Diophantine Approximations And Diophantine Equations

Lecture Notes in Mathematics Editors: A. Dold, Heidelberg B. Eckmann, Ztirich F. Takens, Groningen 1467 Wolfgang M. S...

Author: Wolfgang M. Schmidt

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

1467

Wolfgang M. Schmidt

Diophantine Approximations and Diophantine Equations

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

Author Wolfgang M. Schmidt Department of Mathematics. University of Colorado Boulder, Colorado, 80309-0426. USA

Mathematics Subject Classification (1991): 11J68, 11J69, 11057, 11D61

ISBN 3-540-54058-X Springer-Verlag Berlin Heidelberg New York ISBN 0-387-54058-X 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 other ways, and storage in data banks. Duplication of this publication or parts thereof is only permitted under the provisions of the German Copyright Law of September 9, 1965, in its current version, and a copyright fee must always be paid. Violations fall under the prosecution act of the German Copyright Law. © Springer-Verlag Berlin Heidelberg 1991 Printed in Germany Typesetting: Camera ready by author Printing and binding: Druckhaus Beltz, Hemsbach/Bergstr. 2146/3140-543210 - Printed on acid-free paper

Preface The present notes are the outcome of lectures I gave at Columbia University in the fall of 1987, and at the University of Colorado 1988//1989. Although there is necessarily some overlap with my earlier Lecture Notes on Diophantine Approximation (Springer Lecture Notes 785, 1980), this overlap is small. In general, whereas in the earlier Notes I gave a systematic exposition with all the proofs, the present notes present a varirety of topics, and sometimes quote from the literature wihtout giving proofs. Nevertheless, I believe that the pace is again leisurely. Chapter I contains a fairly thorough discussion of Siegel's Lemma and of heights. Chapter II is devoted to Roth's Theorem. Rather than Roth's Lemma, I use a generalization of Dyson's Lemma as given by Esnault and Viehweg. A proof of this generalized lemma is not given; it is beyond the scope of the present notes. An advantage of the lemma is that it leads to new bounds on the number of exceptional approximations in Roth's Theorem, as given recently by Bombieri and Van der Poorten. These bounds turn out to be best possible in some sense. Chapter III deals with the Thue equation. Among the recent developments are bounds by Bombieri and author on the number of solutions of such equations, and by Mueller and the author on the number of solutions of Thue equations with few nonzero coefficients, say s such coefficients (apart from the constant term). I give a proof of the former, but deal with the latter only up to s = 3, i.e., to trinomial Thue equations. Chapter IV is about S-unit equations and hyperelliptic equations. S-unit equations include equations such as 2 ~ + 3 y = 4 z. I present Evertse's remarkable bounds for such equations. As for elliptic and hyperelliptic equations, I mention a few basic facts, often without proofs, and proceed to counting the number of solutions as in recent works of Evertse, and of Silverman, where the connection with the Mordell-Weil rank is explored. Chapter V is on certain diophantine equations in more than two variables. A tool here is my Subspace Theorem, of which I quote several versions, but without proofs. I study generalized S-unit equations, such as, e.g. -4-a~1 -4- a~ 2 + ... -4- a~" = 0 with given integers ai > 1, as well as norm form equations. Recent advances permit to give explicit estimates on the number of solutions. The notes end with an Epilogue on the abc-conjecture of Oesterl$ and Masser. Hand written notes of my lectures were taken at Columbia University by Mr. Agboola, and at the University of Colorado by MS. Deanna Caveny. The manuscript was typed by Ms. Andrea Hennessy and Ms. Elizabeth Stimmel. My thanks are due to them. January 1991

Wolfgang M. Schmidt

Table of Contents

Chapter

Page

I. S i e g e l ' s L e m m a s a n d H e i g h t s . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1. 2. 3. 4. 5. 6. 7. 8. 9.

Siegel's L e m m a . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . G e o m e t r y of N u m b e r s . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Lattice Packings ....................................................... Siegel's Lemma Again .................................................. Grassman Algebra .................................................... Absolute Values ....................................................... H e i g h t s in N u m b e r F i e l d s . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Heights of Subspaces .................................................. A n o t h e r V e r s i o n o f Siegel's L e m m a . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

II. D i o p h a n t i n e A p p r o x i m a t i o n . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1. 2. 3. 4. 5. 6. 7. 8. 9. 10.

Dirichlet's Theorem and Liouvilte's Theorem .......................... Roth's Theorem ....................................................... Construction of a Polynomial .......................................... U p p e r B o u n d s for t h e I n d e x . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . E s t i m a t i o n of V o l u m e s . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . A V e r s i o n of R o t h ' s T h e o r e m . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . P r o o f of t h e M a i n T h e o r e m s . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Counting Good Rational Approximations .............................. T h e N u m b e r of G o o d A p p r o x i m a t i o n s t o A l g e b r a i c N u m b e r s . . . . . . . . . . A G e n e r a l i z a t i o n of R o t h ' s T h e o r e m . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

III. T h e T h u e E q u a t i o n . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1. M a i n R e s u l t . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2. P r e l i m i n a r i e s . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 3. M o r e o n t h e c o n n e c t i o n b e t w e e n T h u e ' s E q u a t i o n a n d Diophantine Approximation ....................................... 4. L a r g e S o l u t i o n s . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 5. S m a l l S o l u t i o n s . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 6. H o w t o G o F r o m F ( x ) = 1 to F ( x ) = m . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 7. T h u e E q u a t i o n s w i t h F e w Coefficients . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 8. T h e D i s t r i b u t i o n of t h e R o o t s of S p a r s e P o l y n o m i a l s . . . . . . . . . . . . . . . . . . 9. T h e A n g u l a r D i s t r i b u t i o n of R o o t s . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 10. O n T r i n o m i a l s . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 11. R o o t s of f close to ~ . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 12. P r o o f o f 7 A . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 13. G e n e r a l i z a t i o n s of t h e T h u e E q u a t i o n . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

1 1 3 7 9 11 18 22 28 32 34 34 38 42 45 48 49 52 57 63 69 73 73 76 83 85 86 91 99 100 106 111

116 119 124

Viii Table o f C o n t e n t s (cont.) Chapter

Page

IV. S-unit Equations and Hyperelliptic Equations . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1. 2. 3. 4. 5. 6. 7. 8. 9. 10. 11. 12.

S-unit Equations and Hyperelliptic Equations . . . . . . . . . . . . . . . . . . . . . . . . Evertse's Bound . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . More on S-unit Equations . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Elliptic, Hyperelliptic, and Superelliptic Equations . . . . . . . . . . . . . . . . . . . . The N u m b e r of Solutions of Elliptic, Hyperelliptic, and Superelliptic Equations . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . On Elliptic Curves . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . The Rank of Cubic Thue Curves . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Lower Bounds for the Number of Solutions of Cubic Thue Equations .. Upper Bounds for Rational Points on Certain Elliptic Equations in terms of the Mordell-Weil Rank . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Isogenies . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Upper Bounds on Cubic Thue Equations in Terms of the Mordell-Weil Rank . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . More General Results . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

V. Diophantine Equations in More T h a n Two Variables . . . . . . . . . . . . . . . . . . . . . . 1. 2. 3. 4. 5. 6. 7. 8. 9. 10. Epilogue.

The Subspace Theorem . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Generalized S-unit Equations . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Norm Form Equations . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . A Reduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . An Application of the Geometry of Numbers . . . . . . . . . . . . . . . . . . . . . . . . . P r o d u c t s of Linear Forms . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . A Generalized Gap Principle . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Small Solutions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Large Solutions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . P r o o f of Theorem 3B . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . The a b c - c o n j e c t u r e

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

127 127 129 134 137 142 147 159 163 165 169 173 175 176 176 180 182 186 188 192 193 199 200 203 205

Bibliography . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

209

Index of Symbols . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

216

I. Siegel's L e m m a a n d H e i g h t s §1. S i e g e l ' s L e m m a . Consider a system of homogeneous linear equations a l l x l + " " + al,~xn = 0

•

(1.1)

amlX 1 _.}_. . . qt_ amnXn = 0

If m < n and the coefficients lie in a field, then there is a nontrivial solution with components in the field. If m < n and the coefficients lie in Z (the integers), then there is a nontrivial solution in integers. (Just take a solution with rational components and multiply by the common denominator.) It is reasonable to believe that if the coefficients are small integers, then there will also be a solution in small integers. This idea was used by A. Thue (1909) and formalized by Siegel in (1929; on p. 213 of his Collected Works). L E M M A 1. S u p p o s e that in (1.1) the coet~cients aij lie in Z and have [alj[ < A (1 Q1 > (r~P) m = r f " P m > r f " A m/k = r f ~ A ' ' / ( " - ' ~ ) .

Therefore every integer solution ( X l , . . . , xk, yl , . . . , Ym ) # ( 0 , . . . , O) has

max(Ix, l,..., Ixkl)

>

~TmA'~1("-').

3

Here 77 may be taken arbitrarily close to 1. Another approach is as follows. When m = n - 1, consider the system of equations A x i - Xi+l = 0

(i = 1 , . . . , n - 1).

Every nontriviM solution, in fact every nontriviM complex solution, has x , / x l Thus if we set q(x_) = max Ix,/x¢ t,

= A ~-1.

with the maximum over i, j in i < i, j __= c ( n ) A 11("-1) = c , ( n , m ) A ml('~-'O > O.

This approach can be carried out for general n, m. See Schmidt (1985). §2. G e o m e t r y o f N u m b e r s . The subject was founded by Minkowski (1896 ~¢ 1910). Other references are Cassels (1959), Gruber and Lekkerkerker (1987), and Schmidt (1980, Chapter IV). A lattice A is a subgroup of R " which is generated by n linearly independent vectors __bl,... ,b n (linearly independent over Rn). The elements of this lattice are clb 1 + .-. + c , b with Ci E Z .

////////

/

J

/

-////)// -

~

The set b l , . . , ' =bn is called a basis. A basis is not uniquely determined. For example, bl,b I + b2,=b3,... , b is another basis. How unique is a basis? Suppose __b'l, b' is another basis. Then n

6' Z

=i =

j=l

cijb j

and

cij C Z

and

n

b = E cijb ' ~ and =j

ctjiEZ.

i=1

So the matrices (cij) and (c~i) are inverse to each other and cij, c~i E Z, so det (c/j) = det (c~i) = +1. Thus the matrix (cij) is unimodular, where by definition a unimodular matrix is a square matrix with integer entries and determinant 1 or - 1 . L E M M A 2A. A necessary and sumcient condition/'or a subset A of R n to be a lattice is the foI1owing: (i) A is a group under addition. (ii) A contains n lineaxly independent vectors. (iii) A is discrete. For a proof, see e.g., Schmidt (1980, Ch. IV, Theorem 8A). Consider R n with the Euclidean metric and A a lattice with =ba,... ,b n as basis. Let II be the set of linear combinations Alb 1 + ..- + A,b n with 0 =< A~ < 1 (i = 1 , . . . , n). T h e n H is called a fundamental parallelepiped of A.

h, T h e fundamental parallelepiped does depend on which basis is chosen. T h e volume of II is given by V(H) = [det ( b l , . . . , bn) I where the right-hand side involves the matrix whose rows are respectively made up of the coordinates of b l , . . . ,b n with respect to an orthonormal bases of ~n. This volume is independent of the chosen basis of the lattice, since different bases are connected by unimodular transformations. It is an invariant of the lattice. We define det A = V(II). Notice that when __--| b. = (hi1,... , bin), then

V 2 = det

bll b21 .

\ bnl

b12 b22

.."'"

bn2 ... bib

bin ~ b2n I bnn

)

bib 2

det

iz bll b21 "" [ b12 b22 ""

/

.

\ bin .-.

b2n

° • ,

bnl "~ bn2 ) bnl.

l

=bibn )

= det

where the inner product of vectors x_, y is denoted by x y.

(2.1)

Every x_ in R " may uniquely be written as x = x' + x__"where x_' E H and x_" E A. X_ =

~ib i = i=1

{ ~ i } b i 71i=1

[~ilb i . i=l

~"n

eA

Here we used the notation that uniquely = [~] + {~} where [~] is an integer, called the integer part of ~, and {~} satisfies 0 < {~} < 1 and is called the fractional part of ~. i A

Z " is a lattice with basis ___el,... ,_e where___ei = ( 0 , . . . , 0, 1, 0 , . . . , 0), (i = 1 , . . . , n), and with det Z " = 1. If A is an arbitrary lattice with basis b l , . . . , b , then there exists a linear transformation T such that Te__i = b i (i = 1 , . . . ,n). So T Z " = A. Is T unique? Suppose T Z " = T ' Z " . Then ( T ' ) - I T Z " = Z", so det ( ( T ' ) - ~ T ) = 4-1 and ( T I ) - I T is unimodular. Call it U. Then T = TIU. Observe that det A = l d e t

T I.

T H E O R E M 2B. (Minkowski's First Theorem on Convex Sets.) Let B C R " be a convex set which is symmetric about the origin (i.e., x_ 6 B if and only if -x_ 6 B ) of volume

V(B) >

2" aet A

(2.2)

where A is a lattice. Then B contains a non-zero lattice point. C o m m e n t s . The volume here is the Jordan volume, i.e., the Riemann integral over the characteristic function of the set. Every bounded convex set has such a volume. Let g_ E B be a non-zero lattice point in B. Then - g ~ 0 and -g_ E B by symmetry, so B contains actually at least two non-zero lattice p~nts. P r o o f . (Mordell, 1934). V ( B ) / d e t h is invariant under non-singular linear maps. Therefore, after applying a linear transformation, we may assume that A = Z". Then the theorem reduces to: If V ( B ) > 2 n, then B contains a non-zero integer point. Let Bm be the set of rational points in B with common denominator m. Then IS~l

, V(B)

as

m--* oo

Tt~ n

where [ [ denotes the cardinality. For m sufficiently large, IBm[/m" > 2" and thus [B,n[ > (2m)". So there are two points a_ = ( a l / m , . . . , a , / m ) , b = ( b l / m , . . . , b a l m ) in Bm with ai - bi ( m o d 2 m ) (i = 1 , . . . ,n). Then 1

- b)

B

since the m i d p o i n t of___aa n d -=b is ½( a - b) E B. Let _g = ½(a - _b). Clearly g is a n o n - z e r o integer point. E x e r c i s e 2a. If B is s y m m e t r i c , convex, a n d V(B) > 2'~k det A w h e r e k E N, then B contains at least k pairs of non-zero lattice points. A convex body is a c o m p a c t , convex set containing 0 as an interior point. Such a b o d y clearly has 0 < V(B) < oo. R e m a r k 2 C . If B is a s y m m e t r i c , convex b o d y a n d V(B) > 2 n det A, t h e n B contains a n o n - z e r o lattice point. It is easy to show t h a t 2C follows f r o m 2B, a n d vice versa. R e m a r k 2 D . T h e o r e m 2B is best possible. Take A = Z n, B the cube with [xi[ < 1, (i = 1 , . . . , n). T h e n V(B) = 2" = 2 n det A a n d there are no n o n - z e r o integer points in B. Minkowski defines successive minima as follows: Given B, A where B is s y m m e t r i c , convex, b o u n d e d , a n d with 0 in its interior, let hi = inf {h : h B contains a non-zero lattice p o i n t ) . * M o r e generally, for 1 < j < n, Aj -- inf{h : h B contains j linearly i n d e p e n d e n t lattice p o i n t s ) . T h e n 0 2 n det A. So 3A is a strengthening of Minkowski's T h e o r e m 2B. R e m a r k 3B. For B convex, symmetric about '_0, one can show that 5(B) < 1 except for certain polyhedra. E.g., the cube has density 5 = 1. So do regular hexagons in the plane.

