-Lang correspondence preserves closures and interiors, by the Finiteness Theorem, so V(g) = U(g)* holds in SpecR(K[xl--. x,l), as well as in K " . Consider fl E W. Since W is closed, fl 74 a, where -* indicates specialization of points in the real spectrum. Thus, either a ~ fl or a and fl are not compa~ble, that is, neither specializes the other. In the second case, there exists g# E k[X1 ... X,] such that

172

g#(a) > 0 and g#(fl) < 0. In particular, fl E U(-g#) C U(-g#). In the first case, that is, if a ~ fl, then we claim that there exists g# E k[xl... Xn] such that again g#(a) > 0 and fl E U(-g#). (See Lemma 4.7 of [4].) Assume this claim for the moment. Then we have a cover of W h y constructible sets

U(-g#) with a E U(g#) C V(g#). Extract a finite subcover, say W C U U ( - g i ) . Now, obviously, U(-gi) fl V(gi) = ~ , since V(gi) = U(gi) ° and gi > 0 on U(gi). Therefore, a E NY(gi) = V{gi} is disjoint from W, as desired. It remains to prove the following separation lemma, which is an improvement of Lemma 4.7 of [4]. L e m m a : If X C K n has boundary of codimension greater than 1 and if a ~ fl in .~ with a ~ fl, then there exists g E k IX1 ... X , ] with a E U(g) and fl E U(-g). P r o o f : We claim that because of the boundary codimension assumption, there exists 7 E X with 7 "* fl, but with a and 3' not comparable. Then choose g E k[xl . . . x , ] with g(a) > 0 and g(7) < 0. Since -y E U(-g) and "7 "-~ fl, we have fl E ~ C U(-g). To find 7, triangulate X so that X n Supp (a) and X V) Supp (fl) axe subcomplexes. Here, Supp indicates the irreducible varieties in K " corresponding to points of the real spectrum. The triangulation decomposes )[ into disjoint "constructible simplices". Say fl E and a E "~. Certainly a is a proper face of r, since a ~ fl and dim(r) = dim(Supp (a)) > dim(Supp (fl)) = dim(a). Now, find a simplex/~ ~ r , with a also a face of/~ and with dim(#) = dim(r). This may require retriangulating X slightly if the local picture is like one of the following. /

The first barycentric subdivision of a closed simplicial complex which contains X as a union of simplices will induce a suitable new triangulation of X. In any case, here is where we use the boundary codimension greater than 1 hypothesis. It is now possible to find 7 E p with 7 -* fl and dim(Supp (7)) = dim(/J) = dim(r) = dim(Supp (a)). In particular, a and 7 are not comparable. The point is, semi-algebraically h o m e o m o ~ h i c sets correspond to homeomorphic constructibles, so the structure of ~ C U/2 in X is exactly like the corresponding structure for a pair of linear affine simplices. The linear simplex case is the same locally as a pair of positive coordinate orthants, as below.

lfl !!!!l : Then the existence of 7 is just the observation that any order on a field/5 extends to /5(x), with x infinitesimally small and positive, relative to/5.

173

References

1. E. Becker, On the real spectrum of a ring and its application to semi-algebraic geometry, Bull. A.M.S., Vol. 15, No. 1, (1986), 19-60. 2. J. Bochnak, G. Efroymson, Real algebraic geometry and the 17th Hilbert problem, Math. Annalen 251 (1980), 213-242. 3. G. Brumflel, Partially ordered rings and semi-algebraic geometry, Cambridge University Press, 1979. 4. G. Brumflel, The ultraYilter theorem in real algebraic geometry, Rocky Mountain J. Math. 19 (1989), 611-628. 5. M. Coste, Ensembles semi-alg6briques, Proc. ConL on "Gdom6trie Alg6brique R6ele et Formes Quadratiquds", Rennes 1981, Lecture Notes in Mathematics, vol. 959, Springen (1982), 109-138. 6. C. Delzell, A finiteness Theorem for open semi-algebraic sets with applications to Hilbert's 17th problem, Contemporary Math 8 (1982), 79-97. 7. L . v . d . Dries, Some applications of a model theoretic fact to (semi-)algebraic geometry, Indag. Math. 44 (1982), 397-401. 8. H. Hironaka, Triangulation of algebraic sets, Proc. Syrup. in Pure Math. 29 (A.M.S. 1975), 165-185. 9. T.Y. Lam, An introduction to real algebra, Rocky Mountain J. Math 14 (1984), 767-814. 10. T. Recto, Una decomposici6n de un conjunto semi-algebrdico, Actas del V Congresso de la Agrupacion de Matem£tieos de Expresi6n Latina, Madrid (1978). 11. D. Sullivan, Combinatorial invariants of analytic spaces, Proc. of Liverpool Singularities Symposium I, Lecture Notes in Mathematics, vol. 192, Springer 1871, 165-168.

MEROMORPHIC

SUMS OF 2n-th POWERS OF FUNCTIONS WITH COMPACT Ana Castilla

ZERO SET

1. I n t r o d u c t i o n Geometric characterizations of sums of 2 n - t h powers by means of analytic arcs have been thoroughly studied in different contexts: for polynomials, [Pr], for meromorphie functions on a compact analytic surface, [Kz], and finatly for meromorphic germs and meromorphic functions on compact analytic manifolds of arbitrary dimension, [Rz2]. In all the situations they are based on the so called Becker's valuative criterion, [Be]:

T h e o r e m . Let K be a ~eld and f C K. Then f is a s u m o £ 2 n - t h powers of elements of K i f and only i f f is a s u m of squares in K and for any reM valuatlon v of K , we have 2 n l v ( f ).

The aim of this short note is to give a geometric condition for certain meromorphic functions on an arbitrary paracompact analytic manifold to be sum of 2 n - t h powers, and which generalizes the resull of [Rz]. To be more precise, let M be a connected paracompact real analytic manifold, let O ( M ) be the ring of analytic functions on M and let K be the field of meromorphic functions on M, that is, the field of fractions of O ( M ) . T h e n we show

T h e o r e m 1. Let f E O ( M ) be suc~ that its zero set is compact. Then f is a sum of 2 n - t h powers in K i£ and only if t'or every analytic curve c : [ - e , e] ~ M such that f ( c ( t ) ) = at m + . . . , with a ¢ O, it holds a > 0 and 2nlrn.

Our proof is based on Becker's criterion, Ruiz's result for the local case, as well as the attachment to each ordering ~ E S p e c r ( K ) of an ultrafilter U# of closed semianalytie subsets of M , which allows to give some geometric criteria for a function f to be positive in 3. This m e t h o d was used in JAn-Be] and is similar to the one exposed by P. Jaworski in his contribution to this volume. In fact the results he presents were also independently obtained by us, as he kindly points out. Therefore we omit the background concerning the attachment of N#, which can be seen in either of these references, and will appear in my dissertation, [Ca].

175

2. P r o o f o f t h e r e s u l t . Let Oh(M) denote the ring of b o u n d e d analytic functions on M . Let /3 be an order of K , and let W E be the convex hull of R in K . Then, W E is a valuation ring containing Oh(M) and with residue field R . Let m E denote its m a x i m a l ideal, and let S be the family of closed global semianalytic subsets of M . Thus the set /4E = {X • S i X F1 f - 1 ( [ - - 1 , 1]) 7~ 0 for all f • m E VI Ob(M)}. is an ultrafilter a m o n g the filters of sets of the family S, cf. [Jw, P r o p 1]. The main feature we are going to use is the following

L e m m a 1. Let f • Oh(M) and assume that there is X • Lt~ such that from below over X . Then f is a unit in W E.

If[

is bounded

Proof: [Jw]. Before entering in the proof of T h e o r e m 1, we need a technical lemma.

L e m m a 2. Let X C M be a dosed globM semianalytic set, and let f • Ob(M) be such

that fl X > O. Then for each n • N there are u, g • Ob(M) such that Ulx > O, u is bounded from below over X and u f = gn.

Pro@ Let v* : M ~ R be a continuous function such t h a t v*(x) > 0 for all x E M , and V i x = f n - 1 , which exists by Tietze's Theorem. Now let v : M ~ R be an analytic a p p r o x i m a t i o n of v* such that Iv(x) - v*(x)l < l v * ( x ) . We set ul = f ' ~ - l / v and u = u l / ( 1 + u21) • Oh(M). Since for x • X we have t h a t u l ( x ) > 4 f n - l ( x ) / h v * ( x ) = 4/5, we get t h a t u is b o u n d e d f r o m below on X . Now, u f = f'~ ( l / v ( 1 + u~)), and since v(1 + Ul2) is strictly positive, we have t h a t v(1 + u 2) = h '~ for some h • Oh(M). Altogether we get u f = gn, where g = f / h .

Proof of Theorem 1: Necessary condition. If f is a sum of 2n-powers, obviously f is nonnegative on M , so t h a t the only thing to be shown is the condition on the divisibility of the exponent m by 2n. Take any representation of f as sum of 2n-th powers:

f ~ E gi2nl~2n /n ,

withgi, hEO(M),h¢O,

M be an analytic curve. We set x = c(O). If h(c(t)) # O, since hZn(c(t))f(c(t)) = E gi(c(t)) 2n then

and let c : [O,s] ~

2n ord(h(c(t))) + ord(f(c(t))) = 2n ord E

gi(c(t))

176

and easily we get the result. In general, we need to use suitable valuations of the field K associated to the point x. Let m be the m a x i m a l ideal corresponding to x. The curve c defines a sequence

O(M)~, ¢--+O , ( M ) C-~R{t}, where ¢ ( f ) is the expansion at 0 of f o c and O , ( M ) denotes the ring of analytic germs at the point x. Let q = (ker¢) n O ( M ) m = ker(¢ o i). Thus, the h o m o m o r p h i s m ¢ induces an embedding k(q) ~ R({t}), where k(q) is the quotient field of O(M)r,/q and R ( { t } ) denotes the quotient field of R { t } . W e restrict the ordinary valuation of R ( { t } ) to a valuation ~ of k(q). In particular the class i in k(q) verifies 9 ( i ) = o r d ( f o c) = rn. Since O(M)q is regular, there is a valuation w of K with residue field k(q). Finally let v be the composite valuation of 9 and w. We have F~, = rv/r . Since f is a sum of 2n-th powers in K , by Becker's criterion 2n divides b o t h v(f) and w ( f ) , and therefore 2n divides ~ ( f ) = m = ord(f(c(t))).

Su~cient condition: Assume t h a t f ~ ~ I ( 2n. If f(x) < 0 for some x 6 M , then for any curve c: [0, c] --* M with c(0) = x, we have f(c(t)) = at m + . . . with a < 0. Therefore we m a y assume t h a t f is nonnegative over M. Then, by Artin-Lang, see [Jw], [Ca], f is a sum of squares. In particular, it follows from Becker's criterion, t h a t it exists a real valuation w of K such t h a t 2n does not divide w(f). Let fl be an ordering of K compatible with w, and let V = W E ( R , K ) be the convex hull of R in K with respect to ft. Thus, V is contained in the valuation ring of w. In particular Fw = F v / A where v is the valuation defined in V and A is an isolated subgroup of Fv. Therefore, 2n can not divide v(f) . Now, let m E be the m a x i m a l ideal of the ring V and let L/E be the ultrafilter of global closed semianalytic sets defined by

ldE = { X 6 S i X FI f - 1 ( [ - 1 , 1]) # 0 for all f 6 m E F10b(M)}. If

f--1 (0) ~ ~'~E,there

exists X e / 4 E such t h a t f [ x >

0. T h e n we can apply L e m m a

2 to the function f : there exist a unit u in V, and g 6 Oh(M) such t h a t u f = g2n. In particular 2n divides v(f), contradiction. Next, assume that f - l ( 0 ) 6 b/E. Since f - l ( 0 ) is compact, necessarily/~E is the principal ultrafilter defined by a point, say p 6 M . In particular ~9 is centered at p, t h a t is, the m a x i m a l ideal mp of p is convex with respect to }9. Now, we have :

O(M),,

O,(M)

177

Since, V dominates O ( M ) m , we can argue as in [Rz] to find and analytic curve germ c: [0, ~] --* M with c(O) = p and such that f ( c ( t ) ) = at m + . . . , where 2n does not divide m.

3. R e f e r e n c e s .

JAn-Be] C. Andradas, E. Becker: "A note on the Real Spectrum of Analytic functions on an Analytic Manifold of dimension one". Proceedings of the Conference on Real Analytic and Algebraic Geometry (Trento 1988), Lect. Notes in Math. no. 1420, 1-21, Berlin-Heidelberg-New York, Springer, 1990. [Be] E. Becker: "The real holomorphic ring and sums of 2 n - t h powers", in Lect. Notes Math. 959, Berlin-Heidelberg-New York, Springer, 1982. [Ca] A. Castilla: Dissertation, in preparation. [Jw] P. Jaworski: "The 1 7 - t h Hilbert problem for noncompact real analytic mangolds", this volume. [Kz] W. Kucharz: "Sums of 2 n - t h powers of real meromorphic functions", Monatshefte fiir Mathematik. 107 (1989) 131-336. [Pr] A. Prestel: "Model theory applied to some questions about polynomials", Proceedings of the Salzburg Conference, May 29 - June 1, 1986. [Rzl] J. Ruiz: "On Hilbert's 1 7 - t h problem and real NullsteUensatz for global analytic functions", Math. Z. 190, 499-514 (1985). [Rz2] J. Ruiz: "A characterizatization of sums of 2 n - t h powers of global meromorphic functions", Proceedings of the A.M.S. 109, 915-923 (1990).

P~zuDOOR~KOGONALITY OF POWERS OF ~ OF A HO~RPHIC

[~wt]gPING I N T W O

WITH

I,.K ~

COORDINATES VARIABLES

JACOBIAN

Zyn~,ut Charzy6ski a n d P r z e m ~ s l a w S k i b i 6 s k i

Summary.

In the paper the e q u i v a l e n c e

bian of holomorphic mapping a characterization mapping

is showed.

of holomorphic m a p p i n g

There is

obtained

w i t h the constant

This note is d e v o t e d

Jacobian of

the

to

h o i o m o r p h i c mappings with

In section 1 there is given an auxi ~liary infor-

In section 2 it is showed that the c o n s t a n c y

is equivalent coordinates coefficients

to the integral p s e u d o o r t h o g o n a l i t y

of the mapping

(theorem I).

Also

ficients are investigated

of

of

the

of the inverse mapping are derived

3 more general expressions

powers

presenting the coefficients

section are

and auxiliary

later precised.

a

obtained, reduced

to

those re-

(theorem 4).

These

Here a characterization

under consideration by one of its coordinates

be a number fixed for the sequel and

Er

is

in particular,

of the inverse m a p p i n g

in the next section.

notations

the

the mentioned coef-

values of the second one on one of the axes is obtained I. Terms,

the

(theorem 2). In section

some integrals of functions of two variables

the mappings

of for

(lemmas i, 2 and theorem 3). In section 4

some integrals of functions of one variable,

facts are applicated

the Jacobian

formulae

than those r e p r e s e n t i n g

simplification of the formula from the previous

{1.1)

Jaco-

to one of the axes.

Introduction.

namely,

also

and the restriction of the other coordinate

the constant Jacobian. mation.

of

in two v a r i a b l e s with the pseudoorthogon-

ality of powers of its coordinates by one coordinate

of the constancy

infqrmation, r > 0

and

Dy

of the

(theorem 5).

i. Let

0 < ~ < 1

a number

which

lyl

~
O. Alors il existe une constante C telle que, pour tout b C B, le volume k-dimensionnel de Xb soil majord par C × (diam (X~))k

199

Ddmonstration. On peut supposer que l'origine de R'* est dans Xb pour tout b. La fonction , p ( b ) = (diam (Xb)) -1

b,

est semi-algdbrique (pas continue a priori), et en appliquant l'homoth6tie de rapport p(b) ~, la fibre Xb on peut se ramener au cas oh Xb C [-1, 1] n pour tout b. Le tMorSme 4 donne alors le rdsultat voulu. [ ]

3. Prolongement de trivialisations Nous avons d6j~ 6voqu6 le th6or6me de triviatit6 semi-algdbrique de Hardt [HA]. Formulons en une cons6quence dans un cas tr~s particulier, qui nous occupera dans cette section. Soit X un ensemble semi-alg6brique, et f : X ~ R u n e fonction semi-algdbrique. Alors il existe un ~ > 0 et un hom6omorphisme semi-algdbrique h : f - l @ ) × ] 0 , C]

, f - l ( ] 0 , c])

tel que f o h e s t la projection sur ]0,~] et que h(., e) est l'identit6 sur f - l ( e ) . Peut-on prolonger continfiment cette trivialisation au-dessus de ]0, e] en 0 ? I1 faut ddj~i supposer que la fonction f est propre, mais ceci ne suffit pas. Prenons pour f la projection de [0, 1] x [0, 1] sur le deuxiSme facteur, et pour h la fonction de [0, 1] ×]0, 1] dans R 2 d6finie par

(x,

t), {(x,t) '

six+t>l

(tl_--Xx ,t)

s i x + t _ < 1.

Ce h ne peut passe prolonger par continuit6 ~ [0, t] x [0, 1]. Cependant, il y a toujours moyen de choisir la trivialisation pour que le prolongement soit possible. Th6or~me 6. Soit X un ensemble semi-algdbrique, et f : X --+ R u n e fonction semialgdbrique propre. Alors il existe u n ¢ > 0 et une application semi-algdbrique h : f - l @ ) × [0,~]

, /-1([0,¢])

tel que f o h est la projection sur ]0,@ que h(., e) est l'identitd sur f - 1 ( ¢ ) et que h e n restriction (i f - 1 (¢) x ]O, e] est un homdomorphisme sur f - x (] 0, ¢] ). Ddmonstration. Comme f est propre, on peut se ramener au cas off f - l ( [ 0 , 1]) est contenu

dans un pav6 [ - N , N] '~, avec N entier. On applique alors le thdorSme, qui nous donne un ¢ > 0 et un hom6omorphisme semi-alg6brique

g:K×]o,c]

, f-l(]0,c])

off K est un ensemble semi-alg6brique compact, tel que f o 9 est la projection sur ]0, 1] et que, pour tout t E]0,e], g(-, t) : K - - + f - l ( t ) est P-lipschitzien pour un entier P ind~pendant de t. Soit y E K. Puisque f est propre, alors l'intersection de {(V, 0)} x X avec l'adh6rence du graphe de 9 est non vide. Cette intersection est l~duite "a un point. En effet, soient (V, 0, x) et (y, 0, x') deux points de cette intersection. Par le lemme de s61ection des courbes, il existe deux chemins semi-alg6briques 7 , 7 ' : [0,5[ ) K

200

tels que 7(0) -- 7'(0) = y, que limt_.o(g(7(t), t)) = x et que l i m t ~ o ( g ( 7 ' ( t ) , t)) = x'. Alors la propri6t6 de lipschitzianit6 implique que x = x'. On a ainsi montr6 que g se prolonge par continuit6 ~t K x [0, e], ce qui nous donne te rtsultat. [ ] Le th6or~me prtctdent montre que f - l ( [ 0 , e ] ) se rttracte par dtformation semialg6brique sur f - l ( 0 ) , tandis que f - l ( ] 0 , e ] ) est semi-algtbriquement homtomorphe f - l ( c ) × ] 0 , e ] . On a donc ainsi des morphismes de l'homologie de f - 1 ( ¢ ) dans celle de f - l ( 0 ) (et ceci sur n'importe quel corps rtel clos de base). On peut interprtter le thtor~me prtc6dent comme une g6n6ralisation du th6or~me de structure conique locale pour les ensembles semi-algtbriques : soit Y un sous-ensemble semi-alg6brique ferm6 born6 de l'ensemble semi-algtbrique X ferm6 darts R n, et soit f une fonction semialgtbrique continue de X dans R, positive ou nulle, propre, et telle que y f - l ( 0 ) ; par exemple, on peut prendre pour f la distance ~ Y. Alors le voisinage compact f - l ( [ 0 , c]) de Y, pour ¢ suffisamment petit, est semi-alg6briquement homtomorphe au "mapping cylinder" d'une application semi-algtbrique continue de f - l ( c ) dans Y. A. Durfee [DU] a consid6r6 les voisinages semi-algtbriques d'ensembles semi-alg6briques : ce sont les f - l ( [ 0 , c ] ) comme ci-dessus. I1 obtient l'unicit6 ~ hom6omorphisme prts de ces voisinages (ils ne d6pendent essentiellement ni du choix de ¢, ce que l'on a vu, ni du choix de f). Mais les hom6omorphismes sont obtenus par utilisation du premier lemme d'isotopie de Thorn, et ils ne sont pas semi-algtbriques. I ~ thtor~me 6 s'obtient comme constquence facile du rtsultat de triangulation des fonctions semi-alg6briques (h valeurs dans R) de Shiota [SH]. Ce rtsultat dit que, si f : X --. R e s t une fonction semi-alg6brique continue bom6e, alors il existe un complexe simplicial fini K sur R, une fonction g : K ~ R affine sur chaque simplexe de K , et un hom6omorphisme semi-alg6brique h d'une rtunion U de simplexes ouverts de K sur X tels que f o h = g t u . =

4. Trivialisation de families de fonctions semi-alg6briques Dans cette section, nous exploitons le rtsultat de triangulation des fonctions semialg6briques ci-dessus, et en particulier le fait que la dtmonstration de [SH] est 616mentaire, c.-h-d, valide sur n'importe quel corps r6el clos. On peut alors lui faire subir le mSme traitement que celui qui donne le th6ortme de trivialit6 de Hardt h partir du thtor~me de triangulation des ensembles semi-alg6briques, dans [BCR] 9.3.1. Voici le rtsultat qu'on obtient : Th~or~me 7. Soient B C R m e t X C R ~ × B des ensembles semi-algdbriques. On note p la projection de X sur B. Soit f : X ~ R une fonction semi-alg~brique telle que p o u r tout b E B , la fonction fb : X B --+ R est continue. Alors il existe une partition semi-algdbrique

201

finie B = U i e i Bi, et pour chaque i E I un carrd commutatif

($1xi ,Plx~) R × B >

X i = X f l p - : ( B i)

i

hi

Fi

R

oft F i e s t un ensemble semi-algdbrique, h i, ¢i et gi sont semi-algdbriques continues et v&ifient que pour tout b G B i, h~ : Xb ---+F i e t g~ : R ~ R sont des homgomorphismes. Ddmonstration. Sans perte de g6n6ralit6, on peut supposer que f est born6e, 7t valeurs dans [-1, 1]. Soit a E B. La fibre f~ : X ~ ~ k ( a ) est une fonction semi-alg6brique continue bom6e, et on peut lui appliquer le th6or~me de triangulation des fonctions semi-alg6briques continues [SH] pour le corps r6el clos k(a) : la fonction f,~ est semi-alg6briquement 6quivalente ~ la restriction d'une fonction affine par morceaux sur un complexe simplicial fini. On insiste ici pour que le complexe simplicial et la fonction affine par morceaux sont des extensions ?: k(a) d'objets d6finis sur R. De fagon pr6cise, il y a un complexe simplicial fini K d6fini sur R, une fonction ¢ : K --+ R, affine sur chaque simplexe, une r6union de simplexes ouverts F C It', et un carr6 commutatif

fa

,

k(o,)

Go,

Ck(,,)

ofa ¢ est ~[F, et off H a et G~ sont des hom6omorphismes semi-alg6briques. On r6cup6re ainsi un sous-ensemble semi-alg6brique B' C B, tel que a C B', et un carr6 commutatif

X'=XMp-I(B

')

(,fix' ,PIX' ) ,

H

RxB' G

CxB j

F x B'

>

R x B'

202

off toutes les applications sont compatibles avec les projections sur B ' , et off H et G sont des hom6omorphismes semi-alg6briques dont les fibres en c~ sont H a et G,~ respectivement. Pour obtenir t'6nonc6 du th6or~me, il ne reste plus qu'h prendre en compte la compacit6 de pour avoir la finitude de la partition. [ ] On obtient aussi des r6sultats de finitude et de calculabilit6 pour la triangulation des fonctions. Ces r6sultats ont 6t6 montr6 dans [BS] en r6examinant le preuve de [SH]. Ici, on les obtient par un artifice logique. C'est moins fatigant, mais le d6faut est que l'on n ' a aucune pr6cision sur les bornes obtenues, ~ part le fait qu'eltes sont r6cursives. Th6or~me 8. Etant donnds n et d, il existe s fonctions semi-algdbriques continues ~i:RnDxi

~R,

i = 1,...,s

de degrd d, et un entier ~, tels que pour toute fonction q~:W~ D X

~R

de degrd d, il existe un entier i, 1 < i < s, et des hom~omorphismes semi-alg~briques h:X

)X i

et

g:R

)R

de de g r & au plus ~, et qui vdrifient ~ i o h = g o~. De plus s e t t sont bornds p a r des fonctions rdcursives de n e t d. D~monstration. On peut param6trer, par un ensemble semi-alg6brique B ( n , d), la famille de toutes les fonctions semi-alg6briques continues, ~t valeurs darts R, et dont le domaine est un sous-ensemble semi-alg6brique de R '~. On obtient ainsi une fonction

f

X

~ R

'0 h qui l'on applique le th6orSme pr6c6dent. On r6cupSre une partition semi-alg6brique finie B ( n , d) = gls=1 B i, et en choisissant bi E B i, on a pour chaque i E I un carr6 commutatif

X i - - - - - X A p - I ( B i)

(flx~,plx~)

R×B

hi X i = Xb~

R

i

203

Oh h i et g i sont semi-alg6briques continues, v6rifient que pour tout b E B i, h~ : Xb ~ X i e t g~ : R ~ R sont des hom6omorphismes et clue h~ = Idx, et g~, = Ida. En prenant pour t le maximum des degr6s des h i et des 9 i, on obtient la premi6re partie de l'6nonc6. Pour obtenir la r6cursivit6 des bomes, on remarque que, une fois que l'on a fix6 n, d, s e t t, la th~se du th6or6me 8 "I1 existe s fonctions semi-alg6briques.., qui v6rifient ¢i o h = g o ¢." s'6crit comme un 6nonc6 ~ ( n , d, s, t) du premier ordre de la th6orie des corps r6els clos, que l'on peut produire algorithmiquement ~ partir de n, d, s e t t. L'algorithme pour calculer une borne pour s e t t en fonction de n e t d consiste alors ~ appliquer un algorithme de d6cision pour la th6orie des corps r6els clos aux 6nonc6s q~(n, d, 1, 1), ~5(n, d, 2, 2 ) , . . . et en s'arr&ant quand on trouve une formule vraie. La premiere partie du th6or~me dit que l'algorithme s'arrate bien. [ ]

5. U n c o n t r e - e x e m p l e p o u r les f a m i l i e s d ' a p p l i c a t i o n s s e m i - a l g 6 b r i q u e s d6finies sur R ~

Le r6sultat de Benedetti et Shiota formul6 darts le th6or~me 8 g6n6ralise un r6sultat de Fukuda [FU] qui concerne les fonctions polynomiales. On conna~t aussi un thEor~me d'Aoki [AN] qui dit que, le nombre de types topologiques de gennes d'applications polynomiales de R 2 dans R 2 de degr6 donn6 est fini. Mais dans ce cas, on ne peut pas esp6mr une g6n6ralisation aux families d'applications semi-alg6briques continues, comme le montre le contre-exemple suivant inspir6 par Nakai [NA]. Soit a = (al,a2,aa,a4) E R 4,avec 0 < al < a2 < a3 < a4. Soit fa : R 2 ~ R 2 la fonction semi-alg6brique continue d6finie par fa : ( x , y ) '

' (Ixl,ailxl lYl)

Oh i est le num6ro du quadrant dans lequel se trouve (z, y) (les quadrants sont num6rot6s darts l'ordre direct). On va montrer que pour deux quadruplets diff6mnts a et b, les applications fa et fb ne sont pas en g6n6ral semi-algdbriquement 6quivalentes. Avec un petit peu plus de travail, on aurait le m~me r~sultat pour l'6quivalence topologique. Examinons d'abord les fibres de la fonction f~ : - f~-l(0,0) est l'axe des y, - f [ l ( u , 0), pour u > 0, comprend les deux points (u, 0) et ( - u , 0),

- f~l(u,v),

?)

pour u , v > 0, comprend les quatres points ( u , ~ ) ,

V

--Y

( - u , a--~u), ( - u , a---~u),

--V

(u,;S),

- dans les autres cas la fibre est vide.

