Number Theory IV: Transcendental Numbers

A. N. Parshin I. R. Shafarevich (Eds.) i‘ Number Theory IV Transcendental Numbers Springer i‘, Preface This book w...

Author: A.N. Parshin | I.R. Shafarevich | N. Koblitz | N.I. Fel'dman | Yu.V. Nesterenko

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A. N. Parshin I. R. Shafarevich (Eds.)

i‘

Number Theory IV Transcendental Numbers

Springer

i‘,

Preface This book was written over a period of more than six years. Several months after we finished our work, N. I. Fel’dman (the senior author of the book) died. All additions and corrections entered after his death were made by his coauthor. The assistance of many of our colleagues was invaluable during the writing of the book. They examined parts of the manuscript and suggested many improvements, made useful comments and corrected many errors. I much have pleasure in acknowledging our great indebtedness to them. Special thanks are due to A. B. Shidlovskii, V. G. Chirskii, A. I. Galochkin and 0. N. Vasilenko. I would like to express my gratitude to D. Bertrand and J. Wolfart for their help in the final stages of this book. Finally, I wish to thank Neal Koblitz for having translated this text into English. August

1997

Yu. V. Nesterenko

Transcendental N. I. Fel’dman

Numbers

and Yu. V. Nesterenko

Translated from the Russian by Neal Koblitz

Contents Notation

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

Introduction 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 Chapter

11

................................................... .................................. Preliminary Remarks Irrationality of Jz ..................................... The Number 7r ........................................ ............................... Transcendental Numbers .................... Approximation of Algebraic Numbers Transcendence Questions and Other Branches ..................................... of Number Theory The Basic Problems Studied in Transcendental ....................................... Number Theory ................... Different Ways of Giving the Numbers Methods .............................................

1. Approximation

of Algebraic

9

Numbers

..........

..................................... 51. Preliminaries 1.1. Parameters for Algebraic Numbers and Polynomials ..................... 1.2. Statement of the Problem ............ 1.3. Approximation of Rational Numbers 1.4. Continued Fractions .......................... ....................... 1.5. Quadratic Irrationalities ...... 1.6. Liouville’s Theorem and Liouville Numbers

11 11 13 14 15 16 17 19 20 . . . .. ..

22

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

22 22 22 23 24 25 26

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

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

2

Contents

1.7. Generalization of Liouville’s Theorem .................... 52. Approximations of Algebraic Numbers and Thue’s Equation ..... 2.1. Thue’s Equation ....................................... 2.2. TheCasen=2 ....................................... 2.3. TheCasen>3 ....................................... 53. Strengthening Liouville’s Theorem. First Version ofThue’sMethod .......................................... ................................ 3.1. AWaytoBoundqO-p ........ 3.2. Construction of Rational Approximations for $$ 3.3. Thue’s First Result .................................... 3.4. Effectiveness .......................................... 3.5. Effective Analogues of Theorem 1.6 ...................... 3.6. The First Effective Inequalities of Baker .................. 3.7. Effective Bounds on Linear Forms in Algebraic Numbers ... §4. Stronger and More General Versions of Liouville’s Theorem and Thue’s Theorem ....................................... 4.1. The Dirichlet Pigeonhole Principle ....................... 4.2. Thue’s Method in the General Case ...................... 4.3. Thue’s Theorem on Approximation of Algebraic Numbers . . 4.4. The Non-effectiveness of Thue’s Theorems ................ $5. Further Development of Thue’s Method ....................... 5.1. Siegel’s Theorem ...................................... 5.2. The Theorems of Dyson and Gel’fond .................... 5.3. Dyson’s Lemma ....................................... 5.4. Bombieri’s Theorem ................................... $6. Multidimensional Variants of the Thue-Siegel Method .......... 6.1. Preliminary Remarks .................................. 6.2. Siegel’s Theorem ...................................... 6.3. The Theorems of Schneider and Mahler .................. 57. Roth’sTheorem ........................................... 7.1. Statement of the Theorem .............................. 7.2. The Index of a Polynomial .............................. 7.3. Outline of the Proof of Roth’s Theorem .................. 7.4. Approximation of Algebraic Numbers by Algebraic Numbers ................................. 7.5. The Number k in Roth’s Theorem ....................... 7.6. Approximation by Numbers of a Special Type ............. 7.7. Transcendence of Certain Numbers ...................... 7.8. The Number of Solutions to the Inequality (62) and Certain Diophantine Equations .......................... 58. Linear Forms in Algebraic Numbers and Schmidt’s Theorem ..... 8.1. Elementary Estimates .................................. 8.2. Schmidt’s Theorem .................................... 8.3. Minkowski’s Theorem on Linear Forms ................... 8.4. Schmidt’s Subspace Theorem ...........................

27 28 28 30 30 31 31 31 32 33 34 36 39 40 40 41 44 45 45 45 48 50 51 53 53 53 54 55 55 56 57 60 61 61 62 63 65 65 66 67 68

Contents 8.5. Some Facts from the Geometry of Numbers ............... 59. Diophantine Equations with the Norm Form ................... 9.1. Preliminary Remarks .................................. 9.2. Schmidt’s Theorem .................................... §lO. Bounds for Approximations of Algebraic Numbers in Non-archimedean Metrics ................................... 10.1. Mahler’s Theorem ..................................... 10.2. The ThueMahler Equation ............................. 10.3. Further Non-effective Results ........................... Chapter 2. Effective Constructions in Transcendental Number Theory ............................................... Sl. Preliminary Remarks ....................................... 1.1. Irrationality of e ....................................... 1.2. Liouville’s Theorem .................................... 1.3. Hermite’s Method of Proving Linear Independence of a Set of Numbers ........................................... 1.4. Siegel’s Generalization of Hermite’s Argument ............. 1.5. Gel’fond’s Method of Proving That Numbers Are Transcendental ........................................ $2. Hermite’s Method .......................................... ..................................... 2.1. Hermite’s Identity 2.2. Choice of f(x) and End of the Proof That e is Transcendental ........................................ 2.3. The Lindemann and Lindemann-Weierstrass Theorems ..... 2.4. Elimination of the Exponents ........................... 2.5. End of the Proof of the Lindemann-Weierstrass Theorem ... ..................... 2.6. Generalization of Hermite’s Identity .................................. 53. Functional Approximations 3.1. Hermite’s Functional Approximation for e+ ............... 3.2. Continued Fraction for the Gauss Hypergeometric Function .............................. and Pade Approximations ............ 3.3. The HermitePade Functional Approximations 54. Applications of Hermite’s Simultaneous .................................. Functional Approximations 4.1. Estimates of the Transcendence Measure of e ............. 4.2. Transcendence of e” ................................... 4.3. Quantitative Refinement of the Lindemann-Weierstrass Theorem ........................ 4.4. Bounds for the Transcendence Measure of the Logarithm of an Algebraic Number ................................ 4.5. Bounds for the Irrationality Measure of K and Other Numbers .................................... 4.6. Approximations to Algebraic Numbers ...................

3

71 73 73 75 76 76 76 77

78 78 78 79 80 80 82 83 84 85 88 90 92 92 93 93 95 98 99 99 100 103 104 106 110

4

Contents

§5. Bounds for Rational Approximations of the Values .. . of the Gauss Hypergeometric Function and Related Functions 5.1. Continued Fractions and the Values of ez . . . . . . . . . . . . . . . . . 5.2. Irrationality of 7r . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 5.3. Maier’s Results . . .. . .. . . .. . .. . . .. . . . .. . .. . . .. . . . .. . .. . 5.4. Further Applications of Pade Approximation . . . .. .. . . .. . . . 5.5. Refinement of the Integrals. . .. . . .. . . .. . . .. . . .. . . . . . .. . 5.6. Irrationality of the Values of the Zeta-Function and Bounds on the Irrationality Exponent. . . . .. . .. . . .. . . . $6. Generalized Hypergeometric Functions .. . . .. . .. . . .. . .. . . .. . .. . 6.1. Generalized Hermite Identities . . .. . . .. . .. . . .. . . .. . . .. . . . 6.2. Unimprovable Estimates .. . .. . . . .. .. .. . .. . . .. . .. . . .. . .. . . .. .. . . . .. . .. . . .. . . .. . . . . .. . . .. . 6.3. Ivankov’s Construction $7. Generalized Hypergeometric Series with Finite Radius . . .. . . .. . . . .. . .. . . .. . . .. . . .. . . .. . .. . . .. . .. . . of Convergence 7.1. Functional Approximations of the First Kind . . . . . . . . . . . . . . 7.2. Functional Approximations of the Second Kind . . . . . . . . . . . . 58. Remarks . . .. . . . .. . . . .. . .. . . . .. . . .. . .. . . .. . .. . . .. . . .. . .. . . .

