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The Collected Papers of
STEPHEN SMALE Volume 3
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ected Papers o
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The Collected Papers of
STEPHEN SMALE Volume 3
Edited by
R Cucker R. Wong City University of Hong Kong
SINGAPORE UNIVERSITY PRESS NATIONAL UNIVERSITY OF SINGAPORE
W h World Scientific ■ T
Singapore *New Jersey London 'Hong Kong
Published by World Scientific Publishing Co. Pte. Ltd. P O Box 128, Farrer Road, Singapore 912805 USA office: Suite IB, 1060 Main Street, River Edge, NJ 07661 UK office: 57 Shelton Street, Covent Garden, London WC2H 9HE
Library of CongrtM Catalogkig-in-PabHcaUon Data Smale, Stephen, 1930[Works. 2000] The collected papers of Stephen Smale / edited by F. Cucker, R. Wong. p. cm. ISBN 9810243073 (set) - ISBN 9810249918 (v. 1) - ISBN 9810249926 (v. 2) -- ISBN 9810249934 (v. 3) 1. Mathematics. 2. Computer science. 3. Economics. I. Cucker, Felipe, 1958- II. Wong, R. (Roderick), 1944- III. Title. QA3 .S62525 2000 510~dc21
00-031992
British Library Cataloguing-in-Publication Data A catalogue record for this book is available from the British Library.
Copyright © 2000 by World Scientific Publishing Co. Pte. Ltd. All rights reserved This book, or parts thereof, may not be reproduced in any form or by any means, electronic or mechanical, including photocopying, recording or any information storage and retrieval system now known or to be invented without written permission from the Publisher.
For photocopying of material in this volume, please pay a copying fee through the Copyright Clearance Center, Inc., 222 Rosewood Drive, Danvers, MA 01923, USA. In this case permission to photocopy is not required from the publisher.
Printed in Singapore.
V
Contents
VOLUME I Research Themes
1
Luncheon Talk and Nomination for Stephen Smale (R. Bott)
8
Some Recollections of the Early Work of Steve Smale (M. M. Peixoto)
14
Luncheon Talk (R. Thorn)
17
Banquet Address at the Smalefest (E. C. Zeeman)
20
Some Retrospective Remarks
22
Parti. Topology The Work of Stephen Smale in Differential Topology (M. Hirsch)
29
A Note on Open Maps
53
A Vietoris Mapping Theorem for Homotopy
56
Regular Curves on Riemannian Manifolds
63
On the Immersion of Manifolds in Euclidean Space (with R. K. Lashof)
84
Self-Intersections of Immersed Manifolds (with R. K. Lashof)
106
A Classification of Immersions of the Two-Sphere
121
The Classification of Immersions of Spheres in Euclidean Spaces
131
Diffeomorphisms of the 2-Sphere
149
On Involutions of the 3-Sphere (with M. Hirsch)
155
The Generalized Poincar6 Conjecture in Higher Dimensions
163
On Gradient Dynamical Systems
166
Generalized Poincar6's Conjecture in Dimensions Greater Than Four
174
Differentiable and Combinatorial Structures on Manifolds
190
On the Structure of 5-Manifolds
195
On the Structure of Manifolds
204
vi
A Survey of Some Recent Developments in Differential Topology
217
The Story of the Higher Dimensional PoincanS Conjecture (What actually happened on the beaches of Rio)
232
Part II. Economics Stephen Smale and the Economic Theory of General Equilibrium (G. Debreu)
243
Global Analysis and Economics, I: Pareto optimum and a generalization of Morse theory
259
Global Analysis and Economics, IIA: Extension of a theorem of Debreu
271
Global Analysis and Economics, III: Pareto optima and price equilibria
285
Global Analysis and Economics, IV: Finiteness and stability of equilibria with general consumption sets and production
296
Global Analysis and Economics, V: Pareto theory with constraints
305
Dynamics in general equilibrium theory
314
Global Analysis and Economics, VI: Geometric analysis of Pareto Optima and price equilibria under classical hypotheses
321
A Convergent Process of Price Adjustment and Global Newton Methods
335
Exchange Processes with Price Adjustment
349
Some Dynamical Questions in Mathematical Economics
365