Soit maintenant 3' : [0, 1] --+ I~2 un chemin semi-alg6brique avec 3'(0) = (0, 0) et 3"(]0, t]) c {(u,v) ; u > 0, v > 0}. Ce chemin arrive ~ l'origine avec une tangente de coefficient directeur t(7 ). Si 0 < t(7 ) < +c~, alors S(7) = fd -1 (7([0, 1])) contient les quatre points remarquables d'abscisses (dans l'ordre croissant) t ( 7 ) / ( - a 3 ), t ( 7 ) / ( - a 4 ), t(7)/a2 et t(3")/al sur l'axe des y. Ces points, que l'on d6signera par P3(7), P4(3"), P2(3") et P1(3") se remarquent dans S(3") parce que ce sont ceux dont le compl6mentaire a trois composantes. Ce pMnom~ne ne se produit pas si t(3") = 0 ou +oo. Choisissons alors un chemin semialg6brique 3' comme ci-dessus, avec 0 < t(7 ) < +e~ . On peut trouver deux suites de chemins semi-alg6briques (6,) et ( ¢ , ) , telles que 60 = ¢0 = 3", clue P3(6n+1) = P4(6,,) et

204

que P1(¢,~+1) = P2(¢n). DEfinissons s ( n ) comme le plus petit entier s tel que Cs soit en dessous de 5,, pros de l'origine. Les remarques faltes plus haut montrent que

(:,)°

et

t(¢n) = t(7 )

al

On en d6duit donc que n

lira s ( n ) = log(a3) - log(a,) ~ n log(a~) - log(~:)

On suppose maintenant que f~ et fb sont semi-alg6briquement 6quivalentes, c'est-~-dire qu'il existe des hom6omorphismes semi-alg6briques g e t h de I~2 sur lui-m~me tels que fb o g = h o f , . D'apr~s ce que l'on a vu pour les fibres, h envoie l'origine sur l'origine, le demi-axe des u > 0 sur lui-m6me, et le premier quadrant ouvert sur lui-m~me ; g envoie l'axe des y sur lui-mSme. La situation avec les suites de chemins (~5.) et (¢,~) se transporte par h en une situation semblable, avec la m6me fonction s ( n ) , et on doit donc avoir log(a3) -- log(a4) log(a1 ) - log(a2)

log(b3) - log(b4) log(b1 ) - log(b2)"

Cette derni~re 6galitE montre que l'on ne peut pas avoir un nombre fini de classes d'Equivalence semi-algEbrique dans la famille des fonctions fa quand a varie.

R6f~rences [AN]

Aoki K., Nagachi H. : On topological types of polynomial map germs of plane to plane, Memoirs of the School of Science & Engineering 44 (1980), 133-156, Waseda University

[BS]

Benedetti R., Shiota M. : Finiteness of semialgebraic types of polynomial functions, para~tre

[BCR]

Bochnak J., Coste M., Roy M-E : G6om6trie alg6brique r6elle, Springer (1987)

[DU]

Durfee A. : Neighborhoods of algebraic sets, Trans. A.M.S. 270 (1983), 517-530

[FU]

Fukuda T. : Types topologiques des polyn6mes, Publ. math. I.H.E.S. 46 (1976), 87-106

[GR]

Gromov M. : Entropy, homology and semialgebraic geometry (after Y. Yomdin), Ast6risque 145-146 (1987), 225-240

[HA]

Hardt R. : Semi-algebraic local-triviality in semi-algebraic mappings, Amer. J. Math. 102 (1980), 291-302

[NA]

Nakai I. : On topological types of polynomial mappings, Topology 23 (1984), 45-66

[SH]

Shiota M. : Piecewise linearization of subanalytic functions II. Dans : Real analytic and algebraic geometry, Lect. Notes Math. 1420, 247-307, Springer

STIEFEL ON

A REAL

ORIENTATIONS ALGEBRAIC

VARIETY

A . I . DEGTYAREV ABSTRACT. Some natural Stiefel orientations on the normal bundle of the fixed point set of an involution on a smooth manifold are constructed. This result is applied to nonsingular real algebraic varieties in order to generalize Rokhlin's construction of complex orientations on a separating real algebraic curve

INTRODUCTION 0.1. T h e main purpose of this p a p e r is to generalize Rokhlin's construction of complex orientations on a separating real algebraic curve. T h e original construction is the following (Rokhlin [6]): Denote, respectively, by CA and RA the sets of real and complex points of curve A. Since R A separates CA, the complement CA \ XA consists of two c o m p o n e n t s C A ± . T h e complex orientation of complex manifolds C A ± defines a pair of opposite orientations of their c o m m o n b o u n d a r y ]RA. A direct generalization of this construction is obvious: Let F be an m-dimensional s m o o t h submanifold of a smooth n-dimensional manifold X . Suppose t h a t the fundamental class [F] of F vanishes in H,,,(X). T h e n the linking coefficient m a p lk : H n - m - l ( X \ F) ~ Z2 restricted to the b o u n d a r y of a tubular neighborhood of F in X is an (n - m - 1)-dimensional Stiefel orientation on the normal bundle vF of F in X . (The notion of Stiefel orientation is a generalization of the ordinary orientation and Spin-structure on a vector bundle; see Definition 1.4.1 for details.) This orientation is well defined over the subgroup Ker[inclusion. : H , , - m - I ( F ) --* H n - m - l ( Z ) ] . 0 . V i r o [8] proposed a further generalization of this construction (see 4.2) for the case when pair (X, F ) is supplied with the following additional structure: X is a manifold with an involution c : X --* X , and F = F i x c is the fixed point set of c. In this p a p e r we give a n o t h e r approach to Viro's construction which shows t h a t the arising Stiefel orientations are, in fact, also the linking coefficient functionals on homology groups of some a p p r o p r i a t e spaces S s ' - l X \ s k - l F associated with pair (X, c). These orientations exist if for some k /> 1 class IF] vanishes under some "higher" inclusion m a p s k era+k_1 : H . ( F ) --+ H m + k - l ( X ) , and it is defined over the subgroup K e r e .k. . . 1 (see T h e o r e m 4.1.1 for details).

Remark. The advantage of the approach of this p a p e r is t h a t the construction is connected with the general theory of Z2-spaces (i.e. spaces with involution). In particular, if F is an M - s p a c e (i.e. d i m H , ( F ) = d i m H , ( X ) ) , this theory enables to express the Key words and phrases. Stiefel orientation, involution, fixed point set~ real algebraic variety.

206

condition necessary for existence of the orientations in terms of characteristic classes of X (Degtyarev [3]). 0.2. Applying Theorem 4.1.1 to an n-dimensional real algebraic manifold A yields a series of partial i-dimensional Stiefel orientations on the tangent bundle of RA, n - k 0}; On is the group of orthogonal transformations of Nn; V,,k is the Stiefel manifold, which can be defined as either the quotient space O n / O n - k , or the space of all orthogonal k-frames in 1Rn; it is well known that ~ri(v~,k) = 0 for i < n - k, and ttn-k(Vn,k ) = 2;2 (see Steenrod, Epstein [7]); for an On-bundle ( we denote by S(() the associated Sn-l-bundle; S(() is called the sphere bundle of ~. Note that S(~) = Vn,l(~).

208

Let R = (~tl,... ,zk+l) be an orthonormal (k + 1)-frame in R n+l. Denote by Ak(R) the singular simplex A k --. S ~, ~ t~z~ , , ~ tixi/ ~ ( ti) 2. We will use the following singular chains in S":

Ek = An(_~l,...,ek+l)

k+l ~t- E ~ n ( ' ~ I ' ' ' ' ' - - C i ' ' ' ' ' " k + I ) ' i=2

k+l =

i=2 +

E~: will also be considered as singular chains in D k. If c is the antipodal involution on S n, then obviously c#E~_ = E k,_ and OE~: = E k-l+ + E k-a_. 1.4. S t i e f e l orientations. 1.4.1. D e f i n i t i o n . A (reduced) k-dimensional Stiefel orientation on an On-bundle ~ is a class xk E Hk(Vn,n-k(~)) (resp., ;¢k C Hk(Vmn-k(~)) ), such that for any fibre Vn,n-k of fibration Vn,n-k(~) ~ X the restriction of xk (resp., 5tk) to ffIk(Vn,n-k) is non-zero. The set of all (reduced) Stiefel orientations on ~ is denoted by Stk(~) (resp., Stk(~)).

Remark. A 0-dimensional Stiefel orientation is an ordinary orientation of vector bundle ~. A 1-dimensional orientation is a Pin+-structure on ~. A pair consisting of a 0- and 1-dimensional orientation is a Spin-structure. Remark. If k > 0, obviously Stk(~) = Stk(~). If k = 0, a reduced orientation is a pair of opposite ordinary orientations on ~. (So this notion is not trivial only if the underlying space X is not connected.) 1.4.2. T h e o r e m (well known). Let ~ be an On-bundle over X . Then Stk(~) and Stk(~) are not empty if and only i f w k + l ( ( ) = O. In this case Stk(~) (resp., fftk(~)) is an afl~ne space over H k ( X ) (resp., Hk(X)). The reduction map Stk(~) ~ gtk(~) is af[ine. 2. Z2-SPACES 2.1. S i n g u l a r S m i t h h o m o l o g y . Throughout this section X is a fixed Z2-space, i.e. a topological space supplied with an involution c : X ~ X. We denote by Fix c the fixed point set of c, and by X / c the quotient space. For simplicity we assume that pair (X, Fix c) allows a c-invariant cell partition. (Note, though, t h a t most results can easily be extended to general Z2-spaces using an equivariant cell approximation of (X, c).) 2.1.1. N o t a t i o n . Denote A = Z~[c]/(c 2 = 1). Since Z2 acts on S . ( X ) via the induced involution c#, this complex can be considered as a complex of A-modules. We will also consider the Z2-algebra g * ( R p k) = Z2[h]/(h k+l = 0) and the H*(Rpk) module H.(Rpk). The generator of Hi(Rp k) = Z2 is denoted by hi; both hi and h i are assumed to be zero if either i < 0 or i > k.

209

2.1.2. Definition. The Smith homology groups S H , ( X ) are the homology groups of the Smith complex S m , ( Z ) = Ker[(1 + c#) : S , ( X ) --~ S,(X)]. The inclusion homomorphism S H , ( X ) --~ H , ( X ) is denoted by smx. 2.1.3. P r o p o s i t i o n . (I) Let C , ( X ) be the cell complex corresponding to some cinvariant cell partition of pair (X, Fixc). Then S H , ( X ) = g,(Ker[(1 + c # ) : C , ( X ) ---,

c.(x)]); (2) denote by p : X ~ X / c the projection. Then the map x ~ relp#x induces isomorphism H . ( S . ( X ) / S m . ( X ) ) ~- H . ( X / c , Fix c); (3) S m . ( X ) naturally splits into S.(Fixc) @ S . ( X ) / S m . ( X ) , due to (2) this induces an isomorphism S H , ( X ) ~ H,(Fix c) ~ H , ( X / c , Fix c). Proof. The last two statements are well known for cell Smith homology groups (cf. Bredon [1]), so they follow from the first one. Proof of the first statement is absolutely anMogous to the standard proof of similar statement about ordinary homology groups H , ( X ) . We omit the details since they would require developing from the very beginning the theory of functor SH,, which is absolutely analogous to the standard singular homology theory. [] 2.1.4. C o r o l l a r y . Let (X, Fix c) is an n-dimensional cell pair. Then SH~(X) = 0 for i > n, and S H , ( X ) = Ker[(1 + c , ) : Hn(X) -~ H,(X)]. 2.1.5. C o r o l l a r y . Let X be an n-dimensional smooth manifold, and c be a smooth involution. Then there exists one and only one class [X]s E SHn(X) such that smx[X]s = [X]. We call this class the Smith fundamental class of X . ) For (smx)n : S H , ( X ) -* Hn(X) is a monomorphism, and [X] is a c.-invariant class. [] 2.2. S p e c t r a l s e q u e n c e o f an involution. 2.2.1. Definition. Denote by g k x , 0 ~ k ~ c~, the twisted product S k ×z2X supplied with the induced involution and filtration by subspaces 0 C S ° X C S1X C ... C SkX. (Here inclusions are induced by the standard inclusions 0 C S O C S 1 C -'- C Sk.) The homology and cohomology spectral sequences of filtered space S k X are denoted by kE~q(X) and kEPrq(x ) respectively and called the spectral sequences of involution c. (Here mad in other similar notations space X will often be omitted.) Note that if X is a trivial Z2-space, i.e. c = idx, then $ k X = Rp k x X, and H , ( S k X ) = H , ( R p k) ® H , ( X ) . In particular, both the spectral sequences degenerate in term E 1.

2.2.2. Proposition. (1) kE.~. = H . ( R p k) ® H . ( X ) , and kE~'* = H*(Rp k) ® H*(X); (2) d (hp ® x) = ® (1 + and ® x) = h, ® (1 + e*)x; (3) if r >1 2, both the sequences coincide with the Serre spectral sequences of ~bration S k X --, Rp k = Skpt; (4) ~E** is a bigraded H*(Rpk)-algebra, and kE:, is a bigraded kE**-module; if r >1 2, differentials d r and dr are derivatives; (5) kE~q and kE~q converge to Hp+q(SkX) and HP+q(SkX) respectively; the convergence observes both the multiplicative structures;

210

(6)

l 1 2; (4) provided that X is a tlnite CW-complex, Er, and E* converge to H,(Fix c) and H*(Fixc) respectively; the convergence observes the multiplicative structures, but it does not observe the grading of H,(Fix c) and H*(Fix c); (5) homology maps ¢~ are injective; cohomology maps ~b~ are surjective. 2.3.3. N o t a t i o n . The filtration on H,(Fixc) which arises due to convergence E,~ =:~ H,(Fixc) is denoted by f , ( X ) . The composed maps ~q ~ .~q/Fq+l - ~ E ~ are denoted by eq. Note that filtration 9rq can as well be defined if E,~ does not converge to H,(Fix c) (Kalinin [5]). By definition, the class of cycle x -- ~ x i , xi C Si(Fixc), belongs to 5rq if for some p the image of ~ hp+q-i ® xi E Hp+q(£ °° Fix c) in Hp+q(S°°X) comes from

Hp+q(SPX). 2.4. G e o m e t r i c d e s c r i p t i o n . Let C, (resp., C*) be a chain (cochain) complex of A-modules. Define filtered complexes s k c , and SkC* as follows: as a graduated group g k c , = H , ( R p k) ® C, (resp., S k C * = H*(Np k) @ C*); the boundary (coboundary) operator is the map hi ® x ~-+ hi ® Ox + hi-1 ® (1 + c)x (resp., h i ® z ~-~ h i ® 5x + h I+1 ® (1 + c)z);

211

the filtration is, respectively, Fp = Ira[inclusion, ®id : SvC, --* SkC,], and FV = Ker[inclusion* @id : SkC * -~ SPC*]. The spectral sequences of filtered complexes SkC, and SkC * are denoted by kE;q(C,) r and kE~q(C*) respectively. 2.4.1. T h e o r e m . There exists a natural chain map O : $ k S . ( X ) --+ S . ( $ k X ) which induces isomorphisms of (1) spectral sequences kE~.(S.(X)) ~- kEr**(X) and kE**(S*(X)) ~ kE**(X); (2) homology groups H . ( S k S , ( X ) ) ~ H . ( S k X ) and H*(SkS*(X)) ~ H*(SkX); (3) Smith homology groups H , ( R p k) ® S H , ( X ) -~ SH,($kX).

Proof. Let p be the projection S k x X --~ S k X , and ~ be the diagonal involution on S k x X . Fix some natural chain homotopy equivalence # : S, ( X ) ® S , ( Y ) ~ S, ( X x Y ) and define 0 as the map hi ® x ~-~ p##(Ez+ ® x). Since p#5# = p#, and # is natural, O0(hl ® z) - Oe(hi ® ~) = -~- p##(:~'-']~b- 1 ® C#X -~- ~iq_ ® OX) - p # # ( [~,i--1 i ®Ox)= + ® x + ~i--1 _ ® x + E+

= p##(~-~

® (1 + c#)x -- r i --- ~ ® x) = p#(1 - 5 # j~~ / E i+- 1 ® c#~) = o.

This proves that 0 is a chain map. The subquotient map

ep: sps.( x ) / s p - l s . ( x ) = hp ® S,( X)

S.($vX, Sv-IX)

can be decomposed through the map

~p : hv ® S . ( X )

~ S,(D~ x X , S v-1 x X), h v ® x ~ # ( E i+ ® x ) ,

which induces an isomorphism of terms E 1 of the corresponding spectral sequences. Hence 8 induces isomorphisms of the spectral sequences and homology groups. To prove the last statement, we need the following lemma: 2.4.2. L e m m a . Let f : C, -* D, be a chain map which induces isomorphism f , : H , ( C , ) --, H,(D,). Then ( i d ® f ) , : H , ( S k C , ) -~ H , ( S k D , ) is also an isomorphism for any k.

Proof. i d Q f induces an isomorphism of terms E 1 of spectral sequences kEr**(C,) and kE:,(D.). [~ Lemma 2.4.2 and Proposition 2.1.3(2) imply that 0 induces isomorphism

H.(,.qk(S.(X)/Sm.(X))) ~ H . ( S . ( $ k X ) / S m . ( S k X ) ) , and, due to Five Lemma, isomorphism

H (SkSm ( Z ) ) ~ SH ( S k X ) Since c# acts trivially on S m , ( X ) , the first group is equal to H,(ll~p k) ® SH,(X).

[]

212

2.4.3. C o r o l l a r y (of 2.4.1 and 2.4.2). Theorem 2.4.1 remains valid i[ S . ( X ) is replaced with the cell complex corresponding to some ¢-invariant cell partition of pair ( X , Fix c). (Note that for (1) and (2) it even is not necessary that Fix c be a cell subcomplex of

X.) The remaining statements provide a clear geometric description of spectral sequences kE.~. and E .r. (The dual description of cohomology spectral sequences is also valid.) The term "chain" ("cycle") means either singular or cell chain (resp., cycle). 2.4.4. C o r o l l a r y . Let x be a q-dimensioned cycle in X . Then (1) dl(hp ® x) = hp-i ® (1 + c#)x; (2) let r q--r+l

there exist some chains Yi E S i ( X ) , q - r + 2 q + 2 (note that in this case Eq = E ? .

r--1 = xq-1 + (1 + c#)yq-1 = Oyq Proof. If x E Sh"q, then obviously x E ~T'r--I - q - l , and eq_ix vanishes in S~q-1. Hence S~-; C K e r e q - I. Let x E Ker eq_ 1~-1. This means that Xq-1 + (1 + c#)yq-1 = dr-2zq-r+2 + Oyq for some cycle Zq-~+2 E Sq_r+2(X) and chain y'q E Sq(X). Due to Corollary 2.4.5 this is equivalent to existence of some chains zi, q - r + 3 ~< i ~< q - 1, such that Ozi = (1 + c#)zi-x, and Xq_ 1 -~- (1 -'~ e#)Yq--1 = (1 + c#)zq--a + Oy'q. Denoting y~ = Yi -k zi for

q -- r + 2 ~< i ~< q - 1, we see that x E S~-~. This completes proof of (1). Statements (2)-(4) immediately follow from the definitions, and (5) follows from comparing the definition of ,95"~ and Proposition 2.4.6. [] 2.5.5. D e f i n i t i o n . Define map ~q : $5"~ ---* Hq( s r - I x )

_";~

r %,"

2 - r--l r

as the product Hq(Sr-IX)

(here the third map is invertible due to 2.2.2(6); the last map is the edge homomorphism of spectral sequence "-lE~q).

214

2.5.6. P r o p o s i t i o n . Le~ x = ~ xl, xi E SHI(X), and r ~ O. Then i>~q s m s , x ( E hq+r-i ® xi) = 0 in Hq+r(S~X) if and only if x e Sa~q+~+a. In this case s m s . + t X ( E hq+r+l-i ® xi) = -r+2 'r+2 Proof. According to the definition the condition x C s T-q+r+l is equivalent to the equality ~ hq+r-i ® xi = 0 ( ~ hq+r+l-i ® Yi) for some yl C Si(X) (cf. Corollary 2.4.4). In this case ~ hq+~+l-i ® xi = 0 ( ~ hq+r+2-i ® yi) + ho ® (xq+~+l + (1 + c#)yq+~+l); =r+2 _ E Hq+~+I(S~+aX). [] hence this element projects to eq+~+l,

Remark. Since Ca and the edge homomorphisms ~-XE~q -+ Hq(S~-IX) are injective, Proposition 2.5.6 can as well be taken for definition of S~-~ and e~. 3. THE BASIC CONSTRUCTION 3.1 G r o u p s H n - I ( S k S ( ( ) ) . Throughout this section ~ denotes some fixed On-bundle over space X. 3.1.1 Definition. Denote by ~k the tautological linear bundle over Rp k, and by ~k ® the exterior tensor product, which is a bundle over ]l{pk x X. 3.1.2. P r o p o s i t i o n . (1) There exists a natural tJbrewise homeomorphism S(r]k ® ~) = SkS(~), S(~) being considered as a Z2-space via the standard antipoded involution; (2) if space X is connected and k 0, const# EJ+ = 0. Hence ~ k , , - k projects to the map a ~ hk-1 ® #(p#a ® idpt), which induces isomorphism

Hn-k(X)

) hk-1 ® Hn-k(X). k

3.2.4.

+ ~ Hn-i(Vn,i(~)) as the direct

D e f i n i t i o n . Define Xk : H " - a ( S k - I S ( ~ ) )

i=l

sum of composed maps restriction

Hn-I(Sk-lS(~))

)

Hn-a(si-IS(~))

, H rt--i (Vn,,(~)).

3.2.5. P r o p o s i t i o n . For any l n -- k. Proof. Consider an equivariant tubular neighborhood Tk-1 of S k - l V in s k - l x and denote by OTk-1 its boundary, by pr : OTk-1 --* S ~ - I V the projection, and by in : Tk-1 ~ S k - I X the inclusion. The restriction pr[ pr-~ Fv obviously coincides with the sphere bundle associated with qk-1 ® j * u v , so the orientation )0 the number of limit cycles of X~. of period < T

is

locally uniformly bounded.

228

P r o o f. The

proof is the

same

as that

of

[FP],

here because of the explanatory character of this paper. Take the flow ¢:[0,x]xWxU ---> Q, with x 0};

(n-l-sphere);

topology open

Ily-xl[ < r }

The

the be

open

always

semialgebraie

of subsets of R n containing

those of type U(f) = {x where and

e

Rn: f(x)

f E R[Xl'''''Xn ] is a polynomial, intersection,

between

semlalgebraic

its graph Gr(f) function

and

X c R n and

is a semialgebraic

f:U .......>. R,

and

complementation. subsets

where

> 0}

A

stable

under

continuous

Y c Rm

is

finite map

union

f:X

semialgebraic

) Y if

subset of R n+m. For a s e m i a l g e b r a i c

U is an open

semialgebraic

copy from the case R = ~ the notion of derivability.

(~) P a r t i a l l y s u p p o r t e d by C.I.C.Y.T PB 89/0379/C02/01 Plan ERB 4002 PL 910021 - (91100021)

and

subset

of R n, we

The ring S~(U)

Science

is

241

the

set

of

derivatives

semialgebraic of

semialgebraic

each

order

functions.

function.

A

map

coordinate

function

functions and

An

J

introduced

called

semial~ebraically

in

p:XxZ Y

and

g

and

9.4

the

in

Sm(U)

192]:

proper

closed,

identity

if

for

i.e.

semialgebraic

p maps

on

also

This

Y,

[Br.,

together

characterization

f : X ......~ Y

subsets

ma~

J

a

Nash

if

each

is a polynomial semialgebraic

map

f:X

> Y

semialgebraic

f(x) = g(y)} every

semialgebraic

map

[D-K 1] -see

Let

Nash

every

it

is map

~ Z closed

follows

X c Rn,

a

immediately closed.

[D-K I, Thm

12.5]

that

Also,

from

that

subset X of R n is proper

with

the

if and

provide

proper maps,

us

which

in this note:

semialgebraic

Y c Rm.

of Z. Taking

it follows

of semialgebraically

be

semialgebraic

subset p(C)

8.13.5]-

is the starting point of our considerations

T h e o r e m 1. -

called

If each f

proper maps are semialgebraically

if X is bounded.

the following

a

(continuous)

is

a semialgebraic

constant map on a closed semialgebraic only

is

are

partial

projection

onto a closed

semialgebraically

in

have

The notion of proper

= {(x,z) ~ XxZ:

is semialgebraically

9.1

[DKl,pg.

> Y, the canonical

Z = Y

f

is a Nash function.

map was

subset C of X x Z Y

them

> Rm

map.

which

of

f = (fl ..... fm):U f

> R

all

element

we say that f is a polynomial

g:Z

f:U

The

map

following

between

the

statements

are

equivalent: (1)

f

is

semialgebraically

proper.

(2)

f

is

semialgebraically

closed

and

its

fibers

are

closed

i n Rn a n d

bounded. Our polynomial

main

results

mappings:

are

refinements

of

this

theorem

for

Nash

or

242

T h e o r e m 2.- Let Nash map.

f:R n

> R m be a n o n - c o n s t a n t

Then f is semialgebraically

T h e o r e m 3.- Let f:R n semialgebraically

semialgebraically

proper.

> R be a non-constant

proper

if and only

polynomial

if there exists

map.

obtain

some

consequences

of

these

results.

Then f is

some M E R ÷ such

that the fibers f-l(t) are bounded for every t ~ R with We

closed

Itl > M.

For

example,

from

Thm 3 it follows: Corollary is

I.- Let

f:R n

semialgebraically

that

for

every

) R m be a n o n - c o n s t a n t

proper

if and

t ~ R + with

t>M,

only

the

if

polynomial

there

inverse

map.

exists

image

Then f

M E R ÷ such

f-l(sm-l(O,t))

is

bounded.

The r e s u l t s . proof

of

For

an easy

simple proof Proposition

the but

convenience useful

is omitted I.- Let

semialgebraic

> Y,

proper.

Proof.-

prove

g:Z---) Y

is

projection is

must

closed

a

Y c R m,

Then, that

if

Z.

map,

begin pg.

But

Xx Z Y

Our

that

says closed.

Proof of theorem 2.- Using

semialgebraic

is

the

Hence p(C)

Thm.

a

p:XxyZ

semialgebraic

= hog(z)}

semialgebraically

[D-KI,

T c Rp

Z c Rq

XxTZ = {(x,z) E XxZ:hof(x) hypothesis

in

we

with

192],

the

whose

,

such

maps

between

that

hof

f is also s e m i a l g e b r a i c a l l y

semialgebraic

and C is a closed in

reader

stated

h:Y ......)...T be

X c R n,

semialgebraically We

result

the

there:

f:X

subsets

of

subset

is

a

is

closed

projection

1 we must

prove

subset, canonical then p(C)

subset closed

p':XxTZ

is closed

only

the

of XxyZ,

and so C is also

= p'(C)

proper.

semialgebraic

....>.. Z

is

of

in XXTZ. ....). Z

is

in Z.

that

the

fibers

243

of

f

are

bounded.

b ~ R m.

Then,

Let

us

for

assume

every

Mr = f-l(B (b,I/r))\Bn(0, r) since

f is non-constant,

Prop.

8.1.13].