Let 5. = 5(B) where B is a ball in R". Consider the following picture.

The "triangle" lattice A has det A = 1 V~. It is easy to guess that

det A

This had already been proved implicitly by Gauss in his theory of positive definite binary quadratic forms. We know the values of ~2, g3,-. • , 68; see Cassels (1959, Appendix) and Exercise 3b below. The estimation of 6n for large n remains among the central unsolved problems in the Geometry of Numbers. Blichfeldt (1929) proved that ~, < 2-n/2(1 + ~). Also, ~,~ > (½ - e) n if n > n0(e). See Cassels (1959, p. 249). More recently, G.A. Kabatjanskii and V.I. Leven~tein (1978) have shown that ~, -< 2 -°'599"(1-~) for n > One may in a fairly obvious way define a general (not necessarily lattice) packing of a set B, and the maximum general packing density. For a disk in R 2, Thue (1892) had shown that the maximum packing density is in fact achieved for a lattice-packing. It is not known whether a similar result holds for a ball in R 3. It is generally believed that the densest packing density of a ball in R " where n is sufficiently large is less than the smallest lattice packing density. Now V(B)A~ < 2 ~ det A~(B), so that A1 < 2(det A)I/n(~(B)/V(B)) a/". For the unit ball B, V(B) = Y(n) = ~rn/2/F(1 + 2), so that by Stirling's formula we have the asymptotic relation v(nU"

= V(BU"

~

-* oo. n

We define Hermite's constant 7- to be least such that for any lattice A

ha = 5

n

( v ~ A) m/("-'O

if[a~i[ < A for e v e r y i , j . (ii) Also, there is a non-trivial integer solution z_1 with

I~=11 =< (la11...1_%1)1/("-r")

=< (v~A)'/("-').

In the first inequality we used t h a t 7 - - " < ~ ( n - m ) < -2n,~ if n - m => 2, a n d 7 - - - , = 1 < 3-n if n - m = 1, so t h a t n > 2. It is clear t h a t we do not have to restrict to t h e case w h e n the m equations are i n d e p e n d e n t . R e m a r k 4 E . If Minkowski's Second T h e o r e m (2E) is used, (ii) can be s t r e n g t h e n e d to get the following: there are n - m linearly independent solutions x l , . . . ,X__n_r. of our s y s t e m of equations such that

)~1 )1~1 - -. Ix=._ " r _5 I~1 I - " )~" ). T h e first assertion can be s t r e n g t h e n e d in the s a m e way, b u t this is not so obvious.

§5. G r a s s m a n

Algebra.

) W e t h i n k of a__x,... , a m as v e c t o r s w i t h m c o m p o n e n t s in t e r m s of a n o r t h o n o r m a l coordinate s y s t e m in S k±

12 (Also presented in Schmidt (1980), Ch. IV.) Notation: Let K be a field, K " a vector space, and __ei = ( 0 , . . . , 0 , 1 , 0 , . . . . ,0) be basis vectors. Suppose 0 < p < n. Let C ( n , p ) i be the set o f p - t u p l e s a = { i l , . . . , i p } w i t h i i E Z and 1 < il < i2 < - " This set has cardinality

< ip < n.

(;)

Let E

be the formal expression E

~O"

= e~--t . 1 Ae. A . . . A e l.l .l t p ~i 2

There are ( vn ) such expressions. For p = 0, let E• = 1. Let K~rt be the vector space over K generatedby E with a e C ( n , p ) . Then d i m ( K ; ) = ( ,ltl) . Elements of K~rt are called p-vectors. S p e c i a l cases: K p = K n, K~ = K , and K~ is spanned by the single vector e_l h ~ A . . - A e = ~ . We now introduce a more flexible notation. For any p and any integers i l , . . . , i v between 1 and n, the symbol e h - . . A e , should be 0 i f i i = ij, for s o m e j ~ jl. Otherwise, if { i , , . . . ,iv} = { j , , . . . , j , } , (considered as unordered sets), where j l < j2 < "'" < jp, then e. A . . - A e . =-t-e. A . . . A e . with the + sign if we get the i's from the j ' s by an even permutation, and with the sign otherwise. Set Gn = K g {9 I~/~ @ . . - @ K 2. n Then dim Gn = ( o ) + ( 1n ) + " " + (,,) = 2n" We are going to make G , into an algebra over K . We need a product, or wedge A. By linearity, it suffices to define products of the basis vectors E . We set

1A1=1, . , 1 A (_eil A- "" A_eip ) = e .=Ix A . - - A e=zp (_ei A . . . A e :.1 1

~

)Al=e.

~1 1

A . - . A e:.1 p ,

(e_i A...A_e__ip)A(e_j~_ A...A=.he. )----=,:e, A...A=eip A=3~e....A=jqe . Initially, this is given for il < .-- < ip and j l < "" "jq, but clearly it remains true in general. We have -----s e. A ----J e. = - e=.J A ----t e.. This algebra is associative. Note that this fits in with the original notation e. A ... A e . . The resulting algebra is called the Grassman algebra, or ezterior algebra. If x_:,... ,x_v E K n, then x A - . . A x~ p E K ~ ~.

~1

Such a p-vector is called decomposable.

13 LEMMA

5A. Suppose x_i = (~il,... ,~i,) = )-~'=I ~ije__j (i = 1,... ,p). Then

$1^-..^x =

~

GE

u6C(n,p) where ~ is the p x p determinant [~ulwith 1 < i < p, j 6 o. For example, when n = 3,p = 2,

~1A~2 =

~11 ~12]

~1 ~22 ~ 2 +

~11%~13

T h e linear map with E12 ~-~ _~, =_Ela ~-~ - ~ , wedge produced with the cross product. When n = 4 , p - - 2 __z1 ^ x_2

:

I

~12 ~13]

(21 ~23 ~13 + ~22 ~23 ~3" ~2a ~-~ ~1 identifies K 3 with K 3 and the

[11 621

I

~13 ~21 ~22 E12 ~- "'" "~ ~23 ~24 & 4 '

which has six terms. When p = n,

-z_1

A...Ax

=p ----

• " "

~ln

• " "

~nn

"

E12...n

P r o o f . T h e left-hand side is linear in each vector =zi, the right-hand side is linear, too. So it suffices to consider the case when X_l,... ,x___p E {__el,... ,_e }. If two of the =,z"s are the same, then both sides vanish. So without loss of generality =,x = __ejl, (i = 1 , . . . ,p) with ji 6 { 1 , . . . ,n} distinct. Since both sides behave in obvious ways under permutations of vectors, we may suppose j l < j2 < "'" < jp. T h e n the left-hand side is E . = E where 7" = { j l , - . ,jp}. On the right-hand side ~ = 1 if a = r , ~-21""Jn

=r

but ~ = 0 if ~r # % since ~ is the determinant of the submatrix of (~ij) with columns

jl,...

,jp.

A consequence of L e m m a 5A is Laplace's expansion of a determinant after a set of rows. For simplicity we will deal only with expansion after the first p rows. Given p,q with p + q = n, and given a 6 C ( n , p ) , let # E C(n,q) be the complement of ~r in { 1 , . . . ,n}. Let e(a,#) be 1 or - 1 , depending on whether (a,~Y) is an even or an odd permutation of { 1 , . . . ,n}. Let =Xl,... ,_z_p,~ 1 ' " " ' y----q be in K " , and write •

=

~p

=

trEC(n,p)

--

=q

=

-rEC(n,q)

14 Then 271 A . . .

A x p A y I A " ' " A 1/ = =q

o,7 -E

^E

~'6C(n,p) r6O(n,q)

=

aEC(n,p) =

.

a6C(n,p)

By L e m m a 5A, X__1 A

"'"

AX

=P

Ay I

A""

Ay

=q

=

(detM)E12...n

where M is the matrix with rows X l , . . . ,xp, Y l ' " " ' y " We therefore have =q

LEMMA

5B. (Laplace expansion of a determinant.) detM=

~

e(a, # ) ~ a .

aeC(n,p)

Note that by L e m m a 5A the ~, are the (p × p)-determinants from the rows x l , . . . ,xp of M, and the r/a are the (q × q)-determinants from the complementary rows

Yl'''"

' 1/ " =q

LEMMA

5C. x I A . . . A =px = =0if and only if =x1' . . . , x_p are linearly d e p e n d e n t .

P r o o f . It is an immediate consequence of Lemma 5A. LEMMA

5 D . Ilia__1,... ,X__p are linearly i n d e p e n d e n t and Y=I'"" '1/

=p

i n d e p e n d e n t , then £1 A . . . A x

=P

is proportional

to

Yl A

"'"

are linearly

A 1/ if and only i f x l , . . . =p

=

,z__p

and Y=I' " " ' 1/ span the s a m e subspace of K " . =p

Proof.

If the x__'s and _y's span the same subspace, then each !/. (i = 1 , . . . ,p)

is a linear combination of =xl,"" ,_x_p, so that Y l " " . ,yp is a multiple of _xI A -.. A =px. The factor is the determinant of the coefficient matrix for the y.'s in terms of the ~t

x.'s. Conversely, suppose that ~--1 x A ... A :xp = A(y 1 A .-. A =1/p ). For any _x, the vector =.1 = -x A (x I A .-. A__xp) = x A_£1 A - . . A x is zero precisely when x lies in the space spanned __xp)= )~(1/i A Yl A - . - A y=p ) = 0 s i n c e t w o y . ' =t soccur. by x__l, .. . ,z__p. But ~i A (-£1 A .-. A __ SO ~i is in the space spanned by z_l,... ,x__p(i = 1 , . . . ,p). Therefore the spaces are the same. Let S p C_ K " be a subspace of dimension p. Let X__l,... , x___pbe a basis of S p. Then let X = __xt A . . . A _xp, which is a vector with ~ = ( pn ) components and which lies in K ~ ~ K t. T h e components of X are called the " G r a s s m a n coordinates of S / ' . By the lemma, the Grassman coordinates are given up to a factor. Grassman coordinates of distinct p-dimensional subspaces are not proportional. Incidentally, the Grassman

15 coordinates in general do not fill all of K ; , i.e., not every p - v e c t o r is decomposable. A heuristic a r g u m e n t is t h a t the p - d i m e n s i o n a l subspaces of K n constitute a "manifold" with p ( n - p ) degrees of freedom, so t h a t the G r a s s m a n coordinates should be a manifold of dimension p ( n - p) + 1, and for most cases with 0 < p < n we have p ( n - p) + 1 < ($) = dimg;. Now suppose t h a t K = R or C. Make R " into a Euclidean space or introduce a H e r m i t i a n metric on C n with e . e . = gij (i < i, j < n). T h u s in the H e r m i t i a n case, if _x = ( ~ 1 , . . . , ~,), Y = ( r / l , . . . , r/,,), then _xy = ~ / 1 + . . - + ~,~/,. Introduce a similar metric on K ; with E E = =

=

0

= ~,

otherwise.

say. LEMMA

hE. (Laplace identity.) Given

Xl,... ,_Xp,--Yl'''"' _---p y in

have

^...

(=y ^ . . . A g )

=

IX

R " (or C n), we

.-

•

Here the inner p r o d u c t of the left-hand side is in R~ (or C~), b u t each inner product on the right-hand side is in R " (or C"). E x e r c i s e ha. Prove L e m m a hE, using linearity. A consequence of the Laplace identity is t h a t

ii= ~1 """ I_X_lA . . . A ~pl --

-Z-l--%

"

X1

"

"'"

1/2 •

X =p X ~-p

As we have seen, this is the volume of the p - d i m e n s i o n a l parallelepiped spanned by

~1,'" ,~=p" LEMMA

5 F . For a n y ~ l , . . .

(i) 1~1 A " " Axp A~I A " " A y [ ~ Ix I ^ " "

ifx.~uj = 0 ( 1 (ii) For a n y u l , . . .

=q

,z__p, =Yl'"" ' y '

^m, lly1 ^ . . . A ~ l, where equality holds ***

_< i _< p , 1 < j O.

P r o o f . One proves b y i n d u c t i o n on positive integers n > 0 t h a t In] < n, so t h a t also I - n I < n, a n d

Inl =< Inl~.

20 L e t a, b E Z with a > i, b > 1. For u > 0, we can write b" = co + c l a -4- " " -t- Cna n with 0 __< ci < a and c,, # 0. T h e n

IbV = Ib~l _5 Ic01 + I~111~1 + . . . + Ic, llal" < (n + i)aM"

__
]oq Iv for each v, which implies H K ( ~ ) iAIvl=~lv,so HK(),~=) = HK(~_)for A # 0 by the product formula. Example.

> 1. Also

IA_~lv --

Take K = Q. Let x = ( x l , . . . ,x,,) be a primitive integer point. We

have I~1~ = I~I (the usual Euclidean norm) and i~lv = m a x ( i x l l v , . . . , Ix-lp) = 1 for every p # oo, since x is primitive. So HQ(x___)= Ix_I.

E x a m p l e . Take K = Q(v~). Let ~ = (1, 3 + v ~ ) . We have I~lv, = ~/12 + (3 + v ~ ) 2 = v ~ , / 2 + V~, la[v~ = ~ , / 2 v~, I~-i~1 = t~[~, = 1, in fact i-~i- = 1 for ~ nonArchimedean in M ( K ) . So HK(~_) = [~_[,~I~_1,,, = 6 V~.

23 Suppose K C / ~ and ~ E K n, ~ # 0. Then how do HK(~=) and HR(a=) compare?

0EM(R}

veM(K)

0E~(R')

And (v) of section 6 gives

HR(~)= H

I~I.~[~:~

vEM(K)

= (HK(~)) tR:K] . The absolute height H(~) is defined by

H(~) = H~(~_)~/tK:Q1. Then H(a__) does not depend on the field K. R e m a r k . If K -- L and a : K ---* L an isomorphism, ~ E K " , a=~ 6 L " , then by (vii) HK(a__) = HL(a~)and H(~) = H(a~). So conjugate vectors have the same height. E x e r c i s e 7a. Is it possible to estimate H ( S + ~) in terms of H(_q) and H(_fl)? You would need to supppose a.q_# _0, fl # O, a + fl # O. If P ( X ) --- a , X " + . . . + a l X + ao with-ai E K is a nonzero polynomial, define the height by

HK(P) = H g ( a , , . . . , Oil, O~o). We can define H(P), the absolute height of a polynomial, in a similar fashion. LEMMA

7A.