An Approach to the Analysis of Dynamic Processes in Economic Systems On Comparative Statics and Bifurcation in Economic Equilibrium Theory The Prisoner's Dilemma and Dynamical Systems Associated to
368 373
Non-Cooperative Games
380
Global Analysis and Economics
398
Gerard Debreu Wins the Nobel Prize
438
Global Analysis in Economic Theory
440
vu
Part III. Miscellaneous Scientists and the Arms Race
445
On the Steps of Moscow University
454
Some Autobiographical Notes
461
Mathematical Problems for the Next Century
480
VOLUME II Part IV. Calculus of Variations (Global Analysis) and PDE's Smale and Nonlinear Analysis: A personal perspective (A. J. Tromba)
491
A Generalized Morse Theory (with R. Palais)
503
Morse Theory and a Non-Linear Generalization of the Dirichlet Problem
511
On the Calculus of Variations
526
An Infinite Dimensional Version of Sard's Theorem
529
On the Morse Index Theorem
535
A correction to "On the Morse Index Theorem"
542
What is Global Analysis?
544
Book Review on "Global Vanational Analysis: Weierstrass Integrals on a Riemannian Manifold" by Marston Morse
550
Smooth Solutions of the Heat and Wave Equations
561
Part V. Dynamics On the Contribution of Smale to Dynamical Systems (J. Palis)
575
Discussion (S. Newhouse, R. F. Williams and others)
589
Morse Inequalities for a Dynamical System
596
On Dynamical Systems
603
Dynamical Systems and the Topological Conjugacy Problem for Diffeomorphisms
607
Stable Manifolds for Differential Equations and Diffeomorphisms
614
viii
A Structurally Stable Differentiable Homeomorphism with an Infinite Number of Periodic Points
634
Diffeomorphisms with Many Periodic Points
636
Structurally Stable Systems Are Not Dense
654
Dynamical Systems on n-Dimensional Manifolds
660
Differentiable Dynamical Systems
664
Nongenericity of ^-Stability (with R. Abraham)
735
Structural Stability Theorems (with J. Palis)
739
Notes on Differential Dynamical Systems
748
The ii-Stability Theorem
759
Stability and Genericity in Dynamical Systems
768
Beyond Hyperbolicity (with M. Shub)
776
Stability and Isotopy in Discrete Dynamical Systems
781
Differential Equations
785
Dynamical Systems and Turbulence
791
Review of "Catastrophe Theory: Selected Papers, 1972-1977" by E. C. Zeeman On the Problem of Reviving the Ergodic Hypothesis of Boltzmann
814
and Birkhoff
823
On How I Got Started in Dynamical Systems
831
Dynamics Retrospective: Great problems, attempts that failed
836
What is Chaos?
843
Finding a Horseshoe on the Beaches of Rio
859
The Work of Curtis T. McMullen
865
Part VI. Mechanics Steve Smale and Geometric Mechanics (J. E. Marsden)
871
Topology and Mechanics, I.
889
ix
Topology and Mechanics, II.
916
Problems on the Nature of Relative Equilibria in Celestial Mechanics
936
Personal Perspectives on Mathematics and Mechanics
941
Part VII. Biology, Electric Circuits, Mathematical Programming On the Mathematical Foundations of Electrical Circuit Theory
951
A Mathematical Model of Two Cells via Turing's Equation
969
Optimizing Several Functions
979
Sufficient Conditions for an Optimum
986
The Qualitative Analysis of a Difference Equation of Population Growth (with R. F. Williams)
993
On the Differential Equations of Species in Competition
997
The Problem of the Average Speed of the Simplex Method
1000
On the Average Number of Steps of the Simplex Method of Linear Programming
1010
VOLUME in
Part VIII. Theory of Computation On the Work of Steve Smale on the Theory of Computation (M. Shub)
1035
The Work of Steve Smale on the Theory of Computation: 1990-1999 (L. Blum and F. Cucker)
1056
On Algorithms for Solving/^) = 0 (with M. Hirsch)
1076
The Fundamental Theorem of Algebra and Complexity Theory
1108
Computational Complexity: On the geometry of polynomials and a theory of cost, Part I (with M. Shub)
1144
On the Efficiency of Algorithms of Analysis
1180
Computational Complexity: On the geometry of polynomials and a theory of cost, Part II (with M. Shub) On the Existence of Generally Convergent Algorithms (with M. Shub)
1215 1232
X
Newton's Method Estimates from Data at One Point
1242
On the Topology of Algorithms, I.