Thus,

that r ~ R ÷,

is open

Rn\f-l(b)

the

f-l(b)

and

is

the

unbounded

intersection

some

semialgebraic

non-empty.

is a dense

for

On

the

subset

other

of R" by

M n(Rnkf-l(b))

set hand, [B-C-R,

is non-empty

for

r

every r ~ R ÷. In other words, S = {(r,x) projects

onto R ÷ under

apply now Hardt's the

triple

that

of R+kV.

than

points

all

component

h:TxF

or,

Then,

In particular, in

V

and

arbitrary

[B-C-R,

a finite

T

set

point

the

map

H:S

be

F and

H :R÷xR n 2

> Rn

= ~2 o ~ : [ r o , ~ )

from Tarski

is

) R n we

semialgebraic

contradiction

> R÷

take the

a point

proves theorem,

are

subset the

9.3.2]

V c R ÷ such

~ R + bigger

o

semialgebraic

of ff over T means

TxF

i:T

to

that

homeomorphism

) T. The set F is and

we

pick

an

> TxF by i(r) = (r,p)

map TxF h

~-I(T ) C S c R+xR n

the

canonical

going of

theorem.

see e.g.

r

connected

surjective,

a map

) r. We

semialgebraic

a semialgebraic

is

p ~ F. Let us define

the semialgebraic

subset

over each connected

r . The triviality o

~:[ro,--> ) c T i

there

its corollary

let us

let

a semialgebraic

since

and consider

closed

> R÷:(r,x)

) ff-l(T) such that ~oh is the projection

non-empty

If

map ~:S

exists

trivial

subset

x ~ M r , f(x)~b}

better,

there

of R+kV containing

exists

r>0,

the semialgebraic

ff is semialgebraically

component

there

e RxRn:

theorem

(S,~,R÷).

the semialgebraic

to

R n,

but

The

[B-C-R,

is some point a ~ Adh(N)\N.

show

projection

that f(N)

N = ~([ro,~)) is

not

semialgebraicity

Prop.

and

2.2.7].

closed. of

N

a

This

follows

Let us assume

For some r>r O, a E Bn(0, r).

is

that

Then,

if

244

s>r,

the

point

(s,~(s))

= ¢(s) e S

and

so

In

~(s) ~ M .

particular

s

~(S)~Bn(O,s).

Thus ~(s)~Bn(O,r).

Consequently

Bn(O,r) But ~([ro, r]) 2.5.8]

and

is a closed

it does

n N c ~([ro, r])

semialgebraic

not

contain

a.

subset

Thus

of R n, by

[B-C-R,

A = B (O,r)\~([ro, r])

Thm. is

an

n

open

semialgebraic

a~Adh(N).

To

b ~ f(N),

but

since

neighborhood

prove

that

f(N)

b e Adh f ( N ) . .

(r,~(r))

e S.

of is

For

Hence b ~ f ( N )

a

which

not

closed

every

e B (b, 1 / r )

meet

shall

it

N,

check

is

hand,

i.e.,

if

that

f(~(r))vb, ¢ e

R ÷

there

(r,~(r)) ~ S, we get

¢ B (b,¢)

m

and so the intersection

we

other

exists r ¢ [r0,~) such that i/r ~:t

>;

0 if t if

mappings. nature

c l o s e d map b u t f - 1 ( O )

follows

However

we

is

false

maps

for

semialgebraic

whose

arbitrary maps.

For

map

t is

it

closed

shall

be

we state first

t~O t>O

i s unbounded•

mainly

concerned

an easy proposition

with

polynomial

of very general

to be used later -compare with Example 7.7 in [D-K2]- We denote

by e :R nc

> S n the

inverse

of

the stereographic

projection

from

the

n

north pole

en(X 1 ..... x n) =

Proposition

2.-

Let

z× IIx +l '"

f:R n .....) ....R m

be

•

a

.

,

zx I1×11+1

~

ilxll _l] N +lJ

semialgebraic

map.

•

Then

f

is

245

semialgebraically proper if and only if the map f:S n

> S m defined by

fOPS) = pmN and flen(R n) = emOf is semialgebraic. Proof.-

Consider

the

cartesian

square

sn

f

> Sm

Te

I°m

Ra

f

) R m

If f is a semialgebraic map it is also semialgebraically closed since S n is a closed and bounded semialgebraic subset of Rn÷1-use Thm I and [B-C-R, Thm 2.5.8]. Then, also f is semialEebraically closed. if

C is

e (C)

a

in

closed

S ~,

semialgebraic

then

f(C)

subset

= e-l(f(K))

n

of

is

Rn a n d

closed

in

K is

the

Rm. A l s o ,

In fact,

closure let

of

b be

a

m

point

in R m and let gn:Snk{P~ } N

from

p~.

Since

> Rn be the stereographic projection

f-l(b) = H~(en(f-l(b)))

and

e (f-l(b)) = (f)-l(e (b)) N

m

is closed and bounded in S n, we get that also f-I(b) is bounded and by Thm. l,

f

is

semialgebraically

proper.

For

the

converse,

note

that

Gr(f)\Gr(f) is a unique point, and so Gr(f) is semialgebraic. Hence we must

only

check

semialgebraic is open.

the

subset

continuity

of

S m.

of

f.

If pN~u,m then

Let

U

be

an

open

(f)-1(U) = en(f-I(e-1(U)))m

m

If PN~U, then e-l(sm\U)m is a closed and bounded subset of R m.

Therefore,

by

[A, Prop.

1.3], f-l(e-1(SmkU)) is a closed and bounded m

subset of R n and so (f)-1(U) = snke (f-l(e-1(SmkU))) is open in S". m

Let us denote by ~ (R) the projective space over R of dimension n

n, whose points are denoted x = (Xo:X1:...:xn)" We fix the embeddin E Un:Rn( We

can

relate

extendibility".

) ~n(R):x = (xI ..... Xn ) semialgebraic

> (l:xl:...:xn)"

properness

with

"projective

246

Proposition

3.- Let

f:R n

~ R m be

a semialgebraic

that there exists a semialgebraic

map F:~ (R)

Let

us

assume

) ~ (R) such that:

n

(i) F(H n) C H ,m

map.

m

where H n = {x = (Xo: x :...:x ) ~ ~ (R):x I

n

n

= 0}. 0

(ii) U of = Fou . m

n

Then f is s e m i a l g e b r a i c a l l y Proof.By

First

[B-C-R,

we check

that

P (R)

3.4.6],

proper.

the fibers

is

a

of f are bounded.

closed

and

bounded

Let

b ~ R m.

semialgebraic

(in

n

fact

algebraic)

subset

of

some

R s. Thus,

e (f-l(b)) = F-l(u (b)) n

closed

and

bounded

subset

of

~ (R),

is a

m

and

so

f-l(b)

is

bounded.

n

Secondly,

if

the closure

C is

a

closed

of u (C)

in P (R),

n

that

semialgebraic

subset

it follows

from

of

Rn a n d

conditions

K denotes i) and

ii)

n

f(C) = u -I(F(K)),

and

this

is a closed

subset

in R m since

F is

m

semialgebraic

and ? (R) is closed and bounded.

Now we apply Thm.

I.

n

C o r o l l a r y 2. Let f:R semialgebraically

> R m be a non constant polynomial

= ]] of i

i

. th

i

is

non

-projection.

for

f

b y P r o p . 1.

natural

Then f is

proper.

Proof.- We can assume m = I. In fact, f

map.

If

constant, f

l

Thus

is

where

for some II :R m l

semialgebraically

we c a n

n u m b e r n-~l a n d some a ° .

write . . . .

f(t) an e R,

l- R

is

proper,

the

the

same

= a t n+ a t n - l + . . . + a O

ao40.

1

Now t h e

canonical

n

is

true

for

some

semialgebraic

map F:PI(R) verifies

conditions

semialgebraically

Remarks.-

> Pa (R):x

= (xo: xa)

i)

and

ii)

n-1

> (Xo:ao xn+a I Ix I in

Prop.

~ x o +.. " +anX )

3,

and

so

f

is

Consider

fcr

proper.

(i) The converse

of p r o p o s i t i o n

3 is not

true.

247

example

the polynomial

is no extension Prop.

3.

we

C1n : ni: i)

an =

> ~2:(x,y)

) Pz(~)

F:P2(~)

Otherwise

sequences

map f:~2

take

> (x2+yZ, x(x2+y2)).

verifying

the

~ P2(~),

point

bn =

conditions

a = (0:0:I)

(!.n" 0 : I ) ,

both

i)

There

ii)

and

g P2(~)

and

convergent

in the

to

a.

Then lim u2f(l,n)

= lim Fou2(l,n)

n-)~

= lim F(an) = F(a),

n-~

lim u2f(O,n)

= lira Fou2(O,n)

n->Oo

and also

n-)Ca

= lim F(bn ) = F(a).

n->~

Consequently

n-9~

lira (l:l+n2:l+n 2) = lim (l:n2:0), n-)o0 n-~o

i.e.,

(0:I:I) = (0:I:0) a

contradiction. On the other hand, consider

g:~2

> ~2: (x,y)

enough to prove

that

H:~2(~)

(2)

mappings.

and

proper.

h = gof.

semialgebraicatly

In fact,

let us

Prop.

1

it

But i t

is

clear

By

proper.

is

map

> [P2({R): (Xo:Xl:X2)---> (x:: (x2+x2)3:x2(x2+x2)2)l 2 1 1 2

condictions

Corollary

> (x3, y 2)

h is

that the semialgebraic

fulfills

f is semialgebraically

2

i) and ii) with respect

is

not

For example,

true

for

the regular

larger

to h, and so we are done. classes

function

f:~

of

semialgebraic 1

>

> R:t

is not

l+t 2 semialgebraically

Corollary

3.-

Let

proper.

f = (fl ..... f ):Rn

> Rm be

a

polynomial

map.

For

m

every Islam, that each

let Pi be the form of highest

fl is non-constant

and

degree of fi" Let us assume

that PI ..... Pm have

common zeros in R n. Then f is semialgebraically Proof.-

Let

d I = deg

fi'

el = ~

di'

for

proper.

l~imm.

Let

e

semialgebraic 1

it

map ~:R m

suffices

to

> Rm by ~(Yl check

that

. . . . .

the

not non-trivial

Ym) = (Yl 1 . . . . . composition

us

define

a

e

Y:)'

By Prop.

g = ~of

is

248 semialgebraically

proper.

For

that

we

apply

Prop.

3.

Let

us

write

m

g = (gl .... ,gm) and d = ~ d i. Then deg gl = d >0, for every Isism, I=I

and

we write

gi = g i , d ( X ) + g i , d - l ( X ) + ' ' ' + g l , O ( X ) ' where

each

gij

is

the

homogeneous

X = (X 1 ..... Xn )

component

of

degree

/] of

gi"

e

Moreover,

gl,d = Pii' and so the set of commmon

is the origin.

Therefore,

zeros

of gl,d' " " "'gm, d

the semialgebraic map

d ) ~m(R): (Xo:X:...I :Xn ) ..... )(xd: ~- glj(x)xd-J: : 0 "" "

G: IF' (R) n

J=O

satisfies conditions

d ~. gmj(X)X: -])

J=O

i) and ii) in Prop.

3 with respect

to g, and so g

is s e m i a l g e b r a i c a l l y p r o p e r . Remark.

The converse of the last corollary

example, verify

the the

map

f: ~2

hypothesis,

(x,y) ~ R 2, it follows we get

a converse

) ~2: ( x , y )

but

since

is not true in general.

> (x4+x 2+y 2 ,x 4+x 2+y 2,)

]l(x,y)]I2~ ~f(x,y)II

that f is in fact topologically

if we restrict

ourselves

for

does every

proper.

to homogeneous

For not

point

However

polynomial

mappings:

Corollary 4. L e t

fl . . . . . fm

be non-constant

homogeneous

R[x I ..... Xn ]' let Z be the set of common zeros let

f = (fl ..... f ):Rn m

) Rm"

Then,

f

is

polynomials

in

in R n of fl ..... fm and

semialgebraically

proper

if

and only if Z = {0}. Proof.

The "if" part follows

from Cor.

3, and for the "only if" part

note that the cone Z = f-1(O) is unbounded

if Z~{O}.

Proof of theorem 3. We must prove the "if" part and we may assume that f is a polynomial

of even degree and

non-negative

on R n. In fact,

let

249

us define of

even

degree,

Isl>M 2, for g,

g = ~.f,

the

where

fiber

for

every

g-1(s) Prop.

each r ~ R +, R n k B

n

point

x ~ Rn

Hence,

that

to apply Cot.

the

set M

in R, with M

c [O,e).

if

s e R

prove

map with

the

theorem

f is s e m i a l g e b r a i c a l l y

proper.

2. Thus we assume na2 and so, for

= g(RnkB(O,r))

F

and

if we

is a s e m i a l g e b r a i c a l l y

(0, r)

of R n. Thus

> t 2. Then g is a polynomial

is bounded. I, we get

If n = I, it is enough

unbounded)

> R:t

g(x)aO

and using

subset

~:R

connected is an

Consequently

semialgebraic

interval

we have

(perhaps

a well

defined

map d:R ÷ The

map

d

has

> R:r

r ~ R +,

the

fiber

We

g-1(c)

are

has

the hypothesis.

x I g RnkBn(O'r)

square,

r

graph

going

to

choose a point c ~ R, c>b,

contradicts point

infimum of M .

semialgebraic

b = sup d(R +) ~ R u {m}. Otherwise,

>

there

exists

such

c>M.

points

Since

a point

prove

so

that

in

it

exists

fact

b = m.

We shall see that for every

with

norm

c>bmd(r)

that g(xl) Rm be a non-constant map. Let A be the polynomial by

~

the

sums

of

semialgebraically g ~ A and SI,S 2 ~ [

an

Let

Cor.

1.

true

for

if and

only

if

in

there

A.

exist

Then,

f

is

C ~ R +, p e ~,

1=1

1

> R.

m

If

such

F - l ( t 2) c B (O,V/C[I+t2) p) n

Conversely,

if

f

is

an

for

identity

every

semialgebraically

F a n d s o F(R n) = [ a , e )

a non-empty,

polynomials

j

F = f2+...+f2:Rn

inclusion

of

such that

[J~l

Proof.

ring A = R[x I..... Xn, T] and let us denote

squares

proper

polynomial

for

bounded and closed

t • R,

proper,

some p o i n t

algebraic

exists,

the

a • R.

subset

and

we g e t we

apply

same h o l d s

Hence F - l ( t )

o f Rn f o r

every

is tea,

and the well defined map

has

C

1

semialgebraic

e

R÷

and p ~ ~

preimage

r: [a,->): t

> max ~11×11 FOx) = t>

graph.

[B-C-R,

such

F-l([o,b])

F-l([O,t]) c Bn(O,C 2) get

By

Prop.

that

r(t)~C .tp for

is

a

for

1

bounded

n

for

every

there

every

tab.

subset

of

C 2 ~ R +" Then,

some

F-l(t 2) c B (O,C(l+t2) p)

2.6.1]

if

t ~ R,

exists

By Thm. R n,

C = max and

bza,

1 the i.e.,

{CI,C z}

we

the

set

so

n

{(x,t) ~ Rn+l: ~ x~-C(I+T2) p z O, i=l

F(x)-T 2 = O}

is

empty.

Applying

[B-C-R, 4.4.2] we get the desired identity. Let follows

i = ~ we

denote

and C = R(i)

be the a l g e b r a i c

x = (X 1..... Xn )'

Y =

closure

(Yl ..... Yn )'

o f R.

Z =

In what

(Zl ..... Zn )--

253

x+iy, can

and we get

identify

some

C n with R 2n via

insight

on

f = (fl ....

f ):Cn * n

> Cm'

polynomials

gj,hj

E R[x,y]

identify

f

F=(g I . . . . .

with

gm, hl . . . . .

) R 2", z = x+iy proper

~ C[z I ..... z ]. m

J

such

) (x,y).

....

semialgebraically f

the

C n

that

fl(z)

polynomial

map

polynomial

There

F: R 2n

maps

exists

= gj(x,y)+ihj(x,y)

We

real

and

) R 2m,

we

given

by

f = (fl ..... fm ): C A .......>. Cm

is

hm). S i n c e m

we g e t f r o m Thin.

Corollary

= Z If J cz, l 2

J

j=l

]=1

3:

7.

The

semialgebraically

polynomial

proper

for every r ~ R, r>M,

map

if and only

if there

exists

M E

such

R÷

that

the subset {Z E cn:

~ Ifj(z)I 2 = r} J=l

is bounded. In the one variable Corollary

8.

Every

semialgebraically

case we have an analogous non-constant

polynomial

As in the proof of Cot. 2, we d = Z aj(z) d-j with aj ~ C, a 0 40 J=0

P" = (g,h):R 2

> R 2 is given

whose homogeneous

by

components gd(x,y)

Thus,

real

F is semialgebraically

between

the

f:C

2:

) Cm

is

other

may

assume

and

polynomials

the g

and

m = I.

Then

induced

map

h of

degree

d

gd and h d of degree d verify

+ i hd(X,y)

gd and h 4 have not non-trivial

On

map

to Cor.

proper.

Proof. f(z)

result

= a0(x+iy) d.

common

zeros and by Cot. 3 the map

proper.

hand,

for

nz2,

the real and the complex

there

cases:

is

a

significant

difference

254

Proposition map f:C n i.e.,

4.

The

maps

open

(semialgebraic) For

part

it is not

Each a l g e b r a i c

compact.

subset

by s e p a r a t i n g

of real

order

from"

lemmal]

onto

that b o u n d e d

algebraic

subset of C n w h i c h that W k Z ~ o and Z is

and so Z is not closed

the result

V I ..... V s .

in the general [B-C-R,

of C n and

such

fact,

number

following

formula

in

Z(R):

R = ~ and

so

particular

bounded algebraic

is also

of R 2n

For every

algebraic

subsets

expressible

R 2n

of

to

C = C,

Thm.

such

some

as

V

]

that and

"V ~ B for

every d

~

and

the

can

card(V)

arbitrary

real

closed

a

then exists

V

Ba

in

is

as M(n,d) components

integer

subsets of C n are finite.

first

exists

take

~ M(n,d)"

set

there

connected

positive

the

of

~2d.

V ~ Ba,

every

we

in

there

9.9.4],

semialgebraically for

true

Moreover,

if

[B-C-R, of

homeomorphic

in R.

that,

by

V I, . . . , V s

the

5.2.4].

subset

as a formula

in

case

in R[xl,...,Xn, Y 1 ..... ym ] of degree

Consequently,

it

of bounded

parameters

M(n,d) In

semialgebraically of

with

open

the field of complex

of C n can be seen as an a l g e b r a i c

subsets

open,

f is surjective.

algebraic

"V ~ B " is expressible d

subsets

the m a x i m u m

§2.1,

Cn

for real closed fields -IT] or

algebraic

integer

algebraic

of

closure W verifies

be the collection

Z(R)

card(V)sM(n,d).

of

d

fact

language

positive

polynomial

the real and imaginary parts of its equations. let B

the

to see

To obtain

zeros of a polynomial

Clearly,

subsets

is true for C = C,

[S, Ch.7,

we use Tarski P r i n c i p l e

R 2n "coming

This

proper

then f is s e m i a l g e b r a i c a l l y

it is enough

its p r o j e c t i v e

a dense subset of W,

integer d>0,

If m = n,

if Z c C n is an irreducible

is not a point,

W, hence

semialgebraically

subsets of C n and in p a r t i c u l a r

the first

since

a

semialgebraic

subsets of C n are finite. numbers

of

> C m are finite.

f

Proof.

fibers

is

d,

the

true

for

field

R.

In

255

Assume now n = m. Once

the openess

an open and closed s e m i a l g e b r a i c denote and ~d

F = (F 1 ..... F2n): R 2n

by

let deg F = max be

the

polynomial

map

Cn

of

" For the

every

set

formula F ~ ~

) R 2n the

F(B2n(a,r))

maps

R 2n

degree

in ~(R)

is

that an

map

every

f

induced

integer

) R 2n

- C n with

then F is an open map the f o l l o w i n g

subset of C n, i.e.

{deg F I ..... deg

collection

of f is proved,

d>O,

"coming

has

F

let

from"

finite

iN, Prop 4, pp

by

a

fibers,

132]-

and so

is true for R = ~ and every d>O: card

open

F-1(b)

subset

s M(n,d)

of

for

R 2n for

every

every

b e R 2n,

a ~ R 2" and

every r E R, r>O". Therefore closed

field R and since

the e u c l i d e a n Remark.

map

is not

of

map f:C n f:C 2

implies

f

but

is

polynomial

the

point

map

f:R n

) Rm

and

of open

of

induces

maps

maps C n

with

finite

f

sets

condition

finite for

another

in

for

a

For example, fibers every

one

but

it

natural

f :C n c

semialgebraically

c

proper,

proper

is

a

closed

converse

is

not

true:

> x2+y 2, which has not finite

fibers

but

fibers

since

) Cm

Rn

C n = R 2n,

) ~: (x,y)

real

open map.

proper.

but

every

is in f(C2).

Clearly,

semialgebraical ly subset

has

(l,O)~f(C 2)

by Thm 3, s e m i a l g e b r a i c a l l y

semialgebraic

d>O

a sufficient

) (xy, x-y(xy-1))

(l,I/n) = f(I/n,n)

the map f:~2

it is,

every

form a basis

is not

the same polynomials.

semialgebraie consider

the fibers

In fact

polynomial

by

for

) C m to be s e m i a l g e b r a i c a l l y

number n, the point

defined

true

the open balls

) C2: (x,y)

proper.

Each

is also

topology of R 2n, F is a s e m i a l g e b r a i c a l l y

Finiteness

polynomial the

the formula

the

proper. -e.g.

To f i n i s h we show semialgebraically

) C m - have some nice properties:

that

proper

256

Proposition fibers.

5.

Let

Rnkf-l(F)

Let F < ~m

From

9.3.1]

it follows

the

be

a

the

that

be

a

semialgebraic

semialgebraic

finiteness

13.2 of

Acknowledgement:

shortening

> Rm

is semialgebraically

Proof.

Theorem

f:R n

set

map

with

with

finite

dim F~n-2.

Then

connected.

of

the

fibers

dim f-l(F) ~ n-2

and

and

Hardt's

now

theorem

it suffices

[B-C-R

to apply

[D-K] with M = R n and N = f-l(F).

We

wish

proofs

to

of

thank

Thm.3,

the

Cot.4

referee

and

for

Prop.5

his

and

contrlbutlon

some

other

In

useful

comments.

References

[A]

Alonso,

M.E.

Real proper morphisms.

pp. 237-243 [B-C-R]

Bochnak,

Vol 43,

(1984).

J.;

Rdelle.

ArchLy der Math.

Coste,

Ergebnisse

M.; der

Roy,

Math.

Gdomdtrie

M.F. 3

Folge,

Band

Algdbrique

12.

Springer.

(1987).

[Br]

Brumfiel,

Partially

G.W.

geometry. Cambridge: [D-K ] 1

Dells,

H.;

Zeit. [D-K 2 ]

1173.

[N]

175-213

H.; Knebusch,

Springer

Narasimhan,

spaces. L.N,M. Shafarevich,

[T]

Tarski,

A.

geometry. Berkeley

Univ.

and

Press

Semialgebraic

semialgebraic

(1979).

Topology

over

a Real

Spaces.

Math.

(1981).

M. Locally semialgebraic spaces. L.N.M.

(1985). Introduction

R.

[s]

M.

rings

If: Basic Theory of Semialgebraic

178, pp.

Delfs,

Cambridge

Knebusch,

Closed Field

ordered

25, Springer I.R.

to

the

theory

Basic algebraic geometry. Springer

(1951).

for

analytic

(1966).

A decision method for elemmentary Prepared

of

publication

by

J.C.C.

(1974).

algebra and Mac

Kinsey,

S U R LES O R D R E S DE NIVEAU

2n

ET S U R

U N E EXTENSION DU 176me PROBLEME DE HILBERT

DANIELLE GONDARD-GOZETTE UNIVERSITE PARIS VI, 4 p l a c e J u s s t e u , 75252 PARIS c e d e x 05, FRANCE.

ABSTRACT. L e t K be a c h a i n a b l e f i e l d , w h i c h m e a n s a f i e l d a d m i t t i n g o r d e r i n g s o f e x a c t level

2n

field there exists

f o r any ~ e K

n B 1 , f r o m [G1] a n d [G2] w e k n o w t h a t in s u c h a such that

z

~ E K¢ ; w e t h e n d e f i n e t h e n o t i o n

o f ~ - c h a i n o f o r d e r i n g s and u s e it t o o b t a i n e x p l i c i t f o r m u l a s f o r o r d e r i n g s of e x a c t level

2 n in s o m e s p e c i a l f i e l d s . We a l s o o b t a i n f o r s o m e

c h a i n - c l o s e d f i e l d s a n e w e x t e n s i o n o f H i l b e r t ' s 17th p r o b l e m .

INTRODUCTION. Les c o r p s o r d o n n a b l e s s o n t d e d e u x s o r t e s : les c o r p s n o n c h a ~ n a b l e s (i.e. il n ' e x i s t e p a s de c h a i n e d ' o r d r e s d e n i v e a u s u p E r i e u r de niveaux exacts

2 n ) off t o u t e s o m m e de c a r r E s e s t une s o m m e de p u i s s a n c e s

quatriEmes (par exemple

~ , © et ses extensions algEbriques ordonnables), et

les c o r p s c h a i n a b l e s off il e x i s t e un ElEment ~ t e l que

z

ne s o i t p a s s o m m e e s t un c o r p s

de p u i s s a n c e s q u a t r i E m e s ; d a n s ce c a s n o u s d i r o n s que

K

o~-cha~nable ( p a r e x e m p l e ~(X) e t ~((X)) s o n t d e s c o r p s

X-chainables) .

L a n E c e s s i t 6 de f a i r e i n t e r v e n i r une c o n s t a n t e

cc

a 6t6 m i s e en Evidence

par les a x i o m a t i s a t i o n s des t h e o r i e s des corps cha~nables et des corps c h a i n e - c l o s que n o u s a v o n s donnEes d a n s [G1] ( p r e u v e s darts [G2]). Ce p o i n t de vue n o u s a dEj~ p e r m i s de d E m o n t r e r d e s r E s u l t a t s du t y p e 17~me p r o b l E m e de H i l b e r t d a n s [D-G] e t d e c r E e r un a n a l o g u e ~ l ' a t g E b r e r E e l l e d a n s [B-G] , e t n o u s le c o n s e r v o n s ici p o u r o b t e n i r une e x p r e s s i o n d e s o r d r e s de n i v e a u d e c e r t a i n s c o r p s (~(X)

2n

p a r e x e m p l e ) e t une n o u v e l l e f o r m e de g E n E r a l i s a t i o n

du 17~me p r o b l E m e de H i l b e r t ; l ' i n t r o d u c t i o n de la p a r t i e III e x p l i q u e r a c o m m e n t la g E n E r a l i s a t i o n p r E c E d e m m e n t 6tudiEe d a n s [D-G] e t [B-B-D-G] e s t d i f f E r e n t e de c e l l e p r E s e n t E e d a n s c e t a r t i c l e .

258

I-NOYAUX DES CHAINES D'UN CORPS CHAINABLE.

D6finition 616ment

I-1. Un c o r p s ~

t e l que

d ' 6 1 6 m e n t s de

D6finitions

K

2

sera dit ~-cha}nable s'il existe dans

K

un

ne soit pas somme de puissances quatri6mes

K .

I - 2 . Un p r 6 o r d r e

T

de n i v e a u

2n

(i.e.

T + T ~ T

, T

. T c

T

,

n-I

n

K2

_c T ) sera dit un

a-pr6ordre si

2

~ T

(i] est alors de niveau exact

n~l

2n

car

~ K2

n ' e s t pas contenu dans

T ). De m~me un ordre

P

de niveau

n-l

exact

2n

sera dit un

~-ordre si

Definition

I - 3 . Nous a p p e l e r o n s

sup6rieur

(PI)I~

t e l l e que

au moins une

~ p .