H(PQ) < v/~ + 1 H(P)H(Q) when deg P = n, ( or n = min(deg P, deg Q)). P r o o f . Write P = a n X " + ".. + ao. Associate P with _q.a= ( a , , . . . ,a0). Define [PI~ = I~[v. Then

HK(P)=

H

IPl:~"

vEM(K)

Suppose v is non-Archimedean. Then

IPQI. =

IPIdQI~.

This is essentially Gauss' Lemma. We leave its proof as E x e r c i s e 7b. Now write Q = fl,~X m + " " + flo and PQ = 7,+,~X n+m + . . . + 70, where 7i = ~ a + b = i Ota~b. If V is Archimedean, then

2 [PQI2v : E h'il2v: ~ i i

"

E a+b=i

Ota~b

"

24

Cauchy's inequality implies t h a t N

2

N

__ 2. (Otherwise the result is clear.) Write P ( X ) = X d + Old_l X d - 1 -~- '' " -~ Ol0 = Q(X)(X

- "rd)

~:I a m i n d e b t e d to Prof. H a l b e r s t a m for simplifying m y original proof.

26

w h e r e Q ( X ) = X d-1 + ]~d--2x d - 2

+ ''" -~- ~0" T h e n i = O,i,... ,d-i,

O~i----~i-1 -- "/dfli,

with

fl_]

= o,

fld-1 = i.

Writing c = lTd], we get I~il 2 > (c]Zil ~ - I Z ~ - I ])~ = a l Z i l ~ + ]Z~-ll ~ - 2 c l Z ~ - , Z ~ ] _> c~]Zil ~ + ]Z~-,I ~ - ~(IZ~] ~ + IZ~-x] 2) = (c ~

-

c)l#~l

i)lfli_] 12,

(c-

= -

and, summing over i, d-1

d-2

d-2

1 + ~--~' Io.I 2 _> 1 + a - c + (c~ - ~)])-]~ In, l~ - ( c - 1) ~ i=0

i=0

I,a,I 2

i--0

d--2

= (~ - i)~ ~

In,l~ + ~ - ~ + i d--2

= (c - 1)2{i + ~

I/kl ~} +

c,

i=0

so that

IPI,, > (c - 1)IQI,, and

i

IQI. < ~_--:-i-IPI.. By induction, d-1

1-[(1 + hd~) 1/2 _5 5('i-~)/21Q1., i=1

so that

d 1 - I ( 1 .~_

2,1/2

(1

C2)1/25(d_1)/2

i

i=1

=< 5d/2JPiv, since c > 2. L E M M A 7C. Given n, d and B , there are only finitely m a n y non-zero vectors ( a_ = ( a l , . . . , a , ) with each ai of degree < d and H ( ~ ) < B , if you consider proportional vectors the same. P r o o f . Without loss of generality, consider vectors (1, a 2 , . . . , a,,). Then Hg(1,a2,...

,an) > HK(1,ai)=hK(ai)

(i=2,...,n),

27

since I(1, a s , . . . , ~.)1. => I(1, a~)l.. Thus it suffices to show there are only finitely many a of given degree d with h(a) >1, the total number of ~'s is

< 36.22dB 2d.

§9. A n o t h e r Version of Siegel's Lemma. Let K be a number field of degree d with embeddings U l , . . . , O"d into C. Given ~ 1 ' " " '-~m in K " , let S be the subspace of K " which they span. We have ~ i ) , . . . , a~) in K (i) (i = 1 , . . . ,d), where ai is the isomorphism K -+ K (i). Let /~ denote the compositum of K(D,. K (a), and S the subspace o f / ~ spanned by ~ 1 ) , a(1) _~d),. a (a) If rh d i m S , then rh < rod. Let wl, so that each a . has the form "

"

} ~

m

"

~

~---

~_j = wlx__il + . . . + wax=ja

"

°

"

,Wd be a field basis for K / Q ,

(1 =< j =< m)

33

with x_jt 6 Q" (1 < g < d). Then a(. i ) = w ( i ) x

-----3

1 rail

+...+w(i)x

d =jd

(1 < j < m , 1 < i .

.

.

.

0 (i = 1 , . . . , n). This system defines a parallelepiped of volume 2"A1 --~ An i det(aij)l . (1.2)

35 Suppose A a . . . A , > [ det(ai/)[. T h e n the volume of the parallelepiped is greater than or equal to 2", and the result would follow by Minkowski's T h e o r e m (2C) of Chapter I if we had a compact set. However, we have L E M M A 1C. S u p p o s e Ai > 0 (i = 1 , . . . , n ) and A1 . . . A n (1.1). T h e n the s y s t e m o f inequalities has a solution x_ E Zn\O.

> Idet(aii)[ > 0 in

E x e r c i s e l a . Prove L e m m a 1C. P r o o f (of T h e o r e m 1B). In R "+1, consider the system of inequalities J a l y - xl[ < N -1/"

loony -- xn[ < N - 1 / n

_-< N. By L e m m a 1C, there is a non-trivial solution. If we had y = 0, then x l , . . . , z , would all be zero, too. Thus y ¢ 0. Then there exists a solution with y > 0, therefore I < y < N. T h e seeond assertion of Theorem 1B follows just like in T h e o r e m 1A. T h e o r e m 1A was improved by Hurwitz (1891). He showed, for a irrational, that there exist infinitely m a n y fractions x / y with
0 a n d discriminant D = b2 - 4ac. T h e n for A > V ~ , there are only t~nitely m a n y fractions x / y with < a - Y

1 Ay----~ .

E x a m p l e . Consider the polynomial equation a 2 - a - 1 = 0. Here D = 5 and a = (1 + v/'5)/2. Using L e m m a 1D, we see that for A > x/5 there are only finitely many solutions to [o~ - ( x / y ) [ < 1 / ( A y 2 ) . Thus nurwitz's result is best possible.

36 P r o o f . Writing f ( X ) = a X 2 + bX + c = a ( X - a ) ( X - a') gives D = a~(a - a ' ) 2. Then if ]a - ( x / y ) l < 1/(Ay2), we have y2 =

o)

=

< - - {2 Ay 2 v~ < ~

O¢

-

O/

+ - -X y

OL

a + A~y----~.

Subtracting v/-D/Ay 2 from both sides gives 1(._~__~) y2 1 -which becomes

a < A2y 4 , a

y2
y~.

37 (c) Expanding P into a Taylor series at a, we get

P (y)

=

-

t P(i)(Ol) a

i=1

i!

'

since P ( a ) = O. We may assume that

(Otherwise, we're done.) T h e n

1 < ]p ( ~ ) [ ya

=

< o~ y [ ~ [e(i)(°~)[ i! i=1

=

Let c ( a ) be defined by

IP(i)( )I i=1

i~

1 - 2c(a) ;

then the result follows. C O R O L L A R Y 1F. (Liouville) T h e n u m b e r a = Eve¢=I 2 -~: is t r a n s c e n d e n t M . Liouville was first to exhibit transcendental numbers, in fact first to prove the existence of such numbers. P r o o f . Write y ( k )

_

2 k' a n d z ( k ) = 2 k' zX-~k . , v = , -9-u! . T h e n x ( k ) , y ( k )

and

e Z (k = > 1)

o~

y(k

- =k+ 1 = 2-(k+1) ! + . . . < 2- 2 -(k+l)! = 2 / y ( k + 1) < c/y(k) d

for any given c, d, provided that k > ko(c, d). Hence, for any d, we have cr not algebraic of degree d by Liouville's T h e o r e m (1E). T h e numbers which can be proved transcendental by Liouville's T h e o r e m are called "Liouville numbers". T h e y form a set of measure zero. This explains why Liouville's T h e o r e m is not enough to prove the transcendence of classical numbers such as e or ~r. E x e r c i s e l b . Given a E R and N > 0, there exist x, y E Z, not both 0, with N[c~y - x[ + N -1 lY[ < v'~.

38 Now use the arithmetic-geometric inequality to show that given ~ E R \ Q , there are infinitely many rationals ~ with a

x[ 1 --y 2 and p > d, then a-y

1 (d/2) + 1. Siegel (1921) in his thesis improved this to p > 2 v~. Dyson (1947) and Gelfond (1952) showed that the result holds for p > v ~ . In 1956, Roth received a Field prize for his 1955 result with # > 2. Dirichlet's Theorem shows that Roth's result is best possible. T H E O R E M 2A. (Roth (1955).) Ira is algebraic and 5 > 0, there are only finitely many rationals *with y -

1

< y~+~.

Remarks.

(i) Roth's result is correct but trivial for a E C\R. (ii) If deg a = 2, then Lemma 1D is better. (iii) We know that there are infinitely many ~ with

¢X--y < - and only finitely many ~ with

tO~- ~xt < y2-{-----' 1 -~ with 5 > 0. For any given ~ with deg a > 3, it is still unknown whether o~ is badly approximable, i.e. whether there exists a c > 0 so that

>5 for every rational -.~ The conjecture is that this holds for no algebraic a of degree >3.

39

(iv) Another conjecture is that Roth's Theorem holds in the following strengthened form: the inequality

Oe-y
1. The following theorem gives heuristic grounds for the conjectures in (iii) and (iv). T H E O R E M 2B. (Khintchine (1926).) Suppose ¢(y) > 0 is defined on the positive integers and ¢, is nonincreasing. Consider the inequality

]~-

~

¢(Y)

qd. T h u s r/q > d, and we get no i m p r o v e m e n t over Liouville's Theorem. This a r g u m e n t can be modified by using a polynomial in m variables. T h u e used a polynomial in 2 variables of the form P ( X 1 , X 2 ) = X2Q(X1) - P(X~). Siegel used a m o r e general polynomial in two variables, and so did Dyson and Gelfond. Only Roth was able to overcome the difficulties involved in dealing with more t h a n 2 variables. To see why a polynomial in m variables offers an advantage, consider P ( X 1 , . . . , X m ) E Z [ X 1 , . . . ,Xm] of degree at most r in each variablefl Such a polynomial is m a d e up of m o n o m i a l s X~ 1 . . . X~,"* with 0 < i l , . . . , z,, < r. The numbers of such monomials is (r + 1) m, so the n u m b e r of possible coefficients is also (r + 1) m --~ r '~ as m --+ oc. Try to m a k e P vanish at ( a , . . . , a ) of order q. T h e n P ( J ' ..... J ~ ) ( a , . . . , a ) = 0 for j l + ' " + j m

C(O/)

yd-Ct ( C*)

where c(a) > 0 and c~(a) > 0 are effective. Unfortunately, cl(a) so obtained is usually very small. Then further improvements for special numbers were obtained by Baker and Stewart (1988), Bombieri (1982), Bombieri and Mueller (1983), Chudnovsky (1983).

42

§3. C o n s t r u c t i o n o f a P o l y n o m i a l . We will follow Bombieri and Van der Poorten (1987). We will construct a polynomial P ( X 1 , . . . Xm) E Z[X1,... ,Xm]. For any such polynomial P, define Iel to be the m a x i m u m absolute value of its coefficients. If I = ( i i , . . . ,ira), then let

pZ _

1 Oq +'"+i"~P il!i2!'" i,d OX~ ~... OXi~' '

Then pZ E Z[X1,... ,Xm] for P E Z[Xx,... ,Xm]. If P has degree ___2 am

(h = 1,... , m )

and rh

Vh = V[~

(h = 1,... ,m).

Let p ( z ~ , . . . , x , , ) • z[x~,...xm] be such that p # o and

Suppose P is of multidegree < R. Then the index of P at ~Yl ~ with respect to R is at most ~. See Roth (1955), Cassels (1957), or Schmidt (1980) for a proof. Roth's Theorem m a y be proved either by using Roth's Lemma or Theorem 4B below. Neither of these will be proved in these Notes. The proof is by induction on m. Here we will only consider the (trivial) case m = 1. Let P ( X ) E Z[X] and ~Yt a rational with gcd(xi,yl) = 1. We may write " " "

where M (~-~) w ~ 0 and t is the order of vanishing of P at ~-~. vl We P(X)

=

(ylX

--

'

Ym

can

also write

Xl)tQ(X)

where Q (~--~) v, # 0. Since P ( X ) E Z[XI and ( y l X -- X I ) has integer coefficients a n d content 1, we get Q ( X ) E Z[X] by Gauss' Lemma. Thus the leading coefficient of P is divisible by y~ and

yf_ t. Thus P I ( a l , . . . , a m ) = 0 for every I with -~ • ~ ( t , R ) . The n u m b e r of such conditions is given asymptotically by r ~ r 2 . - , r m W (t,-~). Suppose __al,... ,_a_k • C m and the index o f P at_a m with respect to E i s > th (h = 1 , . . . , k ) . T h e n the total n u m b e r of conditions is approximately k h=l

If these conditions are independent and P ¢ O, then ~ =k1 W (th, R) should not be much larger than 1. THEOREM 4 B . (Esnault and Viehweg (1984).) Suppose r l => r 2 => . . . and let a_q,... , a_k E C m with the condition that if a = (O~il , a i m ) , then ---:1

air#air

if

i•j

'

"

"

= > rm,

°

(1 < g __<m).

Suppose P ( X I , , . . , X m ) E C ( X , , . . . X m ) of multidegree < R with P • O, and the index of P at a_h with respect to E = ( O , . . . e m ) is > th ( h = 1 , . . . , m ) . Then

h=l

j----1

i=j+l

where k' = max(2, k). Bombieri (1982) did the case m = 2 before the general case was done. He called this Dyson's Lemma, in reference to work done by Dyson in 1947. For the m = 2 case, the bound is slightly better, namely,

EW

h=l

th,

=1+

-rl

Viola (1985) gave another argument for the m = 2 case. He removed the condition ail 7~ a j l , ai2 7~ aj2 for i 7~ j and imposed the condition that P ( X 1 , X 2 ) have no factor of the form X1 - c or X2 - c. T h e o r e m 4B is algebraic in nature. The proof involves a lot of algebraic geometry and will not be given here. Now suppose K is a number field of degree d. Let _.a = ( a l , . . . , a m ) E K m with Q ( a i ) = K (i = 1 , . . . , m ) and fl = (fll,..-,/~,~) E Qm. We will apply T h e o r e m 4B of Esnault and Viehweg with k = d + 1 and _aO),... ,~(d),/~. T h e gth coordinates are a ~ l ) , . . . ,a~ d), fit, which are all distinct. We will set tl . . . . .

td = t and td+l = r.