1254
Algorithms for Solving Equations
1263
The Newtonian Contribution to Our Understanding of the Computer
1287
On a Theory of Computation and Complexity over the Real Numbers: NP-completeness, recursive functions and universal machines (with L. Blum and M. Shub)
1293
Some Remarks on the Foundations of Numerical Analysis
1339
Theory of Computation
1349
Complexity of Bezout's Theorem I: Geometric aspects (with M. Shub)
1359
Complexity of Bezout's Theorem II: Volumes and probabilities (with M. Shub)
1402
Complexity of Bezout's Theorem III: Condition number and packing (with M. Shub)
1421
Complexity of Bezout's Theorem IV: Probability of success; Extensions (with M. Shub)
1432
Complexity of Bezout's Theorem V: Polynomial time (with M. Shub)
1453
The Godel Incompleteness Theorem and Decidability over a Ring (with L. Blum) Separation of Complexity Classes in Koiran's Weak Model (with F. Cucker and M. Shub)
1496
On the Intractability of Hilbert's Nullstellensatz and an Algebraic Version of ' W/VP?" (with M. Shub)
1508
Complexity and Real Computation: A Manifesto (with L. Blum, F. Cucker and M. Shub)
1516
1477
Algebraic Settings for the Problem "P*NF?" (with L. Blum, F. Cucker and M. Shub)
1540
Complexity Theory and Numerical Analysis
1560
Some Lower Bounds for the Complexity of Continuation Methods (with J.-P. Dedieu)
1589
XI
A Polynomial Time Algorithm for Diophantine Equations in One Variable (with F. Cucker and P. Koiran)
1601
Complexity Estimates Depending on Condition and Round-off Error (with F. Cucker)
1610
1035
28
On the Work of Steve Smale on the Theory of Computation MICHAEL S H U B *
The theory of computation is the newest and longest segment of Steve Smale's mathematical career. It is still evolving and, thus, it is difficult to evaluate and isolate the more important of Smale's contributions. I think they will be as important as his contributions to differential topology and dynamical sys tems. He has firmly grounded himself in the mathematics of practical algo rithms, Newton's method, and the simplex method of linear programming, inventing the tools and methodology for their analysis. With the experience gained, he is laying foundations for the theory of computation which have a unifying effect on the diverse subjects of numerical analysis, theoretical com puter science, abstract mathematics, and mathematical logic. I will try to capture some of the points in this long-term project. Of course, the best thing to do is to read Smale's original papers; I have not done justice to any of them. Smale's work on the theory of computation begins with economics [Smale, 1976]. Prices p = (p t ,...,p t ) e Rf+ for { commodities give rise to demand and supply functions D(p) and S(p). The excess demand function f(p) = D(p) — S(p) e K' has as the ith coordinate the excess demand for the ith good at prices p. A price equilibrium is a system of prices p for which f(p) = 0, that is, supply equals demand. f:R'+-+R' and the problem is to find a zero of/ 1 . Given a C2 function f:M-*R" defined on a domain M c R", Smale suggested using the "global Newton" differential equation Df(x)jt
=
-Xf(x)
* Partially supported by an NSF grant. 1 Actually, f is assumed to be scale invariant f(XP) = f{P) for k > 0 so / may be restricted to the unit sphere intersect R+, S+"1. Walras' law is pf(P) = 0, so / is tangent to S+"1 and the problem is to find a zero of this vector field. 281
1036
282
M. Shub
Rm
fix)
/
FIGURE 1
where X is a real number depending on the sign of Det(£>/(x)). The solution curves of this equation through points x with f{x) # 0 are inverse images of rays pointing to zero. or what is the same, inverse images of points x e S""1 for the function
g-.M-E^S*-1 defined by g(x) = /(x)/||/(x)|| and where E = {x e M\f(x) = 0}. With the right boundary conditions, the "global Newton" vector field is transverse to the boundary and g is nonsingular on the boundary. By Sard's theorem, almost every value in S""1 is a regular value, and for almost every m e dM, g~1{g(m)) is a smooth curve. If Af is compact, this curve must lead to the set of zeros £. Smale proved the existence of price equilibria this way. Differential equa tion solvers can then be used to locate these points. Smale pursued these ideas with Hirsch in [Hirsch-Smale, 1979] where they suggested various ex plicit algorithms for solving f(x) = 0. In particular, their work included poly nomial mappings / : CM -»C m
(or Rm -► Rm)
which are proper and have nonvanishing Jacobian outside of a compact set. An interesting feature of these algorithms is that their natural starting points are quite far from the zero set; for example, in the discussion of global Newton above, they are in dM. The global Newton differential equation had been considered previously and independently by Branin without convergence results. The polynomial system g: R2 -► U2 proposed by Brent, 0l(*l>*2)=4(*l +*z).