~-cha}ne une chaine d'ordres

2

I - 4 . Un c o r p s

Proposition

2

de n i v e a u

t~ P2"

K

est

a - c h a } n a b l e si e t s e u l e m e n t s ' i l e x i s t e

~-chaine.

Ceci r 6 s u l t e d e s d 6 f i n i t i o n s I-1 e t I - 3 e t du r 6 s u l t a t qui d o n n e l ' e x p r e s s i o n

s u i v a n t e s d e s s o m m e s de p u i s s a n c e s :

K# = ( ~ Pl2 ) a ~ K 2 q u e l c o n q u e de

de B e c k e r - H a r m a n

o~ a

d 6 s i g n e un o r d r e de n i v e a u e x a c t

4

W e s t p a s u n e s o m m e de p u i s s a n c e s q u a t r i ~ m e s 2 j tel que ~ ~ P . D ' a p r ~ s le c o r o l l a i r e J2 I - 4 de [H] il e x i s t e a u m o i n s une c h a i n e d ' o r d r e s de n i v e a u s u p ~ r i e u r p a s s a n t

darts

K . Donc si

P12 2

K , il e x i s t e a u m o i n s u n

par ce

P

j2

Proposition I-S. Dans un corps exact

2n , n -~ 2 , d'une

n-I 2

En e f f e t si P

6tant de

~

ct-cha}ne

nl_1

a2 2p

a

l

~ Pl ~ P

que si alors

~ P

1

et

a2

on a

a

x ~ K°

K

~-cha~ne

~-eha~nable tout ordre (PI)I~N

n

de niveau

n-I 2

, alors ~ P

P

est un ~-ordre.

P est bien un ~-ordre. Si ¢~ e P n n , on en d6duit qu'i! existe i -~ 2 tel que

2

~ Pl+l ( p u i s q u e d ' a p r ~ s ta r e l a t i o n de c h a ~ n e d~s que 2 p+q

et si

x ~ P u -P

e P xz

pour tout

q -~ 1 ) ; o r il e s t c o n n u (cf. [Bel])

l+q appartient & un ordre

: en effet si

x ~ P

alors

P

de niveau exact

2n

P + Px est un pr6ordre propre

259

contenant

strictement

P

et donc

P + Px = K , d'oO

- 1 -- a + b x a v e c a e t b |

dans

P

et

entra~ne

x = (-1-a)/b

que

~

2

~ - P . On d6duit de ce rdsultat

que

2

a

e P

1-1

e P

u - P

, puis par

I+i

la relation

1+1

de cha~ne

I+I I-1

P

u

1+1

- P

contraire

=

1+1

( P

P

) u

t

- ( P

~ P

0

)

l

que

c¢2

e P

ce qui est

1

& i'hypoth~se.

Lemme I-6. Soit n

T

~

0

K

un corps

2 n- 1

= ~ K 2 - c~

~-chainable.

Pour tout

n e ~* ,

n

~ K2

est un

~-prdordre propre (i.e.

) de niveau

-i ~ T

n

n

exact

~n.

C ' e s t t o u t & f a i t c l a i r pour

0c2 ~ ~ K 4

~-chalnable

(Pl)l~

; par

I-5 tout

la relation

de chaine

qUne 1Pn_ 1

contient

P P

de

n

~-cha~ne

u - P

c~

e P

u - P n

2

- c~

d'dl6ments

de niveau

n'appartenant

pas & & T

de

0

P

qui est

n

k-1

o n a T k : ~. K 2 - 0c2

clair

Proposition

I-8.

2n

et

n

0

donc

T

que

T

n

exactement

K

~-chalnable

~ Pk ' p o u r t o u t

la preuve

&

clairement

est

n

n-1

car

T

; de m~me

n

a

2

2n

- 1

pour

toute

a-chaine

(Pi)le~

k -> 2 .

du lemme I-6.

Dans un corps

K

~-chainable, n

exact

~ P

au fait

k

~ KZ

d'aprds

n-I

donc propre.

Proposition I-7. Dans un corps k

2

c o n t e n a n t t o u t e s les

ne peut donc pas appartenir

n

6tant

n

contient

2 n , e t le n i v e a u e s t

~

n P ) jointe

montre

; P

K

~-chaine

donc

n-i

2n-l-~mes e P

K

une

~-ordre

n

2n-6mes

appartenir

un

n P ) u - (P n-I

n -> 3

I - 4 il e x i s t e

est

l e s puissanceSn_l

e t donc que

un prdordre

C'est

n = 2 ; pour

n

puissances

peut

= (P n

toutes

et

la proposition

n

2

n = I

et par

n >- 2 , q u i c o n t i e n t

T

tout

P

n-i

= ~ K 2 - ccz

, ordre

de niveau

nn

~ K2

est un

P

n

de n

a-chaine.

n-I

En effet

seulement

l ' u n d e s d e u x dldments

z

n-I

ou

- c~z

appartient

n-1

P

n

.Donc

une ~-chaine

(z

2

~ P

puisque

et

n

si

on

2

en

~ p

ddduit 2

que

toute

, en utilisant

chaine

passant

la condition

par

de chalne

P

It

(qui

est

ne

260

s'exprime

au niveau 3 par

:

P3 u - P 3 = ( P z r~ Po ) u - ( P 2 ~ P o ))

on obtiendrait n-I

2

e P

u - P

3

ce qui est

3

d'o~

~

impossible.

4

~

P

Toute

; on d6duirait

3

chaine

(PI)I~N

en it6rant passant

que

par

~

Pn

2

e P

n-I

est donc une

~x-chaine .

n

I-9. D a n s un corps

Th4or6me

a-cha[nable

T

2n- 1

= ~ Kz - ~

n

~ K2

est dgal

n

l'intersection de t o u s l e s a-ordres de niveau exact

au moins

une a-cha£ne).

D'apr~s est

le t h 6 o r ~ m e

6gal & l'intersection

q u i le c o n t i e n n e n t

1 de Becker de tousles

; un ordre

[Bell, un pr6ordre ordres

propre

de m6me niveau,

qui contient

T

n-1 2

- c~

2 n (qut apparttennent

est

un

de niveau

exact

~-ordre

2n

ou non,

car

n-1n

~ T

~ P n

ce qui entra~ne

que

~

2

~ P

n

; enfin

P

n

est bien de n

n-I

niveau

exact

2n

puisque

~ K2

n'est

pas contenu

dans

P n

n

C o r o l l a i r e I - I 0 . Dans un c o r p s

K

a-cha~nable , K

2n - 1

= ~. K2 - a

n

~ K2

n

est 4gal ~ l'intersectton

des

P

, ordres de ntveau exact

des

2n

n

~-cha£nes.

Cela r6sulte

imm6diatement

de I-7,

I-8 et I-9.

n

Remarque. toujours niveau

Dans un corps

¢~-chainable

T" = ~ K 2

2 n- i

+ ~

n

~ K2

est

n

un pr6ordre exact

D4finition

K

propre

de niveau

2n

mais

il n ' e s t

pas forc6ment

de

2n .

1-12. Appelons

, pour

n

-> 2

,

n o y a u a u n i v e a u _n d e s a - c h a i n e s .

T n

I I - O N O R D E R I N G S OF SOME C H A I N A B L E F I E L D S .

II-1. St un corps

Th4or~me (PI)I~

, ~i 4 c h a n g e

de

P0

K

~-cha£nable n'admet

et

PI

qu'une

est pour tout

n -> 2 ,

a-cha~ne

p r e s , a l o r s l ' o r d r e d e nLveau e x a c t n

de celle-ci

seule

P

= T n

= ~ K2 - ~ n

2 n- 1

n

~ K2

.

2n

261

En e f f e t p a r l a p r o p o s i t i o n I - 7 t o u t e Qn ~ T seule

pour tout

n

a-chaine

Exemple

le c o r o l l a i r e 1-10 d o n n e a l o r s

:Dans

trois ordres,

de n i v e a u e x a c t 2 n intersection

(Ql)le~

n

= T

avec

archim6dien, et admettant

n -> 2 donc p o u r t o u t

A(P o) = A(P I)

de Pasch) que d'apr~s

2n

pour chaque P

616ment d e

et

sdparant

II-2. Soft

le ddbut

les o r d r e s

= ~. ( P

d'une

n P 0

P

K un corps

n >- 2

)2

- a2

~. ( P a-ordres

2n

de

et sont

PI

sont

K

n'admet qu'un

; d ' a p r 6 s le t h d o r ~ m e I I - I n-I

n

~ K2

off

a est un

1

et

P

; pour

n P

, P

0

deux

1

ordres

qui

n >- 2 , s o f t

)2

, alors

C

I

P

n

est

~ g a t ,~

n

le contenant

(de

tels

P

n

sont

afors

a-chatne).

e s t a n a l o g u e ~ c e l l e de 1-10 d6s que l ' o n a m o n t r d q u e

e s t u n p r ~ o r d r e p r o p r e de n i v e a u e x a c t

n

n >-- 2 . De

n-I 0

La d6monstration

P

(P{)IE~

de tousfes

exact

o

n

n-i

I

l'tntersectton

avec et

= ~ K2 _ 2

a-cha~nable

a-chatne

n-I

C

K

e t Po = P1 " On e n d d d u i t (ce c o r p s d t a n t

cet ordre est explicitement donn~ par : K

2n P0

n

de niveau

des ordres

n : il s ' a g i t d ' u n c o r p s

le t h 6 o r 6 m e IV (if) d e [Bell c e c o r p s

s e u l o r d r e de n i v e a u e x a c t

n

qu'une

d ' u n e c l o t O r e r d e l l e de O(X) p o u r u n o r d r e a r c h i m 6 d i e n a v e e u n e

co-chainables

C

n'admet

n

[G4] il d 6 c o u l e q u e l e s d e u x o r d r e s (non a r c h i m d d i e n s )

sotent

K

[Bel] p a g e 61 e s t d o n n ~ u n e x e m p l e de c o r p s a y a n t

dont l'un Pest

pour un

P

e l 6 t u r e r 6 e l l e g ~ n d r a l i s S e p o u r u n o r d r e de n i v e a u e x a c t

Thdor~me

est telle que

n ~- 2 , d ' a p r ~ s l ' h y p o t h ~ s e f a i t e q u e

d'application

exactement

a-cha/ne

2 n . Si

(Pi)leN

est une

rl-1

a-chaine n = 2

commen~ant par

Po r~ P1

d'apr~s la relation

alors

(Po a P1) 2

c Pn : c ' e s t v r a i p o u r

P2 u - P2 = (Po a P1) u - (Po a P1) , e t p a r

n-1

rdcurrence

si

(Po n P1 )2

-¢ P n

a l o r s p a r l a r e l a t i o n de c h a i n e , a u n i v e a u n

n

+

1 ,

Pn+l u - Pn+l = (Po n Pn) u - ( P o n

Pn)

on d d d u i t q u e

(Po n P1)2

n-I

est contenu d a n s

d o n c on a

P

n+l

P

n

6 t a n t de

a-chaine

~

2

n-I

~ P

n

et

- a

2

~ P

C

n

a Pn . On e n d d d u i t qUne IC n , qui e s t c l a i r e m e n t u n p r ~ o r d r e d e n n n-I n i v e a u a u p l u s 2 n (Vx x 2 = (X2) 2 E (P n P )2 , est un pr~ordre 0

1

p r o p r e d e n i v e a u e x a c t 2 n . I1 e s t d o n e 6 g a l ~ r i n t e r s e c t i o n ordres

P

de tousles

qui le c o n t i e n n e n t , e e u x - c i s o n t d e n i v e a u a u p l u s

2n

c a r si le

n

n i v e a u d t a i t s u p d r i e u r on a u r a i t

~ K2

non contenu dans

P

ce qui e s t

262

n-i

impossible puisque

(P

r~ P )2 0

K2n

_~

. Le niveau e s t 6 v i d e m m e n t au

1 ri-1

moins

2n

puisque

~ K2

n'est pas contenu dans

P

qui c o n t e n a n t

n-I

peut contenir ces ordres

2

P

C

ne

ri

n-I

. E t a n t de n i v e a u e x a c t

contenant

(2

ri

2n

e t t e l s que

_ 2

s o n t d e s o r d r e s de n i v e a u e x a c t

2n

~ p

,

ri

de

e-chatnes.

Exemple

d'application

: la p r o p o s i t i o n I I - 2 p e r m e t d ' o b t e n i r une e x p r e s s i o n

d e s o r d r e s de n i v e a u s u p E r i e u r de En e f f e t

~(X) :

~(X) e s t un c o r p s de P a s c h e t , c o m m e dEj& u t i l i s 6 p l u s h a u t , il

dEcoule de [Bel] e t de [G4] q u ' u n c o u p l e d ' o r d r e s c o - c h a i n a b l e s ne l ' e s t que p a r une s e u l e c h a i n e . E n f i n il e s t m o n t r 6 d a n s [Di] que les c h a i n e s de ce corps sont c o n s t r u i t e s sur des paires d ' o r d r e s "symEtriques" qui p l a c e n t

X

i n f i n i m e n t prEs de a ~ ~ a v e c

r e s p e c t i v e m e n t , ou s u r

P

CO+

et

P

130-

X-a

qui r e n d e n t

P

et

P

,

p o s i t i f ou n 6 g a t i f X

infiniment grand positif

ou n 6 g a t i f r e s p e c t i v e m e n t . II y a d o n c d a n s d'ordres

P

0~(X) d e u x

et

&+

de n i v e a u e x a c t

P

&-

2n

( X - a ) - c h a i n e s b&ties s u r les d e u x c o u p l e s

d'une part et

= ~. (P na

r~ P a+

)2

riCO

= ~ (P

CO+

CO-

n P a+

)2

P

o0-

d ' a u t r e p a r t . Les o r d r e s

n-1

~ (P

a-

r~ P

et

n-I

- (X-a) 2

)2

- (X-a) z

et par

a-

n-I

n-1

P

o0+

de c e s d e u x c h a i n e s s o n t a l o r s d o r m , s p a r n-I

P

P

n-1

~ (P

CO+

r~ P

o0-

)2

respectivement •

c e c i r 6 s u l t e 6 v i d e m m e n t du f a i t que c e s d e u x e x p r e s s i o n s s o n t d e s ( X - a ) - p r 6 o r d r e s p r o p r e s de n i v e a u e x a c t

2n

distincts .

III-UNE AUTRE GENERALISATION DU 17~me PROBLEME DE HILBERT AU NIVEAU 2 n

A l ' o r i g i n e le 17~me p r o b l ~ m e de H i l b e r t s e p r 6 s e n t a i t s o u s la f o r m e :

solt

f ~ ~[X] t e l l e q u e V ~c ~ o~P f >- 0 , a - t - o n

f ~ ~. ~ ( ~ ) 2 ? A r t i n le

r 6 s o l u t en 1927 (cf. JR]). n ' a y a n t q u ' u n s e u l o r d r e d o n t les 616ments p o s i t i f s s o n t

p = R2 = Z ~2

une

premiere extension possible est :

soit

K

un c o r p s o r d o n n a b l e ,

soit

f ~ K [ X ] t e l l e q u e V ~ ~ K p f ~ ~ K z,

a - t - o n f ~ ~. K ( X ) 2 ? Ceci a 6t6 Etudi6 d a n s [G-R], [McK], [P2] e t [Z-G]. Une a u t r e e x t e n s i o n e s t le p a s s a g e a u x s o m m e s de p u i s s a n c e s

2n-~mes :

263

soit

K

un c o r p s o r d o n n a b l e e t s o r t II

f

f

~ K[X]

telle que

V ~c • K p

n

• ~ Kz

a-t-on

f

• ~. K ( X ) 2

? C e t t e e x t e n s i o n a 6t6 r 6 s o l u e p o u r une

v a r i a b l e e t c e r t a i n c o r p s c h a i n e - c l o s d a n s [D-G]. P u i s clans [B-B-D-G] ce p r o b l ~ m e e s t r 6 s o l u d a n s un c a d r e p l u s g 6 n 6 r a l , p o u r les p u i s s a n c e s

2n-Omes

et plusieurs variables. Nous p r o p o s o n s ici une e x t e n s i o n d i f f 6 r e n t e du l?-&me p r o b l ~ m e de H i l b e r t a u x o r d r e s de niveau sup6rieur sous la f o r m e : Soit P

K

~. K 2 ~ ~ K 4 ), s o i t

un c o r p s chatnable (i.e. ordonnable et tel que

un o r d r e d e n i v e a u e x a c t

2n ( n a 2 ) d e

K , sort

~ K[)(I t e l l e que

f

n

V ~¢ • K p

f(~C) ~ P

, peut-on

caractdrlser

f

?

n

Nous a p p o r t o n s une r 6 p o n s e d a n s le c a s de

K

chalne-clos (-r6el-clos dans

l a t e r m i n o l o g i e de [ B - B - D - G ] ) n ' a d m e t t a n t q u ' u n e s e u l e v a l u a t i o n h e n s e l i e n n e c o r p s d e s t e s t e s r 6 e l - c l o s . I n s i s t o n s s u r le f a i r que darts [D-G] ou [ B - B - D - G ] l ' o b t e n t i o n d ' u n r 6 s u l t a t n 6 c e s s i t e de c o n s i d 6 r e r l e s off

L

f(x)

e s t une e x t e n s i o n a l g 6 b r i q u e o r d o n n a b l e de

consid~re f(x)

que p o u r

Th6orSme l I I - 1 .

Soit

K

n >- 2

( l ) f ~ E K(X)2

- ~ f

~ corps des restes rdel-clos,

et sort

f • K(X)

:

E K(X)2 f(x)

est ddf~nie

2n

on ne

n

~ P

, oa

P

n

de n i v e a u e x a c t

,

un corps chatne-clos ~-chatnable n'admettant qu'une

2n-I

( H ) Yx ~ K p 00

K , alors qu'ici

les propridtds su$vantes sont dqulvalentes

n

x e Lp

x ~ Kp .

seule valuation hensdlienne alors pour

pour

de

K

dds~gne

l'unique ordre

n

; n

n-I

( l i t ) Yx • K p

o0

f

est ddflnte

f ( x ) • E K2

a2

(iv) Yx • Kp

oO

f

est d d f t n t e

f ( x ) e K Z U - a,

-

n

2n-I

n

E K2

K2n

La p r e u v e u t i l i s e le c o r o l l a i r e 1-10 du p r 6 s e n t a r t i c l e e t le l e m m e s u i v a n t o b t e n u a v e c Delon d a n s [D-G] :

L e m m e I I I - 2 [D-G]. S o i e n t et

K

K

et

L

d e u x c o r p s c h a I n e - c t o s t e l s que

n ' a d m e t q u ' u n e s e u l e v a l u a t i o n hens61ienne & c o r p s d e s t e s t e s r 6 e l - c l o s

alors sont 6quivalents : (i) (ii) (iii)

K _c L

K t~ L z = K z

;

K est relativement alg6briquement clos dans K ~ L

( off

"~"

L .

e s t une i n c l u s i o n 6 1 6 m e n t a i r e ).

;

264

De l ' e x p r e s s i o n

d ' u n c o r p s chaYne c l o s

K = K 2 u - Kz u a K z u - a Kz qu'une

a-chainable

sous la forme

et du lemme pr6c6dent

seule valuation hens61ienne & corps des restes

dans un autre

corps chaine-clos

a-chainable

it r 6 s u l t e

r6el-clos

L , alors

on a

que si

K

n'a

et est contenu K ~ L

Preuve de III-1. (ii) ~ (iii) o (iY) ~-chainable

K

est imm6diat

: En e f f e t

l'unique ordre

dans un corps chaine-clos

de n i v e a u e x a c t

2n

est donn6 par

les

n 2n-I n n 2n-i K2n = ~ K2 - a ~ K2 = Kz u - ~

expressions suivantes : P

; la

n

premi&re f o r m e r6sulte du fait qu'un corps chaine-clos uniquement

a-chainable est

a-chainable et du th6or&me II-2 , et la seconde vient de

]'expression des ordres de niveau sup~rieur d'un corps n'admettant que deux ordres usuels donn6e par Becker dans [BI], II suffit de montrer le th~or&me pour n

f ~ K[X]

n

f = gh 2 -1/

f = g /

car si

h

alors

n

h2

et l'on sait que d'une part

n 2n-I n T CKCX)) = ~ K(X) 2 - ~ ~ K(X) 2

~ K2

e s t un

_c p

et que d'autre

n

pr6ordre

part

(voir § I } .

n

(i) ~ (iii) : il s u f f i t n

~. K(X)2

- a

n ~, K 2

de v 6 r i f i e r

2n-I

~ K(X)2

2n-I - a

f , appartenant

&

, est ddfinie en

x , alors

f(x)

appartient

n ~ K2

nous autoriser

q u e si

n

. C e t t e p r e u v e e s t d u e ~ B e c k e t e t n o u s le r e m e r c i o n s

~ la reproduire n

Notons

f ~ ~ K(X)2

o4 les

r I , s3 ~ K(X)

ici :

2n-1

- ~¢

n

~ K(X)2

; soit

de

zn

sous la forme

x = (Xl, ..., xp) ~ Kp

f = ~ rI

2n-1

- ~¢

2n

~. sj

et soit

Ox= K[X](x~x,i..,x_x ) l'anneau local en x ; soit A : K(X) - - > K u {00} p une place telle que A(Xl) ~--> x i et soit VA l'anneau correspondant ; alors si

f

est d6finie en

x ,

f ~ O

x

a VA . II s u f f i t de montrer que

r i , sj ~ VA d'o~ l'on d6duit =~

Mf) = f(x I . . . . . x ) = p ~(rl)2n - ¢~2n-i ~ A(SJ)2n ~ P n

Soit p a r exemple

rI

tel que

alors on a

f = r2 I

v(r I) = min { v(r i) , v(sj) } n-I

n

[ I + ~ ( r l / r l ) 2n - ~2

f 6 VA et le crochet , not6

, si r I 6 VX ,

n

~. ( s j / r l ) 2

]

• on s a i t que

z dans la suite , d a n s l'expression ci-dessus

est une unit6 ce qui donne une contradiction ; en e f f e t si

z

n ' 6 t a i t pas une

265

unit& a l o r s d a n s le c o r p s r ~ s i d u e l r/

r1

et

s/r

1 n

;~(z) = I + E yl z

VA / m A = K

appartiennent 2 n-I

- ~

&

on aurait,

p u i s q u e les

VA :

,2 n

~. yj

= 0

ce qui est impossible puisque - 1 ~ Pn

L'autre cas o~ l'on a par exemple

s

d4fini par : I

v(s I) = m i n ~ v ( r I) , v(sj) } s e t r a i t e

de mani~re analogue.

(iv) ~ (i) : On c o n s i d ~ r e la t h 6 o r i e d e s c o r p s c h a i n e - c l o s

a-chainables

u t i l i s e le l a n g a g e d e s a n n e a u x a u g m e n t 6 d ' u n s y m b o l e d e c o n s t a n t e corps par

K

chalne-clos

P / Q

avec

a-chainable

l ' h y p o t h ~ s e (ii) se t r a d u i t ,

= 0

v

f(x) = y

2n- 1

v

qui est ~-ehainable on fixe une ct-chaine cl6tuve chaine de

K(X)

l'hypoth~se faite sur

f(x) = - ~ (PI)I~I

K

on a

K ~ L ,donc

= 0

a u s s i que a-chaine

de niveau

II

f de

appartient

v

f(x) = y x = X

2n &

de

2

dans

L

f(x) = - ct

dans

L

P

n

2n

des

de

Ce r a i s o n n e m e n t f

n

c o r o l l a i r e 1-10 q u ' e l l e e s t 4 g a l e & y. K(X) z

formule

z

2

) " f

L ; cet ordre prolongeant

K(X) , n o u s a v o n s

& l'intersection

K(X)

de t o u s l e s

d o n t on s a i l p a r le

zn-1

- a

est

e s t f a i s a b l e p o u r r o u t e s les

appartient

~-chaines de

une

n

et on obtient que 2n

L

K(X)

III-2 et

la m ~ m e 2

v

; dans

et on consid6re

I' ~ - c h a i n e c h o i s i e s u r

K(X) , p a r c o n s 4 q u e n t

ordres de niveau exact

) "

n-i

appartient ~ l'unique ordre de niveau exact

P

f

2n

pour cette chaine . D'apr~s le l e m m e

satisfaite ; on ehoisit alors

l'ordre

en notant

z

n

" V x 3y 3z ( Q ( x )

ct . D a n s le

P , Q ~ K[X] , p a r l a f o r m u l e s u i v a n t e : 2n

" V X 3y 3Z ( Q ( x )

e t on

n

~ K(X) z

.

REFERENCES

[Bel] E. B e c k e r :"Hereditarily p y t h a g o r e a n f i e l d s and o r d e r t n g s of higher" t y p e s " , I.M.P.A., L e c t u r e s N o t e s # 29 (1978), Rio de J a n e i r o . [Be2] E. B e c k e r : "The real h o l o m o r p h y ring and s u m s o f 2n-th p o w e r s " , in G ~ o m 4 t r i e A l g 4 b r i q u e r 4 e l l e , L e c t u r e N o t e s in Math. # 959, p.138-181, S p r i n g e r - V e r l a g 1982. [ B - B - D - G ] E. B e c k e r , R. B e r t , F. Delon e t D. G o n d a r d : " H i l b e r t ' s 17th p r o b l e m and s u m s o f 2n p o w e r s " , p r e p r i n t . [B-G] E. B e c k e r e t D. G o n d a r d : "On r i n g s a d m i t t i n g o r d e r i n g s and 2 - p r i m a r y

c h a i n s of o r d e r t n g s of h i g h e r level", M a n u s c r i p t a M a t h e m a t i e a , 69, 2 6 7 - 2 7 4 , 1990. [D-G] F. Delon e t D. G o n d a r d :"17~me p r o b l d m e de H i l b e r t au ntveau n dans l e s c o r p s c h a f n e - c l o s " , The J o u r n a l o f S y m b o l i c Logic, vol. 56, # 3 S e p t . 1991, p . 8 5 3 - 8 6 1 .

266 [Di] M. Dickmann :"Couples d ' o r d r e s chatnables", Expos4 au S6minaire " S t r u c t u r e s alg6briques ordonn6es" (D.D.G.), Univ. p a r i s VII 1987-88. [GI] D. Gondard :"Thdorie du premier ordre des corps chatnables et des corps cha~ne-clos", C. R. Acad. Sc. Paris, T o m e 304, if 16, 1987. [G2] D. Gondard : "Chainable fields and real algabratc geometry", in Proceedings "Real Algebraic and Analytic Geometry" (Trento Oct. 88) , Lectures Notes in Math. 1420, Springer-Verlag, 1990. [G3] D. Gondard :"Kernels of chains and chatnable fields", in Abstracts A.M.S., vol i0 #5 Oct. 1989. [G4] D. Gondard :"Sur l'espace des places rdelles d'un corps cha~nable", preprint. [G-R] D. Gondard et P. Ribenboim :"Sur le 17dine p r o b l d m e de Httbert", C. R. Acad. Sc. Paris, tome 277, 20-08-1973, p. 303-304. [H] J. Harman :"Chains o f h i g h e r l e v e l ordertngs", Contemporary Mathematics, vol. 8, 1982, pp. 141-174, A.M.S.. iLl] T. Y. Lam :"The t h e o r y o f ordered f i e l d s " , Proceedings of Alg. Conference, pp 1-152, M. Dekker (1980). [L2] T. Y. Lam :"Ordertngs, Valuations and Quadratic Forms", C.B.M.S. regional conference, # 52, 1983, A.M.S.. [McK] K. McKenna : "New f a c t s about H i l b e r t ' s 17th problem", L e c t u r e notes in Math. 498, S p r i n g e r - V e r l a g , pp. 220-230. [PI] A. P r e s t e l :"Lectures on f o r m a l l y real f i e l d s " , I.M.P.A., Monografias de Matematica, if 22, 1975, Rio de Janeiro. [P2] A. P r e s t e l : "Sums o f s q u a r e s in f i e l d s " , Bol. Soc. Brasil., Rio de Janeiro, 1975. [R] P. Ribenboim : "Arithmdtique d e s corps", Herman, P a r i s 1972. [Z-G] Zeng Guangxin : "A c h a r a c t e r i z a t i o n o f p r e o r d e r e d f i e l d s w i t h the w e a k H t l b e r t p r o p e r t y " , proc. of the A.M.S. 104 (1988), 335-342.