47

If P ( X 1 , . . . , X m ) # 0 of multidegree R = (r~,... ,rm) has index __> t at (i = 1 , . . . , d) and index > r at fl with respect to R, then Theorem 4B gives

dW(t) + W(r) Oq=30212m"

Consider

W(t)e a°'/2m
m

v/(log 2 d ) / m / 6 > - 25

,

we obtain

Io/--/31 < h(/3) -2-12 ~

(4h(o/)) -12x

< hA(o/, fl)-2-12

~l(log2d)/m.

We now proceed as with T h e o r e m 6A: If there is no a p p r o x i m a t i o n / 3 with h(/3) > B2, then we are finished. Otherwise, . . . .

§7. P r o o f o f t h e M a i n T h e o r e m s ,

i.e., T h e o r e m s

6C, 6D.

Suppose (o/1,fll),... ,(Otrn,flrn) are given such t h a t Q ( a i ) = K and fl, E Q 1 , . . . , m). Pick t = t(m, A) such t h a t d W ( t ) = 1 - (l/A), and 7" such t h a t W(7") = We will use L e m m a 3A to construct a polynomial P ( X 1 , . . . , X , , ) of multidegree ( r l , . . . , r m ) , where the ri are large, such t h a t P has index > t at ( h i , . . . , a m ) respect to R. We can choose r l , . . . , rm such that ri+l < _ _ 1

ri

(i = 1,..

. ,m

(i = 2/A. R = with

-- 1).

= 2dmA

T h e n by L e m m a 4C, we know t h a t P will have index < r at (/3t,... ,/3,~) E Q. T h u s there exists an I = ( i l , . . . ,ira) with

i~ + . . - + i., < r rl rm

(7.1)

such t h a t p I ( ~ ) # 0. Set Q ( X ) = P I ( X ) . T h e n by (3.1) we have IP--71 = IQI < 2rlPl where r = rl + . . . + rm. Moreover, by (3.2) we have IQJI < 3~lPI for any J . Since P had index => t at c~, it follows from (7.1) that Q = p I has index => t - r at _a (with respect to R). Since dW(t) 1 - (l/A)

1 - dW(O we have

-

(1/A)

-A-l,

dW(t) _ 1 d W ( t ) (1 + ~) < A

for ¢ > 0 sufficiently small. T h e n by L e m m a 3A, the polynomial P can be constructed such t h a t I p I __< ( ( 4 h ( a l ) ) r , . . . (4h(a,,,))r,~)x.

53

Then IQ"I < a"lPl < 3r((4h(al)) r' " " ( 4 h ( a , ~ ) ) r ' ) a. Writing 13i = x i / Y i (i = 1 , . . . , m ) , we have y[' . . . y~"Q(/3) • g \ { 0 } . Thus uP"-

u,~"Q(~')

=> 1.

Writing the Taylor's expansion for Q about ~ gives

Q(_~) =

~

(/3~ - ~lY' ... (/~m - ~,,)i~ QJ(~_)

J=(/~ ..... jr.)

~+...+~

_>_,-~

where the sum is restricted to ~ + . - - + ~,, => t - r since all lower order partial derivatives vanish at e. By condition (iii), we have

] a i - fli[ < 1/2

(i --~ 1~... ~m)~

which gives 1 Io, d < 1,8'~1+ ~ =

." [

Ix,I + (1/')./I,. I 1 + ~ = ~

(i = 1 , . . . ,m).

Applying Cauchy's inequality to the right-hand side gives

I~,1 =< v'T~h(fl,)

(i =

1,... ,m).

Yi Now we have I q s ( ~ ) l < 1~-71

,-~ + 1

*

-.-

"'

and therefore

y["" "yDqs(a=) < 3rlPl

(r~ + 1)

h(5~) i----1

.

54

The Taylor expansion for Q gives us y[' • • • y~,~Q(__fl)[

max

" ' l~

] ~+...+~_>_,-~

(1/31 - ~, Ij '

IZ~

-~1 j') ri + 1) 2

= IPI

3

max

h(fli)

(Icq - fill i t - . . la,~

i=1

-flmlJ"). So for r i large,

luP -. • u2" Q(__Z)
¢

min

~ot+'--+~m > t--r ~oiEIL

(~lrlL1 +--.¢pmrmLm).

=

Now choose ri = [L/Li] (i = 1 , . . . , m ) where L is large. By condition (iv), we have Li+l > 3 d m / k L i (i = 1 , . . . , m - 1), so that ri+l

1

ri

2dm~

- - < - -

(i=1,

"

.. m - i ) . '

55 Dividing (7.3) by L and letting L --~ co, we get m > ¢

min

(~1 + " " + q0m) = ¢ ( t -

r).

~l'+'"-t-'.Pm > l - - r

Hence

¢
2d. 4 m. This gives 1< 1 W(t) < ~ = ~ . and

w(~) < ~ _ so that we can apply L e m m a 5A in both cases. T h a t is, tm

w ( ~ ) = m--i' d~-]=

1-

,

t = (m!) l/m

1-

,

and T m

W(r)

=

2

i

m!'

vm

)~ m!' r = (m!)'/'(21~) So we have

t - r = (m!)'/md -'Ira > (ml)'i"d-'l" = (m!)'lmd-'l"(1

((

1-

((l-

'/'.

- (2dl,~) '/"

)

~) -(2dlJ)'l")

- '7)

where

= (11,~) + (2dl,~) li"

(7.5)

~, 2(2dl,t )'l"

< 112

(by (ii)).

56

Observe that

1

l-r/

< I + 2 r / ~ 1 + 4(2dlA) 1/m =

Therefore t

-

r ~(m!)ll'*d-'l'~(1

+

(by (7.5)).

4(2dlA)'l')-'.

Now we are in a position to e s t i m a t e ¢ with

IZi - ail < hA(ozi,~i) -¢. B y (7.4) we have

¢5

m t--7"

< (Fr~!)l/rn m 31/.,( ID -, -1 -t- 4(2dlA) l/m) = cmda/m(1 + 4(2d/A)l/m). However, in T h e o r e m 6C (iii), we have

]Oti -- flil 2 gives us W(t) < 1/2. T h e r e f o r e t < m/2, say t = ( r n / 2 ) - 8 with 0 < 8 < m/2. T h e n by L e m m a 5B,

w ( t ) < ~-°2/'', so t h a t

Now we h a v e

eO21m < d ( l _ l )

-1

< 2d

(by (ii)).

T a k i n g the l o g a r i t h m of b o t h sides gives

8< ~

log 2d

and t > (m12)-

4-A-~2~.

So t - ~ > (m/2) - 1 - v/~iog2d.

N o w we may return to our task of estimating ¢. By (7.4), ¢
1, a set S of positive numbers is a C-set, if for any y, y~ in S we have y~ 1 be given. A 7-set is a set of positive real numbers with the following property: if y, y~ are in the set and y < y', then y' > 7Y. Thus a "~-set has a certain "Gap Principle". A set which is b o t h a C - s e t and a 7-set will be called a (C, 7)-set. Its elements are positive real numbers, not necessarily integers. LEMMA

8A. Suppose C > 1 and 7 > 1 are given.

The cardinality of any

(C, 7 ) - s e t is < 1 + ( l o g C ) / l o g 7. P r o o f . Let G > 1 and 7 > 1 be given. Suppose V0 < yl < y2 < "'" < y~ belong to a (G, 7)-set. T h e n

vi _->y0- 7 i

(i = 0 , . . . ,

and

Cyo > y~ > y07 ~. Therefore u __< (log C)/log u,

~)

58 and the cardinality of the (C, 7)-set is =< 1 + (logC)/log3'. Suppose 5 > O. Let L = log(1 + 5). LEMMA

(s.1)

8B. Let a reM number ~ be given. The number of reduced fractions

(~/~) with xI

1

y < 2y2+-------g

-

(8.2)

and y in a window of exponentiM width C is

C)/L.

< 1 + (log

P r o o f . If y, y~ are in a window of exponential width C, then yt < yC. We will call this an exponential C-set. Now, if x/y, x'/y' satisfy the hypotheses and x / y • x'/y', say y~ => y, then

I

(1 + 6 ) l o g y - log2. Now suppose that xo/Yo, x l / Y l , . . . , x , / y , • .. = < y~. T h e n

are such approximations with y0 < yl
(1 + 6)log y0 - log2, logy2 > (1 + 6 ) l o g y l - log2

__>(1 + 6)2 log yo - ((1 + 6) + 1)log 2,

l o g y , => (l q- 6)" logy0 -- ((l + 6) "-1 + . . - + ( l q - 6 ) > (1 + 6)"(log W - (log 2)/6). Since W = > 41/6, we have log W __>(log 4)/6 = 2(log 2)/6 and l o g y . > (1 + 6 ) " ( l o g W ) / 2 . We also have y. < W c, so that C l o g W => l o g y , => (1 + 6 ) " ( l o g W ) / 2 . Thus (1 + 6) ~ =< 2 C

and

u = < (log2C)/L. T h e lemma follows.

T1)log2

60 THEOREM where B > e, is

8D. The n u m b e r of 5-approximations with denominators y < B , < L -1 log log B + 20((1/6) + 1).

Recall, L = log(1 + 6), as in (S.1). This result, as well as T h e o r e m 8E below, is due to Mueller and Schmidt (1989). P r o o f . We will say that "large solutions" are those with e ~/~ __ h(/3) c,,,d ' / " (, +( x / ))((3h(2 a )d)-2 Y x/2)~,,,d'/'~.

Since

u > (3h(oO") ~/x, we have

(3h(ot)d)-2y x/2 > 1, so that

y~' > h(t3)c~t'/'(i+(xt2)).

We therefore obtain

Is -/31 < h(3) -cmd'/~°-(x/2)). Apply T h e o r e m 6A with X/2 in place of X. Then we have either h(fl) < B1, or h(fl) lies in the union of at most m - 1 windows of exponential width C = 6dmA where )~ = 2d(12/X) m. We also know that the first case is ruled out because h(fl) > y > B1. Therefore, by L e m m a 8C, the number of solutions is not greater than 1+

log 2C log(# - 1)"

We know that 2C = 2d2(12/x)mm, so that log 2C > log# >> logd. So the n u m b e r of solutions in such windows is B1 is 41°, then h(a) > (Sh(a)) 1/~. Our interval is a window of exponential width not greater than log B1 = 4d(12/X),,+l. log(Sh(c~)) 1/2 We also have 5 = tt - 2 > 2.1 - 2 = 1/10, so that 41/~ < 4 l°. Then by Lemma 8C, the number of approximations is not greater than

log(Sd(12/X)) m+'

1+

log(~

-

1)

'

which is 3, and 0 < 5 < 1. Then the number of 5-approximations to ~ is less than log + log h(o~) +e(d,5), L

(9.2)

where

-~-(log 2d) log

log 2d

.

This theorem estimates the number of "exceptional" approximations in Roth's Theorem. Davenport and Roth (1956) had given an estimate with a s u m m a n d exp(70daS-2). The latest results are by Bombieri and Van der Poorten (1988) and by Luckhardt (1989). Both use the Theorem of Esnault and Viehweg. In Theorem 9B, the first term in the estimate is best possible (see Theorem 9C below), but the c(d, 5) term can probably be improved. Actually Bombieri and Van der Poorten had 3000 in place of 10 s, but they also had

c~--~
5/2 in (9.2). P r o o f . Put m = [(50/5) 21og2d] and A = 2m m. Then A > d. Consider "small solutions" to be those with

y =
B2. As in the previous proof, if ,~ = x/y, then h(~) < 3h(a)dy. Consider y2+~ =

y2+(~/2)y6/2

> hG3)2+(~/2)(3h(a))-3dyS/2

> h(¢~)2+(~/2~. The last inequality follows because y > B2 and X > d. So y2+~ yields 1

h(¢~)2+(~/2)" Apply Theorem 6B with ~/2 in place of 6. Then either h(~) < B2 (which in our case is ruled out) or h(/~) lies in the union of at most m - 1 windows of exponential width C = 6drnX. Notice that B2 > 4 2/~, so we can apply Lemma 8C with ~/2 in place of 6. This tells us that the number of approximations with h(~) in the given window is not larger than log 2C 4 + 1 =< ~ 1 o g 2 C + 1

log(1 +

< 5 log 2C. o

We will estimate log2C. We know that 2 C = 24din re+l, so that log 2C = < 2m log m + log 24d.

68

Then the number of approximations in a given window is
0 be given. Then there are infinitely many a E K, with K = Q(a) and h(o0 > e,

such that the number of 5-approximations to a is greater than co(K, 5).

loglog h(a) L

R e m a r k . We could drop the condition that h(a) > e if we replace log log h(a) with log + log h(a). This result is due to Mueller and Schmidt (1989). P r o o f . We may choose 7 E K with Q(7) = K and I'rl < 1/2. We also construct

thesequence {P-~n} a s i n T h e o r e m S E . G i v e n N > 1, let b N b e t h e l e a s t i n t e g e r s u e h that bN > q~N +8. Then we may pick an integer aN with aN bN

"Y

[

1

loglogh(aN) _ co(K,~). =

§10. A Generalization

L

of Roth's

Theorem.

Let Q c k C K be algebraic number fields. As in §I.6, let M(K) be an indexing set for absolute values of K. We write

M(K) = Mo(g) U Moo(K),

70

where Mo(K) consists of non-Archimedean, and M ~ of the Archimedean absolute values. In most parts of these Notes, S will denote a finite set of the type M ~ ( K ) C S C M(K). H o w e v e r , in the present section we need only that S is a finite subset o f M ( K ) . Suppose that for each v 6 S we are given a linear form Lv = L~(X, Y ) with coefficients in K. We will study the inequality

IX IL'(x)lT°lxl-o < Hk(x) - ~ - "

(10.1)

yES

in unknowns x = (x, y) with components in k, where Ixl. = max(Ix[r, [y[~) and where n~ is the local degree. Since (10.1) is unaffected if we replace x by a multiple, x in projective space Pl(k). 10n. Given ~ > O, (10.1) has only finitelym a n y solutions x 6 pl(k). This is due to Lang (1962). Earlier generalisations of Roth's Theorem were given by Ridout (1958). Now why does this actually give Roth's Theorem? Suppose a is algebraic, and suppose THEOREM

~_~ < 1 lyl2+ .