136 136 139 143

Chapter

146

3. Hilbert’s

Seventh Problem

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

Problem .................................. 51. The Euler-Hilbert 1.1. Remarks by Leibniz and Euler .......................... 1.2. Hilbert’s Report ....................................... ........................ 52. Solution of Hilbert’s Seventh Problem 2.1. Statement of the Theorems ............................. 2.2. Gel’fond’s Solution .................................... 2.3. Schneider’s Solution ................................... 2.4. The Real Case ........................................ 2.5. Laurent’s Method ..................................... $3. Transcendence of Numbers Connected with Weierstrass Functions ................................................. 3.1. Preliminary Remarks .................................. 3.2. Schneider’s Theorems .................................. ................. 3.3. Outline of Proof of Schneider’s Theorems $4. General Theorems .......................................... 4.1. Schneider’s General Theorems ........................... 4.2. Consequences of Theorem 3.17 .......................... 4.3. Lang’s Theorem ....................................... 4.4. Schneider’s Work and Later Results on Abelian Functions §5. Bounds for Linear Forms with Two Logarithms ................ 5.1. First Estimates for the Transcendence Measure of ab and lna/lnfl ............................................. 5.2. Refinement of the Inequalities (19) and (20) Using Gel’fond’s Second Method ..............................

112 112 113 115 116 120 121 127 128 131 133

146 146 146 147 147 147 149 150 151 152 152 153 155 157 157 158 159 . . 159 161 161 163

Contents Bounds for Transcendence Measures ..................... Linear Forms with Two Logarithms ...................... Generalizations to Non-archimedean Metrics .............. Applications of Bounds on Linear Forms in Two Logarithms .................................... 36. Generalization of Hilbert’s Seventh Problem to Liouville Numbers ....................................... 6.1. Ricci’s Theorem ....................................... 6.2. Later Results ......................................... $7. Transcendence Measure of Some Other Numbers Connected with the Exponential Function ............................... 7.1. Logarithms of Algebraic Numbers ....................... 7.2. Approximation of Roots of Certain Transcendental Equations .............................. 58. Transcendence Measure of Numbers Connected with Elliptic Functions ...................................... ........................ 8.1. The Case of Algebraic Invariants 8.2. The Case of Algebraic Periods .......................... 8.3. Values of P(Z) at Non-algebraic Points ................... 5.3. 5.4. 5.5. 5.6.

5

164 164 165 165 172 172 173 173 173 175 176 176 177 177

Chapter 4. Multidimensional Generalization of Hilbert’s Seventh Problem ....................................

179

31. Linear Forms in the Logarithms of Algebraic Numbers .......... 1.1. Preliminary Remarks .................................. 1.2. The First Effective Theorems in the General Case ......... 1.3. Baker’s Method ....................................... 1.4. Estimates for the Constant in (8) ........................ 1.5. Methods of Proving Bounds for A, Ac, and Ai ............. 1.6. A Special Form for the Inequality ........................ 1.7. Non-archimedean Metrics ............................... $2. Applications of Bounds on Linear Forms ...................... 2.1. Preliminary Remarks .................................. 2.2. Effectivization of Thue’s Theorem ....................... 2.3. Effective Strengthening of Liouville’s Theorem ............ 2.4. The Thue-Mahler Equation ............................. 2.5. Solutions in Special Sets ................................ 2.6. Catalan’s Equation .................................... 2.7. Some Results Connected with Fermat’s Last Theorem ...... 2.8. Some Other Diophantine Equations ...................... .................................... 2.9. The a&-Conjecture 2.10. The Class Number of Imaginary Quadratic Fields ......... 2.11. Applications in Algebraic Number Theory ................ 2.12. Recursive Sequences ................................... 2.13. Prime Divisors of Successive Natural Numbers ............

179 179 180 182 185 187 188 188 189 189 189 192 193 194 195 196 197 199 199 200 201 203

6

Contents

2.14. Dirichlet Series . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 93. Elliptic Functions . . . .. . . . .. . .. . . .. . .. . . .. . . .. . .. . . .. 3.1. The Theorems of Baker and Coates . . . . . . . . . . . . . . . . . .. . .. . . . .. .. .. . .. . . .. .. . . .. 3.2. Masser’s Theorems .. . . .. . . .. . . .. . .. . . .. . . . .. .. . . .. 3.3. Further Results . . .. . . .. . . .. . . .. . . .. . .. . . .. 3.4. Wiistholz’s Theorems 54. Generalizations of the Theorems in $1 to Liouville Numbers 4.1. Walliser’s Theorems .. . .. . . . .. . .. . . .. . . .. . .. . . .. 4.2. Wiistholz’s Theorems . . .. . . .. . .. . . .. . . .. . . .. . . .. Chapter 5. Values of Analytic Functions That .......................................... Differential Equations

Satisfy

. . . . . .

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

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

203 204 204 204 205 206 207 207 207

Linear

............................................... 51. E-Functions 1.1. Siegel’s Results ........................................ 1.2. Definition of E-Functions and Hypergeometric E-Functions 1.3. Siegel’s General Theorem ............................... ...................... 1.4. Shidlovskii’s Fundamental Theorem ............................... §2. The Siegel-Shidlovskii Method 2.1. A Technique for Proving Linear and Algebraic Independence ......................................... 2.2. Construction of a Complete Set of Linear Forms ........... .............. 2.3. Nonvanishing of the Functional Determinant ................. . ................. 2.4. Concluding Remarks §3. Algebraic Independence of the Values of Hypergeometric ............................................... E-Functions 3.1. The Values of E-Functions That Satisfy First, Second, and ...................... Third Order Differential Equations 3.2. The Values of Solutions of Differential Equations of Arbitrary Order ....................................... ............ $4. The Values of Algebraically Dependent E-Functions 4.1. Theorem on Equality of Transcendence Degree ............ 4.2. Exceptional Points ..................................... 55. Bounds for Linear Forms and Polynomials in the Values of EFunctions ............................................... ...... 5.1. Bounds for Linear Forms in the Values of EFunctions 5.2. Bounds for the Algebraic Independence Measure ........... ...... 56. Bounds for Linear Forms that Depend on Each Coefficient ....................... 6.1. A Modification of Siegel’s Scheme 6.2. Baker’s Theorem and Other Concrete Results ............. 6.3. Results of a General Nature ............................. ............................... ‘$7. G-Functions and Their Values 7.1. G-Functions .......................................... ................................... 7.2. Canceling Factorials 7.3. Arithmetic Type ......................................