CURVES OF DEGREE & WITH ONE NON-DEGENERATE DOUBLE POINT AND GROUPS OF MONODROMY OF NON-SINGULAR CURVES

I.V.ITENBERG

Abstract. The paper is devoted to the plane

projectiv~

real

rigid

algebraic

isotopy

curves

of

classification degree 6

of

~ith

a

non-degenerate double point and to ~he calc~lation of the groups

of

monodromy of non-singular curves of degree 6.

Key words and phrases : curve

of degree 6 with a non-degenerate

double

point,

rigid

isotopy c l a s s i f i c a t i o n , group of monodromy,

Address • Oep, E~f Math. and Mechanics Leningrad St~.ite Unive,~sity Bib!iotechnay~, pl. 2, Leningrad~

Petredvoretz,

~98904 ~ UoSR e-ma~!

address

: degt@!omi,spb.su

Introduction

There are two main problems for every class of plane projective real algebraic curves :

the classification up to real

isotopy

and

the c l a s s i f i c a t i o n up to r i g i d isotopy. Two curves-A and B are r e a l l y isotopic i f f

sets ~A and

~B

of

268

their

real

points

are i s o t o p i c

in ~pz

are rigidly isotopic iff there

Two c u r v e s of

exists

a

real

a

fi>:ed

isotopy

class

connecting

these curves and consisting of the sets of real points of the curves of th~ same c l a s s ,

Real isotopy and r i g i d isotopy c l a s s i f i c a t i o n s of

non-singular

curves of degree ~4 ~nd of the curves with a non-degenerate point

of

degree

~4

were known

in

the

last

century.

c l a s s i f i c a t i o n up to real isotop7 of curves of

degree 5

known.

the

In

1969 D,A.Gudkov

classification

of

[I~

obtained

non-singular

curves

double

of

The

was

real

also

isotopy

degree

6,

The

c l a s s i f i c a t i o n of non-singular curves of degree 5 and of degree 6 up to r i g i d isotopy was completed in 1978-198! in works of [2]~ V,V.Nikulin [~] and V,M,Kharlamov [ 4 ] ,

V.A.Rokhlin

The results of Kharlamov

[4] also allow to obtain the r~gid isotopy c l a s s i f i c a t i o n of

curves

of degree 5 ~ith a non-deg~ner~ite double point, in 1991 the author ( [ 5 ] ,

[6]) obtained the

classifications

up

to reai isotopy and up to r i g i d isotopy of the curves

of

with one non-degenerate double point using

scheme ( [ 3 ] )

and the results of E,B.Vinberg

Nikulin's

degree

6

([7]~ [8]) on the groups generated by

reflections, in the present paper we consider the scheme of obtaining

these

c l a s s i f i c a t i o n s , Besides, we discuss the c l a s s i f i c a t i o n up to

rigid

isotopy in the cases of the most interest,

Kharlamov noticed that the scheme developed gives a method for

non-singular

caiculation

of

the

by

groups

([3])

Nikuiin

of

monodromy of

curves of degree 6 ( if the beginning and the end of

rigid isotopy coincide with non-singular isotopy defines some permutation real points of A; we w i l l

curve A,

then

of t h e components

call the group of

all

the group of monodromy of curw~ A ). Kharlamov

this

of

the

a

rigid set

of

s u c h permutations has

calculated

groups of monodromy of non-singular M-curves of degree 6 is a curve with ~a~imal possible number of components

of

(

the

M-curve

the

real

point set for' given degree; this number is equal to 11 for degree 6) in

the

present

paper

we

consider

the

general

scheme

of

269

c a l c u l a t i o n of th~ group of

monodromy

of

curves

non-singular

of

discussed

degree 6. Only the cases of ~-curves and (M-l)-curves are in details,

l.Statements of results

We ~il! consider the curves D~ degree 6 with double point

as

the

results

of

the

a

simplest

non-degene:ate

degenerations

of

non-singular curves, Rigid isotopy

classification

of

these

~urves

is

given

by

Theor;~m 2. t and Propositions 3.4, 3.5 of the present paper. Calculation of the g~'oups of monodromy of

non-singular

curves

oF degree 6 c~n be done using Theorem 3.6. Zn the present part of the p~p~r ~e w i l l

discuss the

cases

of

M-:urves and (M-l)-curves only. For the sc~emem of d i s p o s i t i o n of ovals of non-singular (an oval is ~ connected component of the set of

tea!

curves

points

of

curve which is homeomorphic to c i r c l ~ and embedded two-sidely ~pZ) we w i l l

a

into

use the system of notations suggested by Viro [9] . The

curve consisting of a single oval w i l i

be denoted by symbol , the

empty curve - by symbol . I f

symbol stands for

ovals then the

set

obtained

surrounding a l l

old ovals ~ L i l be denoted by symbol .

of

ovals

by

some set

addition

of

an If

curve is the union of two non-Jntersectino sets of ovals denoted

of oval the by

and r e s p e c t i v e l y with no ova~ of one set surrounding an oval of the other set, then t h i s curve w i l l Besides, i f

A i s the notation

for

be denoted by symbol

some set

of

ovals

A~...UA of mnother notation where A repeats n times w i l l by nxA; parts nxl of notation w i l l

.

then

part

be denoted

be denoted by n.

Let us notice that the c l a s s i f i c a t i o n s up to real up to r i g i d isotopy coincide in the c:ases of

isotopy

non-singular

and

M-curves

and (M-i)-curves of degree 6 ([3]). Thus ~n these cases it is enough to point out the scheme of d i s p o s i t i o n of ovals in order to f i x r i g i d isotopy type of a curve.

the

270

No~ we w i l l present the schemes of

possible

conjunctions

c o n t r a c t i o n s of ovals of non-singuiar if-curves and degre~ 6. Not~ that no

oval

of

these

curves

and

(M-ll-curves

can

conjunct

of

with

itsmlf. An oval of a non-singuiar curve i~ c a l l e d even (odd) i f

it

lies

i n s i d e of ~ven (odd) number of ovals of t h i s curve. For convenience we f i r s t

fortnulate a general statement :

each empty ov~i of a non--singular c~rve contracted and th~r~ exi~to on!y one

rigid

of

degree

isotopy

6

eiass

can of

be the

r e s u l t s of such degeneration.

M-curves There arm three r i g i d isotopy types of non-singular M-curves of degree ~ :

, U 9),

Schemn_ .. , U ~x,~" xl,

Tho

~c hm.~,e of

possibl~

conjunctions

c o n t r a c t i o n s of ovais of a curve of t h i s isotopy type Fig.

I.

kl

i

\

.C

c.

Fig.l

is

shown

and in

271

it is stated that the even empt7 ovals of the to this rigid isotopy typ~

curve

belonging

can be numerated by numbers from I to

9

in such a way that the ¢ i r s t oval cmn conjunct with the second one, the second oval can

conjunct with the t h i r d one and so on with

last nineth oval conjuncting

with

first,

seventh

the

~ourth

and

the

the

first

the

o n e . Moreover,

owls

can

conjunct

the

with

a

non-empty one while the other conjunctions of ovals are impossible. The l e t t e r s standing near two dotted

lines

coincide

iff

the

r e s u l t s of the conjunctions corresponding to these l i n e s are r i g i d l y isotopic.

The l e t t e r s placed inside the empty ovals coincide i f f

the

r e s u l t s of contractions of these ovals ~re r i g i d l y i s o t o p i c . ~e w i l l

nam~ such schem~s the marked schemes of

conjunctions

and contractions of ovals. It

is clear that the even empty ovals of a non-singular M-curve

of degree 6 with the scheme < I < ! ) U g> can conjunct only in a c y c l i c order. There is natural cycli~ order for these ovals ( t h i s order defined up to i n v e r s i o n ) . Let us take a point inside the

odd

is

empty

oval and a l i n e containing t h i s point; we can r o t a t e the chosen l i n e around the chosen point

and

mark the

i n t e r s e c t s the even ~mpt7 mva!s. I t

order

is

the

in

w h i c h the

corollary

line

~rom Bezout

theorem that the c y c l i c order obtained cannot be changed by a isotopy. Thus non-neighbouring (in sense of the given c y c l i c

rigid orderl

ovals cannot conjunct.

Rigid isotopy cl~ssification of the

results

of

the

simplest

degenerations of curves with scheme U 9> can be formulated

as

follows : the numbers of r i g i d isotopy classes contmined

in

i~otopy

classes represented in Fig. 2 are: 2a) - i~ 2b) - I , 2c) - 2, 2d)

But in t h i s statement

some information is l o s t .

The group of monodromy of a curve of under

discussion

is

- 2.

equal

permutations of 3 elements.

to

$3,

the

where S

rigid is

isotopy the

group

type Of

272

b)

o)

c)

d)

Fig.2

Schem~

l•'~,~-" t~ U 5>.

The s~heme of conjunctions

and contractions

ovals is given in Fig.3. This scheme is not marked because case it is convinient

to say that the results of

the

in

this

contractions

of non-coincident ovals are not r i g i d l y isotopic and the r e s u l t s conjunctions of

non-coincidEnt

isotopic as well. We w i l l

pairs

of

ovals

are

of

not

of

rigidly

always present non-marked schemes

in

the

cases like this, It

is necessary to note that there are l i n e a r

the even empty and on the odd ovals of a curve of

orders degree

both 6

on

having

scheme ,

The group of monodromy under discussion is trivial,

of a curve of

the

rigid

isotopy

type

273

Q

Fig.3 Scheme .

The

~arked scheme

of

conjunctions

contrmctions of oval5 is represented in Fig.4,

tE

~c

®°

Q

Fig.4

and

274 Note that there is

a

natural

inversion) on the odd ovals of a . So i t

linear

curve

order

of

(defined

degree

6

with

up

to

scheme

i~ clear that the odd ovals can conjunct only

in

chain. The group of monodromy of a curve of under discussion is equal to

the

rigid

isotopy

type

~/2Z.

(M-1)-curves

There are 6 rigid isotopy types of non-singular degree 6 :

,

(I U 5>,

(M-1)-curves of

(i U 47, (I U I>,

>. Only those degeneration~ are

discussed

whose r e s u l t s

cannot

be obtained as the r e s u l t s of a degeneration of an M-curve.

Scheme . The marked scheme of conjunctions and contractions o~ ovals is given in Fig.5.

/

/

!

,A

i /

f

4' %

/

' I

;}_

-

' %

I /

I

' ^" ~ < - ~'~,

x %

I I

,A

A~ x

I

/

X

!

%

% %

Q

'6

6"

/

A Fig.5

1 I

275 The group of monodromy of a curve of under discussion is equal to A , where

A

the is

rigid

isotopy group

the

of

type even

permutations of 5 elements..

Scheme U B>.

The

marked scheme

of

conjunctions

and

contractions of ovals i s given in Fig,G,

k,

Fig.6 The group of monodromy of a curve of

rigid

isotopy

of

symmetries

of

and c o n t r a c t i o n s

of

The two dotted lines between the non-empty oval and one of

the

discussed is equal to De, where D e is the

the group

type

the regulmr 8-gon,

Scheme (L. The scheme of c o n j u n c t i o n s ovals i s given in Fig,7,

even empty ovals denote the existence of two rigid

isotopy

of the results of conjunctions of the corresponding ovals,

classes Thus

it

276

is riot enough to point out a pair of conjunct ovals for pointing out the rigid isotopy cl~ss of the res,a!ts of conjunctions.

© df /

"'0-

Fig.7

The group of monodromy of a c~rve of this rigid isotopy type is trivial.

Scheme (I U 4>. The scheme of conjunctions and contractions

ovals is given in Fig,8,

/

0

"'Q)' !

Fig.8

of

277

The group of monodromy of a curve of this rigid isotopy type is trivial.

Scheme

,

The

of

marked scheme

conjunctions

and

contractions of ovals i~ given in Fig,9,

#

F F

,

F

A

Fig.9 The group of manodromy of a curve of t h i s r i g i d isotopy type is

equal to Z / ~ ,

Scheme 0. The diagonal forms over ~ of the r e s t r i c t i o n s

L_h,~hav~ one

positive

square.

T h u s the

consists of two connected components. Each of these

space

0

components can

be transformed to one another by i n v o l u t i o n - ~ . We are interested in periods only up to ~quivalenc~ and thus we can consider the quotient space 0 / ( i , - ~ ) spaces

of

spaces.L+~

as the space of periods. Let ~+ and ~-h,~

Lobachevsky obtained

and L _ A , ~ ® ~

by

respectively,

the standard

be

the

procedure

from

It is easy to verify that

281 Period~ ([~+1,[~_]) b~longing to ~ +

x ~-h.~ are the periods of

the marked k~-s~rfaces " "~ with ~ non-degenerate double point is no vector v ~ h

if

that B (~,v)=O, B(v~h')=O, B(v,v)=-2 in

lattice

c

L. Consider a reflection with

respect

to

the

there

hyperplane

in

~+

orthogonal to vector v~L+ with square -2 (this reflection transforms x to x + B ( x , v ) . v ) .

I t can be transposed with ~ leaving vectors h' and

~ ' i n their places. Thus the reflection described gives us equivalent periods. I t also

h o l d s true

for

a

reflection

with

respect

to

hyperplane in ~-h,~ orthogonal to vector v ' ~ L_h,~ having square -2.

Let O+ and O_h,~ be the fundamental domains of groups generated reflections with respect to hyperplanes orthogonal to

by

vectors

squares -2 in ~+ and ~_ respectively. We have shown that the

with image

of the period mapping belongs to ~+x /~-h,~ up to equivalence. Consider element ~ ~ L ~ ~ such that there is vector v~L

with

the properties: B (~,v}=O, B(v,h')=O, B(v,v)=-2. c Let ~=v++v where v+e L÷ ® ©, v_e L_ ® @. We w i l l denote the orthogonal complement in 2~, E L_h. Note that i f

L_ to vector

h"

by

L_h

. Then we have

[~]~+XD_h,G , then v_# 0. Now we consider

t~o case~:

a) v+= 0. i f vector v

does not belong to space L_h.~ ~ ~

projection of vector v

, then the

on L h,~ ® ~ w i l l be denoted by v'

It

is

easy to notice that 2vj~ L_h,~ , B(2~'L,2v~)

0 or B(2v',2v~) = -6,

2~'_~ ~'

(mod 2L).

Consider hyperplanes in L_h,~ orthogonal to vectors v'_~ L_h,B where

B(2v',2v'_}=-6 and 2v_~ ~'

(mod 2L). Let

t h e s e hyperplanes divide

open polytope (~-h,~ int~ .polytopes ~t_h,B ( i = I , 2 , . . . ) (the number

of

these poiytopes may be infinite).

b) v+~m0. I t should be noted that B(V+-v ~v+-v _ ) = B(v++v _~v++v _) _

I f vectors v+-v_,v++v_, ~'

(*) except 6"

=

-2

are linear independent, t h e n class

[c~]

282

belongs to the sertain subset of codimention 2 This

subset is

vectors

the

and 2v÷_= ~ '

~ + x ~-h,G

intersection of two hyperplanes orthogonai

v++v_ and v÷-~'_. Linear

v -v_, ~ '

of space

is possible

only if

(mod 2L}.

Consider

dependence

v = ~'/2.

of

But

vectors

in ~ .

v e c t o r s 2v+~_~+ where B(2v+,~°v+)=-6 and 2v+= ~"

orthogonal

(mod 2L)

the previous case. Let these hyperplanes divide

v++v_

,

B(2v÷,2v+} = - 6

then

the h y p e r p l a n e s

to

open

similar

polytope

J (j=i,2 ...) (the number of these polytopes may into polytopes £~+

to to

~+ be

i n f i n i t e as well}. W.p have proved that the imago equivalence can be obtained

by

of

mapping

removing the

of

periods

closed

up

subset

to with

condimP.nsion at least 2 ( t h i s subset w i l l be denoted by 7.) from the j union of components £)xO . ~ (i=I ;j=l ) I t should be noted +.3 -tl,~ ~... 1. . . . 1, that some components ~+xD h,~ can be transformed to other ones by the automorphisms of quadruple ( I ) . Similar to [33 i t [co]~(u~J~xC~_h,8)\7._

is

possible

defines

only

to

show that

one

class

every

of

the

class

of

projective

equivalence of real K3-s¢Irfaces being t,~o-sheeted covering spaces of

C P z branched over a curve of degree 6 with a non-degenerate

double

point.

It is easy to prove that one of lattices L+ and

L h.~

has

no

vectors v such that v _----~' (mod 2L). Thus one of collections ~+J and O_h,~. consists of the single poiytope. Let O, stand £~+ i f

there exists vector v in L such that v = ~'

for

collection

(mod 2L} and

for

collection ¢)s -t~,~ in the other case. We have proved the theorem. Theorem 2 . 1 .

Consider

tile

curves

of

non-degenerate double point having quadruple

degree

6

with

a

(H2(X,Z),conj,5 h,

~)

isomorphic to quadruple (1). The rigid isotopy types of these curves are in one-to-one correspondence with the elements of collection

~,

considered up to the automorphisms of ( l ) .

We can choose for each curve of degree 6 with a

non-degenerate

283

double point involution conj in such a way that l a t t i c e L+ would not have such vectors v that ~ ~ ~'

(mud 2L), i . e .

collection ~ ,

coincide with collection 0 -h.~" t We suggest that involution

would

conj

is

chosen in this way.

3.Corollaries of theorem 2,1.

Let ~-h be the space of Lobachev.sky obtained in a standard way from space L_h. The group generated by the reflections with

respect

to the hyperplanes orthogor,al to the vectors in L_A having square -2 acts in ~

.. Let ~

stand for ~ polytope b~ing a fundamental

domain

- t%

of this group and having a fa~e orthogonal to ~'. The Coxeter scheme ( see ~or example appears to contain all

information

[7])

of

polytope

required about collection [~, (let

us remind that the involution conj was chosen in

such a

way

that

collection ~ , coincides with collection O_tL~).. p

Let C be the Coxeter scheme of polytope 0_, e be the vertex scheme C corresponding to ~ ' .

Remove all thick

and

dotted

from scheme C (two vertices of a scheme are connected by

of

edges

thick

(or

dottedl edge i f bilinear form B has the vaiue equal (or ~ore than] 2 on corresponding vectors). Let C' stand for

the

scheme obtained.

Consider the symmetries of scheme C' generated by the of (L,~o,h').

The group of these symmetries acts on C'. Let

the corresponding quotient-scheme and e' for vertex e. C''.

it

automorphisms

the

class

C''

be

containing

Let K stand for the connected component of vertex

e'

in

is easy to prove the following statement.

Proposition

3. I.

The number of the polytopes

of

collection

~,

which cannot be transformed one into another by the automorphisms of (1) is equal to the number of vertices of scheme K.

Corollary 3,2. The number of the r i g i d isotopy types of the curves of degree 6 with a

(H2(X,Z},conj,,h,

non-degenerate doubl~ point

having quadruple

~} isomorphic to (I] is equal to the

number

o{

284

of s c h e m e K.

vertices

The construction of s c h e m e s

for

K

all

possible

classes

of

isomorphism of quadruples (H ( ~ Z ) , c o ~ j . , h, 3) cor?esponding to the 2

curves of degree 4 ~ith a

non-degenerate

double

point

allows

to

obtain r i g i d isotopy c l a s s i f i c a t i o n of the curves mentioned.

It

necessary to note that

a!ways

scheme K (unlike c o l l e c t i o n 0 , )

is

is

finite. ~e ~ i l l

reformulate now the above r i g i d isotopy

in terms of conjunctions and contractions of ovals

classification of

non-singular

curves of degree 6. But !~t

~s f i r s t

ma;~ some remarks.

Let A be m curve of degree 6 mith a non-degenerate double o point. There ~re two types of ~moothings of singular point of th'is curve. [.~t At

(t~O) be a smoothing of one

of

these

types

and

At

(t~O) be a smoothing of the other type. Let the number of ovals

of

non-singu!ar ~,~L.... ,~ A

,:~' z .5

~ig.6

3) Keeping double point and degenerate 4-fold point we can obtain the 6-th degree curve G 1=0 shown on fig.6 as a result of sequence of small perturbations shown on fig.7.

Zig.7

298 4) Let us consider the 5-th degree curve

(c t , , ~ ) ( x o : x 1 :x,z)

H I (x0:x I :xz)-

-- o

Xo4Xl 3 where hy(x0:x I :x2) = (x0x I :x12:xOx2) is birational quadratic transformaZion and by -I (xO:x I :x2)=(xO2:xoXl :XlX2). It is shown on fig.8. This curve has 3-fold point in (I "0:O), and 3 points

x~-~°/ k J\ /

~ig.8

Fig.g

of intersection with axis x0=0 one of which is flex point. Tangent to the curve H I=0 in this flex point is axis x0=O. I~t (0:0:1), (0:1 :~" ), (0:1:~) be coordinates of points of intersection ( ~ < ~" ). There exist two numbers a,b ~ ~,b~0 and linear projective transformation li(x0:x I :x2)=(xO:x1:(ax1+bx2)) such that li(0:0:1) = (0:0:1), li(O:1:~') a (0:1:k'), li(0:1:~) = (0:1:k) for any k < k'. So for any 0 < k < k' there exist 5-th curve H(xo:x I :x2)-(H I°li -I )(x0:x I :x2)=0, located in s shown on fig.9 and H(0:x I :x2)= cx I (x2_k, Xl )(x2_kXl ) 3 where c~0. It is not difficult to see ~here exists neig_j~,bm~hood U ~- ~ of number k" such that for any different k I ,k2,k3~U there exists the 5-th degree curve H(x0:x I :x2)=H(x0:x I :x2)-H(0:x I :x2)+

+cx I (x2-kx I )(xz-k Ix I ) (xz-kzx I )(x2-ksx I )=o shown with its chart on fig. 10. 5) Under transformation by -I the curve ~(x0:x I :x2)=0 maps onto 9-th degree curve S(x0:x I :x2)=~ohy)(x0:x I :x2)/cx1=0 that

299

Fig. I0 has a chart shown on fig. 1 I. This curve has N28 germ defined by equasion (I) in point (0:0:1) and has Z15 germ defined by equaZ ion x I (x_ox,2-mlXl 2 ) (xOx,2-m2xl 2 ) (xox2-m3xl 2 ) =0 in the point (I :0:0), where

(21

O<m 1<m2<m3.

FJ~E.11 6") In [3] it was proved that for any three numbers O<m1<m2<m3 any Z15 germ of plane curve defined by equation (2) can be smoothed by gluing the charts shown on fig. 12.

~

(o

2.

0

o

,~

oo

o

o

z

G

Fig. 12 7) iet us glue by Viro method the charts shown on Zig.1 t with ~he charZs s h o e on f i g . 1 2 we obtain the charts shown on

fig.1.

300 Theorem 2. For any two numbers m,m'~ ~ (0<m°<m) there exists neighbourhood U~- ~ of number m" (m~U) such that for any different m I ,m2,m3EU any plane curve NZ8 germ that defined by equation

Xl (xOx2_ml Xl 2 > <xOx2_m2xl 2 ) (xOx2_msxl 2 ) <xOx2_mx 12 ) =0 can be smoothed by gluing the charts shown on fig.13.

o

o

#

#

"7

'1

3

"1

Proof. T et US transform the curves provided with theorem I to transformation (x0:x I :x2)~-,(xO2:X0Xl : (x0x2-tx 12)), where t>k' and then to symmetrization (XQ:XI:XZ)~-~(Xo:XI:(-X2)) we obtain the charts we need. The next two theorems were proved in [4]. Theorem 3. For any two numbers k,k'~ ~ ( O, ~=24-/9-z-&-~, where (/9,;*,6)=(1 ,! ,! ), ~=1,5,9,13,1T,21 (/9,;*,6)=(1,1,3), ~'--3,7,11,13 (#,;*,6)=(1,1,5), ~=5,9,13 (/~,;*,6)=(1,1,7), ~=7,11,15 ( # , ~ , 6 ) = ( 1 , 1 , 9 ) , ~=9 (f~,~',6)=(1,1,11), ~=11 (/9,;*,6)=(1,3,3), ~=5,9 (#,;~,6)=(1,3,5), ~=7,11 (/9,;*,6)=(1,3,7), ~=9 (/9,;*,6)=(1,5,5), ~=5 (/9,;*,6)=(1,5,7), ~=7,11 (#,;*,6)=(1,7,7), ==9 (#,;*,6)=(3,3,3), ~--~3,7 (#,;*,6)=(3,3,5), ~=5 (#,;*,6)=(3,3,7), ~=7 (#,;*,6)=(3,5,5), ~=7 (#,;*,6)=(5,5,7), ~=7 6) <Jt~U1 >, a=26-/9-r, where ;*=1, /9=0, 2- , where 0~o~15 2) <JL~UI>, where a+P~-14, 0 ~ < 1 3 , 3 ) <JUt

~i(Yl,...,yn):=

(Yl,---,Yi_I,Yi+I,---,Yn)Lemma

3.1.

Fix

there e x i s t s

i 6 {I .... ,n}.

an a l g e b r a i c

If

set

Ai(En)

is d e n s e

B i c C n-I

En,

then

has

the

such t h a t

-i BF = gi (Bi)"

(3.2)

Proof. We shall first show that f o l l o w i n g property:

the

(3.3)

nil(=i(y))

if

Indeed,

y ~ e n \ BF, then

let

2.l,the Yi-l'

us

take

set

S

=

y

=

{x

~

complement

(y~,...,yn) e

:

function

components

Pi[Sj

BF,

therefore

the

fact

z E C,

is a b i r e g u l a r

k = d.

that

Hence,

Sl,...,S d

and,

mapping

from

are

if

(3.3)

it

easily

y ~ B F , then

we

obtain

the

an a l g e b r a i c is o b v i o u s . o

set in

follows

(i) F

is a p o l y n o m i a l

(ii) Ai(En)

= En,

of

yl,...,Pi_l(X)

=

into d i s j o i n t

any

Sj

such

Sj,

the

onto e. since

we

obtain

= d.

This

BF

that Bi

the

of

Pi[Sj

that, gives

for

y and any

property

has t h e p r o p e r t y :

automorphism

set

B i := ~i(BF)

so defined,

assumptions

on

of

F,

equality

the

is (3.2)

following

C n,

i = 1 ..... n-l,

(iii) Ai(En)

is d e n s e

in

En,

(iv)

is dense

in

En

AI(En)

=

Proposition

c BF .

For the

T h e o r e m 3.4. U n d e r the above c o n d i t i o n s are equivalent:

By

bijectivity

that

difficulty

C n-l.

BF

for

disjoint,

~il(~i(y))

without

g

decomposes

#F-l(yl,...,Yi_l,z,Yi+l,...,yn)

(3.3) . From

Hence

Sl,...,S k

~\B

c en \ BF .

Pl(X)

Pi+l (x) = Y i + l ' ' ' ' ' P n (x) = Yn }

irreducible

E

in

i = 1 .... ,n-l, and

kerA 1 =

{f(P2,...,Pn)

n : f ~ En_l}.

Proof.

I.

function Propositon

(i) such 1.2,

,

(ii).

that

f

Take

og/0Y i

=

c E . Let g be an e n t i r e _I n . Put h := goF. Then, by

foF

312

A i (h) = a(h°F-l) a¥. 1 2. 3. be

oF = ~ . o F 1

= f.