92(x1,x2) = 4(x, + x 2 ) + (x, + x 2 )((x, - 2)2 + xl-
1),
is illustrated in [Branin, 1972] and reproduced below. The curve | J\ = 0 is Det(#) = 0. Note the region of closed orbits and nonconvergence which are close to zero.
1037
28. On the Work of Steve Smale on the Theory of Computation
283
FIGURE 2
Other algorithms for the location of pure equilibria were known. Scarf, in particular, had developed simplicial algorithms; see [Smale, 1976] for a dis cussion of this. Having competing methodologies to solve the problem led Smale to develop a framework in which to compare their efficiency. Smale's article "The Fundamental Theorem of Algebra and Complexity Theory" [Smale, 1981] is a startling step forward. Consider the first three paragraphs of the paper: The main goal of this account is to show that a classical algorithm, Newton's method, with a standard modification, is a tractable method for finding a zero of a complex polynomial. Here, by "tractable" I mean that the cost of finding a zero doesn't grow exponentially with the degree, in a certain statistical sense. This result, our main theorem, gives some theoretical explanation of why certain "fast" methods of equa tion solving are indeed fast. Also this work has the effect of helping bring the discrete mathematics of complexity theory of computer science closer to classical calculus and geometry.
1038
284
M. Snub
A second goal is to give the background of the various areas of mathematics, pure and applied, which motivate and give the environment for our problem. These areas are parts of (a) Algebra, the "Fundamental theorem of algebra", (b) Numerical analysis, (c) Economic equilibrium theory and (d) Complexity theory of computer science. An interesting feature of this tractability theorem is the apparent need for use of the mathematics connected to the Bieberbach conjecture, elimination theory of algebraic geometry, and the use of integral geometry. The scope of the undertaking is very large and the terrain unclear. It was the discrete theoretical computer scientists who had the most developed no tion of algorithm, of cost, and of complexity as a function of input size. For them, the space of problems, complex polynomials of a given degree, so natural for a mathematician is not natural because it is not discrete. From another direction, numerical analysts have practical experience with rootfinding, algorithms which are fast and algorithms which are sure, algorithms which are stable and those which are not. In [Smale, 1985], we see Smale grappling again with these questions. First, there is the quote from von Neumann quoted again in [Smale, 1990]. The theory of automata, of the digital, all or none type, as discussed up to now, is certainly a chapter in formal logic. It would, therefore, seem that it will have to share this unattractive property of formal logic. It will have to be from the mathematical point of view, combinatorial rather analytical. ... a detailed, highly mathematical and more specifically analytical, theory of autom ata and of information is needed. and Chapter 2, Section 6 6. What is an algorithm? PROBLEM 11. What is the fastest way of finding a zero of a polynomial? This is a kind of super-problem. I would expect contributions by several mathematicians rath er than a single solution. It will take a lot of thought even to find a good mathematical formulation. In some ways, one could compare this problem with showing the existence of a zero of a polynomial. The concept of complex numbers had to be developed first. For Problem 11, one must develop the concept of algorithm to deal with the kind of mathematics involved. Consistent with the von Neumann statement quoted in the introduction, my belief is that the Turing approach to algorithms is inadequate for these purposes. Although the definitions of such algorithms are not available at this time, my guess is that some kind of continuous or differentiable machine would be involved. In so much of the use of the digital computer, inputs are treated as real numbers and the output is a continuous function of the input. Of course a continuous machine would be an idealization of an actual machine, as is a Turing machine. The definition of an algorithm should relate well to an actual program or flow-chart of a numerical analyst. Perhaps one could use a Random Access Machine (RAM, see Aho-Hopcraft-Ullman) and suppose that the registers could hold real numbers.