(10.2)

Then ]o~y- x] < Cl]X[ - 1 - t

with a constant C1. Further if a = a O ) , . . . , a(d) are the conjugates of a (in C), then - x J . . . I (d)y -

xl =< C21xl d-

-t

Set L , ( X , Y ) = a Y - X for each v Then if k = Q and K = Q(a), we have H vEMcc(K)

IL'(x)[~u = f i [a(/)y - x[ < C2lx[ - 2 - " = C2Hk(x) d-2-~, i=1

or

II ] L ' ( x ) I ~ < C2Hk(x)-2-e" v6M~(g) ]xln~ By Theorem 10A this has only a finite number of solutions, x 6 pl(Q), so that (10.2) has only finitely many solutions x/y. A more quantitative version is as follows. THEOREM 10B. Suppose K-is of degree 5. Suppose these are not more than t distinct forms Lv for v 6 S. Define [Lvlv and HK(Lv) in terms of the coe~cient vectors

71

of L~, and suppose that HK(L~) < H t'or v 6 S. Then for given C > 0, the number of solutions

H (' IL~(x)l~

)°o< CHK(X)-~-~

in x E Pl(k) with Hk(x) > Cl(6, t,e)(C + H + 1) c2(6't'')

is less than

c3( , t, where s = Card S. As Evertse, GySry, Stewart and Tijdeman (1988) point out, this theorem can be proved by making Lang's arguments more explicit, and combining them with ideas of Davenport and Roth (1955). But no explicit proof of T h e o r e m 10B has been published. T h e following exercises are not on the material of this particular section but could have been given earlier. E x e r c i s e 10a. Let B be a symmetric convex body in •"

and A a lattice. The

inhomogeneous minimum of B with respect to A is defined as the least # such that A + # B covers R", (i.e., every x 6 R " may be written as g + x with l 6 A and y E #B). Prove that # is well-defined, and that 0 < # < n)~n/2, where )~, is the n t h minimum. For n = 2 and B a disk centered at the origin we have the following picture.

E x e r c i s e 10b. Let c~ 6 R be irrational. We call (x, y) 6 Z 2 a best approximation to a if y is positive, l a y - x [ < 1/2, and if for any other pair (x', y') with 1 < y' < y, we have lay' - x'] > lay - x[. Show that one gets an infinite sequence of best approximations, say ( x l , y l ) , ( x 2 , y 2 ) , . . . , with yx < y2 < "'" and 1

l a y i - x i l _-< - Yi+l

(i = 1 , 2 , . . . ) .

72 E x e r c i s e 10c. Let a E R be irrational, and for N > 1, let H(N) be the parallelogram lay-

zl =< ~1,

lyl =< N .

(10.3)

Since the area of P ( N ) is 4, Minkowski's Convex Body Theory says that the first minimum satisfies ~1 = AI(N) =< 1. Show that there are arbitrarily large values of N with A2(N) < 1. (Hint: This should follow from Exercise 10b.) E x e r c i s e 10d. Combine Exercises 10a and 10c to show that if a, fl E R, where a is irrationM and fl is not of the type fl = m a + n with m, n E Z, then there are infinitely many (x,y) E Z ~ with y > 0 and lay-

~-

x I < l/y.

R e m a r k s . Exercise 10d is a quantitative version of the one-dimensional "Kronecker's Theorem", which only asserts that we can solve l a y - ~ - xl
3 which is irreducible over Q. R e m a r k . Such a form F can never be irreducible over C. First consider

F ( X , 1) = aox d + a i X d-i j r . . . Jr ad = ao(X - (~1)"" (X - (~d) with a l , . . . , ad algebraic of degree d and conjugates of one another. Then

F(X,Y) = ydF (X,1)

= ao(X - - o ~ l Y ) . . . ( X - - a d Y ) .

T h e o r e m 1A. (Thue, 1909). Let F as above and m be given. The equation (1.1)

F(x, y) = m has only t~nitely many integer solutions (x, y). R e m a r k . Today, equations of type (1.1) are called Thue equations. R e m a r k . Theorem 1A is false for d -- 2. Consider, for example, x 2 -- 2y 2 =

1.

This equation factors into (x + v

y)(x - v S y ) = 1.

If D -- Z[x/2], i.e., the ring of elements x + vf2y with x,y E Z, then e = x + v ~ y and g = x - v ~ y axe units in D. In particular, we can take x = 3 and y -- 2. Then ~0 = 3 jr 2x/~ is a unit. For each n > 1, the number ~ is also a unit, which gives a solution e~ = x , jr v ~ y , to x 2n _2y2 = 1. For example E02 = 17jr 12v/2, so that x2 = 17, y2 -- 12. P r o o f . Factoring F(x, y) over C, we can write a0(

-

-

dY) = m .

(1.2)

Then dividing by yd and taking absolute values gives ]a0[]C~l-y "'" ~d--Yl---- ~ddl" We have, without loss of generality, Ix -- aly[

=

min

l 3, Roth's Theorem implies that there is only a finite number of solutions

(~,u).

E x e r c i s e l a . The proof of Liouville's Theorem 1E of Chapter II uses implicitly that

If( x, Y)I > 1 for integers x, y. (Actually, it uses that IP(z/y)l > 1/y d for a polynomial P E Z[X] of degree d which does not vanish at x/y.) Employ Roth's Theorem to show that for F as in Theorem 1A,

IF(x,y)l _->c 0 ( F , e ) ( m a x ( I x l ,

lyl) a - z - "

> 0.

E x e r c i s e l b . In Theorem 1A, one may weaken the hypothesis on F. Rather than supposing that F is irreducible over Q, assume that at least three of the complex numbers a l , . . . , ad in (1.2) are distinct. The methods of Thue, Siegel, and Roth are "ineffective" in the sense that they don't yield a bound A = A(F, m) such that any solution (x, y) satisfies max(]xh lyl) < A. Alan Baker (1967) remedied this situation. Thue's method, however, can be used to give some upper bound on the number of possible solutions. Lewis and Mahler (1961) gave a bound

B = B(d, m, g), where H =

max

lail. For m a n y years, it had been conjectured that a better bound

~ pl1[. The following lemma is now obvious. L E M M A 2C. Let d be fixed and let ~ be a class of norm forms which is closed under nonsingular substitutions F : Z n ~ Z n. Let N¢ be the maximum number of solutions o f F ( x ) -- 1 over all F in ~. Let N¢ (p) be the maximum number of solutions of F(x) = 1 over ~ F in ~ with D ( F ) >= # l . Then

N¢ < (V + 1)No (V). R e m a r k 2D. The lemma can be modified in two ways. Rather than counting solutions of F ( x ) --- 1, we m a y count solutions of F ( x ) E S, where S is a given set of integers. Secondly, the lemma remains true if instead of counting all integer points we count only primitive integer points. Now let us restrict to n = 2 and F ( X , Y ) as in Thue's Theorem, and of degree d. Let WE(m) be the number of integer solutions of the Whue inequality [F(x, Y)I < m and PF(m) the number of primitive integer solutions of the same inequality. Let P~F(m) be the number of primitive integer solutions of the inequality

m/2 ~ < IF(x, u)l _5 m.

(2.4)

Furthermore, let N ( m ) = max.NF(m), and likewise for P ( m ) and P'(m). F P R O P O S I T I O N 2E. Suppose F is a form as in Thue's Theorem and D ( F ) >=(50ma/d) 2111. Then

P V m ) (hK(a(i))) s (i = 1 , . . . , d). By L e m m a 3C, some root a satisfies x o:

-

d 2dhK(a)d-2rn
50din, we have (4.2)

[a - y

3, we have d=~

1d

7d

+8

1

> !d

> -8d + - 2 - d = 8

+

v/-d "

Then (4.2) for large solutions yields x

- y

d

1

(4.3)

< yd/Sy3 V~ /2 < ya Vq /-----~ "

As an application of Theorem 9A of Chapter II, we saw that the number of solutions of (4.3) with y > h(c 0 is h g ( a ) s > h((~). Thus the number of large solutions for the given root ~ is I/~ -/3~llD(xo,x)l --->( I g - ~ l

1

fi

- 2

2)lD(x°,x)[

~)lD(xo,x)h

= (oiif ~/i = ]g - f l i ] + 1. For each i, p u t Q t

r/i=

if

T/i if 1

if

7/i__>Q, Q1/2d
(O'~' - ~ ) Now suppose that ~bi > 1/2d. Then Q~' > ~

(i=1,. ,d)... > 7 by (5.1), (5.2), and

1/ILl(x)[ > Q¢'y/2 > Q¢,/2y

(i = 1,... ,d).

That is, IL,(x)l _5 llQ¢'/2y

(i = 1,... ,d)

and

a ( i ) - - Y l < I/Q¢'/2y2

(i -- 1 , . . . , d ) .

We state this in the following lemma. L E M M A 5C. Suppose F. is a form as above (normalized and reduced) and x = (x, y) is a primitive solution of our inequality, i.e.

m / 2 d < [F(x, y)[ < m

(5.3)

with y > O. Then there exist numbers ¢i = ¢i(x) (i = 1,... d) as above with

]a(i) - yx I
Q¢'("J)l~yj/2 => Q¢,(,,j)/4 yj. (This relationship between yj and Yj+x can be thought of as a "variable gap principle".) Since we have y~,/yl < Y , this gap principle tells us that v--1

YI Q~,(xj)/4 __ 0, and d is even. Then F ( x , y ) = 0 for real x , y would imply that x = y = 0. So F ( x , 1) has no real root. However, F ( x , y ) = 1 has d non-trivial integer solutions, namely +(1, 1 ) , + ( 1 , 2 ) , . . . , 4-(1, d). This example was communicated to me by M. Waidschmidt. §6. H o w t o G o f r o m F ( x ) = 1 t o F ( x ) = m. We now know that the equality F ( x ) = 1 has P-".

(ii) If A1,.. • , A , - 1 and A'1 , - - - , A.-1 ' b o t h occur in anchors and if Ai 0 (i = 1 , . . . , n - 1) and ]z1 "'" ~n--1 => p--U by observation (i). Hence Vl + el

_~_ Yn--1

< U•

en--1

Suppose two different anchors have v l , . . . ,vn-1 and v ~ , . . , v~n-i with vi =< v i~ (i = 1 , . . . , n - 1). By applying observation (ii) to the corresponding A1,... ,A,,-1 and A~, . . . , A In _ i , we find t h a t vi = v i' (i = 1 , . .. , n - 1). Therefore, v n - 1 is unique once V 1, . . . , U n _ 2 a r e given• It remains, then, to count the n u m b e r of non-negative V l , . . . , vn-2 with Vl 71el

+ Vn--2 < U• en--2

99

Since ei 0"

Since all of the roots of f* lie in Izl < 1, we have t h a t all the roots of f lie in ]z] < e ~ = e ~(~(r))+x. T h u s L e m m a 8C (i) is true for k = p. For 0 < k < p, consider instead the polynomial ilk]

:o(z) = ~

a,z d'

i=0

and the corresponding

f;(z) = yo(e ~z) = ~

a~z"

i=0

with a given by (8.4). Applying our previous results to f0, we know t h a t all of the di[k] roots of f~(z) lie in Iz] < 1. We claim t h a t exactly di[k] roots of i f ( z ) are, in fact, in Izl < 1. Once this is proven, we will know t h a t exactly di[k] roots of f(z) lie in [zl < e ~ = e ~(~(k))+~.

106

Using Rouch6's Theorem to prove the claim, it will suffice to show that

lf;(z)- f*(z)l < lf;(z)l for

lzl=

x.

Then the polynomials f0 and f~ will have the same number of roots in the disk ]z I < 1. But for [z I = 1, we have [ f ~ ( z ) ] - If~(z) - i f ( z ) [ >= [ai*[k][- E

i•i[k]

>la,'t 11(1 ---~0.

]a[l

1

1

3

9

'

1

1

3

9

"")

To prove part (ii), the lower bound for the zeros, put ] ( w ) = w a f ( w1) . Then the Newton polygon of ] is obtained from the Newton polygon of f by a reflection through the line x = d/2. Under such a reflection, sharp vertices remain sharp and the bounds follow.

Exercise 8a. Let f ( z ) be a polynomial with coefficients in a field E and I I a non-Archimedean absolute value. Define the Newton polygon as before. Suppose that f ( z ) has all of its roots in the field E. Then it is known (see, e.g., Koblitz (1977)) that one has a mapping from the roots to the segments of the Newton polygon such that roots corresponding to Pi(,,-1) Pi(~) have log Izl = ~(u), where a ( u ) is the slope of this segment. Now prove this statement for a trinomial. §9. T h e A n g u l a r D i s t r i b u t i o n o f R o o t s . In §8, we studied the radial distribution of the roots of a sparse polynomial f ( z ) . In this section, we will consider the angular distribution for binomials f ( z ) = az a + c and trinomials f ( z ) = az ~ + bzq + c. For a binomial, the roots make up a regular d-gon, so that the angular distribution is completely regular. In what follows, A will denote a wedge with vertex 0, i.e., a region bounded by two rays emanating from 0. Part or all of the boundary of A may belong to A. Write IAI = ¢/2~', where ¢ is the angle between the rays. We will consider the whole plane to be a wedge with IAI = 1.

0

W i t h this notation, we have the following result, which also holds (trivially) for b = O, i.e., for binomials.

107

T H E O R E M 9 A . Let f ( z ) be a trinomial of degree d. If Z ( A ) denotes the n u m b e r of roots of f which lie in A, then Z ( A ) - dlA I < 6 ~_

•

P r o o f . We may suppose t h a t b is real since the roots of f ( z ) are not affected when we multiply the polynomial by a suitable constant. P u t t = d - q, so that d = t + q, and write the equation as az t + cz -q = -b. We may assume, without loss of generality, that t > q so that t > d/2. Introducing the notation e(x) = e 2~I~, write a = lale(~), c = ]c]e(7), and z = ]z]e(C). T h e n

lal Izl' e(tC + ~) + Icl Izl -q e(-qC + ~) = - b . The imaginary part of the left-hand side must be zero, so we have lal Iz[t sin (2~r(tC + a)) = [c[ [zl-q sin (27r(q C - 7 ) ) . The left-hand side of this equality vanishes for some ~0, hence precisely for ~0 + ( m / 2 t ) , for m E 2g. The right-hand side may also vanish for one of these values. By a change of notations we can suppose that it vanishes at the same value {0. In that case, the right-hand side will vanish at Co + (m'/2q) for m' E Z. For the time being, we will require the additional hypothesis that gcd(t, q) = 1. 1 Thus, for the values T h e n m / 2 t = m'/2q is possible only when m / 2 t = m'/2q C=~Z. C = Co + (1/2t), C0 + ( 2 / 2 t ) , . . . , C0 + ((t - 1)/2t), ¢0 + ((t + 1 ) / 2 t ) , . . . , (0 + ( ( 2 t - 1)/2t), the left-hand side vanishes but the right-hand side does not. If arg(z) = ¢ and ( is one of the above values, then we see that z cannot be a root of f . T h e rays given by arg(z) = C, where ( is one of the above, are called "forbidden rays" because they contain no solutions. Furthermore, these rays are determined independently of b. Now let a # 0, c # 0 be fixed and denote the d roots of f as functions of b by z l ( b ) , . . . , zd(b). One can arrange this such that the zi(b) are continuous functions of t h e real variable b. By the continuity of zi(b), we see that the values of zi(b) for various b can not cross any forbidden ray.

forbidden ray

)

108

W h e n b = 0, the roots of f form a regular d-gon and

Z(A)-dlA

I = < 2.