(ii) ~ (iii). Obvious. (iii) ~ (i). Let d

the

bifurcation

set

algebraic

set

B* c C

therefore algebraic

B ~ C. Take set

of

the

F.

degree

By

Lemma

of

F

3.1,

and

there

let

BF

exists

B F = cn-lx B * . Since

such that

,

an

B F ~ C n,

,

V := Pnl(z) Since

be

z ~ B

and c o n s i d e r

the n o n s i n g u l a r

= F-l(cn-lx{z}).

(cn-lx{z})

n B F = ~, the m a p p i n g

has the same number, Moreover, FIV is a

equal local

FIV : V

) en-lx{z}

to d, of e l e m e n t s in each fibre. b i h o l o m o r p h i s m . H e n c e we e a s i l y get

that FIV is a covering. Since en-lx{z} is simply connected, then F is a b i h o l o m o r p h i s m on each t o p o l o g i c a l c o m p o n e n t of V. So, the n u m b e r of t h e m is equal to d. Let Vl,...,V d be these components.

Since

gradP n

:=

(aPn/aXl,...,aPn/aXn)

vanishes

nowhere, therefore, from the c o n n e c t i v i t y of the i r r e d u c i b l e components of a l g e b r a i c sets ([6], Coro4.16) we o b t a i n that Vl,...,V d are also the irreducible components of V. The mappings Hence

FlVi,

and

FIV i

from

i

=

is a b i r e g u l a r Let

Qi(x)

=

Since

be

regular

of

Vi

([6],

onto

polynomials

gradP n

Pn - z = ~QI...Qd,

that

nowhere,

and

biholomorphic. we get that

en-lx{z},

such

vanishes ~ ~ e,

and

Th.3°20)

i = l,...,d.

Vi =

{x

therefore,

QilVj

= const,

~ ~n: by

the

for any

Qi = TijQj + ~ij w h e r e Tij ~ C[X], ~ij ~ e. T h e n we degTij = O. In consequence, there e x i s t s a p o l y n o m i a l

i,j. H e n c e e a s i l y get H ~ e[Z]

are

Main T h e o r e m

mapping

QI,...,Qd 0}.

above,

l,...,d,

Zariski's

such

that

because, otherwise, g r a d P n vanishes.

degH = d there

and

would

Pn - z = HoQ I. H e n c e

exist

points

in

en

d = 1

at w h i c h

Since d = i, t h e r e f o r e F is injective. H e n c e F is a p o l y n o m i a l a u t o m o r p h i s m (see [8], Th.l.4, [i], Th.2°l). 4. (i) ~ (iv). By the i m p l i c a t i o n (i) ~ (ii) it s u f f i c i e s to show only the e q u a l i t y ker~ 1 = { f(P2,...,Pn) ~ E n : f ~ En_l}. D e n o t e the set on the right hand side of this e q u a l i t y Obviously, sition

we have

Enc

ker~ I. Let

depends

E n-

By Propo-

i.2, a(g°F-l) 8Y 1

Then

g ~ kerA I.

by

B(goF-l)/aY 1

=

O,

and

only on the v a r i a b l e s

oF = 0

hence

the

Y2,...,Yn.

entire

function

Denoting

goF -I

goF -I by

f,

313

we have 5.

g = f(P2,...,Pn). (iv) ~ (i). Put

dominating

mapping.

So,

ring

C[YI,...,Yn_I]

ring

e[X]

of

So, g ~ En" F~= (P2 ..... Pn ) it

of

regular

induces

regular

C[X]

then

P

~

functions

functions

assumption kerA 1 = E ~, n F (C[YI,...,Yn_I]) =C[P2,...,Pn] Indeed, if an element P ~ C[X] kerA 1

: C n --~ e n-l.

a monomorphism on

on

C n.

C n-I

We

implies

It

F

claim

that

is

a

of

the

into

the

that

the

the

ring

is integrally closed in e[X]. is integral over e[p2,...,pn] c

(because

P

is

constant

on

each

irreducible component of any fibre of F). Hence P = f(P2,...,Pn) for some f ~ En_ I. Since P'P2' .... Pn are polynomials, therefore, f is also a polynomial. This means that P ~ ~[P2,...,Pn]. From the fact that e[P2, .... Pn] is integrally c l o s e d in C[X]

it

easily

follows

that

the

field

e(P2,...,Pn)

is

a l g e b r a i c a l l y closed in C(XI,...,Xn). Hence F is primitive, i.e. there exists an a l g e b r a i c set V c ~n-l, V ~ C n-l, such that,

for

any

y

~

en-l\v,

the

algebraic

set

irreducible ([4]). So, for any y e en-l\v, c o n n e c t e d ([6],Cor.4.16). On the other hand, the a s s u m p t i o n that

the

En

each

implies

mapping

(by

Proposition

PlIF-l(y)

F-l(y).Hence,

for

2.1)

is b i r e g u l a r each

that

for

on each

y E Cn-l\v,

the

topological mapping

automorphism

Remark. 3.5. = En in that

is

~-l(y)

AI(En)

injective. This implies that in the open set mapping F is injective. Also, degF =i. This polynomial

~-l(y)

set

is

is dense

in

~ c n-l,

the

y

component

of

Pl[F-l(y)

is

F-I(cx(cn-I\v)) the gives that F is a

([8],Th.I.4,[I],Th.2.1).D

In the above theorem,

(iv) can be omitted

for

because

n = 2

it follows

the c o n d i t i o n

kerA 1

from the a s s u m p t i o n

J a c F = i. G

4. O p e r a t o r s Let

A[

k = ~

polynomial

mapping

operators

A~l

in the polynomial

rings

or

(PI,...,Pn)

C.

such

Let that

to the ring

F = JacF

k[X 1 t ' "

= °

i.

,Xn]

The

: k n -~ k n restriction

we shall

be of

also denote

a

the by

1 T h e o r e m 4.1. Under the above assumptions, the f o l l o w i n g are equivalent: (i) F is a polynomial a u t o m o r p h i s m of k n,

conditions

(ii) A[(k[X 1 ..... Xn] ) = k[X 1 .... ,Xn], i = 1 ..... n-l, (iii) ~ & ~~( k [ X 1 ..... Xn]) = k[XI,...,X n ] and kerA~TM

=

314

k[P2,...,Pn]. Proof. i. (i) ~ (ii). A n a l o g o u s l y as the i m p l i c a t i o n in t h e o r e m 3.4. 2. (i) ~ (iii). A n a l o g o u s l y as the i m p l i c a t i o n in t h e o r e m 3.4. 3. (ii) ~ (i). a) k = C. From (ii) it follows that is d e n s e

in

En

for

i = l,...,n-l.

a polynomial automorphism. b) k = ~. F has a :C n

--~ C n.

We

shall

Hence,

canonical

show

that

by t h e o r e m

extension ~[(C[X])

to

=

(i) ~

(ii)

(i) ~

(iv)

A~l(e[X]) 3.4.,

the

C[X]

F

is

mapping for

i

=

t

l,...,n-l. T a k e i ~ {l,...,n-l} and P ~ C[X]. Let P = P' + iP'', P',P'' ~ ~[X]. By the a s s u m p t i o n there exist Q',Q'' ~ ~[X] such t h a t =

P.

From

automorphism the t h e o r e m

A[(Q') the of in

= P'

and

case

a)

we

en [2]

Hence F is

~[(Q'') obtain F = also

= P''. that

Hence F

~[(Q"

is

a

+ iQ")

polynomial

Fi~ n is an injection. a surjection. So, F

bijection. Since F -I = F -ll~n, t h e r e f o r e F -I p o l y n o m i a l mapping. 4. (iii) ~ (i). a) k = e. A n a l o g o u s l y as the (iv) ~ (i) in t h e o r e m 3.4.

is

a

By is a real

implication

b) k = ~. As above, let F : Cn > Cn be a c a n o n i c a l e x t e n s i o n of F. From the a s s u m p t i o n s it f o l l o w s easily that ~l(e[X])

= ~[X]

and

kerA~

= C[P2,...,Pn].

From

the

case

a)

we

have that F is a p o l y n o m i a l a u t o m o r p h i s m . P r o c e e d i n g a n a l o g o u s l y as in 3b) we o b t a i n that F is a real p o l y n o m i a l automorphism, a Acknowledgement. We thank Mr K. R u s e k for our a t t e n t i o n to the p a p e r by Y. Stein [i0].

his

having

called

References [I]

H.Bass, E . H . C o n n e l and D.Wright, The J a c o b i a n C o n j e c t u r e : Reduction degree and formal expansion of the inverse, Bull.Amer. Math. Soc.7(2) (1982) 287-330. [2] A . B i a l y n i c k i - B i r u l a and M. R o s e n l i c h t , I n j e c t i v e m o r p h i s m s of real a l g e b r a i c varietties, P r o c . A m e r . M a t h . S o c . 1 3 ( 1 9 6 2 ) , 200-203. [3] Z . C h a r z y ~ s k i , J . C h ~ d z y n s k i and P . S k i b i n s k i , A c o n t r i b u t i o n to K e l l e r ' s J a c o b i a n C o n j e c t u r e III, Bull. Soc. S c i . L e t t r e s L6d~ 39, N O . 4 ( 1 9 8 9 ) , I - 8 . [4] T . K r a s i n s k i and S.Spodzieja, On the i r r e d u c i b i l i t y of fibres of c o m p l e x p o l y n o m i a l m a p p i n g s (to appear). [5] S . L o j a s i e w i c z , I n t r o d u c t i o n to C o m p l e x A n a l y t i c G e o m e t r y (PWN, Warszawa, 1988) (in Polish). [6] D.Mumford, A l g e b r a i c G e o m e t r y I, C o m p l e x P r o j e c t i v e V a r i e t i e s (Springer, B e r l i n - H e i d e l b e r g - N e w York, 1976). [7] K . R u s e k and T.winiarski, Criteria for regularity of h o l o m o r p h i c mappings, B u l l . A c . P o l . : M a t h . 2 8 ( 1 9 8 0 ) , 471-475. [8] K . R u s e k and T . W i n i a r s k i , Polynomial Automogphisms of C n, Univ. Iagell. Acta M a t h . 2 4 ( 1 9 8 4 ) , 1 4 3 - 1 4 9 . [9] S.Spodzieja, On c o m m u t a t i v i t y of the c o m p o s i t i o n of W h i t n e y operators, Bull. Soc. Sci. Lettres Lodz 39, N o . 1 3 ( 1 9 8 9 ) , I - 6 . [I0] Y.Stein, On lineir d i f f e r e n t i a l o p e r a t o r s r e l a t e d to the

315

Jacobian Conjecture, J.Pure Appl.Algebra 57(1989),175-186. [ii] Y.Stein, On the density of image of differential operators generated by polynomials, J.Analyse Math. 52(1989),291-300. [12] Y.Stein, Linear differential operators related to the Jacobian Conjecture have a closed image, J.Analyse Math. 54 (1990),237--245. [13] D.Wright, On the Jacobian Conjecture, Illinois J.Math.25

(1981),423-440.

On a subanalytic stratification satisfying a Whitney property with exponent 1

by K r z y s z t o f K u r d y k a Instytut Matematyki, Uniwersytet Jagiellofiski, Reymonta 4, Krakdw PL-30059 (Poland)

If A is a subanalytic and closed set in R n, then A satisfies following Whitney property (see [St]): every point of A has a neighbourhood U such that, if x and y are two points in A M U then there exists an are )~ joining x and y in A M U such that length~_< C I x - y I~ ,where C and o~ are constants depending only on U. Actually we have the same property for strata of a stratification obtained by projections (e.g. triangulations or cellular decompositions, see [Ha], [Lo3] ). In this paper we construct a stratification compatible with a given family of subanalytic sets such that on each stratum the above property holds with exponent o~=1 ( corollary B ). Actually the strata we obtain (theorem A) are L-regular in the sense of Parusifiski [Pall i.e. are of the form {(z',zn) E R n-1 x R; / ( x ' ) < x~ < g(x'),x' G A'} ,where A' is L-regular in R ~-1, f and g are subanalytic and continous in --~, A analytic in A', f(x') < g(x') for x' G A'. Moreover I dx, f 1< _ M,I d~,g I< M for each z' ~ A',where M is a constant depending only on n. Other strata are the graphs of aaaalytic functions (with bounded differentials) defined on sets of previous form. Our result is related to the papers of Parusifiski ([Pal],[Pa2l). We use only elementary facts on subanalytic sets i.e. subanalycity of tangent mapping and stratifications of mappings. Hence our result holds in semialgebraic and semianalytic case. This paper was inspired by a question of Prof. S.Lojasiewicz (corollary C), the author is very grateful, to him for interest and encouragement. The author thanks G.Jasi~ski for helpful remarks. The paper was written when the author was associate member of Dept. of Math. of Universit~ de Savoie at Chambfiry and wants to acknowledge people working there (particulary P.Orro) for friendly ambiance and hospitality. The author thanks also the referee for valuables remarks.

0 . P r e l i m i n a r y r e m a r k s . Let M be an analytic manifold. By a stratification of M we mean a locally finite family T of analytic, connected submanifolds of M such that: M = U { T : T E T} (disjoint union), moreover if S M(T \ T) ~ 0, where S, T E T, then S C (T \ T) and d i m s < dimT. We say that stratifcation T is subanalytic if all strata (i.e. members of :r) are subanalytic in M. Let A be a family of subsets of M, we say that stratification 7" is compatible with family ,4 if T M A ~ 0 implies T C A, for each T C T, A E A. Let C be a family of analytic, connected submanifolds

317

of M, subanalytic in M. We say that family C is stratifying if for every locally finite family A of subanalytic subsets of M there exists a subanalytic stratification T of M , compatible with .,4 such that T C C. It follows from the general m e t h o d of constructing of subanalytic stratification (see e.g. [DS],[Lo3]) that C is stratifying if and only if : (*)For every A analytic submanifold of M, subanalytie in M, there exists F subanalytic in M, closed, nowhere dense subset of A such that every connected component of A \ F belongs to the family C. We recall a definiton of angle between linear subspaces. 1 . D e f i n i t i o n . Let X be a linear subspace in R'`, let P be a line in R'`.\¥e define angle between P and X as

5(P, X ) = inf{sin(P, S); S is a line in X } where sin(P, S) is a sine of the angle between P and S. Let Y be a linear subspace in R'~,we put 5(]I, X ) = sup{5(P, X); P is a line in Y } If Y = 0 we put 5 ( 0 , X ) = 0. Function 5 takes values in [0, 1]. In general 5 is not symmetric, however we have: (1.0) if d i m X = d i m Y then 5(X, Y) = 5(]I, X). (1.1) if Y C X then 5 ( Y , X ) = O. (1.2) if dirnZ < dirnY < dirnX then 5(Z, X ) 0. We say t h a t P is s - flat, if for each x, y • F we have 5(T=F, TyF) < ¢. If dimF = 0 we assume t h a t F is s-flat for every e > 0. 5 . P r o p o s i t i o n . Let A be locally finite family of subanalytic sets in R n. Then for given e > 0 there exists a subanalytic stratification T compatible with the family ~4, such that each stratum of ~r is s-flat.

Proof. We use condition (*) of remark 0 to prove the existence of such stratifications. For every k = 1, ...,n - 1 let us take a subanalytic finite partition of Gk,n into disjoint sets E/k, i = 1,..., rk such t h a t 5(X, Y) < ¢ for each X, Y • Eik. Actually we can take a stratification of Gk,n compatible with a finite covering by 5-balls of radius ¢.Let A be an analytic submanifold, subanalytic subset of R ~, d i m A = k. Let r : A ~ Gk,~ rk be corresponding Gauss mapping, then the set F = A \ Ui=l IntA(v--l(E~)) is closed and nowhere dense in A. Clearly every connected c o m p o n e n t of A \ F is s-flat. Hence by (*) of r e m a r k 0 the proposition follows. Remark 5.I. Actually we can require more for the stratifcation 2r. By a theorem of Stasica [St] (see also [KR]), for given M > 0, we can refine T in a such way that every s t r a t u m T of T , dirnT = k < n, is a g r a p h (in suitable coordinate s y s t e m in R n) of an analytic m a p p i n g ¢ : U ~ R n - k , where U is open in R k and [ dz¢ 1 0 (depending only on m) such that ] d~,~ ], F~, F=, F~, contenant des polynomes auxquels on souhaite imposer respectivement les conditions de signes > 0 , >/0, = 0 , :~ 0 , on dira que F = [F> ; F~ ; F= ; F~ ] est fortement incompatible dans K si on a une 4galit4 dans K[X] du type suivant : S + P + Z = 0 avec S c Yvf(F>U F#'2), P c Cp(F~U F>), Z c I(F=) Nous utiliserons la notation suivante pour une incompatibilit6 forte: ,~ [ S1 ) 0..... Si> 0, P1 )/0 ..... Pj,'/0, Z 1 = 0..... Zk= 0,N 1 • 0..... Nh:~ 0 ] ,], I1 est clair qu'une incompatibilit6 forte est une forme tr6s forte d'incompatibilit6. En particulier, elle implique l'impossibilit4 d'attribuer les signes indiqu4s aux polynomes souhait4s, dans n'importe quelle extension ordonn6e de K . Si on consid~re la cl6ture r6elle R de K , l'impossibilit6 ci-dessus est testable par l'algorithme de H6rmander, par exemple. Le th4or~me des z4ros r6els et ses variantes

Les diff6rentes variantes du th6or6me des z6ros dans le cas r6el sont cons6quence du th6or6me g6n6ral suivant : Th4or~me : Soit K un corps ordonn6 et R une extension r6elle close de K . Les trois fairs suivants, concernant un syst6me de csg portant sur des polynomes de K[X], sont 6quivalents : l'incompatibilit4 forte dans K l'impossibilit4 dans R l'impossibilit6 dans toutes les extensions ordonn4es de K Ce th4or6me des z4ros r4els remonte/~ 1974 ([Ste]). Des variantes plus faibles ont 6t6 4tablies par Krivine ([Kri]), Dubois ([Du]), Risler (IRis]), Efroymson ([Efr]). Toutes les preuves jusqu'~ ([Lom a]) utilisaient l'axiome d u choix. Degr6 d'une incompatibilit4 forte Si nous voulons pr6ciser les majorations de degr4 fournis par notre preuve du th6or6me des z6ros r6el, nous devons pr6ciser la terminologie. Nous manipulons des incompatibilit6s fortes 6crites sous forme paire, c.-~-d.: S + P + Z = 0

avec S c g , f(F>"2UF#'2), P c C ~ F ~ U F > ) ,

Zc/(F=)

(la consid6ration des formes paires d'implications fortes a pour unique utilit6 de faciliter un peu le calcul de majoration des degr4s). Q u a n d nous parlons de degr6, sauf pr4cision contraire, il s'agit du degr6 total maximum. Le degr~ d'une incompatibilitd forte est par convention au moins 4gal ~ 1, c'est le degr4 m a x i m u m des polynomes qui ~composenb~ l'incompatibilit6 forte. Par exemple, si nous avons une incompatibilit4 forte : ~, [ A > 0 , B > 0 , C > / 0 , D > , , 0 , E = 0 , F = 0 ] ~, explicit6e sous forme d'une identit6 alg6brique :

327

h k A2.B6 + C. ~ pi.Pi2 + A.B.D. ~ qj.Qj2 + E.U + F.V = 0 i=1 i=1 le degr6 de I'incompatibilit6 forte est : sup { d(A2.B6), d(C.Pi2) (i = 1,...,h), d(A.B.D.Q2) (j = 1,...,k), d(E.U), d(F.V) }. Le calcul de majoration Nous allons expliquer dans cet article comment peut ~tre men4 un calcul de majorations primitives r4cursives pour le th6or6me des z6ros r6els. Les d6tails des calculs sont dans [Lom d]. Les donn4es sont trois enfiers d, n, k qui majorent, dans un syst6me de csg incompatible H , respectivement les degr4s des polynomes, le hombre des variables et le nombre de csg. Le calcul doit aboutir ~ 3 fonctions primitives r6cursives explicites ~(d,n,k), ~(d,n,k) et ~(d,n,k) qui donnent des majorants pour, dans une incompatibilit6 forte ~, H ~, , respectivement le degr6 maximum, le nombre de termes dans la somme, et le nombre d'op4rations arithm6tiques dans K n6cessaires pour calculer les coefficients dans l'incompatibilit4 forte ~ partir des coefficients donn6s au d6part. En fait, chacun des th40r6mes ou propositions qui conduit ~ la preuve constructive du th4or6me des z4ros r6el peut ~tre accompagn4 d'une majoration primitive r6cursive du m~me type. Ces majorations s'enchainent les unes les autres, sans difficult4 majeure. Comme le calcul est tr6s fastidieux, nous nous en sommes tenus aux majorations de degr6s, laissant au lecteur courageux les deux autres majorations. On notera que l'usage de l'algorithme de H6rmander 'sans raccourci', ~ la base de notre m6thode, rend a priori les majorations obtenues sans int6r~t pratique. Constructions d'incompatibilit4s fortes D6finition 2 : Nous parlerons de construction d'une incompatibilit6 forte h partir d'autres incompatibilit6s fortes, lorsque nous avons un algorithme qui permet de construire la premi6re ~ partir des autres. I1 s'agit donc d'une implication logique, au sens constructif, liant des incompatibilit6s fortes. Notation 3 : Nous noterons cette implication logique (au sens constructif) par un signe de d6duction "constructif". La notation

($

$ et $ H2 $)

o.s $ H3 $

signifie donc qu'on a Tan algorithme de construction d'une incompatibilit4 forte de type H 3 ~ partir d'incompatibilit6s fortes de types H~ et H 2 Cela n'a d'int4r~t que lorsque les incompatibilit6s fortes d4sign6es en hypoth6se et en conclusion comportent des 616ments variables. Un exemple fondamental aidera a mieux comprendre.

328 Le raisonnement par s~parafion des cas (selon le signe d'un polynome) N o u s donnons ici un ~nonc~ d~taill~ des Dans le sens direct : soit H u n

syst~me de csg et une incompatibilit6 forte ,], ( H , Q a ' 0) ,~ de degr6 d , on peut construire l'incompatibilit6 forte ,], ( H I , H ) ,~ en raisonnant cas par cas. Dans te cas Q ~ 0 on utilise l'incompatibilit6 forte .J, ( H I , Q o 0 ) ,~ de degr6 d 1 et dans le cas Q G' 0 on utilise l'incompatibilit4 forte ~, H , Q o ' 0 ,], , on conclut en utilisant la proposition 4. R6ciproque : on a une incompatibilit6 forte sous forme paire de degr6 2.dQ : ~, ( Q o 0 , Q ~' 0),~, obtenue en lisant convenablement l'identit6 Q2+Q.(_Q) = 0, on applique alors la d6finition de l'imptication d y n a m i q u e en prenant pour H la seule condition Q 0 0. O v e r s i o n a l g 6 b r i q u e d y n a m i q u e d e la d i s j o n c t i o n D ~ f i n i t i o n et n o t a t i o n 9 :

Soient H 1 , H 2 . . . . . H k et K~, K 2 ..... K m des syst~mes de csg portant sur des polynomes de K[X] . Nous disons que le syst~me H~ implique dynamiquement la disjonction K I V K 2 V ... V K m lorsque, pour tout syst6me de csg H portant sur des polynomes de K[X,Y], on a la construction d'incompatibilit6 forte : {,~[ K~(X), H(X,Y) ]~, et ... et ,],[ KIn(X) , H(X,Y) ],~, } ~ons ,~[ HI(J0 , H(X,Y) ],~ NOUS noterons cette implication-disjonction dynamique p a r : " ( H i ( X ) ~ [K~(X) V K2(X) V ... V KIn(X)] ) ' . Lorsque le syst6me H 1 est vide, nous utilisons la notation " ( K I ( X ) V K2(X) V ... V Km(X))'. Enfin, la notation : " ( [ H I V H 2 V ... V Hk] ~ [ K I V K2V .... V Kin] )" signifie que chacune des implications-disjonctions dynamiques ° ( H i ( X ) ::~ [KI(X) V K2(X) V ... V Kin(X)] )° est v6rifi6e

(i = 1,...,k)

Remarque : Toute formule sans quantificateur de la thdorie du premier ordre des anneaux totalement ordonn6s dicrets/~ param6tres dans K est 6quivalente a une formule en forme normale disjonctive et donc ~ une formule d u type KI(X) V K2(X) V ... V Kin(X) oh les Ki(X) sont des syst~mes de csg portant sur des polynomes de K[X]. Les implications-disjonctions d y n a m i q u e s consituent une forme de raisonnement p u r e m e n t ~ddentit6 alg6brique~> concernant les formules sans quantifi-

332 cateur, oh la logique a 6t6 6vacu6e au profit d'algorithmes de constructions d'identit6s alg4briques. Fonction-degr4 d'une implication-disjonction dynamique

Une implication-disjonction dynamique " ( H I ( X ) ::~ [KI(X) V K2(X) V ... V KIn(X) ] )* signifie par d6finition un algorithme fournissant la construction : {~,[ KI(X), H(X,Y) ]~, et ... et ,~[ Km(X), H(X,Y) ],~ } ~ons ,~,[ HI(X), H(X,Y) ]~, Chaque fois que nous 6tablissons une implication-disjonction dynamique particuli6re, nous devons 6tablir des 'majorations primitives r6cursives de degr6' pour cette construction d'incompatibilit6s fortes : le degr6 de l'incompatibilit6 forte construite est major6 par une fonction A(d t .... d m) off d i est le degr6 de l'incompatibilit4 forte initiale n°i. Nous disons qu'il s'agit d'une fonction-degr6 acceptable pour l'implicationdisjonction dynamique consid6r6e. Exemples : La proposition 4 peut ~tre relue comme affirmant des disjonctions ou implications-disjonctions dynamiques : Proposition 4 bis : On a les implications-disjonctions dynamiques suivantes :

" ( Q e a 0 ==~[Q>O v Q < O ] ) * "( Q,, ~ O = ~ [ Q > O v Q=O]) ° "(Q~0 V Q=0) ° "(Q>O v Q~O) ° "(Q=O v Q)O v Q0,AB 0 , A+B > 0 ] ~ fonction-degr6 : (d;5,5'), 8 = sup(d(A),d(B)). f) L'implication A 2k _ 0 , B > 0 ] accepte p o u r

* d.3' + 25 off 5' = d ( A B ) / i n f ( d ( A ) , d ( B ) ) , A= 0

accepte p o u r fonction-degr6 :

" 2k.d

D e m ~ m e l'implication [ A > 0 , A < 0 ] ~ degr6 : d :

A = 0 accepte p o u r fonction-

," 2d.

g) L'implication (d;8').

accepte p o u r fonction-degr6 :

ot~ ~ = d ( A ) , 3' = d(A.B) / d(B).

c) L'implication [ A > 0 , A.B > 0 ] ~ (d;8):

accepte p o u r fonction-degr4 :

P(X,U) • P(X,V) ~ U :~ V accepte p o u r fonction-degr6 : ~ d.3' o£l 5' = d(P(X,U) - P(X,V)) / d(U - V)

Par e x e m p l e p o u r le a) : on multiplie, t e r m e a terme, l'implication forte p a r A 2, en p r e n a n t soin d e r e m p l a c e r les B.A 2 p a r (BA).A. Le p r i n c i p e de s u b s t i t u t i o n P r o p o s i t i o n 15 : O n consid6re des variables X1,X2..... Xn, UI,U2,...,U h, Z1,Z2,...,Zk, et des p o l y n o m e s P1,P2 ..... Pn d e K[Z]. N o t o n s P(Z) p o u r PI(Z) .... , Pn(Z) . Si on a

"(~,-~I(X,U) :~ ~--~2(X,U))"

(a)

alors o n a aussi ° ( H I ( P ( Z ) , U ) ~ H2(P(Z),U) ) ° (b) Si A 1 est u n e fonction-degr6 acceptable p o u r (a), u n e fonction-degr6 acceptable p o u r (b) est donn6e p a r :

d ~

~ g + Al(g + 8.d) of1 8 = sup(deg(Pi))

et X et g sont explicit6s dans la p r e u v e .