1039
28. On the Work of Steve Smale on the Theory of Computation
285
Then one might with some care expand the list of permissible operations. There are pitfalls along the way and much thought is needed to do this right. To be able to discuss the fastest algorithm, one has to have a definition of algorithm. I have used the word algorithm throughout this paper, yet I have not said what an algorithm is. Certainly the algorithms discussed here are not Turing machines; and to force them into the Turing machine framework would be detrimental to their analysis. It must be added that the idealizations I have suggested do not eliminate the study of round-off error. Dealing with such loss of precision is a necessary part of the program. Problem 11 is not a clear-cut problem for various reasons. Factors which could affect the answer include dependence on the machine, whether one wants to solve one or many problems, time taken to write the program, whether polynomials have large or small degree, how the problem is presented, etc. So in [Smale, 1981] he has launched into a discussion of the total cost of an algorithm without a precise notion of cost or algorithm available; these will come later! To have enough conviction that such a long-range project will work out is not uncharacteristic of Smale. When Smale was awarded the Fields Medal in 1966 for his work in differ ential topology, Rene Thom wrote (translation my own): ... Smale is a pioneer, who takes his risks with calm courage, in a completely unex plored domain, in a geometric jungle of inextricable richness he is the first to have cleared a path and planted beacons. [Thom, 1966] Returning to the 1981 paper, Smale restricted the class of functions for which roots are to be found to complex polynomials of one variable and degree d, normalized as f(z) = Y,1=oaiz' with at; e C ad = 1 and \at\ < 1 for 1 0, so global Newton can be taken as ^
=
-Df{x)-1f(x).
fro)
FIGURE 3
Now the solution curve through z 0 is the branch through z0 of the inverse of image of the ray from f(z0) to 0, and because / is proper, it does lead to a zero
1040
286
M.Shub
of / with the exception of a finite number of rays which contain the critical values of / Incidentally, this argument proves the fundamental theorem of algebra. To turn the proof into an algorithm, Smale considers the Euler approximation to the solution of the differential equation, for h a positive real, N„(f)(z) = z -
hDf-\z)f(z).