Now let A be an angular domain bounded by two forbidden rays. We call such an A a "special angular domain". In that case, the continuity of the zi(b) gives

Z(A)-

diAl]

1 = N _

_

(tMizl q

-

-

d)

.

If z is a large root of g, then M lit 3 -4 < Izl, so that Ig'(z)l

> M1/t

'

34

( t M l + ( q / 0 3 -4q - d )

o

Furthermore, t M l+(q/t) 3 - 4 q > M > 2d, so that we have

Ig'(z)l >

Ml+tq/t)

= 2.3 4.3 4d.M1/t

~. M ( d - 1 ) I t .

If z is a small root of g, then ~ = 1 / z is a large root of the reciprocal polynomial ~(z) = q-z d -4- # z t -t- 1. In that case, I~1--< Ma/q 34 and I~'(~)1 >- M ( d - 1 ) / q • The original polynomial g ( z ) and its reciprocal polynomial ~(z) are related by the equation

114

g(~) = z~ get

~(~).

F r o m the p r o d u c t rule a n d the fact t h a t ~(~) = 0 for a root z of g, we

g'(z) = _~d-~ ~,(~) = _ ~ - d ~ , ( ~ ) .

Then

Ig'(z)t ~- M (2-d)/q MCd-1)/q-~

Ml/q

•

Now we look at case (ii). F r o m the first p a r t of the proof, we have gt(z)

~-- - - t # z q - 1

d

T-

g

w h e n z is a root of g. S t a r t i n g with the original p o l y n o m i a l g a n d differentiating twice, we get g " ( z ) = d(d - 1)z d-2 + # q ( q - 1)z q-2 = d(d-

1) g ( z ) -7

+ # ( q ( q - 1) - d(d - 1))z q-2 T

d(d-

z2

1) •

As previously, the first t e r m vanishes if z is a root. In t h a t case, g " ( z ) = # ( q ( q - 1) - d(d - 1))z q-2 T

d(d - 1)

z2

Now consider t h e expression z g ' ( z ) ( d ( d - 1) - q(q - 1)) - z 2 g " ( z ) t = ~:d(d(d - 1) - q(q - 1) - t ( d - 1)) = ~ : d q t .

T h i s e q u a t i o n implies t h a t either [zg'(z)(d(d-

1) - q ( q - 1))1 > d q t / 2

or

[z2g"(z)tl

>= d q t / 2 .

In o t h e r words, either

Izg'(z)l ~- 1

or

Iz2g"(z)l >- 1

B y L e m m a 10A, we have Iz] -~ 1, so t h a t ]g'(z)l ~- 1

or

[g"(:)i >'- i.

T h e two p r e c e d i n g l e m m a s dealt with t r i n o m i a l s of t h e f o r m g ( z ) = z d + p z q 4- 1. In general, we have f ( z ) = a z d + bzq 4- c E Z[z], where a, c > 0. P u t # = ba - q / d c - t / a

and

M=[#I.

T h e n f ( z ) = c g ( w ) , where w = ( a / c ) l / d z . Now the various cases will d e p e n d on H = m a x (a, Ibh c). We will s u p p o s e t h a t a, b, c are integers, so t h a t in p a r t i c u l a r a = > 1, c__> 1.

115 LEMMA

1 0 C . / . f M > 3ad and H = c, then every root o f f has

If'(z)l

~- n 1-(1/d) •

P r o o f . As we have seen, f ( z ) = cg((a/c)l/dz), so

lY'(z)l =

¢ ( a l c ) 1/d I g ' ( ( a l c ) ' / d z ) l

.

If z is a root of f , then (a/c)l/dz is a root of the special polynomial g, and by L e m m a lOB, we know that ~- 1 . Therefore,

Ig'((a/c)l/dz)l If'(z)l

~- c(alc) "lId = a l l d c l - ( i l d )

> H 1-(1/d),

since a > 1 . LEMMA

I O D . If M > 34d and H = Ibl, then every large root o f f has

If'(z)l where

~- P q H 1-(2/a) ,

p=(~b_~a)O-1/d)/t

( L ~ ) (Ud)/q

P r o o f . As before, we have f ( z ) = cg((a/c)l/dz) and

l/'(z)l

c(a/c) lid

=

Ig'((a/c)l/dz)l

N" c(a/c) lid M (d-i)/t

(Ibl~-~~/* ....

= \ ~-zzr-, ] = pqal/d

ibll-(~ld) ella

> pq HX-(21d) . LEMMA

1 0 E . Suppose M < 34d and a < c. Then every root o f f has either

If'(z)l ~-

or

n 1-v/a)

P r o o f i Since M -4 1, we have with w = (a/c)I/dz,

If"(z)l ~-

[b[ -< a q/a c t/d -.4 c

n ~-(2/a) •

and H -~ c. From the chain rule,

If'(z)l =

a lid c 1-°/d)

Ig'(w)l

If"(z)l =

a ~/d c 1-(2/d)

Ig"(w)l.

and By L e m m a 10B, either

If'(z)l >" a l / d

cl-(1/d)

~"

H 1-(l/d)

116 or If"(z)l ~-

a =/a c 1 - ( 2 / a )

>'- H 1-(2/a).

§11. R o o t s o f f c l o s e t o ~. We now return to the T h u e inequality

IF(x,y)l < m,

(11.1)

where F is the homogeneous polynomial given by

F ( X , Y ) = a X d + b x q Y t -4- cY d , with a, c > 0. We will see t h a t either there exists a root a of f ( z ) = F ( z , 1) which is close to ~, or there exists a root/3 of the reciprocal polynomial ] ( z ) = F(1, z) which is close to ~. We m u s t distinguish several cases.

Let H = m a x (a, c). Suppose that M > 3 4a and (x, y) is a solution to (11.1) with x ¢ 0, y ¢ O. Then either there is a root a o f f ( z ) -- F ( z , 1) with LEMMA

llA.

x ]

m H (1/d)-I

or there is a root fl o f f ( z ) = F(1, z) with

fl - -x y '< mHO/a)-l]x[ d

P r o o f . We m a y suppose t h a t H = c. In this case, we will see t h a t the first alternative holds. By L e m m a 3A, with the p a r a m e t e r u = 1, there is a root a of f with

O/

X[ -~

m ~ if,(a)l lyl ~ •

F u r t h e r m o r e , by L e m m a IOC, we have [if(a)[ ~- H 1-(l/d)

, so

that

x ] mH (1/d)-I T h e second case follows similarly when H = a. LEMMA l l B . Suppose M >__ 3 4d and H : ]b], and (x,y) is a solution to (11.1) with ~ # o, y # o. T h e . either there is a l ~ e root ~ o f f ( z ) = F(z, 1) with

o~

x - y

-~

m H (2/a)-1

I#

117

or there is a

large root/3 o f / ( z ) = F ( 1 , z ) with ?1 x

mH(2/d)-I Ix[ d

P r o o f . T h e polynomial f(z) has t large roots, say a l , . . . , a t , and q small roots, say 1/fll,... , 1//~q. T h e n the reciprocal polynomial ] has large roots /31,... ,/~q and small roots 1 / a l , . . . , 1/at. Let L = min (Ix -

a l y h . . . , Ix - atyl)

L = min ( l Y -

fllxh'.. , ] Y - flqxD,

and and consider the two real n u m b e r s

L(~b]) (I-lId)It

(.C._C '~(1-1/d)/q

L\lbl ] By s y m m e t r y , we m a y suppose t h a t

( a "~(1-1/d)/t < L ( c "~(1-1/d)/q L\lblj = \Ibl) We will see t h a t the first case follows. We have

(1
h(a) is 3, but noting t h a t factors 7? with T]a do not affect x = ~i(XTl) a and y = ~j(y,)d, we see t h a t the total n u m b e r of solutions is < 62 (4s) 2~ (4d) 268 < d2,(4s)2~ (4d) 26~. For instance, if d = 3, we get the b o u n d 32s (12) 26s (4s) 2~ =< s2~(12) 28s 42~ < s 2~ 1230s . So we have used the fact t h a t a Thue equation has only finitely m a n y solutions, to show t h a t an S-unit equation has only finitely m a n y solutions as well. One can see t h a t the converse is also true, i.e. t h a t the finiteness of the n u m b e r of solutions of S-unit equations implies the finiteness for Thue equations. We consider the T h u e equation F ( x , y) = m , where F ( X , Y ) E D s [ X , Y] is homogeneous with at least three non-proportional linear factors and m E D s . In some extension, F factors as F = L1 . . . L d , where the Li are linear with coefficients in a field N D K D Q • Let S ~ C M ( N ) consist of the extensions of the absolute values of S to N. We can assume t h a t m and the coefficients of Li lie in D8,. Consider the equation Ll(~_)...

Ld(x)

= m

(1.1)

and seek solutions x, y E D s , • Take linear forms L1,L2, L3 which are non-proportional. Since these are three linear forms in two variables, they must be linearly dependent. We have La = 11 L1 + 12L2

with

X1,~2 E N × ,

and Ll(X_) "~1 ~

L2 (x_) -~- )~2

--

La(x__)

- 1 .

129 By (1.1) above, we know that Ll(__x) I m in iDs, . Up to equivalence, there are only finitely m a n y divisors of m. Say LI(__x) = pu , where u 6 Ust, and there are only finitely many possibilities for p. T h e n Ll(x__)

where ul 6 Us, and there axe only finitey m a n y choices for pl- Similarly,

L~(Z) L3(~)

- p2 u2.

This gives A,plu, + A2p2u2 = 1

with u l , u 2 6 Us, • This is an S'-unit equation in u l , u 2 , and it has only finitely m a n y solutions. Thus there are only finitely m a n y possibilities for L l ( x ) / L 3 ( x ) and L2(x_)/L3(x). But these quotients determine x up to a factor of proportionality. It now follows immediately that (1.1) has only finitely m a n y solutions x.

§2. Evertse~s B o u n d . Let K be a n u m b e r field with [K : Q] = 6 and S C M ( K ) with card S = s, as before. Evertse (1983) gives a bound for the number of solutions of the S-unit equation a l x + a2y = 1

which is independent of the coefficients a l , a2 and which does not require that a l , a2 6

us. THEOREM

2 A . (Evertse (1984)) The equation O:lX + o~2y = 1,

where a,, OL2 6 K

are non-zero has at most

3 • 7 ~+2s solutions x, y in Us. Here, we will prove less, namely that the number of solutions is =< c , ( 6 ) e ~ .

For instance, if K = Q, then the number is < c~. First we reformulate the problem in projective space. T h a t is, we look at solutions to the equation ~ l x + ~ 2 Y - z = 0,

130 where (x, y, z) 6 P2(Us). O r let aa = - 1 and consider OllX + a 2 y + a s Z = O, which m a y be r e w r i t t e n as yl + y 2 + y a

=0.

T h e n we consider solutions where y l / a i E Us are considered the same. Let _a = ( a l , a 2 , a 3 ) and y = (yl,y2,ya). define 1-T A = !!

(i = 1, 2,3), and p r o p o r t i o n a l solutions As in C h a p t e r I, let (()~ = [~[~"~, and •

v~S

Since [yi/ail. = 1 for v ~ S, we have

H

v6S

(YlY2Y3)V 1 V~S 1 V~S ((~-~->3 (YlY2Y3)V) (y___)3 -- 1~ ~ (YlY2Y3)v ~,(y)3 (O/10/20/3)v _

v6S

=

= A Hk(y_) -a . Given y and v, let ( i , , j , , k,) be the p e r m u t a t i o n of (1, 2, 3) with the p r o p e r t y t h a t

[Yi~ [~ < [yj~ [~ < lYkv[.- If v is n o n - A r c h i m e d e a n , t h e n lYj~ tv = [Ykv[, since Yi~ + Yj~ + Yku -- 0. If v is Archimedean, t h e n [Yk~I- < 2[y/~ [. and [_y]. < v~[Yi~ ],. This gives

I_yl

'

where 1 a t , ~--

T h e n we have

if v is n o n - A r c h i m e d e a n , if v is Archimedean.

(Yi~ (y)~).._._~v < 36 A v6S = YI

Hk(y) -a .

(2.1)

This ties in with R o t h ' s T h e o r e m in its generalized form, t h a t is, T h e o r e m 10B of C h a p t e r II. We consider the variables as yl, y~, and we have the linear forms

L.=

Yl

if

i~ = 1

Y2

if

it,=2

-yl-y2

if

i.=3

We know t h a t m a x (lYll~, lY21~, ]Yl + Y2[~) =< 2 ~'(~) m a x (lYl[~, [Y2[~),

131 where

: { oi

if v is n o n - A r c h i m e d e a n if v is A r c h i m e d e a n .

L e t t i n g x = (yl, y2), the inequality b e c o m e s

T h e n (2.1) yields H

(L.(x_)>~ 6~ _ (L.>~x__>. < A HK(X_) -a

(2.2)

v6S

since (Lv)v => I. To count solutions, we apply Theorem 10B of Chapter If. W e get s

solutions with

gg(x__) > c1(6,t,c) (C + H + I) c2(~'''*) , where t is the number of distinct linear forms L., so that t = 3, and 6 = I, C = 12~A.

So we h a v e

< c3(*)4 solutions w i t h

HK(X=) > c1(6) (12~A + 2 + 1) c:(~) . Now we are left to count small solutions. We have

HK(X_) < Hg(y) < 2~HK(x) , a n d there are two possibilities.

If A < 12 n~, t h e n HK(y_) < Ca(6) . T h e r e are only

finitely m a n y solutions in this case, say < cs(6). T h e n we are left with the case where A > 1212~, a n d we have to count solutions with H K ( y ) < A c~(~). To do this, we need the following l e m m a . LEMMA 2 B . Suppose 0 < 7 < 1 and s 6 Z + are given. Then there is a finite set G of s-tuples (P1,.-. ,l'~s) o f non-negative reals with F1 + . . . + Fs = 7, such that for e v e r y x = ( x l , . . . , x s ) with xi > O, there is an s-tuple from 6 with xi > F i ( x l + . . . + x , ) (i = 1 , . . . , s). Fhrthermore, card ~ < c8(7)*.