preuve> N o t o n s X = P(Z) p o u r : X 1 = P I ( Z ) ..... X n = Pn(Z). Soient Y1,Y2 ..... Yn d e s nouvelles variables. P o u r t o u t R f i g u r a n t d a n s H2(X,U) consid6rons P6galit6 qui p e u t ~tre o b t e n u e p a r d i v i s i o n s s u c c e s s i s v e s ties degr6s d e s R i sont tous inf6rieurs o u 4 g a u x au degr6 d e R) : R(X,U) = R(Y,U) + (X 1 - Y1) RI(X,Y,U) + ... + (Xn - Yn) Rn(X,Y,U) (1) En s u b s t i t u a n t Pi(Z) /~ Yi on obtient u n e 6galit6 : R(P(Z),U) = R(X,U) + (X 1 - P I ( Z ) ) R1(X, PfZ),U) + ... + (X n - P n ( Z ) ) Rn(X,P(Z),U) (2) Ces 6galit6s f o u r n i s s e n t u n e implication s i m p l e :

337

• ( [ H2(X,U), X = P(Z) ] ::# H2(P(Z),U) )" (3) d o n t le d e g r 4 absolu est major6 par ~t = s u p ( 0 , sup{ deg((X i - Pi(Z)) Ri(X,P(Z),U)) - deg(R(P(Z),U)) ; R figure d a n s H 2 , R(P(Z),U) ¢ cte } D e la me, m e manii~re, p o u r les R figurant d a n s H 1 o n a des 6galit6s : R(X,U) = R(P(Z),U) - (X1 - PI(Z)) RI(X,P(Z),U) - ... - (Xn - Pn(Z)) Rn(X,P(Z),U) (4) qui f o u r n i s s e n t u n e i m p l i c a t i o n s i m p l e • ( [ HI(P(Z),LO, X = P(Z) ] ::#

[ H I ( X , U ) , X = P(Z) ] ) •

(5)

d o n t le degr4 absolu est major6 par k = s u p ( 0 , sup{ deg((X i - Pi(Z)) Ri(X,P(Z),U)) - deg(R(X,U)) ; R figure d a n s H 1 } Par ailleurs l'implication d y n a m i q u e : • ( [ H I ( X , U ) , X = P(Z) ] ~

[ H2(X,U), X = P(Z) ] )•

accepte la m 6 m e f o n c f i o n - d e g r 6 q u e l'implication d y n a m i q u e En c o m p o s a n t (5), (6) et (3) o n obtient : • ( [ H I ( P ( Z ) , U ) , X = P(Z) ] ::~ Enfin, c o m m e les v a r i a b l e s l'implication dynamique :

X

H2(P(Z),U) ) •

ne f i g u r e n t pas d a n s

• ( H I ( P ( Z ) , U ) :::) [ H I ( P ( Z ) , U ) , X = P(Z) ]) )•

(6)

(a). (7) HI(P(Z),U),

on a

(8)

o b t e n u e en r e m p l a q a n t les X i par les Pi d a n s l'incompatibilit6 forte initiale. Elle accepte p o u r fonction-degr6 : d t * d.5 avec ~ = sup(deg(Pi)). En r6sum6, si a 1 est u n e fonction-degr6 acceptable p o u r (a), u n e f o n c t i o n - d e g r 6 acceptable p o u r ( b ) e s t d o n c : d. * ~,+Al(~t+&d) Q F o r m u l e s de T a y l o r m i x t e s

O n consid6re d e u x variables U et V e t o n p o s e A := U - V . O n consid6re u n p o l y n o m e P a coefficients d a n s u n corps o r d o n n 6 K o u plus gdndralement dans Un

commutatif A qui est une Q-alg~bre.

flnnffflu

Si degfP) x( 4 , o n a l e s 8 formules d e Taylor mixtes suivantes: P(U) P(V) = A.P'(V) + (1/2).A2.p'(v) + (1/6).A3.p(3)(V) + (1/24).A4.p (4) P(U) - P(V) = A.P'(V) + (1/2).A2.p"(V) + (1/6).A3.p(3)(U) - (1/8).A4.p (4) P(U) - P(V) = A.P'(V) + (1/2).A2.p"(u) - (1/3).A3.p(3)(V) - (5/24).A4.p (4) P(U) - P(V) = A.P'(V) + (1/2).A2.p"(U) - (I/3).A3.pC3)CU) + (1/8).A4.P (4) -

P(U) - P(V) = A.P'(LO - (1/2).A2.p'(v) - (1/3).A3.p¢3)(V) - (1/8).A4.p (4) PfU) - P(V) = A.P'CU) - (1/2).A2.p'(v) - (1/3).A3.p(3)0d) + (5/24).A4.P (4) P(U) - P(V) = A.P'(U) - (1/2).A2.p'0d) + (1/6).A3.p(3)(V) + (1/8).A4.P (4) P(U) - P(V) = A.P'CU) - (1/2).A2.p"(U) + (1/6).A3.p(3)(U) - (1/24).A4.p (4) C o m m e toutes les c o m b i n a i s o n s de signes possibles se pr4sentent, o n obtient : supposons que u et v attribuent la m ~ m e suite de signes (au sens large) p o u r les d6riv6es successives d ' u n p o l y n o m e P n o n c o n s t a n t de d e g r 6 < 4 , n o t o n s e 1 = 1 o u -1 selon que P'(u) et P'(v) sont tous d e u x > 0 o u tous d e u x < 0 , alors le fait q u e P(u) - P(v) a m S m e signe q u e e p ( u - v) est r e n d u 6vidnet

338

par l'une des formules ci-dessus, ce qui d o n n e l'implication sous f o r m e d ' u n e implication simple (u et v peuvent ~tre des 614ments de K mais aussi des variables, ou des polynomes) si u et v n'attribuent pas la m~me suite de signes p o u r un p o l y n o m e P de degr6 < 4 et ses d6riv6es successives, alors on a une identit6 alg6brique qui donne le signe de u - v a partir des signes des P(i)(u) et des ~(i) r (V) : la formule de Taylor mixte a utiliser est avec p(i) ( i = 0, 1, 2, ou 3) off i e s t le plus grand indice p o u r lequel tes deux signes ne sont pas identiques Plus g4n6ralement on a : Proposition 16 : (formules de Taylor mixte)

Pour chaque degr6 s , il y a 2 s- 1 formules de Taylor mixtes et toutes les combinaisons de signes possibles apparaissent. Formules de T a y l o r g6n6ralis6es (le l e m m e de T h o m sous f o r m e d'identit4s alg4briques) Le l e m m e de T h o m affirme (entre autres) que l'ensemble des points o~ un p o l y n o m e et ses d6riv6es successives ont chacun un signe fix6, est un intervalle. Une p r e u v e facile, p a r r6currence sur le degr6 d u p o l y n o m e , est bas6e sur le th6or~me des accroissements finis. Nous pouvons, grace aux formules de Taylor mixtes, traduire ce fait g6om6trique sous forme d'identit4s atg6briques, que nous appellerons des f o r m u l e s de T a y l o r g6n6raIis6es. Plut6t que de risquer un 6nonc6, nous d o n n o n s un exemple. Un exemple : Consid4rons le polynome g6n6rique de degr6 4 P(X) = co X~ + c I X3 + c2 X3 + c3 X2 + c4 X4 + cs Consid6rons le syst6me de conditions de signe portant sur le p o l y n o m e P e t ses d6riv4es successives par rapport a la variable X : H(U) : P(U) > 0, P'(U) < 0, P(2)(U) < 0, P(3)(U) < 0, P(4)(U) > 0. Consid4rons 6galement le syst6me de conditions de signe g6n6ralis6es obtenues en relachant toutes les in6galit6s, sauf la derni6re : H'(U) : P(LO > 0, P'(U) < 0, P(2)(U) ~ 0, P(3)(U) _ 0. Le lemme de T h o m affirme (entre autres) : [H'(U),H'(V), U. On note H(C,T) ou H(T) le syst6me de csg : P'(C,T) c~ 0 ..... P(i)(C,T) c~i 0, ..., P(s)(C,T) crs 0 (les d6riv6es sont par rapport h T ). Soit H'(T) le syst6me de csg obtenu ~t partir de H(T) en relachant toutes les conditions de signe sauf celle relative h p(s). Soit HI(T) le syst~me de csg : P(s)(C,T) > 0, P(i)(C,T) )/0, i = 1, ..., s - 1 . Soient enfin trois variables U , V , Z distincte des C i . On a alors les implications dynamiques suivantes : "( [ H'Od), H'(V), U o 1 V ] ~ POd) > P(V) )" (a) "( [ HI(U), V > U ] ~ P(V) ) POd) )" (b)

"( [ H'(U), H'(V), U < Z < V ] ~

H(Z) )"

(c)

Ce sont des implications simples qui ne cofitent rien. preuve> L'implication d y n a m i q u e (a) r6sulte de formules de Taylor mixtes. L'implication d y n a m i q u e (b) r6sulte de la formule de Taylor ordinaire au point U. Les formules de Taylor g4n6ralis6es 6tablies pour l'implication dynamique (c) r6sultent des formules de Taylor mixtes. On constate qu'il s'agit d'implicafions simples qui ne cofitent rien (ceci parce que U, V, Z sont des variables et non des polynomes). Q

340

4)

Existences p o t e n t i e l l e s

N o t a t i o n s et d6finitions Elles sont tout h fait analogues ~ celles donn6es pour les implications dynamiques.

D6finition et notation 18 : Soient H I un syst~me de csg portant star des polynomes de K[X], H 2 un syst~me de csg portant sur des polynomes de K[X,TvT2,._,Tm] = K[X,T]. Nous dirons que les hypotheses H I autorisent l'existence des Ti vdrifiant H 2 lorsque, pour tout syst~me de csg H portant sur des polynomes de K[X,Y], les variables Yi et Tj 6tant deux ~ deux disfinctes, on a la construction d'implication forte :

$ [ H2(X,T) , H(X,~9 15 ~ons$ [ Hi(X) , H(X,Y) ] $ . d'existence potentielle des Ti vdrifiant les hypotheses H~

Nous parlerons 6galement

H 2 sous

Nous noterons cette existence potentielle par : "(H1CX) ::~ B T H2(X,T) ) ' . Lorsque le syst~me H 1 est vide, nous utilisons la notation * ( 3 T H2(X,T) )*. La notion de fonction-degr4 acceptable pour une existence potentielle peut ~tre elle aussi directement recopi4e du cas des implications dynamiques.

Remarques : 1) La notion d'existence potentielle est une notion d'existence faible. L'existence potentielle signifie qu'il n'est pas grave de faire comme si les T i existaient vraiment, parce que cela n'introduit pas de contradiction: on peut paraphraser la d6finition en disant : pour construire l'incompatibilit4 forte $ [ HI(X), H(X,Y) ] $ il suffit d'avoir construit $ [ H2(X,T) , H(X,Y) ]$ 2) On pourrait 4tendre la d6finition de l'existence potentieUe en remplaqant le syst&me de csg H2(X,T) par une disjonction de syst6mes de csg, comme on a fait avec la notion d'implicafion-disjonction dynamique.

Quelques r6gles de manipulation des 6nonc6s d'existence potentielle Transitivit4 La transitivit6 des existences potentielles est imm6cliate. Voici l'6nonc6 pr6cis6 en termes de fonctions-degr6 acceptables.

Proposition 19 : (transitivit6 dans les existences potentielles) On consid~re des variables X1,X2,...,Xn,TvT2,...,T m, U1,U2,...,Uk et des syst~mes de csg Hi(X), H2(X,T) et H3(X,T,U).

341

Les existences potentielles ° (HI(X) ~ 3 T H2(X,T) ) ° et ° (H2(X.T) ~ B U H3(X,T,U) ) ° impliquent l'existence potentielle : °(HI(X) ~ 3 T,U Ha(X,T,U) )" Supposons que la premiere existence potenfielle admette comme foncfiondegr6 acceptable Al(d;p) off d est le degr~ de l'incompafibilit~ forte ~, [ H2(X,T), H(X,Y) ] ,]. et p repr~sente certains param~tres d6pendant de HI(X) et H2(X,T), supposons de mSme une fonction-degr6 acceptable A2(d;q) pour la deuxi~me existence potentielle, alors une fonction-degr6 pour l'existence potentielle construite est donn6e par : A(d;p,q) = A1(A2(d;q);p) Preuves cas par cas Voici maintenant u n 6nonc6 corresponclant aux preuves cas par cas d'une existence potentielle, cons6quence imm6diate de la proposition 4 . Proposition 20 : (raisonnement cas par cas) Soit Q un polynome de K[X]. a) Pour d6montrer une existence potentielle "( [ HI(X), Q ~: 0 ) ~ B T H2(X,T) )" il suffit de d6montrer chacune des existences potenfielles " ( [ H I ( X ) , Q > 0 ] ~ B T H2(X,T))* e t ' ( [ H I ( X ) , Q ( 0 ] ~ B T H2(X,T))" Si A i (i = 1,2) sont les deux fonctions-degr6 des existences potentielles suppos6es, une fonction-degr6 pour l'existence potentielle d6duite est donn6e par : A1 + A 2 a'), b), c), d), e) : 6nonc6s analogues d6calqu6s de la proposition 4 Le principe de substitution Le principe de substitution pour les existences potentielles se d6montre comme pour les implications clynamiques. L'existence implique l'existence potentielle Un autre principe utile est le fait que l'existence implique l'existence potentielle. II s'obtient facilement : on remplace les variables T i ~existentielles~ par les polynomes concrets Pi qui r6alisent l'existence. On reconnait l~ une analogie formelle avec la r~gle d'introduction du quantificateur existentiel en calcul naturel par exemple (cf. [Pra]). Proposition 21 : ( l'existence implique l'existence potentielle) Soient PI,P2,...,Pm E K[X] et notons P(X) pour PI(X) ..... Pro(X). On a l'existence potentielle : "(H2(X,P(X)) ~ 3 T H2(X,T) )" . Si 8 majore les degr6s des Pi • l'existence potenfielle accepte pour fonctiondegr6 : (d;8) ~ ~ d.sup(1,8)

342

CoroUaire : (m~mes hypoth6ses)

Si *(HI(X) ::~ H2(X,P(X)) )" alors °(HI(X) ~

3 T H2(X,T)) °

Si A1 est une fonction-degr6 acceptable pour l'implication forte de l'hypoth6se, une fonction-degr6 acceptable pour la conclusion est donn4e par : (d;8) ~ , Al(d.sup(1,8)) off 8 majore les degr6s des Pi •

E.xistences p o t e n t i e l l e s f o n d a m e n t a l e s On sait d6montrer les existences potentielles correspondant aux deux axiomes existenfiels de la th4orie des corps r4els dos. Th4or~me 22 : (autorisation de rajouter l'inverse d'un non nul) On a l'existence potentielle de l'inverse d'un non nul. Ce qui s'6crit: " ( U ~ : 0 ~ 3 T I = U . T )* Soit ~ le degr6 de U, une fonction-degr6 acceptable pour l'existence potentielle est (d;8) : - d + d.3 + R e m a r q u e : La preuve de cette existence potentieUe recopie ce qu'on fait, dans la preuve d u th6or6me des z4ros de Hilbert, pour passer d u th6or~me des z6ros faible au th6or6me des z4ros g6n6ral (c'est le ~~, par exemple dans l'expos6 classique de van der Waerden). La notion d'existence potentielle de l'inverse d ' u n non nul est donc en filigrane dans les classiques. Th4or~me 23 : (autorisation de rajouter une racine sur un intervalle oh un polynome change de signe) Soit P(C,X) un polynome de degr6 s en X et de degr6 global 5. On a l'existence potentielle d'une racine sur un intervalle off ce polynome change de signe. Ce qui s'6crit, en notant P(X) pour P(C,X) : " ( [ P ( X ) . P ( Y ) ( 0 , X d z r n z ~

(3)

+ trdeg K I F

_.

~(g*)

>_e,m~

2v(-F6¥ w

--

2,(K*)+w

where F is the residue field of VIE. Also, 2v(g*)+v(T*)

(4)

,. 2v(K*) + W a,m: 2 ~ - ~ ~ - ~ )

~ ~--

w 2W+v(T*) SO we have

W W ,. v(F*) = dim2 2W + v(T*) < ~ i ' ~ -< a'm~ 27--~)

where the last inequality follows from Brbcker's Index Formula [7, 3.2]. Since Q is v-compatible, P is compatible with VIE so the place ,~p : F --+ ~ [.J oo factors through the place p : F -~ F U c~ associated with V[F. Thus, we have (5)

v( F * )

6"

zrn2 ~ )

,.

vp(F * )

< azm22vp(F,) =

re(P).

Combining (2)-(5), we have (6)

,.

v(g*)

azm2~

< trdeg K]F + re(P).

This leaves the case where IV I = 2 and we have equality at (6). To prove (1) in this case, we must show e(P) = 1.

350 N

w

Since we have equality at (6), we must have equality at (3) so F C K is algebraic. Fix a finite extension F1 of F such that F1 C_ K and the two orderings in V C Sper K remain distinct when restricted to F I . W'e also have equality at (5) so re(P) = a*rn2 " v(F*) Thus the place F ~ IR U eo 2v-ggp~. induced by factoring Ap through p has 2-divisible value group. By the definition of the place hop : F ~ F ( P ) U oe, p factors through hop inducing a place F ( P ) -+ F U ~ . Let F1 be a finite unramified extension of F(P) having F1 as residue field. Then F1 has at least two distinct orderings so F ( P ) is not hereditarily Euclidean and therefore, e(P) = 1. This completes the proof. We return to the proof of (1.7). By [13, 5.1],

s(X) = sup{s(X(p)) I p C_ A is a real prime with X(p) ~ (~}. Let p C_ A be a real prime with X(p) ¢ 0. Since X = SperTA, X(p) = SperTF(p) so we have

(*)

s(X(p)) < trdeg F ( p ) l f + ~ = dim A/p + m

by (1.9). Since dim A/p 1, and spaces of orderings

Zi @ Z~ C_ SperpF(pl),

i >_ O,

such t h a t Zi, Z~ are fans of order 2 i + ~ and Zi is the pull-back of Z~_ 1 along ,ki if i > 1. (We assume ~ < oo.) T h e n we take i

r,= U(zj e z;). j=0

i

s,=

U({P,} e z.b j=o

where Pj is a fixed element of Zj. To obtain Z~, i >_ 1, we want to choose additional primes qi-1 D Pi, i = 1 , . . . , d , with qi-1 ~ P0 such t h a t the natural m a p pi : A/pi --* A/qi-~ lifts to a discrete rank 1 place pi : F ( p i ) ~ F ( q i - 1 ) U oo and s(SperpF(qi-1)) = i - 1 + ~ . T h e n Z~ can be obtained by pulling back a fan of order 2 i - 1 + ~ in SperpF(qi-1) along pi. By our construction, the places pi, /~i a r e independent so Zi, Z~ lie in distinct Brgcker classes and therefore Zi U Z~ = Zi @ Z~ holds automatically if i >_ 1 (see [6], [14, 3.2].) We use the notation and theory in [13] to show t h a t Y/ is s a t u r a t e d and Si is basic closed in Y/. Let 2 i = {a E A I a > 0 on Yi}. T h e n a 2 E Ei for a E A \ P0. Thus,

352 if Q E cl(Yi), Q ~ ~, then supp(Q) D_ q j-1 for some j _< i so supp(Q) ¢ Po, and consequently Q ~ U(E2). This proves that Yi is closed in U(E 2) = Sper E~-IA. Since Yi(Pj) = Zj • Z} is saturated in SperpF(pj) for all j < i, this implies Yi is saturated (see [13, 2.3].) Also, it is clear that Si is closed in Yi and S(pj) = {Pj} @ Z} is basic in Y,(pj) = Zj ® Z} for all j < i so, by [13, 3.21, s~ is basic closed in Yi. The argument in [16] shows that at least ~ = 0 ( J + m) = i(i + 1)/2 + (i + 1)N inequalities (>) are required to describe Si in Yi. Namely, i + ~ of these inequalities, say f~ >_ 0 , . . . , fi+m >_ 0, are required just to describe {Pi} = Si A Zi in Zi = Yi N Zi (since Zi is a fan of order 2i+m.) Moreover, each fk is strictly negative at some Q c Zi and, of course, fk _> 0 at Q' where Q' D Q is the specialization of Q in Z~_ 1 (since Z~_ 1 c_ Si.) Thus fk = 0 at Q', that is, fk c Pi-1. This is true for k = 1 , . . . , i + ~ . Thus, none of the inequalities fl >_0,..., fi+-~ >_0 can contribute to the description of Si-1 = Si 71Yi-1 in Yi-1 = Yi C?Yi-1. Therefore, by induction on i, ~ j =i--1 0 ( 3 - + ~ ) additional inequalities i

are required for the description of Si-1 in Yi-1 so, altogether, ~ j = 0 ( ) + ~ ) inequalities are required to describe Si in Y/. Thus g(Yi) _> i(i + 1)/2 + (i + 1)~. In particular,

"g(SperpA) > 2(Yd) >_d(d + 1)/2 + (d + 1)~. To be able to apply this in the case V = R n, we assume

(,)

There exists a finite extension L of F with Zo • Z~ C_SperpL where Z0, Z~ are fans of order 2 m.

Assuming this, we can realize the above set-up in A = / ~ [ X I , . . . , Xd] as follows: take Pa = (0), Pa-1 = ( X a ) , . . . , P l = ( X 2 , . . . , X a ) , P0 = (f, X2,...,Xa) where f E F[X1] is such that F[X1]/(f) ~ L. The choice of qi-1, i = 1,...,d, is fairly arbitrary, for example, we could take qa-1 = (Xa + 1 ) , . . . , ql = (-X'2 + 1, X 3 , . . . , Xd) and qo = (I(X~ + 1 ) , X 2 , . . . , X d ) . The point is, with this choice of qi, qi-x ~ P0, r(qi-1) F(X1,...,XI_I) for i _> 2, and F(q0) ~ L so s(SperpF(qi_l)) = i - 1 + ~ , i = 1 , . . . , d . This proves: Theorem

2.3. If (F, P) satisfies (*) then 5F(R") = n(n + 1)/2 + (n + 1)~.

If (*) does not hold, we are still able to get a somewhat weaker result by replacing L by the field K constructed in (2.1) and taking Z0 = ~ (so we only require that SperpF(po) contains a fan Z~ of order 2m) and taking i

Yi = U ( Z j ® Z}), j=l

i

Si = U ( { P j }

® z;).

j=l

i Then, starting our induction at i = 1 instead of i = 0 yields-g(Y~) _> ~j=1(2 +-~) = i(i + 1)/2 + i ~ . Thus we have:

Theorem

2.4. For any F, n(n + 1)/2 + nm < -gF(Rn) < n(n + 1)/2 + (n + 1)~.

353

Remarks 2.5. (1) If ~ -- 0, (F, P) is hereditarily Euclidean so cannot satisfy (*) but, nonetheless, by (2.4) the conclusion of (2.3) is valid. (2) We have been assuming ~ < co but this is unnecessary. If ~ -- co, then 7F(R n) SF(R ~) = n + ~ = co, so the conclusion of (2.3) is true in this case as well.

>__

(3) For 0 < ~ < co, (F, P ) may or may not satisfy (*). Fixing an integer k, 0 < k < co, we have the following examples: (i) Take F to be the iterated formal power series field ~ ( ( t l ) ) . . . ((tk)) and denote by A : F --* R U co the naturally associated discrete rank k place. Any ordering P E Sper F is compatible with ), and ~ ( P ) = k. But (F, P ) does not satisfy (*) since, for any finite extension L of F , L has at most 2 k orderings.

(ii) Take F to be the rational function field R ( t l , . . . , tk) and take P E Sper F to be the restriction to r of any ordering on R ( ( t l ) ) . . . ((tk)). Again, ~ ( P ) ---- k, but now, according to [15, §4], (F, P) does satiny (*). Of course, having (*) fail does not necessarily imply that the conclusion of (2.3) fails. We know of no example where the conclusion of (2.3) fails. REFERENCES 1. C. Andradas, L. BrCcker, J.M. Ruiz, Minimal generation of basic open semi-analytic sets, Invent. Math. 92 (1988), 409-430. 2. E. Beeker, On the real spectrum of a ring and its application to semi-algebraic geometry, Bull. Amer. Math. Soc. 15 (1986), 19-61. 3. J. Bochnak, M. Coste, M.-F. Roy, Gdomdtrie Algdbrique RdeUe, Ergeb. der Math. 3 12, SpringerVerlag, Berlin Heidelberg New York, 1987. 4. L. BrScker, On the stability indez o] Noefherian rings, preprint. 5. _ _ , On basic semi-algebraic sets, Expositiones Math. (to appear). 6.--, Uber die Anzahl der Anordnungen eines kommutativen KSrpers, Arch. Math. 29 (1977), 458-464. 7. _ _ , Zur Theorie der quadratischen Formen iiber formal reelen KSrpern, Math. Ann. 210 (1974), 233-256. 8. M. Knebusch, C. Scheiderer, Einfiihrung in die reele Algebra, Fiedr. Vieweg & Sohn, Braunschweig/Wiesbaden, 1989. 9. T.Y. Lam, A n introduction to real algebra, Rocky Mountain J. Math. 14 (1984), 767-814. 10. _ _ , Orderings, valuations, and quadratic forms, CBMS Regional Conf. Ser. in Math. no. 52, Amer. Math. Soc., Providence, R.I., 1983. 11. _ _ , The theory of ordered fields, Ring Theory and Algebra III (ed. B. McDonald), Lecture Notes in Pure and Applied Math., col. 55, Dekker, New York, 1980, pp. 1-152. 12. L. MahC, Une dgmonstration Jlementaire du thdor~me de BrJcker-Scheiderer, preprint. 13. M. Marshall, Minimal generation of basic sets in the real spectrum of a commutative ring, preprint. 14. J. Merzel, Quadratic forms over fields with finitely many orderings, Ordered Fields and Real Algebraic Geometry (eds. D. Dubois and T. RCcio), Contemporary Math., col. 8, Amer. Math. Soc., Providence, R.I., 1982, pp. 185-229. 15. C. Scheiderer, Spaces of orderings of fields under finite extensions, Manuscripta Math. 72 (1991), 27-47. 16. _ _ , Stability index of real varieties, Invent. Math. 97 (1989), 467-483. DEPARTMENT OF MATHEMATICS AND STATISTICS, UNIVERSITY OF SASKATCHEWAN~ SASKATOON, CANADA STN 0W0

CONFIGURATIONS

O F AT M O S T 6 L I N E S O F R P a

V. F. MAZUROVSKII Ivanovo Civil Engineering Institute

An unordered (n; k)-configuration of degree m is defined to be an unordered collection of rrz linear k-dimensional subspaces of R P n. We associate with each configuration its upper and lower ranks, i.e. the dimensions of the projective hull and intersection respectively of all the subspaces of the configuration. The combinatorial characteristic of a configuration is, by definition, the list of upper and lower ranks of all its subconfigurations. Two configurations are said to be rigidly isotopic if they can be joined by isotopy which consists of configurations with the same combinatorial characteristics. It is obvious that the property of being rigidly isotopic is equivalence relation. The equivalence class of a configuration by this relation is called its rigid isotopy type. The space SPC~k of unordered (n; k)- configurations of degree rn is naturally isomorphic to m-th symmetric power of Grassmanian manifold G,,+l,k+~. A configuration is said to be non-singular if all its subspaces are in general position. The set GSPC~k of non-singular configurations is an open subset of manifold SPCnm,~,in Zanski topology. The set of all non-singular configurations of the same rigid isotopy type forms a connected component of GSPC~,,k (in strong topology). These connected components are called cameras of manifold SPC~k. In the paper [6] O. Ya. Viro enumerated the cameras of the spaces SPC~,,1 for m _< 5 and showed that non-singular (3; 1)-configurations of degree ___5 are determined up to rigid isotopy by the linking coefficients of the lines of a configuration. In [4] the author enumerated the cameras of SPC~,1 and proved that non-singular configurations of 6 lines of R P a are not determined up to rigid isotopy by the linking coefficients. The main purpose of the present paper is to describe the mutual position of the cameras in SPC~,1 for ra < 6. In particular we give a detailed proof of classification of non-singular (3; 1)-configurations of degree 6 up to rigid isotopy (in [4] this proof was only outlined). The author is grateful to O. Ya. Viro for posing the problem and fruitful discussions. § 1. BASIC CONSTRUCTIONS 1.1. L i n k i n g coefficients. In the next two sections we describe the constructions of O. Ya. Viro (see [61, [7]). Two disjoint oriented lines L~, L~ in the oriented space R P a have linking coefficient Ik(L~, L~) equal to +1 or - 1 (here the doubled linking coefficient of cycles L~ and L~ in the oriented manifold NP a is considered). Let L = {L1,L2,La} be an unordered non-singular configuration of three lines of the oriented space R P a, and let L~, L~, L~ be the same lines equiped with some orientations. The product lk(L~,L~) x