209 209 209 . 210 213 214 215 215 218 219 221 222 222 225 229 230 231 234 234 237 241 241 243 244 246 246 250 252

Contents 7.4. Global Relations ...................................... 7.5. Chudnovsky’s Results .................................. Chapter 6. Algebraic Independence of the Values of Analytic Functions That Have an Addition Law ............................ $1. Gel’fond’s Method and Results .............................. 1.1. Gel’fond’s Theorems ................................... 1.2. Bound for the Transcendence Measure ................... 1.3. Gel’fond’s “Algebraic Independence Criterion” and the Plan of Proof of Theorem 6.3 ........................... 1.4. Further Development of Gel’fond’s Method ............... 1.5. Fields of Finite Transcendence Type ..................... , .......... $2. Successive Elimination of Variables ................ ............... 2.1. Small Bounds on the Transcendence Degree 2.2. An Inductive Procedure ................................ $3. Applications of General Elimination Theory ................... 3.1. Definitions and Basic Facts ............................. 3.2. Philippon’s Criterion ................................... 3.3. Direct Estimates for Ideals .............................. 3.4. Effective Hilbert Nullstellensatz ......................... $4. Algebraic Independence of the Values of Elliptic Functions ...... 4.1. Small Bounds for the Transcendence Degree .............. 4.2. Elliptic Analogues of the Lindemann-Weierstrass Theorem 4.3. Elliptic Generalizations of Hilbert’s Seventh Problem ....... $5. Quantitative Results ........................................ 5.1. Bounds on the Algebraic Independence Measure ........... 5.2. Bounds on Ideals, and the Algebraic Independence Measure 5.3. The Approximation Measure ............................ Bibliography Index

7

253 255

259 260 261 262 264 267 268 271 271 272 276 276 278 284 287 290 291 . 295 296 302 302 . 304 306

. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 309

. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 344

Notation

9

Notation

W is the set of natural numbers No = NU{O) Z is the set of integers Q is the set of rational numbers IR is the set of real numbers C is the set of complex numbers A is the set of algebraic numbers ZA is the set of all algebraic integers Zz is the set of all algebraic integers of the field lK , z,) is the set of all rational functions in the variables ~1,. . . , z, K(Zl,... over the field K , z,] is the set of all polynomials in the variables ~1,. . . , z, over K[Zl,... the field IK H(P(z)) = H(P) is the height of the polynomial P(z) E @[zr, . . . , z,], i.e., the maximum absolute value of its coefficients L(P(z)) = L(P) is the length of the polynomial P(z) E C[zr, . . . , zm], i.e., the sum of the absolute values of its coefficients deg,, P is the degree in Zi of the polynomial P deg P is the total degree of the polynomial P t(P) = deg P + In H(P) h(1) is the rank of the homogeneous ideal I C iZ[ze, . . . , z,] deg I is the degree of the homogeneous ideal I C Z[ZO, . . . , zm] H(I) is the height of the homogeneous ideal I C Z[ZO, . . . , z,] t(I) = deg I + In H(I) (I(w)] is the magnitude of the homogeneous ideal I c Z[ZO, . . . , z,] at the point w E C*+l degcY is the degree of the algebraic number cr H(o) is the height of the algebraic number & L(a) is the length of the algebraic number o Norm(a) is the product of all of the conjugates of the algebraic number a m is the maximum absolute value of the conjugates of the algebraic number a IQ],, is the p-adic norm of the algebraic number LLI ]]a]] is the distance from the number a E IR to the nearest integer ]]zr]] = maxr 1 -b’

which contradicts the fact that anbn -+ 0. These computations are related to the continued fraction expansion of fi. From (1) we obtain %+I = WI + %I,

%I+1

= 2%

+%I,

The recursive relation implies that w,/u, continued fraction approximation to fi: l+2+1

1

211 = 0,

Wl =

‘112= 02 = 1 .

is the (n - 2)-th convergent of the

= [l; 2,2, . . .] = [l; 2] = 1 + p11 + p11 + . . . *

2+“. The fact that this is an infinite continued fraction - i.e., the Euclidean algorithm for the side and diagonal of the square goes on indefinitely - is equivalent to the irrationality of fi. 0.3. The Number 7r. The mathematicians of antiquity were always in search of ways of computing the area of a circle or the volume of a sphere or ellipsoid. It was this that led to the famous problem of squaring the circle and to various rational approximations to X. The first question that arises is whether 1 and K are commensurable. That they are not, i.e., that x is irrational, was proved by J. II. Lambert in 1766. There was a gap in his proof, which was later filled in by Lagrange. The problem of squaring the circle is the following question: Starting with the radius of a circle and using only ruler and compass, is it possible to construct the side of a square that has the same area as the circle? It was not until two thousand years later that this question was answered in the negative - when F. Lindemann in 1882 proved that x is transcendental. We now explain the connection between this result of Lindemann and the squaring of the circle.

14

Introduction

We choose a coordinate system that has the radius of the given circle as its unit of measurement. The problem is to construct a segment of length fi or to prove that such a construction is impossible. Now any construction with ruler and compass is a sequence of elementary steps of one of the following types: (1) construct the point of intersection of two (non-parallel) lines; (2) construct the points of intersection of a line and a circle; (3) construct the points of intersection of two circles; (4) draw the line passing through two given points; (5) draw the circle with given center and radius. In addition, one sometimes has to make an arbitrary choice - say, of a point on some line or circle that has already been constructed. We shall say that a point, line, or circle is algebraic if the coordinates of the point are algebraic numbers, the line can be given by an equation with algebraic coefficients, or the circle has center at a point with algebraic coordinates and radius equal to an algebraic number. By working with the equations of lines and circles, we easily see that when the elementary steps listed above are applied to algebraic points, lines, and circles, the results are also algebraic. Note that whenever one has to make an arbitrary choice of a point in the plane or on an algebraic line, one can always choose such a point to have algebraic coordinates. Thus, if it is possible by ruler and compass to construct a square of the same area as the given circle, then the corners of that square will have algebraic coordinates. This would mean that fi - and hence also x - are algebraic numbers. We conclude that the impossibility of squaring the circle follows from the theorem that 7r is transcendental. 0.4. ‘lhnscendental Numbers. References to the existence of transcendental numbers go back many centuries. Certainly Leibniz believed that, besides rational and irrational numbers (by “irrational” he meant algebraic irrational numbers in modern terminology), there also exist transcendental numbers. Euler conjectured that log, b = In b/ In a is transcendental if a, b E Q and this ratio is irrational (the latter assumption excludes trivial cases,such as log, 4). Euler’s conjecture turned out to be an extremely fruitful one for transcendental number theory. Its proof led to the development of a powerful analytic method, which we shall describe in Chapter 3. Incidentally, the first examples of transcendental numbers were not found until 1844 - almost a century after Leibniz. Liouville’s construction of these examples will be given in Chapter 1. It should be clear by now that our field has an impressive geneology. In its early years, transcendental number theory was a collection of diverse facts sometimes quite elegant, often rather difficult to prove, and in general isolated from the rest of mathematics and even from other parts of number theory. It is only in the twentieth century that transcendental number theory has become an important branch of number theory with its own problems and methods. At the same time its applications have become far more numerous.

Introduction

15

0.5. Approximation of Algebraic Numbers. We have already mentioned that the approximation of algebraic numbers by rational numbers is part of transcendental number theory. This statement might seem odd, and even presumptuous, since there is a separate area of mathematics called “algebraic number theory,” and transcendental number theory is supposed to be the study of numbers that are not algebraic. However, there are some important reasons for including this topic in transcendental number theory: (1) In order to prove that certain numbers are transcendental, one often usestheorems on approximation of algebraic numbers; and to prove results on approximation of algebraic numbers, one frequently makes use of theorems on transcendental numbers, as we shall see in Chapter 4. (2) The construction of many examples of transcendental numbers has been based on theorems on approximation of algebraic numbers. For instance, that is what Liouville did in 1844; and K. Mahler’s beautiful example 0.123456789101112131415.. . (whose decimal expansion consists of the sequenceof natural numbers) is also of this type. We now discuss the role that theorems on approximation of algebraic numbers play in the study of Diophantine equations. We begin with a simple example. Suppose that we want to determine whether or not the equation x3 - 2y3 = m,

mCZ,

(2)

has infinitely many rational integer solutions. Let (X, Y) be such a solution. We may suppose that mXY # 0. If we factor X3 - 2Y3, from (2) we find that

It is well known that any real number has infinitely many approximations X/Y, X,Y E Z, to within Y- 2; however, almost all real numbers (in the sense of Lebesgue measure) do not have infinitely many approximations to within Y-2--6, 6 > 0. From (3) it follows that fi is approximated by X/Y to within O(IY le3). If we show that (3) cannot hold for IY I sufficiently large, then this will imply that there are only finitely many solutions to the Diophantine equation (2). The same argument holds for the entire class of Diophantine equations of the form f(x, Y> = m , (4) where f(x, y) is a form with rational integer coefficients. After obtaining a lower bound for the absolute value of the difference between an algebraic and a rational number, the Norwegian mathematician A. Thue in 1908 proved that any such Diophantine equation (in particular, (2)) has only finitely many solutions, provided that the form is irreducible and has degree greater than two. Sixty years later, W. Schmidt extended Thue’s theorem to Diophantine equations of the form

Introduction

16

7% N

Q1(d q -t ... + a,Cd xm

j=l

a!j) E Kc

A,

i = l)...,

m, j = l)...)