E x e r c i s e 2 a . P r o v e L e m m a 2B. In the case s = 2, the e l e m e n t s o f ~ following picture.

lie on the line F1 + F 2

= 7. We have the

132

~

a ( r l , r~)

3'

1

It is easy to see that one may cover the line Xl + x2 = 1 with a finite n u m b e r of sets o = ~ ( r l , r ~ ) consisting of (xl,x2) with xi > Fi (i = 1,2). The lemma then follows for the case s = 2. Remark. As a consequence of this lemma, we have that for xi < 0, (i = 1,...,n),thereissome(F1,...,F,)•®suchthatxi 3, n > 2. Using the lemma, write x - ~1 = / ~ 1 y l d,

Then /~,yl - Z~y~ = ,~ - ~, ~

o,

which is a Thue equation in the variables Ya, Y2 since d > 3. So we have only finitely m a n y solutions yl, y2- Since x is determined by the/~i's and yi's, we have only finitely m a n y possibilites for x.

141

T h e remaining ease is when d = 2 and n > 3. As above, write X - - O ~ 1 -----

X --OL 2 =

X--

OL3 =

We need to solve this system of equations in x, yl, y2, y3 E First, we extend K so that it contains v/-~l, x/~2, sides will be squares, i.e., let zi = v/-~i yl so that x - ai 73 = ~2 -- or1 #- 0, and permuting the indices to get 71,72,

L~s. v ~ 3 . T h e n the right-hand = zi2 (i = 1,2, 3). Letting we have

zl2 - z~ : 73, -

71,

=

Z 2 __ Z 2

~=72.

Now the left-hand sides can be factored. We have, for instance, (Zl -- Z2)(ZI

"~- Z 2 ) : - "~3"

We write zl

-

z2

=

(4.2)

p3u3,

where u3 is a unit and (since Zz - z2 divides 73) where we may take P3 from a finite set. We also have Z 2 - - Z 3 ----- P l U l , Z3 -- Z1 =

/)2152.

Adding these last three equations gives plUl

-{- p 2 U 2 + p 3 U 3

== 0 ,

an S-unit equation. Hence there are only finitely m a n y solutions (u,, u2, u3) G P2(Us). We would like to know that there are only finitely m a n y possibilities for the zi (i = 1, 2, 3). T h e n it will follow that there are only finitely many solutions to the originM hypereUiptic equation in this case. So we consider zl+z2--

73 , p3u3

which in conjunction with (4.2) gives zl

=

pau3

+

p3U3

Similarly, by cyclic permutation, z2 =

plUl

+

plul

(4.3)

142 and

z3 = x

p~u2 +

p2u2

We also have directly from (4.2), (4.3) that

Z2 =

-~

paU3

-- psU3

•

Now the "Yi are fixed and we have only finitely many choices for the pi- There are finitely m a n y possibilities for (ul, u2, u3) up to equivalence in p2(Us). So suppose that we replace ui by Aui (i = 1, 2, 3). Equating the two expressions for z2 gives

( ")'1 (fllUl -[- f13U3))~ = -- P'~I

"f3 )~ PSU3 '

so A is determined (up to + ) unless plul + p3u3 = 0, which is impossible. In the next section, we will obtain estimates on the number of solutions. §5. T h e N u m b e r o f S o l u t i o n s o f E l l i p t i c , H y p e r e l l i p t i c , a n d S u p e r e l l i p t i c

Equations. Here we discuss relatively explicit bounds on the number of solutions of the various equations. These results are the joint work of Evertse and Silverman (1986). Let K be a number field of degree 6 and K × the multiplicative group of K . Let S be a finite set of absolute values which contains all of the non-Archimedean ones, i.e. Moo(K) C S C M ( K ) , and let s = card S. As above, let Ds denote the S-integers in K and Us the S-units. Consider polynomials f ( X ) E Ds[X] with discriminant A ( f ) E Us. Notice that this last requirement is not much of a restriction, since we may enlarge S to force A ( f ) E Us. Then the cardinality s will reflect the number of prime factors of A(f). In what follows, L is an extension of K with degree [L : K] = t. We will also have d > 2, and hd(L) will denote the order of the subgroup of the ideal class group of L consisting of elements [9.1] with [9/]d = 1. We will count solutions of the superelliptic equation yd = f ( x ) , (5.1) with x E Ds, y ~ 0, y E K . (Then automatically, y E Ds).

T H E O R E M 5A. (a) Suppose d > 3, n >__2, and L contains at least two roots of f . Then the number of solutions of(5.1) with x E D s and y E K* is < 17 t(6~+~) d 2ts ha(L).

(b) Suppose d = 2, n > 3 and L contains at least three roots of f. Then the number of solutions is 7t(46+9s) h2(L) 2.

143

R e m a r k . We may pick L with g < n ( n - 1) in case (a) and g < n ( n - 1)(n - 2) in case (b). Aside from the choice of L, the coefficients of the polynomial f do not enter into the estimates. In the case of an elliptic equation, one may conclude that the number of solutions is < c(¢)H 2+e, where H is the height of the equation. See Schmidt (to appear). Here we will prove a weaker form of the case (a). We will show that the number of solutions in (a) is

=< (c,-dU" hd(L). We need several lemmas first. L E M M A 5B. Suppose [ [ is a non-Archimedean absolute value on a field E. Let the polynomial f(X)

= anX n +...

- al)...(X

-I- ao = a ( X

- an),

be given with ai, ai in E and la, I _5 1 (i = 0,... , n ) , and also Ih(f)l = x where A denotes the discriminant. Then for every x E E with Izl 1, then ci ~-- 1. If, on the other hand, la, l~ _-< 1 and I~jl. _-< 1, then I(x - ai) - (x - % ) Iv = lai - a j l . = 1 by Lemma 5B. So only one of Ix - a i h Ix - %l can be strictly less than 1, thus only one of ci, cj can be strictly less than 1. Therefore, ci = 1 with one possible exception, and each ci E G d. T h a t is, I X - - Olll v

m a x ( l , I,~1~)

e a~,

(i = 1 , . . . , ~)

as desired. As in Chapter III, Section 13, suppose there are t non-Archimedean elements of S. These absolute values correspond to prime ideals ~ 1 , . . . , ~ t . Given fractional ideals ~t, ~ , we write 92 =__~3 (mod S)

145

if ~/fl~ is of the type ~3~' ... ~ ' with integers c l , . . . , ct. We write ~1 - ~B (mod S, d) if ~i/~B is of the type ~ t ... a t ' E~ where ~: is any fractional ideal. Consider the congruence in the variable z given by (z) - m (mod S, d) (5.4), where (z} is the principal ideal with generator z. If z is a solution and z' = zw d, then z' is also a solution. So it is valid to count solutions z E K X / ( K X ) d. LEMMA

5 D . The number of solutions of the congruence (5.4) in g × / ( g x ) d is

< d t hd(K).

P r o o f . Suppose that there exists a solution z0. T h e n for any other solution z, we have

(z/zo} = (1} ( m o d S , d). Thus, it suffices to count solutions z of

( z / - (1) (mod S, d). Suppose z is such a solution. By the definition of the congruence relation, we have

and without a loss of generality, 0 < ci < d (i = 1 , . . . ,t). We will count solutions z with fixed c l , . . . , ct. Say zt is a fixed such solution,

(zll = v ? . . . v ? ¢ f and z is an arbitrary such solution, (z)

=

. . . V?

•

Then

(z/z,i = (¢/¢,)d . T h e ideal class of ¢ / ~ , , say [~/~,]~ has [ ~ / ~ 1 ] d = [1]. Also, if ~:, ¢1 are in the same ideal class, then ( z / z , ) E ( K × ) a. So (since we only want solutions modulo ( K × ) d) all that remains is to count ideal classes whose d th power is [1]. But their n u m b e r is hd(K) by definition. Allowing for all the possibilities for c t , . . . ,ct with 0 _-< ci < d (i = 1 , . . . ,t), we have < dt hd(K) solutions in K x / ( K x )a.

146

LEMMA

5E. Suppose that d > 3 and ~ is a fractional ideal. Consider solutions

a E K x to the paJr o f congruences

(rood ,9, d),

(a) =_ ~

(1 - a) = (1, a) (rood S). T h e n u m b e r o f such a is < (cld) 2" h a ( K ) ,

where cl is an absolute constant. P r o o f . Write a = w z ~, where w runs through a complete residue system in K × / ( K X ) d. T h e n by hypothesis,

- w z d) (-17 ~ --(1)(roodS),

(1

which m a y be written as

< 1 - wz d)

3, n > 2, and L contained at least two ]roots o f f , say a l , a 2 E L. Let S I be the set of absolute values of L which extend absolute values of S. For x E O s , put Z(

x)

--

X -- Oll . X - - OL2

For v ~ S', we have IA(f)I. = 1, so by Lemma 5B, we have 141 - ~21~ = m ~ ( I x

-- ~ x k , Ix - ~21,~)

147

and 1

0~I -- 012

xX - a l ~2 v

X

~

O~ 2

: max

(1,

v

This means t h a t for every v ~ S'

I1 - z ( = ) l ~ = m a x (1, I Z ( x ) l . ) , or in terms of prime ideals,

(1 - Z(x)) _= (1, Z(z)) (roodS'). Also, by L e m m a 5C, for v E S', we have

max (1, lail~)

(i = 1, 2),

E a~

so that IZ(=)l~

=

ma~(1, I~1~) a m a x O , Io~=1~) " 9~,

with g~ E G~. Now we have

(Z(x)) = 93(modS', d), where 93 is a certain ideal. By the last lemma, the number of possibilities for Z(x) is Cl(n 2 - ~ . . . -~ r ~ )

where c 1 ) 0. T h e n u m b e r of integers n l , . . . , a n with h(nlP1 + ... + nRPR) =c5 log

=

say, by Lemma 9C. We use exercise 2b of Chapter I. Let ]C be the set of all _x with F(x_) _ vl. So the first minimum A~ satisfies .~l ~_- v f ~ • We count the number of points x_ with E(_x_) < v, i.e. the number of integer points in the set v/viC. By the exercise, the number of such points is =

-El

Since for m > mo(n) we have v __ 1, there are only finitely many rational points. Another proof, with ideas closer to diophantine approximations, was given by Vojta (to appear), with a more elementary version given by Bombieri (to appear). There is every hope that this will lead to bounds on the number of rational points. However, when g > 1, effective results on the size of integer points or rational points (the size of numerators and denominators) seem at present to be quite beyond reach.

V. Diophantine Equations in More than Two Variables. References: Evertse, Gy6ry, Stewart and Tijdeman (1988), Schmidt (1980) §1. The Subspace Theorem. T H E O R E M 1A. (Subspace Theorem, Schmidt (1972)). Suppose that L I , . . . , L~ are linearly independent linear forms in n variables with algebraic coefticients. Suppose 5 > 0 is given. Then the integer points x_ 7~ O_with

ILl(__x)... L,-,(_x_)l < Ix__1-6 lie in a finite number of proper subspaces of Qn. The reader may find a proof in Schmidt (1980).

C O R R O L L A R Y l B . Suppose a l , . . . ,otn are algebraic and 1 , a l , . . . ,an are linearly independent over Q. Then there are only finitely m a n y rational n-tuples ( ~ , t y , . . . , ~ , l y ) with y > 0 and a~ - Y

(9.1)

1 < yl+(1/n)+~,

(i ----1, " " ' n).

In the special case n = 1, we get Roth's Theoerem. Also, the exponent 1 + ( l / n ) is best possible by Dirichlet's Theorem (Theorem 1B of Chapter II). P r o o f . Multiplying together all of the inequalities in (9.1), then multiplying by yn+l gives y l ~ l y - x , I . . . I~ny - ~ n l < 1/y ~. Now ptlt x - - ( X l , . . . , Xn, y) E Z n+l a n d let X--~ ( X l , . . .

L~(X) = a , Y - X ,

, Xn, Y).

Let

(i = 1 , . . . , n )

and

L,-,+i(X) = ]I. Then we have

]L~(__z)... L,,+~(_z)l

< 1/y 6
O, lie in finitely m a n y proper subspaces. R e m a r k . It is reasonable to consider x E P n - I ( K ) , since both sides of the inequality are invariant under multiplying x by a scalar. LEMMA such that

1F. A n y element x_ of P n - I ( K ) has a set of coordinatesx_ = ( x l , . . .

(i) (ii)

, x,)

[__x[v~ 1 for v non-Archimedean, H

IxI~ >=1 / c l ( K ) ,

v non-Archimedean

(iii)

[X[v =< c2(g)[x[w for any Archimedean v, w.

R e m a r k . While (i) bounds Ixlv from above for v non-Archimedean, (ii) says that Ix_Iv can not be too small. Result (iii) says that all of the Archimedean absolute values are about the same. P r o o f . Consider the ideal 2(x_) generated by X l , . . . , Z n . There is some integral ideal P2 in the same ideal class as 2(x), and Af(g2) __ n this is a generalization of the T h u e equation. For if n = 2, the linear form L(X, Y ) = X - a Y gives norm forms d

F(X,Y) =a H (X-a(i)Y). i=l

If deg a = d > 3, then F(x, y) = m is a Thue equation. As x_ runs through Z n, the linear expression L(x) will run through a set ff)I C K . This set ff)t is a free Z-module of rank n with basis a l , . . • , an. So we could rewrite the norm form equation as = m, where p E fiR. Let Q O:R be the set of products q# with q E Q, # E 9)t. T h e n Q giR consists of alXl+...+anXn withxi EQ ( i = 1 , . . . , n ) . Let E be a subfield of K and let ff)IE be the set of p E flY[ such that A~ E Q ~ for every A E E . T h e n ff~E is a submodule of ~ . If E C E ' , then fir E' C ff~s; and we have ~ = frye. T h e module ff~ is called degenerate if there is a field E C K with E ~ Q and E not imaginary quadratic such that ffy~E ~ {0}. We say that F is degenerate if the corresponding ~ is degenerate. E x a m p l e . Take K = Q(i, V~), which has d = 4, and take L(X, Y, Z) = X + i Y + i v ~ Z . Let E = Q(i). T h e n { x + i i t : x, it E Z} = fiRE. For i f x + i i t + i ~ z E ff~E, then we would need ix - It - v/2z E QffR, which forces z = 0. This does not show that ~ is degenerate, though, since E is imaginary quadratic. We could also take E = Q ( i ~ ) to see = + # {0}, or t ke E = Q(v ) to get __ { i I t + iv z} # {0}. So we see that ff~ is, in fact, degenerate. E x a m p l e . Suppose d = p where p is a prime > 2, and 9~ is a Z-module of rank n, where n < p. T h e only subfields of K are K and Q, so we just need to consider 92tK. Suppose # E K, p ~ 0. Notice that Q ~ is a vector space over Q of dimension n. As A runs through K , then A# runs through K , a vector space of dimension d. So K # = K D QffR and thus p ~ 9/I K. T h e n 92lK = {0} and 92t is not degenerate. E x a m p l e . I f n = d, then K -- Q ~ a n d f f R K = 9~. I f K ~ Q and K is not imaginary quadratic, then 9Jl is degenerate. Degeneracy is i m p o r t a n t , for if F is non-degenerate, then the norm form equation F ( x , It) = m has only tlnitely many solutions. (Schmidt (1972) and (1980) Lecture