When h = 1, this is Newton's method; we denote this also by N{f). Newton's method gives quadratic convergence near simple zeros, so Smale defines an approximate zero of / as a point where Newton's method is converging quadratically. Definition. z0 is an approximate zero for / iff \zk — zt_! | 0. If it does, find a point x which minimizes it. Taking a Gaussian distribution on J?™" x Rm x R" and letting p(m, n) be the number of steps of Dantzig's self-dual method to solve LPP, Smale [1983a] proves that p is sublinear in the number of variables; see also [Smale, 1983b]. Theorem 2. Let p be a positive integer. Then depending on p and m there is a positive constant cm such that for all n P(m,n)zcmnlip. In [Smale, 1985a] which won the Chauvenet Prize of the Mathematical Association of America in 1988, Smale confronts some new issues. First is the problem of ill-posed problems. In the fall of 1983, Lenore Blum was visiting New York and we studied the problem of the average loss of precision (or significance) in evaluating rational functions of real variables. Let 5 be the input accuracy necessary for desired output accuracy e. Then |ln5| — |lne| is the loss of precision (or significance for relative accuracy). We showed this loss was tractable on the average [Blum-Shub, 1986]; Smale focused on linear algebra. There the condition number, KA = \\A\\ \\A_11|, of a matrix A measures the worst-case relative error of the solution x of the equation Ax = b divided by the relative error of the input b. Thus, log KA measures the worst-case loss of significance. Smale wrote an explicit integral for the aver-
1043
28. On the Work of Steve Smale on the Theory of Computation
289
age of log KA and Ocneanu, Kostlan, Renegar, and others made progress toward its estimation. Finally, Edelman [1988] has shown that up to an additive constant the average is Inn. This result helps explain the success of fixed precision computers in solving fairly large linear systems. Demmel [1987a; 1987b] interprets the condition number as the inverse of the distance to the determinant zero variety, i.e., the singular matrices. Thus, there is an analogy between the success of fast algorithms and the intrinsic difficulty of robust computation. They are both measured in terms of distance to a subvariety of bad problems. This theme surfaces again in Smale's work, although the precise relationship remains somewhat mysterious. The same 1985 paper dealt with two other problems. The efficiency of approximation of integrals: I will not say much about this except that Smale showed that the tfapezoid rule is more efficient than Riemann integration on the average forfixederror on Jfl functions, similarly Simpson's rule is more efficient for 3>C2 functions. Newton's method is an example of a purely itera tive algorithm for solving polynomial equations. A purely iterative algorithm is given as a rational endomorphism of the Riemann sphere which depends rationally on the coefficients of the polynomials (offixeddegree d) which are to be solved. A purely iterative algorithm is generally convergent if for almost all (/, x) iterating the algorithm on x, the iterates converge to a root of /. Smale [1986] conjectures that there are no purely iterative generally conver gent algorithms for general d. McMullen [1988] proved this for d > 4 and produced a generally convergent iterative algorithm for d = 3. For d = 2, Newton's method is generally convergent. Doyle and McMullen [1979] have gone on to add to this examining d = 5 in terms of a Galois theory of purely iterative algorithms. In contrast, Steve and I showed in [Shub-Smale, 1986b] that if complex conjugation is allowed, then there are generally con vergent purely iterative algorithms even for systems of n complex polyno mials of fixed degree in n variables. Smale has devoted a lot of effort to understanding Newton's method. These are recounted in [Smale, 1986], but let me mention a few of the results of this paper and [Smale, 1985b]. To generalize the one-variable theory, Smale considers the zero-finding problem for f: E-+ F, where / is an analytic map of Banach spaces. Newton's method is the same z' = N(f)z = z — Df(z)~lf(z), and the definition of ap proximate zero is the same. Definition. z0 is an approximate zero for/iff \\zk-zk_l || c||/(z0)l|A#(r0,A
A
- 1/n.
Let wt = (1 — «'A)/(z0), i = 0,...,n. Then, inductively, z, = A//_W((z,_,) is welldefined and z„ is an approximate zero of f. Renegar and I [Renegar-Shub, 1992] use a version of these theorems to give a simplified and unified proof of the convergence properties of several of the recent polynomial time linear programming algorithms. For one vari able, Kim [1988a] also considered the algorithms of Theorem 3. Smale [1986] also estimates y(f,z) for a polynomial / in terms of the norms of the coefficients, the norm of z, and the norm of the derivative of/ at z. With this estimate, he was able to prove: Theorem 4 [Smale, 1986]. The average area of approximate zeros for f e Pd{l) is greater than a constant c > 0, where c is independent of d. Smale [1986] also dealt with a regularized version of the linear program ming problem which I will not consider here. Smale [1985a] asserts: A study of total cost for algorithms of numerical analysis yields side benefits. It forces one to consider global questions of speed of convergence, and in so doing one intro duces topology and geometry in a natural way into that subject. I believe that this will have a tendency to systematize numerical analysis. This development could turn out to be comparable to the systematizing effect of dynamical systems on the subject of ordinary differential equations over the last twenty-five years. We have already seen geometry. In [Smale, 1987], he turns his attention to topology. To begin with, Smale defines a (uniform) algorithm for a problem. An algorithm is a rooted tree, the root at the top for input. Leaves are at the bottom for output. There is an input space J a state space Sf, and an output space 0 which are finite-dimensional real vector spaces.