12,

>

= M )

degK=n>m,

MEZ.

We shall discuss the theorems of Thue and Schmidt in more detail in Chapter 1. Thue’s theorem was ineffective: it did not give a method of finding all of the solutions of (4). This deficiency was removed 60 years later by A. Baker using lower bounds for the absolute value of linear forms zilnai 3,*..,

We will describe

Baker’s

%I

work

+.-.+z,lna,, E 4

in Chapter

Xl,...,Xm

E z.

4.

0.6. Transcendence Questions and Other Branches of Number Theory. Transcendental number theory has many connections with other areas of number theory. Not only do the methods of transcendence theory influence those of certain other fields, and vice-versa; but even the statements of the problems are often related. Earlier we explained how the purely geometrical problem of squaring the circle reduced to the transcendence of K. We now give two more examples. One of the achievements of transcendental number theory is connected with a problem that goes back to Gauss - that of determining all quadratic imaginary fields of class number 1 (i.e., in which the algebraic integers have unique factorization). In 1948, A. 0. Gel’fond and Yu. V. Linnik indicated how one could use transcendental number theory to attack this problem. Their idea could be implemented once Baker managed to prove the necessary estimates for linear forms in the logarithms of algebraic numbers. This was done in 1969 by Bundschuh and Hock, who proved that there are no other imaginary quadratic fields of class number 1 besides those found by Gauss. Using similar considerations, Baker and Stark obtained an effective bound for the discriminants of imaginary quadratic fields of class number 2. Much later, using methods unrelated to transcendental number theory, Goldfeld, Gross, and Zagier proved an analogous result for imaginary quadratic fields of arbitrary fixed class number. In 1770, Lagrange proved that every natural number can be written as a sum of at most 4 squares. In the same year E. Waring posed the following problem: Prove that every natural number can be written as a sum of at most 9 perfect cubes, as a sum of at most 19 fourth powers, and so on. In other words, for any n 2 2 prove that there exists a number s such that every natural number can be written as a sum of at most s perfect n-th powers. This form of the problem was solved by Hilbert at the end of the last century. As Waring’s problem was studied further, many additional questions were asked. We shall discuss only one aspect of this area of research in analytic

Introduction

17

number theory, namely, the search for a proof of a formula for the smallest number s such that every natural number can be written as a sum of s n-th powers. This value of s is customarily denoted g(n). It turns out that g(3) = 9 and g(4) = 19. Using earlier results of I. M. Vinogradov, in the late 1930’s L. E. Dickson and S. S. Pillai published exact formulas for g(n). In particular, they showed that g(n) = 2n + [(3/2)n] - 2 provided that ]](3/2)“]] > (3/4)n, where ]]z]] denotes the distance of z to the nearest integer. It is conjectured that the last inequality holds for all n 2 5; in any case, it has been verified for all 5 5 n 5 471600000 (Math. Comp. 55, 1990, p. 85). Thus, analytic methods were used to reduce the problem of an exact formula for g(n) to that of proving a fact of transcendental number theory. In 1957, Mahler proved that for any E > 0 there exists C(E) such that ]](3/2)nii > emEn,

n 2 c(e) .

This result implies the formula for g(n) for n > c(ln $). But unfortunately, Mahler’s inequality is ineffective, so we have no way to compute C(E). Effective inequalities have also been obtained, but thus far they have all been too weak to give the formula for g(n). The best effective inequality is due to A. K. Dubitskas [1990], who proved that ]](3/2)n]] > (0.5769)n. 0.7. The Basic Problems Studied in Transcendental Number Theory. The first type of problem is to prove irrationality, linear independence, transcendence, and algebraic independence of numbers. We now give the definition of the last of these concepts. Definition 0.2. Numbers HO. Since qc - p = q(C - t), we could have used S( 0 7

and the problem of transcendence of c, which is equivalent to the inequality lanCn + . . . + a01 > 0

forall

a0,...,

a,EZ,

laol+~..+la,l>O.

Thus, in all caseswe are dealing with lower bounds for linear forms. In order to give a lower bound for a linear form C, it is often useful to find linear forms over Z that are not proportional to L but have coefficients that are close to those of ,C, and that are sufficiently small in absolute value. We illustrate this with a simple exampie. Let Cl = a< - b,

Ls = c( - d,

a, b, c, d E Z,

ad # bc .

Then 1 5 lad - bcl = IaL2 - cLl[ < la&I + IcLll . If la&I < 0.5 and c # 0, then

Introduction

21

Similar remarks apply to linear forms in several variables (these will be studied in detail later). In what follows we shall often want to construct (small) approximating forms in the numbers under consideration. It is no exaggeration to say that most of this book is devoted to methods for constructing such forms, as will be clear to the reader who examines any of the subsequent chapters. Over the years a large number of techniques of proof and types of result have accumulated in transcendental number theory. Of course, it is not possible to cover all of this, and we had to choose just a small part of the available material. One should not assume that the most important topics have all been included in the book, and the rest is of limited interest. For example, we had to omit such areas as the classification of transcendental numbers; Mahler’s work in the early 1930’s on the values of functions that satisfy certain functional equations, and the subsequent development of his ideas (for a survey of this branch of transcendental number theory, see [Nishioka 19961); and many results involving non-archimedean metrics. On several occasions we had to shorten the length of this volume. We ended up deleting Chapter 7, which was devoted to the important question of estimates for the number of zeros of a function. We ask our colleagues not to judge us too harshly for our omission of some topics from the book.

22

Chapter

1. Approximation

of Algebraic

Chapter 1 of Algebraic

Approximation

Numbers

Numbers

51. Preliminaries 1.1. Parameters for Algebraic Numbers and Polynomials. Let p(z) =cp(zl)...)

z,)

. .. 2

= c

Vl =o

A,, ,...,v,zp

. . .z:

E C[zl, . . . , z,] .

“, =o

Definition 1.1. The degree in zk, the height, and the length of p(z) are defined, respectively, as follows: deg,,cp(z)

=

L(v)

H(p)

nk;

=

2

. a.

v1=0

=

2

mu

I&,...,v,,,I

IAV1,...,v,,,I

;

.

v, =o

In the definition of degree we suppose that at least one of the coefficients A with vk = nk is nonzero, k = 1,. . . , m. For every cx E A there is a unique polynomial P(z) = u,zn +. *. + a0 E Z[z] ) called the minimal polynomial of cx, such that (1) a, > 0, (2) (as,. . . , a,) = 1, (3) P(z) is irreducible, and (4) P(a) = 0. Definition 1.2. By the degree, height, we mean, respectively, dega = degP(z) = n;

and length of an algebraic number (Y

H(a) = HP(z));

L(a) = L(P(z))

.

, cxtn) of the polynomial P(z) are called the conjugates Therootscx=a(‘),... of Q. They are distinct, and they all have the same degree, height, and length. Their product is called the norm of (Y: Norm a = a(l) . . . otn). Obviously, Q c A. If cr = u/b with a, b E Z and (a, b) = 1, then dega = 1, H(a) = max(lul, lbj), and L(a) = Ial + Ibl. 1.2. Statement of the Problem. Let (Y be a real algebraic number. In the simplest case, the central problem of this chapter can be stated as follows: determine how small 6 = 6(ct; ;’ = IQ - ;I can be for p E Z,

q

E N. In particular, one might want to

a) find out how much is possible, i.e., how close rational numbers can get to a; or

$1. Preliminaries b)

find out how much is impossible,

23

i.e., find a lower bound for 6.