184 Notes). On the other hand, if F is degenerate, there will exist some m so that F(x__) = m has infinitely m a n y solutions. Before justifying this last remark, we will consider the simplest case which has infinitely many solutions. E x a m p l e . Suppose a l , . . . ,ad form an integral basis for K . Take n = d and L ( X ) = OtlX 1 + . . . + oedXd. Consider the norm form equation ¢Yi(L(x_)) = 1. T h e n we seek solutions to the equation ~ ( e ) = 1 where e = a l x l + . . . + a n x , . Thus e is a unit. By Dirichlet's Theorem, we have infinitely many solutions unless K is Q or is imaginary quadratic. In general, suppose that there exists a subfield E with ffj~E ~ {0}. We claim that if A E E , then not only is / ~ E C Q ~ but /~j~E C Q~e~E. For if S E 9XE, then As E Q ffJI, so that rnA s E 9X for some positive integer rn. Given A' E E , we have A'rnAs E QDJI, since A'rnA E E. This shows that mAS E 9)IE, thus AS E QgJ~E. Now let D E be the set of A E E with , ~ S C ~j~S. T h e set D E has the following properties: (i) It is a ring containing 1. (ii) It contains a field basis of E over Q. For if A1,... , A~ is a field basis, then ,~i~lJ~E C -~ 9XE for some positive m E Z. Then r e a l , . . . , m.k~ E O E form a field basis of E over Q. (iii) There exists an ~ > 0, ~ E Z such that gD E contains only algebraic integers. For if S • 0 is in 9XE then ,ks E 97tE for every A E D E . T h e n ,k E ttz ~'j~E. But ~1ffjtE is a free module with only finitely many generators, so there exists an £ so that -~t*ffJtE contains only algebraic integers. T h e n ~D E C ~ 9XE contains only algebraic integers. Any subring of E which satisfies these three conditions is called an order of E. The set ~E of all the integers in E is an example of an order. It is a fact that may order D in E is contained in the maximal order, D E. See Borevich and Shafarevich (1966) for a more complete discussion. E x a m p l e . Take E = Q(v/2). Then D E consists of x + v/2y, with x, y E Z. Another example of an order consists of x + 2x/~y with x, y E Z. We call D E the ring of multipliers. Take CE to be the group of units of D E of norm 9~E/Q(e) = 1. By Dirichlet's T h e o r e m on units, this group is infinite unless E = Q or E is imaginary quadratic. Now suppose that S0 # 0 is in 9XE. For e E EE , we have 91(es0) = 92(S0) and es0 E ffYtE. T h e n if 92(S0) = m, the norm-form equation 9l(s)

= m,

s E 9n E

has infinitely m a n y solutions. So the condition of non-degeneracy is necessary to ensure the finiteness of the n u m b e r of solutions. E x a m p l e . Let K = Q(i, V/2) and F(x, y, z) = 92(x + iy + iV~z). T h e n ffJt: x + iy + iv/2z. Now let E = Q(v/2). We have ffJtE: iy + i v ~ z = i(y + V~Z) and O E = D E, the ring of integers in E. We know that g E is infinite and we have a unit e = V~ - 1. Any solution of fRE(y + V~z) = +1 gives a solution iy + iv/2z E 99t of 9I(iy + iv/2z) = 1. Let us start with a particular solution of 9tE(y + V/2z) = 9~E(S) = +1, say with #0 = 1 (so that Y0 = 1, z0 = 0). By multiplication with powers of e we

185 obtain further solutions. Setting #t = #0e* = et, we have pa = e = v ~ - 1 (so that yl = - 1 , Zl = 1), #2 = ( x / 2 - 1) 2 = 3 - 2 v ~ (so that y2 = 3, z: = - 2 ) , etc. The reader may find a further discussion, especially about the degenerate case, in Schmidt (Lecture Notes, 1980). We return to the ease where F is non-degenerate and consider bounds on the number of solutions. THEOREM 3 A . (Schmidt (1986b)). I f F ( X ) is a non-degenerate norm form of degree d with coefficients in Z, then the norm form equation F(_x_) = m,

x_ E Z "

has at m o s t

d 2a°'d= el(n, d, m ) solutions where

el(n, d, 1)

= 1 and

(.

1) ~ d,-l(md),

where w is the n u m b e r of distinct pr/me factors of m and dn-l(g) is the n u m b e r of ways ofwritingg=gl...g,_x withgi>0 (i=1,...,n-1). As was seen in Section 6 of Chapter III, the general case follows from the case m = 1. Thus, one may concentrate on the norm-form equation =

1

and the bound (3.1)

d 23°"d2.

In these Notes, we will not prove Theorem 3A and (3.1), but a variation. See Theorem 3B below. Incidentally (Schmidt (1989b)) has also proved another bound in place of (3.1), namely d(2,) "2"+4 " This second bound can probably be improved, removing one of the exponents by using some combinatorial techniques. The second bound is nicer for fixed n, since it grows only like a polynomial in terms of d. How can this be generalized to the degenerate cases7 T h e r e we would have finitely many solutions up to multiplication by certain units. An explicit bound so far has not been derived. Also, what about asymptotic estimates for the number of solutions? The inequality IF(x_] ~ m defines some n-dimensional set of volume c y m '~/d where cy depends on F only. Mahler (1934) has shown in the case n = 2, i.e. for the Thue case, that the number of solutions of this inequality is ~ c y m n/d as m ~ oc. R a m a c h a n d r a (1969) proved this asymptotic formula for a class of norm form equations.

186

Now we will specialize the n o r m forms F somewhat. Let F be a n o r m form given by F ( X ) = a L < I ) ( X ) . . . L n ion. In the sections which follow, we will distinguish large and small solutions. Small solutions will be those with

151 _-< --~(F) 6"d"

=

H(L) 6,~d"+'.

The remaining solutions will be (:ailed large solutions. §5. A n A p p l i c a t i o n o f t h e G e o m e t r y o f N u m b e r s . 'LetL i _< d). Let

=

OqXl-}-...-}-o~nXn withoq E K, a n d w r i t e L (i) / a

=J

\

andA=a

A...Aa

o~i)x,--~

-. .

..a-a(1)X, ~, (1 =
= 2n~3/2

a./:(con t F)(._,)/dS(F),/~,

where V ( n ) is the volume of the unit ball in N". Notice that both sides are invariant under ~. The right-hand side depends only on F , but the left-hand side depends on how we write F = aL O) ... L (d). We could write instead F = a ' L ' 0 ) . . . L '(d), where L' = ,~L. Then a' = a/9l()~), but there is no simple way to express A(L') in terms of )~ and A(L). The matrix (a~ i)) (l_ (cont Y)/lal. T h e A r i t h m e t i c - G e o m e t r i c Inequality gives

=

+... + Iz( )l

> Id > ~/-d ~/(cont F)/[a[.

2

190

We may conclude that (5.1)

•1 => ~

~/(cont

F)llal

.

Now suppose that b l , . . . , b is another basis of A. Say

~!1) ) r 3 b. =$

= mjl

[~!d)

a 1 "4- . . .

-1"- m j n

an,

r.7

where the matrix (mjk) is in

SL(n, Z). Introduce the row vectors

_,(,) = (,i'),... ,,(:)) and

tic,) = (t},),..., t(,)). Then we have 3 t(i)

= m j l o~i)

+ ...

+ m jr,

Ol(ni)

= a~i)mjl + . . . + a~i) m.i,, so t (1) =

~(i)Mt, where M t is the transpose of 2lz/. Let F M' (X) = F(Mt_A') = a 1-I ( ~ ( ' ) M ' X ) i=1 d

---- a I I (flCi)x) i=1

Since

F M' ,~ F, we have H'(F M') >=~(F),

so that

d

lal2 I-~ I-~(i)12 --> "~(F)2" i----1

Using the Arithmetic-Geometric Inequality, we get d

I_~(')1~ = d(.5(F)/lalfl/d. i----1

Recall that

~--

----3

191 Therefore

(5.2)

~

Ibsl ~ > d(D(F)llal) 21d,

j=l which says t h a t a basis b l , . . . ,b n can not be too small. Given our lattice A and any basis =bl,... ,b n of A, there are linearly independent lattice points g , . . . ,g such t h a t 1

=n

Ig, I =/i~, Ig2=l= w,..., where/i1, • • •

,/in

Ig,, I =/i,,

are the successive minima. For n = 2 these g. necessarily f o r m a basis,

but for n > 2, they are not necessarily a basis. However, one can show t h a t there is a basis b l , . . . , b with the p r o p e r t y t h a t

I:bjl< j/ij

(j = 1 , . . . , n).

E x e r c i s e 5 b . Verify this last statement. T h e reader m a y consult Cassels' text on the G e o m e t r y of N u m b e r s (1959). Given such a basis, we have

_

=

=It

j=l

/t n.

j=l

If we combine this with (5.2) we obtain

dl/2

/i. > n-777/2(S~(F)Ilal) lid. F r o m (5.1), we also h a d

#1 > d '/2 (cont F/lal) '/d

(j = 1 , . . . , n - 1).

Taking the p r o d u c t of these inequalities we see t h a t

d./2

~ x m . . . / i , > n3/2 lal,/d (cont F) (n-D/d ~ ( F ) x/d. By Minkowski's Second T h e o r e m (2E of C h a p t e r I), we have

#1 . . . / i n V ( n ) < 2 ~ d e t A , so t h a t

v(,~) det h :> 2" ,~i~ lalni~ d"l~ (toni F) ("-')/d.~(F)a/d.

192 But det A = I det ai_ =ajl1/2

= la, A...

^ a I = A(L),

a n d the p r o o f is complete.

§6. Products of Linear Forms. LEMMA 6 A . Suppose F ( X ) = a L ( 1 ) ( X ) . . . L ( d ) ( X ) is a n o r m f o r m with coefflcients in Z. Suppose x_ 6 Z n is a solution of

F(x=) = ~. Then there exist i x , . . . ,in with 1 4. Given such reduced and distinct x l / y a , . . . , x , , / y , , with ya < ... < y~, then < a--xJ-x + ~-< 2--, Yj - 1 Yj =

therefore

Yj - 1

P Yj - 1

P Y j > "~ Y./-1,

and so y~ > ( p / 2 ) v-1 .

194

T h e n u m b e r of such approximations in reduced form with 1 < y < B is logB < 1 + 2 logB =< 1 + log(P/2~----) = log P " We now want to generalize this. L E M M A 7A. Suppose L1,. •. , Ln are n > 1 linearly independent linear forms in n variables with complex coefficients. Suppose (n!) 4

~j (F) lid > ~ (F) 1/2d > (n!) 4,

where the last two inequalities follow since Y) (F) > n l°nd by (4.1). As we mentioned previously, small solutions will be those with

Ixt 5 ~ (F) 6rid". Let this bound be B. By Lemma 7A, the small solutions satisfying (8.1) lie in no more than n an (log B~ log p ) n - 1 subspaces, and we have nan(log B / log p)n-1

< na n ( 6 n d n log ~j ( F ) ) n-1

g

=

_< 12 n n 4n dn2-1

.

Counting the number of possibilities for the n-tuple 1 < il < . . . < in < d, which is

(:) =
0, there exists a constant c2(e) such that

max(lal, Ibl, Icl)
c3yl/2 for positive integers x, y with x 2 # ya. A weaker version of Hall's conjecture follows from the abe-conjecture. To see this, let d = gcd(x 2, y3), and then set a = x2/d, b = - y a / d , c = (ya _ x 2 ) / d . Then P=

H p (= piabc

• vlv

-

The abc-conjecture gives for any e > 0 that Ibl =

v31d
c,(,) y (1/2)-~. This has the following consequence concerning a particular elliptic equation y2 = x 3 + k,

k # 0,

called the Mordell equation. Hall's conjecture gives

Ikl= Iv= - ~ I > ~ ( ~ ) x (1/2)-'. Thus every solution to a given Mordell equation has

14 < ~?(~)Ikl2+~. Next we consider the Fermat conjecture. T h a t is, we look at the equation x n + yn = z n

where

n >=3,

gcd(x,y,z) = 1,

and

x , y , z > O.

To apply the abc-conjecture, let a = x n, b = yn, c = - z n. T h e n

P=

I I P 5. Math. Scand. 2, 29-32. E. Dubois and G. Rhin (1976). Sur la majoration de formes lindares d coefficients algdbriques rdels et p-adiques. .Demonstration d' une conjecture de K. Mahler. C.R. Acad. Sci. Paris 282, S6rie A, 1211 F. J. Dyson (1947). The approximation to algebraic numbers by rationals. Acta Math. Acad. Sci. Hung. 9, 225-240. P. ErdSs, C. L. Stewart and R. Tijdeman (1988). many solutions. Compositio Match. 66, 37-56

Some diophantine equations with

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I n d e x of s o m e definitions Archimedean absolute value p. 19 base point p. 152 best approximation p. 71 Bezout's Theorem p. 148 binomial Thue equation p. 98 convex hull p. 94 covariant forms p. 169 decomposable form p. 90 degenerate module p. 18] density of lattice packing p. 7 Dirichlet's unit theorem p. 123 divisor p. 148 elliptic curve p. 152 equivalence of S-unit equations p. 133 fundamental parallelepiped p. 4 gap principle p. 57 genus p. 137, 151 Grassman coordinates p. 14 Hall's conjecture p. 203 height of a polynomial p. 23 height of a subspace p. 12, p. 33 Hermite's constant p. 9 ineffective p. 73 index of a polynomial at a point, p. 42 inhomogeneous minimum p. 70 large root p. 111 lattice p. 3 lattice packing p. 7 local parameter p. 149 Mordell equation p. 204 Mordell-Weil height p. 156 Mordell-Weil Theorem p. 156 Neron-Tate height p. 156 Newton points p. 101 Newton polygon p. 101 non-Archimedean absolute value p. 19 non-singular point p. 146 norm form p. 75 normalized form p. 81 order p. 182 Pillai's conjecture p. 205 primitive integer point p. 10 principal divisor p. 151

217

rank p. 156 rational function p. 148, 149 reduced form p. 81 Riemann-Roch Theorem p. 151 ring of multipliers p. 182 semi-discriminant p. 77 singular point p. 146 small root p. 111 successive minima p. 6 Thue inequality p. T4 Thue-Mahler equation p. 124 torsion p. 156 trinomial Thue equation p. 98 ultra-metric p. 91 Weierstrass equation p. 164 window of exponential width C p. 50 C-set p. 57 S-integer p. 123 S-unit p. 123 6-approximations 58 "y-set p. 57

Diophantine Approximations And Diophantine Equations

Diophantine Approximations and Diophantine Equations

Diophantine approximations