1045
28. On the Work of Steve Smale on the Theory of Computation
291
input
compute
Branch
j : y - » y rational
P: if -* R polynomial
'^y
Eoutput
There are two additional types of nodes computation and branching nodes. Computations are rational functions from if to if. The input and computa tion nodes have one existing edge. The branch nodes have two and which is chosen is determined by a polynomial inequality p(x) > 0 or p(x) < 0. The input and output functions are rational. These algorithms are also called tame machines; they are limited, for exam ple, by the fact that there are no loops. A problem is a subset X H*{X) and K{f) = {y e H*(Y)\f*(y) = 0}, where H*(Y) is the singular cohomology ring of Y. The cup length of K(f) is the maximum number of element y1,...,yko{K(f) s.L the cup product y, u • • • u yk # 0. Smale [1987] proves: Theorem 5. Let f:X-*Ybea problem. The topological complexity of f is bigger than or equal to the cup length of K(f\
1046
292
M. Shub
He applied this result to prove, via a complex algebraic topology computa tion, the main theorem of Smale [1987]. Theorem 6. There is a e(d) > 0 such that for all 0 < e < e(d) the topological complexity of the s-all root problem for Pd is greater than (log2 d)2'3. This result has been vastly improved by Vasiliev, [1988]. Levine [1989] has also worked on n-dimensional analogues. In the fall of 1987, Lenore Blum and Steve Smale were visiting at Watson. Steve began extending his model of computation from tame machines to allow loops and to work over ordered rings. Soon all three of us were involved. First, we specify a ring, the functions we compute on the ring, and the branching structure. Our functions are polynomial or rational, involving only a fixed finite number of variables, and having a finite number of nontrivial coordinates. We branch on # 0 or = 0 and ^ 0 or 0 or
DNPw PR
% -*
PAR->EXPw
PARw
where an arrow —> means inclusion, an arrow —► means strict inclusion and a crossed arrow *+means that the inclusion does not hold. Other results in [Cucker, Shub, and Smale 1994] include a proof of the equality BP(PARw) = PSPACE/po/y. Here PSPACE denotes the class of subsets of {0, l } 0 0 decidable in polynomial space.
2
Solving equations
2.1 Algorithms for deciding the feasibility (and finding solutions, if ap propriate) of complex systems of polynomial equations, or real systems of 2
The polynomial hierarchy is an increasing sequence of complexity classes between P and EXP widely believed to be strict, i.e., no two classes in the hierarchy coincide.
1062
polynomial equations and inequalities, have a long history. For the most part, at least concerning feasibility, they rely on algebra or, more precisely, on elimination theory. These algorithms have several virtues (for instance, they show that NP is included in EXP, the class of problems decidable in ex ponential time). But they are slow and they do not appear to be stable when implemented with floating point numbers. A possible reason for these draw backs is that these algorithms solve in exponential time all input systems. Therefore, they have to deal, on equal footing, with a collection of ill-posed systems (e.g. feasible overdetermined systems, systems with multiple roots, etc.). The tradition in numerical analysis suggests a different strategy for the design and analysis of algorithms. A condition number is associated to an input. Several features of the algorithm and of the output corresponding to the input will depend on this number. In particular, ill-posed inputs, those having infinite condition number, may produce exceptional behavior of the algorithm. Condition numbers were originally introduced to measure the sensitivity of a given input (for a specific computational problem) to perturbations. If