Since Q is everywhere dense in JR, it follows that for any 6 E R (in particular, for any real 0 E A) and for any E > 0 there are infinitely many rational numbers p/q such that p - I’ < & . Thus, questions (a) and (b) are trivial unless we impose some additional conditions. But these questions become very nontrivial for irrational a if we bound q from above and refine (a) and (b) as follows: A)

Find a positive the inequality

non-increasing

function

cp(z) = cp(z, a), 2 E N, such that

IQ - iI I 949) B)

has infinitely many solutions (p, q) with p E Z, q E N. Find a positive non-increasing function Q(z) = Q(z:, a), x E N, such that the inequality

IQ- ;I 2 ti,(Q) holds for all p E Z and q E N with p/q # (Y (or at least for all such pairs with q 2 90). Here we would clearly like to find v(x) that decreases as rapidly and $(x) that decreases as slowly as possible.

as possible,

1.3. Approximation of Rational Numbers. It is easy to get complete answers to questions (A) and (B) f or rational cr. Let Q = a/b, a E Z, b E N, a/b # p/q; then = IQ - ;I . (1) Since (a, b) = 1, by assumption, it follows that the equation ax - by = 1 has infinitely many solutions x, y E Z, and so we can take (p(x) = q(x) = l/(bx). These choices are best possible. Definition 1.3. Let 8 E R, and let W(X) > 0 be a function on N that approaches zero as x + co. We say that 0 has a rational approximation of order w(q) if for some c = ~(0, W(X)) the inequality

0 < le - tj < cw(9) holds for infinitely many pairs (p, q) with p E Z, q E N. (Note that one could include the constant c in the function w(x); however, it is more convenient, for example, to speak of an approximation of order qm3 than to speak of an approximation of order 0.0731qm3.)

24

Chapter 1. Approximation

of Algebraic Numbers

Thus, what we just established is that any rational number has a rational approximation of order q-l and does not have a rational approximation of any higher order. 1.4. Continued Fractions. It is well known that every irrational real number 0 can be represented by an infinite continued fraction (rational numbers are actually represented by finite ones)

whereaeEZandui,uz,... and the numbers

E N. The ok are called the partial quotients of 0,

P7l -=uao+2L+...+II 4n IQ

Id

(Pm%) = 1, 4n E w 7

(2)

are called the convergents. If 8 is irrational, then the continued fraction representation is unique. In 1798, Lagrange [1867] proved that for irrational 8 the convergents satisfy the inequality 1 Qn(Qn

+

Qn+l)

1 < tbP” Qt, Qt = qnt. Then from (3) we have

and hence Qt+l > 0.5Qf’

.

(6)

Thus, the denominators of “good” approximations of a real number must grow rather rapidly. 1.5. Quadratic Irrationalities. Lagrange [1867] proved that, if a! is a real algebraic number of degree two, then its partial quotients are periodic from some point on, and, in particular, are bounded. From this result and (5) we have Theorem

that

1.2. Let (Y E A, dega = 2. There exists a constant c = c(a) such

I I a-;

> cq-2.

Of course, the theorem is obvious if Im(a) # 0. Thus, by Theorem 1.1, a real quadratic irrationality has a rational approximation of order qe2; but, by Theorem 1.2, it has no higher order rational approximation. It is sad to have to admit that the above results seem to exhaust the possibilities for using (5) to answer questions (A) and (B) for (Y E A. We have not yet been able to find the full continued fraction expansion for a single

26

Chapter

1. Approximation

of Algebraic

Numbers

algebraic number of degree 2 3. Computations of the first several thousand partial quotients for such numbers as ti and ti support the conjecture that the sequence of partial quotients is unbounded. Thus, to study the order of rational approximation of algebraic numbers of degree 2 3 one needs other methods. The earliest result in that direction was obtained by Liouville (see [1844a], [1844b]). 1.6. Liouville’s Theorem

Theorem

and Liouville

Numbers

1.3. Let a E A, deg Q = n 2 1. If o # p/q, then c(a)

= (1+

~a~)l-“n-‘L(a)-l

.

Proof. When n = 1 the theorem follows from (1). Let n 2 2. If la-p/q1 2 1, the inequality in the theorem holds for any c E (0,l). Suppose that la-p/q1 < 1, in which case Ip/ql < (CX]+ 1. Let P(z) = a& + . . . + au be the minimal polynomial of cr. Then

Hence.

la-;1

>c(a)lq”P(;)lq-?

Since 12> 2, all of the roots of P(z) are irrational, and hence A = qnP(p/q) 0. But since A E Z, we have IAl > 1, and the desired inequality follows.

#

Theorem 1.2 is obviously a special case of Theorem 1.3. A consequence of Theorem 1.3 is that for k > deg a: > 1 the inequality

QI
Y X

yr-1 XP-1

WI

IXI’ ’

$3. First

Version

of Thue’s

Method

33

from which it follows that the rational number Y/X is an approximation of order r to the algebraic number cr. In the past our rational approximations have always been written with positive denominators; hence, if X < 0 we write Y/X = (-Y)/lXl. Th e relation (25) allows us to determine the function w in $3.1. Let Xi,Yi E Z be a solution of (24). Thue took the rational number Yi/Xi as P/Q, and used (22) to construct an infinite sequence of good approximations to cr. Because p,q,+i # pn+lqn, it follows that for any other solution X2, YZ E Z of (24) with 1x21 > 1x11 one can carry out the plan in $3.1 with p/q = Xs/Yz. But when 1x2) is large compared to 1x11, we get a contradiction between (19) and (25); hence, 1x21 cannot be arbitrarily large, and this means that the equation (24) can have only finitely many integer solutions. Although the title of Thue’s paper referred only to approximation of algebraic numbers, it was in that paper that he proved finiteness of the number of solutions of the Diophantine equation ax’ - by’ = c, T E IV, r 2 3, a, b, c E Z. At the same time the paper essentially contains a strengthening of Liouville’s theorem in the case (Y = m. Thue used his theorem to derive several results. Corollary 1.1. Let h E N, k E 74, k # 0. There exists n E N such that for of distinct integers al, . . . , a,, at least one of the products

any n-tuple

al . ..a.,

(al + k) . . . (a, + k)

has no fewer than h distinct prime divisors. Corollary 1.2. Let k E Z, k # 0, n E N, n > 2. The Diophantine equation xn + (x + k)” = yn has only finitely

Corollary equation

many solutions (x, y) E Z2.

1.3. Let h, k E Z, hk # 0, n E IV, n > 2. The Diophantine

x2 - h2 =

ky"

has only finitely many solutions (x, y) E Z2. Corollary 1.4. Let h, k E Z, hk # 0, m,n equation (x + h)m + (-l)“%? has only finitely

E N, m,n > 2. The Diophantine = kyn

many solutions (x, y) E Z2.

3.4. Effectiveness. Theorem 1.6 leads us to the question of effectiveness. An examination of the proof showsthat the last step in the argument, which led to a contradiction, works only if 1x1 I is greater than some constant that depends on a, b, and m. We also needed the number In 1X21/ In 1x1 I to be sufficiently large. From this point of view we can restate the theorem as follows.