355

Ik(L~,L~)lk(L~,L~) denoted by Ik(L1,L2,La) does not depend on the choise of the orientations of Li,i = 1,2, 3, is preserved under isotopies of L, and changes under reversal of the orientation of R P a. Unordered non-singular (3; 1)-configurations L = ! {L1, L 2 , . . . , Lm} and L' = {L1, L 2I . . . L~} are said to be homology equivalent if there exists a bijection q0 : L ---+ L' such that for a fixed orientation of R P a Ik(Li,Lj,Lk) = Ik(~(Li), v(Lj), v(Lk)) with any i,j,k = 1 , 2 , . . . , m , i C j , i # k, j # k. 1.2. C o n s t r u c t i o n o f t h e j o i n o f t w o c o n f i g u r a t i o n s . Let A = { A 1 , . . . , Am} be an unordered configuration of k-dimensional subspaces of R P ~, and let B = { B 1 , . . . , Bm} be an unordered configuration of/-dimensional subspaces of N P ~. We suppose that R P ~ and N P s are imbedded into I~P n+8+1 as disjoint linear subspaces. If n and s are odd we suppose, in addition, that N P n, NP~, and NP "+~+1 are oriented, and linking coefficient of the images o f R P n and R P ~ in R P n+s+l equals +1. Let f : { 1 , . . . ,m} ---* {1,... ,m} be some bijection, and Ci be the projective hull of the images of Ai and Bf(i) in ~P'~+~+~,i = 1 , . . . , m . It is clear that C = {C1,---,Cm} is a ( n + s + l ; k + l + l ) configuration of degree m. The configuration C is called the join of A and B. A configuration is called an isotopy join if it is rigidly isotopic to the join of some two configurations. 1.2.1. L e m r n a . The mirror image of an isototy join is also an isotopy join. 1.3. D e g e n e r a t i o n a n d p e r t u r b a t i o n . Let s : [0,1] --* SPCnm,k be a path with the beginning at a point A such that the restriction s][0,1) is a rigid isotopy of A. If configurations A and A' = s(1) have distinct combinatorial characteristics, then s is called a degeneration of configuration A, and path s -1 is called a perturbation of A'. Two subspaces Ai and A 1 of a configuration A = { A 1 , . . . , Am} are said to be contiguous if either they coincide, or there exists a degeneration such that (1) its restrictions on the configurations { A 1 , . . . , Ai-1, Ai+l,..., Am} and { A 1 , . . . , A j-l, A i + a , . . . , Am } which are obtained by removing the elements Ai, Aj from A respectively are rigid isotopies, and (2) the subspaces corresponding to the subspaces Ai and Aj coincide in the result of this degeneration. 1.4. A d j a c e n c y g r a p h . A configuration is said to be 1-singular if all configurations rigidly isotopic to it form a codimension 1 subset in the configuration space. The set of all 1-singular configurations of the same rigid isotopy type is called a wall. Two 1singular configurations are said to be p-equivalent if they belong to walls which separate the same cameras. The set of all 1-singular comCigurations of SPC~, k will be denoted by GI SPCr,n,k. The mutual position of the cameras in the configuration space can be described by means of the adjacency graph (see [2]), whose vertices and edges are in one-to-one correspondence with the cameras and walls respectively, and two vertices representing some cameras are connected by an edge if and only if these cameras are adjacent to the wall corresponding to this edge. It may happen that the beginning of an edge coincides with its end, as in the following cases: a) if the configuration space has a boundary and the wall is contained in it; b) if the wall is a one-sided subset of the configuration space; c) if the wall is a two-sided subset, but has the same camera adjacent at each side.

356 Each of these cases corresponds to a loop in the adjacency graph. In cases b) and c) the wall is called inner one. 1.5. A f f i n e (3; 1)-configurations. By an unordered afffine (3; 1)-configuration of degree m we mean an unordered collection of m lines of R 3. The canonical imbedding R3 ~ ~ p 3 induces a map of the set of a/fine (3; 1)-configurations of degree m into SPC~, 1. This map is called the projective completion, and the image of an atone (3; 1)-configuration K under this map is called the projective completion of K . A rigid isotopy of an affine (3; 1)-configuration is an isotopy of this configuration which induces a rigid isotopy of its projective completion. Let Oxyz be the canonical Cartesian coordinate system in R 3. The planes of R 3 defined by the equations z = const will be called horizontal planes. The common line of the projective completions of horizontal planes is called the horizontal line of infinity. Consider the affine (3; 1)-configuration S which consists of the following lines f y = 0 [ z 1' {x----0 z=l'

{ y=x

z=-l'

(y=--x

Thepairsoflines {y=0 z=l'

z=-l"

{x=O z=l

and { y = x

z=-l'

y = - x will be called the frames of configuration S. Let S be the projective comple-

x = --1 tion of S. Consider a configuration obtained by adjoining to S several pairwise disjoint lines which have no common points with :~. Such configuration is said to be framed. The subconfiguration S of a framed configuration will be called the skeleton, the other lines will be called free lines. We call the sliding translation the a ~ n e transformation given by the formulas x ~ = x + az, y' = y + bz, z t -- z, where a and b are some numbers. In what follows we suppose that X3 is canonically imbedded into ~ p 3 . § 2. JOIN CONFIGURATIONS OF LINES OF R P 3

2.1. Classification of non-singular isotopy join configurations of 6 lines of R P 3 up to rigid isotopy. Let L 1 and L 2 be two oriented disjoint lines in R P 3 with positive linking coefficient. Let A j be some pairwise different points of LJ, j = 1,2, i = 1, 2 , . . . , m, such t h a t the increase of index i agrees to the orientation of L j. Consider a permutation a of degree m. The lines passing through points A~ and Aa( 2 0 form a nonsingular join (3; 1)-configuration of degree m, which will be denoted by jc(a). It is easy to see that up to rigid isotopy this construction provides all isotopy join configurations of m lines of R P 3. It is also clear that the map Sm ~ rco(GSPC~,I ) which assigns to a permutation a C Sm the rigid isotopy type of jc(a) is well defined. We denote this map by >r. In what follows ( a l , a 2 , . .. am) denotes permutation ( 1 '

0-1

2

...

0"2

• . •

rn) am

"

2.1.1. L e m m a . 1) Let # and ~ be permutations of degree m preserving the natural cycle order. Then ~ ( , . 0 - . v) = ~(0-). 2) x(0- -1) = ~:(0-). 3) x ( ( m , m - 1 , . . . , 2 , 1 ) - a - (re, m - 1 , . . . , 2 , 1 ) ) = x(0-) .

357

4) Suppose that permutation a ---- ( a l , . . . , a i - 1 , a i , . . . , a j , a j + l , . . . , a m ) satisfies the following condition: for any integer p, with rain ak < p < max ak, there exists an ilYk. Pour obtenir les polyn6mes gl,..., gt, il suffit d'utiliser le thfior~me 1.3.3.(ici V~i~g = V~i~g). Remarque: Le re@me r@sultat peut @tre obtenu sur un corps Mg4briquement clos de caract6ristique 0.

1 . 3 . 0 u t i l s utilisds: Nous utiliserons les r~sultats suivants pour construire l'algorithme annonc4. Elimin_ation ,rapide" des qt~antificateurs:(Fitchas/Galligo/Morgenstem[FGM]) Soit d2 une formuIe prdnexe du premier ordre du langage des corps aIgdbriquement clos d n variables et h r blocs de quantificateurs definie par des pol~nomes ?t coej~cient dans un anneau ~. Soit D Ia somme des degrds des polynSmes apparaissant dans la formule ~.

Th ~o r ~ m e 1 . 3 . 1 :

Alors il existe un r~seau arithm6tique sur [4 qui construit avec une complexitd s~quentielle D n°(~) une formule q! sans quantificateurs dquivalente it d2.

A priori l'6timination des quantificateurs est doublement exponentielle en n (cas oh r = n)(voir [H]). Cependant dans de nombreuses situations significatives le nombre de blocs de quantificateurs est ind6pendant de n. Dans ce cas, l'41iminations "rapide" des quantificateurs permet d'obtenir des algorithmes simplements exponentiels. Le calcul de la multiplicit4 d'une vari4t4 V en un point x se fera en sommant les multiplicit6s des composantes 4quidimensionnelles V d de V en x. Aussi le r@sultat suivant nous sera d'une grande utilit@: D4comp0sition d'une vari4t4 V en composantes fiquidimensionnelles V d (Giusti/Heintz[GH]): Th~or~me 1 . 3 . 2 : Soit V un ensemble alg~brique de C n d~fini par des polyn6mes f l , . . . , fs. Soit D := ~i"=1 degfi. Alors il existe un rdseau arithmdtique qui construit avec une complexitd sgquentielle 0 °("2) des polynSmes f~,..., fs°o, ..., fd, .--, f,~, ..., f ? , ..., f , : ddfinissant les composantes dquidimensionneIles V d de V et vgrifiant: * V d = {x e Cn l f d . . . . . fsdn = 0 } pour d = O , . . . , n * E d.= l E i s=d l degfi d

1.1.5. R~merks.

(1) Proofs of theorems 1.1.3, 1.1.4 E8] impl y that local branches of curve f with centres on coordinate ax-

392 es correspond one-to-one to local branches of Ft~,,,; F,v determined by edges of Z~ . (2) In [8 ] it is shown that we can smooth an arbitrary subset of S C F ) in theorems 1.1.3, 1.1.4, while all other singularities are retained. 1.2 Singularities deformation. Let ~l~p ~ * 9 = S 4 ( ~ 4 + 3 ) / ~ j be the space of real curves of degree ~99 in plane. Let F ~ ~ p ~ be a reduced curve and S£ M S ( F ) -- 5,/bJ 5 2, where 51 ~ 5a = ----~ and 51, 5 z are invarlant with respect to complex conjugation COMj . If ~ = { 0 ) O ) in some affine coordinate system { x 1 ~ ) ~ then F C x ~ I ~ ) denotes the polynomial, which defines the curve ~ in this affine plane• The minimal order /< of a not vanishing K-jet of polynomial F[-xj ~/~ ) is called the order OV-~/= (~-) of F at the point ~ . Put

~O(~) = ~/~P ~ [t?,P'~I

o~(~)

~ ozd Fc~)-~].

Let T ( ~ ) ~ E65~}[I::.)~ mean the tangent cone at F to the germ of locus of curves 5m ~ ~ P ~ with a singular point in some sufficiently small neighbourhood of 1.2.1. Lemma (see [9]). For any 2 ~ 5 6 ~ (F) ,

T(~)

= ( ~ ~/~p'~ I

a E

1.2•2• Lemma (see [9]). Linear varieties

intersect transversally in ~ P Put

n4£

= O~Z,~F (2:) -- 2 ~ 2 ~- S ¢

1.2.3. Coro!lary. %vith properties

(i)

For any polynomials

J~P~n4~

P/9~-jet of

&

(x,~),

~e

5f)

~S~

(iii) P~ ~ ~ 6 5~ ~ there exists the close to F following conditions : (1)

~

qb(:~l~

curve

)

are sufficiently close to zero; ~ ~ ~P n satisfying the

P~ (x~)~@51,, neighbourhood of 5 4

is equal to

(2) ~ is non-singular outside some 1.2.4. Theorem [2]. If ~ ~ P ~ is a nodal curve, then there exists the anyhow close to F curve ~ b ~ ~ n with nodes in neighbourhoods of every point ~ ~ ~ ~ and non-singular outside these neighbourhoods.

393 2. Construction of nodal curves Here we consider curves with ~ ( / : 3 consisting of only nodes. Any gluing of such curves is allowed by theorem I. 1.3. The curve ~ with Newton polygon z~ and ~ { ~ ) , consisting of d~ knots, ~ single points and ~ g imaginary nodes, we'll denote as F ~ z~ ~ j ~/ c~ or F ~ ~ / ¢ c~ . If F~ {~/~ 6~ is irreducible and has the degree ~ , then according to the PlUcker formula [14]. 2.2. Theorem. For any non-negative integers &~ ~/ C satisfying (2.1), there is an irreducible curve ~ 6 {&]/~/ C ~ of degree ~ . Proof. According to theorem 1.2.4 it is enough to study the case ~x W- ~ -~- ~ C -: { M 4 - 4 ~ 4 - ~ / ~ . We'll use induction in ~ . All nodal curves of degree ~ Z/ are well-known [15], therefore we suppose ~ ~ ~ . I step. Assume ~ = O ) 6L ~ 2 ~ - ~ . Then

-,- ,I. (c

7-.) =

therefore there is a curve q) ree M4--~ . Using theorem from curve ~O U g ~ t] t ~ ginary conic with C~/ {] ~ p Z 2 step. Assume ~= O) (2 ~ ~ . Since -I- 2. ( c

3.)

~ ( ' ~ / O / g- - 2 ~ + ~ of deg1.2.4 we obtain the desired curve C2, where C 2 is an ima-__ ~ . ~ - 3 -~C--< ~m,!-- ~ . Then

=

.,

there is a curve q9 ~ {61-"/~ ,O, C - ~ 4 + 3 ) of degree M~-~ . Also there is a real conic C 2 meeting @9 exactly at two real points, because g l - { $ "/ . Using theorem 1.2.4 we smooth one of real points in ~ ~ ~ 2 and obtain the desired curve /c ~ {~/ % C ) . 3 step. Assume ~---- O ; C ~ PM-2.3. Lemma. ;or any ~ ~ O there exist curves ~ ¢ ; Z~¢~ (~C9-{3/~2

O2

(0; ~p.f-'f~, {9,.v~,'O]

(9.) ]

with Newton triangles and

/ (O; @3

{ ((9,'0)/ ((9,-~+~2 (~,1 /JJ

resp ec t ively. Proof. We get the curves ~ by gluing the curve ~ F~, where ~ ( ~ ) = ~ - Dg ~ { 5 ~ + ~ ~ ( ~ , ~ is a union of different real straight lines in general position, with curves

,

394

g~'=~H*~fczD+z ~+~

g+= ~(4*~c~3)÷ 4 m

Let H = ~-~ (~-43 (2d-2~)'..." (:3~-144 + ~g ) o There are real straight lines Lfj L a meeting H exactly at ~ [ C / ~ and ~ [ ( d t ~ / ~ _ ] imaginary points respectively. The gluing of 7 ~ _ 4 and H gives a curve ~ E ~ E ~ 4 4 - 3 ) ~ N 4 - ~ p / ~ ; O 1 0 3. Smoothing two real points in ~ / O (L~ U I-2) we get the desired curve F from the curve ~ U L ~ L 2 / _ z " 4 step. Assume ~ ~ j c ~ M4- ~ . Since ,~ . ( t - ~ ) -,- ~- (c - ~ . 3 ) = (~-3.)(~-~)/2 .. there is a curve ~b E {G2 g- % d - ~ t 4 + 3 ) of degree /~/?-2. Let L be a general imaginary straight line. Then we obtain the desired curve F from the curve %~ U L ~ d D ~ Zby smoothing two conjugate points in ? D ( L U co~' L ) . 5 step. Assume ~ ~ ~ ~< M4-~,. C ~ ~ta-c/ 2.4. Lemma. For any triangle U- with integral vertexes

there i~ a c ~ v e

{ E T(O,

~, O )

, and al~o, if ¢, is even,

a curve S E-T"fd~ I O / 9 / ~ ) " Proof. A curve with Newton triangle V- can be obtained from a curve with Newton triangle ~ fO/'2)~ (~-4,' ~ ~" ~ ~ by birational transformation remaining ~ (~3 (or ~ {~ ). Then is an image of

where

P~,.¢ ( g ) ---" ~-O.S6 ~ 0

polynomial, and, ~

~T~g~ ~ )

is the Chebyshev

is an image of

2.5. Remark. It should be noted that sections of polynomials /~ ~ on edge [ ( p g ? - p ; g ) 2 { ~ - f , . d p ~ ] of Newton triangle m are products of real factors, linear in ~ . Introduce the following polygons: ( i ) quadrangle ~E" =" { {~]0),, (~,1 .~)/ (JIM-'l] "/.,)2 ('IM/ OJ ~,; (ii) triangles T ~

-- { ~,~a-,~-~~ ~C'~ or ~ < 0 then stop, (2) if ~ is even and ~ C ~ then glue F / with ~ ( ~ ' ¢ ~ and decrease ~/ by ~ / ~ , other~ wise glue ~ with I [ ~ £ 3 and decrease by ~ (3) substitute F ~ for the obtained curve, ~ for - ~ , and ~ for -~ , and then return to (S). After stopping algorithm we get: If ~ = ~ then ~ is the gluing of F / and a curve

F " ~ ~c'~[cb 6" c? zf ,~, ;> O )

"

l ' q - .~d' ~ ~ -

£

then F

i s the g l u i n g

396

't ec rve E ~ " ( i , i , l - y _ --

~, ~ g'j

Let

E =

q

E COjg~I

O)

"-
a(Plk,P2d,p3n)contradiction.

406

We shall call a s u b s e t J/1 c J / p - l o n g i f p is the maximal n a t u r a l n u m b e r such t h a t for every point z from some p-dimension s u b s e t M 1 of M holds : P~ r~ J/1 ¢ ~#-It is clear t h a t J/1 is p-long iff dim(stl,2(J/1)) = p . L e m m a 1. The set ~ is not p-long. P r o o f . Suppose t h a t ~ is p-long. Choose in ~ (correspondingly - in M ) a pdimensional smooth connected semialgebraic s u b s e t ~1 (correspondingly - M 1) such t h a t for every z e M 1 the intersection P= r~ ~fl ¢ ~ consists of a unique point and this correspondence is bijective. The sets ~1 and M 1 can really be chosen. Indeed, since the n u m b e r of the c o n n e c t e d c o m p o n e n t s of ~f is finite, a m o n g t h e m t h e r e will be a p - l o n g c o m p o n e n t ~'. Choose in the set ~ ' which is open in 7z a p - d i m e n s i o n connected semialgebraic s u b s e t ~" which is a smooth manifold. According to the definition of p-long set there exists a p-dimension open subset M" of M such that for every y e M" the intersection Py c~ ~ " ~ ~. Besides, from the transfer principle follows t h a t for all pairs Y l , Y 2 • M " c M holds : Pyl ¢'~ Py2 ('~ fl.t = d). Therefore we have t h e smooth m a p of smooth manifolds 92 : CA~''

) M", if

z e ~ " c~ Py then 92(z) = y. By Sard's theorem, t h e r e exists a connected p-

dimension semialgebraic s u b s e t M 1 c M" such that for every point y e M 1 the fibre 921(y) consists of the noncritical points of the m a p 92 and t h u s - of finite n u m b e r of points. Let ~1 be such a smooth semialgebraic subset of 921(M1) t h a t 92(~.~1 ) = M 1 a n d the restriction of 92 on -~1 is injective. Therefore we h a d constructed the sets M p ~1 having the necessary properties. Choose a point x 0 • M 1 and denote by L the p-plane t a n g e n t to manifold M at x o. Since the smooth manifolds Vp can be given by a formula of first-order theory of the field Q for every point z e M the intersection Pz n L ~ ~ consists of the unique point and this correspondence 91 "L

> M is bijective.

L e t L 1 = ~0~1(M1). Denote by ~1 : L1 > Q(eo,~l,e 2) a smooth function taking the value of the euclidean distance b e t w e e n a point from L 1 and its image in M 1 in force of the smooth map 91. Denote by V2 : L1 " Q(~o,Q,~2) a smooth function which for y e L 1 is taking the value equal to the s u m of ~I(Y) and the distance between 91(Y) and 921(9](y)). Let us prove t h a t there exists a point a • 5g1 such t h a t euclidean distance form a to M is an element from Q(eo). This will contradict with the definition of J/.

407

According to Lagrange's t h e o r e m on finite i n c r e m e n t s , for every point y e L 1 there exist points $1 e L 1 and 02 e L1 such t h a t ~I(Y) = < grado~(~), (y-x o) > , ~2(Y) - ~2(Xo) = < grade2(~'2), (Y-Xo) > , therefore ~2(Y) -- ~I(Y) -- ~2(X0) = < grado2(tg2) - gradel(V1), (Y-Xo) >Choose a point y 0 e L 1 such t h a t lYo-Xol = e~ for some 0 < 7 e Q, grad=o(~ 2) is c011inear to (yo-Xo) and W2(Yo) _>~2(Xo). Then by definition of the set ~ D ~1 for c o r r e s p o n d i n g O 2 the scalar product < grade2(~g2), (Yo-Xo) > ~ a e~ w h e r e 0 < a e Q. Besides, [ g r a d o , ( ~ l ) [ < e~o for a certain 0 < fl e Q a n d t h u s < grade,(Wl) , (Yo-Xo) > Igrade,(wl)[. l yo-xo I <e~o+r . Let a = ~21(~1(yo)). Then the distance from a e A1 to M 1

v'2(Yo) - ~(Yo) > ~(a-~o) + ~(Xo) > e~. This contradicts the inclusion Stl,2(a) e M 1 which is valid by definition of the set A1 c ~q. The l e m m a is proved. Let us r e m a r k t h a t the l e m m a r e m a i n e s t r u e if we take i n s t e a d of ~ a n a r b i t r a r y s m o o t h algebraic h y p e r s u r f a c e t h e s t a n d a r d p a r t st 2 of which coinsides with V. The property to be algebraic is i m p o r t a n t : consider for example a line on a plane for V and an infinitely close sinusoid h a v i n g an infinitesimal period for the smooth hypersurface. Denote by -¢ : ~ ) S n-1 the Gauss map (see e.g. [5]) o f ~ , here

s--*= { x

Ixl = 1 }

Corollary. Fix a point x e fie°). p For almost all point y e Cep O) c~ ~=(e o) (i.e. for all points with the exception, maybe, of the points of a semialgebraic subset of the dimension less t h e n p) holds : d i m ( s t o & 2 ( J ( ~ • ~y(Q)))) < n - p - 1 . Moreover, the set Sto,l,2(J(~ n ~y(el))) belongs to the intersection of the (n-1)sphere S "-1 with (n-p)-plane, passing through 0. P r o o f . According to l e m m a I, stl,2(5~) c

fie°) p (where ~ appears from x as in

l e m m a I) is a semialgebraic subset of the dimension less t h e n p. Let the point y belong to the complement to stl,2(5~) in 1fiE°) p . Then, by definition of £P, for any point z e (Tz ¢~ ~y(~l)) the normal to ~r at this point is %-collinear to (n-p)-plane which is passing through 0 and is collinear to P=. The corollary is proved.

408

Consider the s t r a t u m Vp for a certain 0 _

n - p - 1 . On the other hand, according to the corollary to l e m m a I, e < n - p - 1 .

S u p p o s e t h a t the

l e m m a is proved for s t r a t a of the dimension g r e a t e r t h e n p. Consider nonmaximal s t r a t u m V (e°) and apply to it lemma 2 (the hypothesis of l e m m a 2 is P

valid by the inductive hypothesis). The lemma is proved.

3. R e p r e s e n t a t i o n s o f s o m e s e m i a l g e b r a i c sets. Consider a semialgebraic set R given by a formula of first-order theory of Q. Let the dimension of R at every point (see the proof of l e m m a 2) be the same and equal to ra. The fact that z • R is a smooth point in R (that is the fact that for a n y two pairs of distinct points in the neighbourhood of z in R, the lines passing through the first and second pair are almost collinear to the same mplan) can be expressed by a first order formula. Note t h a t in R = Vp every point is smooth ; if Vp+i is incident to Vp then the point z • Vp c (Vp+ i u Vp) = R is not smooth. It is hence clear t h a t the s u b s e t S m ( R ) of all smooth points of R is semialgebraic and can be w r i t t e n as a formula of first-order with a c o n s t a n t (i.e. not depending on R) n u m b e r of quantifier alternations. In the algorithm described below an i m p o r t a n t role is playing the set of the form :

412

Kp = stl,2( {x e//" : dim(stl,2(~¢(~ f r~ ~x(el)))) = n-p-l} ), where the external st is taken for all points of the set for which it is defined. L e t us prove t h a t Kp is a semialgebraic set which can be given b y a formula with

a constant

number

(not d e p e n d i n g

on ~ ) of q u a n t i f i e r

alternations. Indeed, the image of a semialgebraic set u n d e r the G a u s s m a p is obviously semialgebraic. The s t a n d a r d p a r t of a semialgebraic set A in the representation of which atomic polynomials belong to Q[Sl,~ 2] [X1..... X n] is, as it was noted in [3], also semialgebraic (consider ~l,S2 as n e w variables, t h e n stl,2(A) coinsides with the intersection of n-planes {Q = £2 = 0} and the closure in euclidean topology of the set A (~ {Q > 0 & E2 > 0}. The proposition dim(.) = t can be w r i t t e n in the following w a y : there exists a linear transformation of coordinates L : (X 1..... Xn) ' > (Y1 ..... Yn) with the matrix

I

I ~'2 1 "'" ... ~'n 0 1

0

0

0

0

(3)

1

such t h a t the projection of the set on the subspace of coordinates Y1 .... ,Yt containes a t-ball and for every linear m a p of the kind (3) the projection on the (t+l)-subspace does not contain a (t+l)-ball.

4. The algorithm and its m, nning ~me. As a n a u x i l i a r y p r o c e d u r e s our a l g o r i t h m

e s s e n t i a l l y involves t h e

effective algorithms for quantifier elimination in first-order t h e o r y of real closed fields proposed in [6] (see also [7]) a n d for finding all c o n n e c t e d components of a semialgebraic set from [2] or [3]. The algorithm works recursively in p. Let p = m = dim(V). The algorithm defines the set Sm({x e V/dim(V) in x equals m}) by a formula (with quantifiers) O~ ) of the first-order theory ofQ(e~,E 2) as it was explaned a t the beginning of section 3. After t h a t using several times the quantifier elimination procedure from [6] t h e a l g o r i t h m p r o d u c e s q u a n t i f i e r - f r e e f o r m u l a (~(2) w h i c h is equivalent to O ~ ). Obviously Vm = {O~)}. A s s u m e t h a t s t r a t a Vm, .... Vp+t a r e a l r e a d y c o n s t r u c t e d a n d given by formulas O m ..... ep+ 1 correspondingly. The algorithm defines the set Sm(Kp) by a formula (with quantifiers) O(p1) as it was explaned in section 3, and then, with the help of procedure from [6] produces an equivalent quantifier-free formula

413

O(p2). Using [2] the algorithm finds all connected components of {@(p2)}and after t h a t with the help of the procedure from [8] for solving systems of polynomial inequalities selects those which have empty intersections with every Vm..... Vp+1. Let the selected components be given by formulas O(p2'1)..... O(p2'sp). For every l < i <sp the algorithm writes a formula (with quantifiers) O(p3'/) defining the closure (in the topology in (Q(Q,~2)) n with the base of all open balls) of the set Op(2,/) and then, eliminating quantifiers in Op(3,/) finds an equivalent quantifierfree formula O(p4'/) . Let o(pb) =

V

O(p4J) and O(p6) be the formula defining Sm({O(pb)}).

l