34



Theorem 1.6. There exist constants C and c which can be computed for given a, b, and m, such that either all of the solutions (X, Y) E Z2 of (24) satisfy the condition 1x1 5 C, or else there exists a solution (Xl, Yl) E Z2 for which 1x11 > c 7

and in the latter case any other solution (X2, Y2) E Z2 satisfies 1x21

5

lXlC

*

From this version of Thue’s theorem it immediately follows that all integer solutions of (24) satisfy the inequality

PI+ IYI 5 co. However, unlike in the case of c and C, one cannot say that Co can be expressed explicitly in terms of a, b, and m. This constant also depends on Xi, about which we usually know nothing. For this reason we say that CO is a noneffective constant, and so Thue’s theorem is non-effective as well. Although it gives us finiteness of the number of solutions of (24), it does not lead to an algorithm for finding the solutions. In what follows we shall frequently return to the question of effectiveness. 3.5. Effective Analogues of Theorem 1.6. In [1918] Thue published the first effective versions of Theorem 1.6. He proved Theorem

1.7. Let r be an odd prime, a, b, (Y,p, y E N, and

ua’ - bfi’ = y , (4aa~)~-2

,

y2r-2T~2(r-1)-‘(aarb-lp-r)2r-4+2/r

(26) .

(27)

If p, q, k are natural numbers satisfying the inequality

I& - br I 5 k , then q < AkB .

(28)

Here A and B are constants that depend only on a, b, Q, p, y, and r. Thue gave explicit formulas for A and B, but we shall omit them, because they are quite cumbersome. Of course, (28) gives an upper bound for the solutions of the Diophantine equation ax’ - by’ = m (29) for any m. From this it is also easy to show that ]axf - byr] is bounded from below by a function of 1x1+ Iy] that approaches infinity as Ix] + ]y] + co, x,y E z.

$3. First Version of Thue’s Method

35

Thus, (28) gives effective bounds on the absolute value of the solutions of (29), provided that for given a and b we can find (Y, /3, y E N satisfying (26) and (27). Thue gave two examples when such (Y, p, y can be determined. 1. Let t E N. For (26) we take the equality (t + 1) . l3 - t . l3 = 1 . Here a = t + 1, b = t, T = 3, (Y = p = y = 1, and condition (27) holds for t > 37. Hence, for any h E Nc and m E Z the integer solutions of the equation (38 + h)x3 - (37 + h)y3 = m can be effectively bounded by (28) with 2. For (26) we take the equality

Ic = m and suitable

A and B.

1. 37 - 17. 27 = 11 . The condition the equation

(27) holds; hence, for any m E Z all of the integer solutions X7

are effectively

bounded.

- 17y7 = m

In particular,

of (30)

Thue proved that from the inequality

Ip7 - 17q71 < 106,

P,QEN,

it follows that q < 14293. Using Thue’s examples and the inequality (28), it was possible to give an effective refinement of Liouville’s theorem for the numbers qm, t E N, t 2 37, and m. In fact, in the case of (30) the inequality (28) takes the form q < AIrnIB which

implies

the following

= Alp7 - 17q71B ,

bound for the irrationality

Here C > 0 is also an effective constant.

It remains

measure of m:

to note that

7-$3,and

WI-

1+n/2

->

XnC2n-2

)

then the inequality axn -byy

SC

has at most one solution in relatively prime natural Siegel gave several concrete examples: 1. The equation 33x” - 32yn = 1

numbers

with n = 7, 11 or 13, has only one integer solution 2.Leta,b,c~Z,n~N,nz3,and

x = y = 1.

Then there is only one pair of relatively the Diophantine inequality

prime natural

x and y.

numbers

that satisfies

axn - bynj 5 c . 3.6. The First Effective Inequalities of Baker. In 1964 a series of papers by A. Baker ([1964b], [1964c], [1967a]) gave effective theorems on approximation of algebraic numbers that involve roots of rational numbers, along with bounds for the solutions of the corresponding Diophantine equations.

$3. First Version of Thue’s Method For any K. > 2 he obtained

the inequality axT - by’1 > CIxl-

where

37

a, b, r E N, r > 3, and b/a is “close”

)

to 1, namely, 10n - 5

a > (a - l1)~(3r)~~-~,

p=m.

Here c = (9ar2/2)-0,

a=(~--)

(

ln(9ar2/2) ln(81r4/2)

(r+a-b+1+(2p-5)

From this he derived an effective bound for the solutions equation axT - byT = Ck,lXkY1, ck,l E c

> ’ (31) of the Diophantine z

.

k+l Qlq-n,

n>2,

where ~0 = 31-‘(2a)-‘H-“(2

+ Ial)l--nc,

H = m~(l4 I4 1x1,IV) ,

and c is given by (31). In particular, cr can be taken to be m. In some cases (32) can be used to find bounds on rational approximations to roots of natural numbers. For example, Baker obtained the inequalities

I

fi _ ; > 10-6q-=‘55; I

m - iif > 10-gq-2.4 9I I

In deriving these bounds an important role was played by relations of the type (26) with small y. In the case of these examples, these relations were 2 . 43 - 53 = 3,

183 - 17. 73 = 1 .

In order to prove his inequalities Baker used the identity A,(x)

- (1 - z)~B,(x)

= x~~+‘C,(X)

where nEN,

v=i,

n>3,

mEN,

O<x 0, m E N, and lnm > 11.4(1+

$3. First Version

of Thue’s

Method

39

Here p and Q are arbitrary natural numbers, and c = c(m,~) > 0 is an effective constant. This inequality is a consequence of a more general theorem of Nikishin on approximation of roots of certain numbers in Q(i). We also mention some results of A. N. Korobov [1990]: 1. Ia - p/q1 > 4-2.5, P, Q 6 N, Q # 1, Q# 4. 2. All solutions (z, y) E Z2 of the Diophantine equation 2x3 - y3 = M,

MEZ,

satisfy the inequality

I4 + IYI I M2 . 3.7. Effective Bounds on Linear Forms in Algebraic Numbers. In [1967a] Baker also obtained lower bounds for the absolute value of linear forms in several roots of natural numbers. Let a,b,mr,... ..,mk3,and ,mk,n E N; hml,. x = a(a - q-“-14-“k(“+l)

> 1.

Thenforanyze,zr,...,zkEjZwehave X0 + X&/b)ml’n

+ *. . + x&%/b)““‘”

> Cx--& ,

where

x = oyjyk I4 ’ 03 -c =

IC= (Iclna-nk(k+l)ln4)ln-‘X,

8-n-‘A-55(n+l)k-(k+l)(n+l)

.

As a corollary he proved that for m E N, m > loll, 1 = m5 - 1, and M E Z, all of the integer solutions of the equation x5 + ly5 + 12z5 + 51x2yz2 - 51xy3z = M satisfy the inequality

14,Ivl, I4 < e500M2. Similar results were obtained by other authors. In particular, effective lower bounds were found for linear forms of the form L = xl(q + alyy

+ . . . + xm(q + a*$”

)

where al,... , a,, q,y E &, K is an imaginary quadratic field, IqI is much larger than (ail,. . . , la,], xi,. . . ,x, is a nontrivial m-tuple of elements of &, and v E Q. For certain values of Y the lower bounds for IL1 made it possible

40

Chapter

1. Approximation

of Algebraic

Numbers

to derive effective upper bounds for the solutions in & analogue of Thue’s equation:

of a multivariable

NormL=f(zi,...,z,). Here f(zi, . . . , 2,) is a fairly low degree polynomial over Zx. E. M. Matveev, who proved several theorems of this type in the nonarchimedean as well as archimedean metrics (see [1979], [198Oa], [1980b], [1981], [1982]) obtained effective bounds on the solutions of certain ThueMahler equations [Matveev 198Oa] Norm L = p”f(zi,

. . . , z,),

f E qxl,...,Gn]~

The method used in the multivariable caseis a generalization of the method described in $3.1. Similar arguments will be used to prove some of the theorems in Chapter 2, so that we will later have the chance to examine this method in more detail. Recently, a series of new effective lower bounds for the absolute value of linear forms of the form

where a,b,Al,... ,A,,B A. K. Dubitskas [1990].

$4. Stronger

E N and x0 ,...,

x,

E Z, have been obtained by

and More General Versions of Liouville’s Theorem and Thue’s Theorem

4.1. The Dirichlet Pigeonhole Principle. In the general case, when o is an arbitrary algebraic number of degree n 2 3, Thue was not able to obtain explicit formulas for polynomials analogous to the ones in (20), so he used a different method. In the last century Dirichlet [1842] showed how to use a very elementary argument to obtain important and rather precise results on Diophantine approximations. This argument, often called the “pigeonhole principle,” is that if n+ 1 objects are placed in n pigeonholes, then at least one of the pigeonholes must contain two or more objects. We shall give several examples of theorems whose proofs use this principle. Theorem 1.8. Let al,. . . , aN E R, XI,. . . , XN E N, and X = maxXj. Then there exists a nontrivial N-tuple x1, . . . , xN E Z (‘nontrivial” means not all zero) such that

54. Stronger Versions of Liouville’s and Thue’s Theorems

Theorem

41

1.9. Let M, IV, X E N, N > M,

Li = 5

ER

aij

&,jXj,

-4 > 5

i = l,...,M.

lai,jl,

j=l

j=l

Then there exist x1,0,. . . , XN,o such that

I-Woo>l I AiX 1 - N/M Theoreml.lO.

) i = l,...,M;

O< lFjyN -- Ixj,Ol 5 x .

Letm,nEN,m 0, N E N,

Using an analogue of Theorem PN(x),

QN(x)

A=II$.~mlAjl. --

7

--

M = [N(S - 1 + n/2)]

.

(35)

1.10, Thue proved that there exist polynomials

Ezkl,

Ez[x, a] ,

RN(x)

such that J'N(x)

d%fk(X),

- ~QN(x) d%QdxC) L(QN),UPN)

= (x -

# 0,

ajNR~(x)

I M + N;

deg,

RN(X)

5 DN ,

where D = D(a, S) is an effective positive constant. Let p E Z, q E N. If Iqa - pl < 1, then, setting PN

=

PN(p/q)q”+N,

qN

= &iv@/q)q”+N

(36) I M ;

(37)

Chapter 1. Approximation of Algebraic Numbers

42

and using (35), (36) and (37), we obtain an infinite sequenceof approximations to (Y. We can also obtain a series of complementary approximations if we use the identities that are obtained from (36) by differentiating several times these are the rational numbers

We already saw that we want to ensure that the condition (16) holds. In the previous case, Thue used (23). He did not have such an identity in the general case, so he developed a different technique, generalizations of which still play an important role in transcendental number theory. Let V&N(X)

=

p;‘(,)Q:‘(x)

-

@‘(x)&$(x),

s,tEN,

s#t.

Vs,t,~(z) is a polynomial with rational integer coefficients and with degree at most K = 2M + 2N - s - t. Thue proved that Vs,i,~(z) is not identically zero and is divisible by a high power (say, the Z-th power) of the minimal polynomial of (Y. Thus, if V~,I,N(Z) is divisible by (qz-p)h, we must have 0 5 h 5 K-nl. This implies that h is not very large, i.e., p/q cannot be a root of VO,~,N(X) of very high multiplicity. It is easy to show that the derivatives of VO,~,N(Z) can be expressed as a linear combination of the polynomials VB,t,~(~), from which we see that there exist a, b E No that are not very large and are such (p/q) # 0. Thus, we can take the pb, qh and pz, qi in $3.1 to be that Va,b,N P$)(p/q)q”+N-“/al

-7 Q$‘(p/q)q”+N-a/a!

P$“(p/q)q”+N-b/b!,

.I

Q;‘(p/q)q”+N-b/b!.

The rest of the argument proceeds as in $3.1. Using the above considerations, Thue [1908] proved the following general theorem. Theorem 1.11. Let f(z, y) E Z[z, y] be an irreducible form Then for any M E Z the equation

f(Xc,Y> = A4

of degreen 2 3. (38)

has only finitely many solutions (x, y) E Z. Remark. The irreducibility condition can be replaced by the weaker requirement that the polynomial f (z, 1) h ave at least three distinct roots. If this condition fails, then the corresponding equation (38) might have infinitely many solutions. Simple examples of this include the equations (z + y)” = 1 and (z2 - 2~~)~ = 1. The latter equation is satisfied by all pairs (xm, ym) E Z2 given by 2, + fiy, = (3 + 2fi)m, m = 0, fl, f2,. . .. Theorem 1.11 has been used to answer several number theoretic questions. Thue [1917] and Mordell [1922] proved, respectively, that the equations

$4. Stronger

Versions

ax2+bx+c=dyn,

of Liouville’s

and Thue’s

a, b, c, d, n E Z,

Theorems

43

n 2 3, a(b2 - 4ac) # 0

and ey2=ax3+bx2+cx+d,

a,b,c,d,eEZ,

ae#O

have only finitely many integer solutions (in the second equation one must assume that the cubic on the right has distinct roots). In particular, the Diophantine equation x2 - y3 = m, m E Z, m # 0, was shown to have only finitely many integer solutions. This equation had been the subject of research for three centuries. Polya [1918] proved that if a,b,c,x E 2 and a(b2 - 4ac) # 0, then the greatest common multiple of the numbers ax2 + bx + c approaches infinity as z + 00. In particular, this is true of the polynomial x2 + 1 (as had been conjectured by Gauss). Thue proved that, if P(x, y), Q(x, y) are forms in Z[z, y], P(x, y) is irreducible, and deg P > deg Q, deg P > 2, then the Diophantine equation P(x,Y> = Q(x,Y) has only finitely many solutions. Here is one more corollary of Theorem 1.11. Suppose that n E N, n 2 3, and the strictly increasing sequence {zk} consists of perfect squares and perfect n-th powers. Then LFa(zk+l - zk) = 00 . We shall give one example with a proof. Let P be the set of natural numbers all of whose prime divisors belong to a certain finite set (~1, . . . ,p,}. We show that the equation x-Y=c,

CE z,

c#O,

(39)

has only finitely many solutions X, Y E P. Suppose that x, y is a solution of (39), with x =pY’ . ..ymy.

yzpp

. ..p$-.

pi,4

E No *

If we set pi = 3ai + ui,

Vj

ai,ui,bj,Wj

=3bj+Wj,

z =p;‘...p”,“,

t =pi’

. ..pk

E No, Ui,wj 5 2, )

then Az3 - Bt3 = c , where A and B are integers taken from a finite set. By Thue’s theorem, each of these equations has only finitely many solutions. Since there are only finitely many equations, this means that the solutions of (39) are bounded. However, the bound on the absolute value of the solutions is non-effective, since Thue’s theorem is non-effective.

44



4.3. Thue’s Theorem on Approximation of Algebraic Numbers. In [1908] Thue did not go beyond proving Theorem 1.11, although he already had the tools needed to strengthen Liouville’s theorem. He published his improvement on Liouville theorem in [1909]. Theorem 1.12. Let Q: E A, n = dega 2 3, and E > 0. There exist effective positive constants qo = qo(a,e) and a = a(a,e) such that ijpp1/ql and pz/qz satisfy the inequality

Q_ ?1< q-l-E-n/2 ) I QI

(40)

where then (41) Remark. In Theorem 3 of [Thue 19091,the inequality qz 5 G = G(q) was given rather than (41); however, from the proof it was clear that In qz/ In q1 was effectively bounded. An immediate consequence of Theorem 1.12 is Theorem 1.13. Suppose that the conditions of Theorem 1.11 are fulfilled. There exist constants QO = Qc(o, E) > 0 and ~0 = ~(a, E) > 0 such that 1) the inequality (40) has no solutions with q 1 Qo; and 2)foranypEZ,qENonehas

Q_; >Qq-l--G. I I

(42)

Thue derived Theorem 1.11 as a corollary of his Theorem 1.12. In fact, one can obtain even more from (42). Suppose that f(z, y) is a form in iZ[z, y], deg f = n 2 3, and (~1, . . . , cr, are the roots of the polynomial f(z, l), where cyi # crj for i # j. Then by Lemma 1.2 If(x:,~)l

2 44 + IYI>“-~ lo.

We use Theorem 1.13 with E = l/4 to bound the last factor from below. We obtain (here cl(f) > 0, x, y E Z):

If(

> s(f>(l4

+ Ivl>“-’ * 64 + lYlF’2)-(1’4)

= Cl(f)(lXl

+ lyl)(2n-5)‘4 .

(43)

Now suppose that the degree of the polynomial g(z, y) is lessthan (2n - 5)/4. Then it follows from (43) that g(z, y) h as smaller order of magnitude than f(z, y) as 2, y -+ 00, z, y E Z. Hence, all of the integer solutions of the equation m, Y> = g(z, Y>

(44)

$5. Further

Development

of Thue’s

Method

45

must satisfy the inequality

I4 + IYI I MO. This observation was published by Maillet in [1916]. 4.4. The Non-effectiveness of Thue’s Theorems. The condition (41) gives an upper bound on the denominators of rational numbers that satisfy (40). However, because this bound depends on q1, Theorem 1.12 can become effective only when we are able to bound q1 from above. So far no one has found any examples where they were able to determine a solution pl/ql of (40) for which q1 2 qo, or at least give an upper bound for such a q1. To be sure, in 1982 Bombieri proved a theorem similar to Gel’fond’s theorem (which we shall give in s5.2) without the condition q1 2 qo. This led to a series of effective inequalities, which we shall discuss in $5.4. Thue himself commented on the non-effectiveness of his theorems (see [Thue 19771,p. 248). Despite the lack of any effective upper bounds for the solutions themselves, it is fairly simple to find effective upper bounds for the number of solutions to equation (38) and inequality (40). We shall describe such results in 57.8.

55. Further

Development

of Thue’s Method

5.1. Siegel’s Theorem. We have used Thue’s identity (36) to set up some good rational approximations to a number CLNow we shall use the identity in a somewhat different way. We first introduce a new argument y, and subtract yQ(z) from both sides (we also omit the subscript N). We obtain the identity (x - c~)~R(z) - (y - a)&(~) E P(z) - yQ(z) - T(z, y) . Suppose that pl/ql

(45)

and pz/qz are “good” approximations to a. If we also have (46)

then, since T(z, y) E Z[z, y], we find that (47) In view of (45) and (47), lo - pl/qll and lo - pz/qzl could not both be very small, and we would be able to prove Theorem 1.12. Although we do not have (46), we can obtain

for some s that is not very large. In fact, Thue proved that there exist some not very large values of a and b for which Va,b(pl,ql) # 0 (see 54.2); hence, one of the two numbers

46


is nonzero, so that s Theorem 1.12 follows Instead of (45), in (where N is an integer


can either be taken equal to a or to b. The proof of once we bound r8 from below. [1921a] Siegel used the following more general identity parameter, as it was for Thue):

(x - 4Nw&

Y) - (Y - a)Vh

Y) = WC&Y>

(48)

I

where

U(z,Y>, Vb, Y> Ea? As before, the main difficulty not very large and for which

Y, 4,

WhY)

E a&Y1 *

was to prove that there exists

s E No that is

(49) where & and 0, there exists cl = C~(CY,E) > 0 such that q E N one has

11 a - f

> c1q-x,

X=

min t4...,n(&+g

for

+e*

any p E Z,

(52)

Here one can replace X and cl by 2&i and c2 = c~((Y) > 0. 2) Let IK be an algebraic number field of finite degree, degK (Y = d > 2, E > 0. Then there exists c3 = C~(CK,l&c) > 0 such that for any primitive element (’ of the field lK one has

Here one can replace p and cs by 24 and cd = C~(CY,K) > 0. 3) Given E > 0, there exists c5 = C~(CY,E) > 0 such that for any [ E A, deg p = u,=$::, , ,

~5H(t)-~,

Here one can replace p and c5 by 2ufi

+e.

and c6 = es(a).

The number 2&i becomes less than the exponent 1 + n/2 in Thue’s inequality (42) as soon as n = 12. Note that (52) is a more precise inequality than (42) for n 2 7; the two inequalities are identical for n 5 6. Siegel derived many corollaries from his Theorem 1.14. We shall give some of them. 1. He extended Thue’s Theorem 1.11 to the case when M, the coefficients of f (2, y), and the solutions to equation (38) are algebraic integers. He did the same for equation (44). 2. In [Siegel 19261 he proved that there are only finitely many rational integer solutions of the equation Y2 = P(x),

P(x)

(5 Q4

P(x)

$0 7

provided that P(x) has at least three distinct roots. 3. Let cr E A, degcr = n. There exists a constant c = c(a) such that for any P(z) E Z[z], P(z) $0, degP(z) = h < n one has IP(

2 ~H(P)-~~‘fi+l

.

48

Chapter 1. Approximation of Algebraic Numbers

For small h (namely, for 2h2 5 fi) this inequality is more precise than (11). However, the constant c is non-effective. 4. Suppose that K is an algebraic number field of finite degree, P(z) E Z,[z], P(z) $0, 0 E Zn 0 [Siegel 1929/1930]. He proved that if lK is an algebraic number field of finite degree, and C is an algebraic curve over Z that cannot be parameterized by rational functions, then C can contain only finitely many points whose coordinates are integers of K. In [1931] Mahler proved Theorem 1.14 for (Y= mm, a, b, m E N, using a method that resembled some ideas of Hermite that we shall discussin Chapter 2. Theorem 1.14 and its corollaries are non-effective, since cl, . . . , cs are noneffective constants. 5.2. The Theorems of Dyson and Gel’fond. The following refinement of Thue’s theorem was proved in [Dyson 19471. Theorem 1.15. Let Q E A, degcu = n > 3, E > 0. There exists a constant c = ~(a, 8) > 0 such that for any p E Z and q E N with q > c one has a--p >q- diik 9 I 1

.

Here also c is a non-effective constant. In [1948] Gel’fond obtained this inequality as a corollary of a more general theorem. We shall give Gel’fond’s theorem in a somewhat more precise form (in Gel’fond’s original version si = s2 = S = 1). This refinement can be obtained without any significant changes in Gel’fond’s original proof. Theorem 1.16. Let crl, (~2 E A, llC = Q(CXI,(~a), deg lK = n, 6 = 1 # K is real and 0.5 otherwise, E > 0, 2 5 t$, e2 < n, &t12 = 2n + E, s E N. There exist efectiwe constants LO = Lo(crl, (Ye,E,&, B2,s) > 0 and a = U((YI, CQ,E,&,&, s) > 0 such that, if two algebraic numbers 0 and for all H 2 Ho = Ho@, E). In [1976a] Galochkin was able to prove bounds with this exponent for certain values of the functions pj(z) (see (87)). Theorem 2.18. Suppose that m 1 2, the algebraic numbers Xi,. . . , A, are not equal to -1, -2,. . . and are such that P(z) = x(z + Xl) *. . (z + X,-l)

9 = m4gll,.

. . , lib-11),

= P + gm-lP-l

+ * * * + g1z E Jl[Z] )

and b E Za, b # 0. Then for any hj E Zp, 1 5 j 5

m, such that mm

l<j<m --

IhjlIH,

HZ3,

one has the inequality Ihlcpl(l/b)

+ ... + hmcPm(l/b)I > cll(lblH)l-”

(88)

where 75 = (m - 1)2g - (m - l)Reg,-i + m(m2 + m - 4)/2 and cl1 e’sa positive constant depending only on m and g. When b, gi, hi E Z, the bound in Theorem 2.18 is, except for the exponent 75, the same as in the result for almost all numbers. The proof of Theorem 2.18 usesthe same construction of functional linear forms as in Theorem 2.17 with A, = 0. The forms one uses are Rj(Z) =

(mn-l+j)! 27ri

cpM)dC‘ r (C 1)mn-1+j ’ f

j = l,...,m,

where r is the circle ]C - I] = 1 taken in the counterclockwise direction. Is it possible in certain cases to further improve (88), i.e., to get a better ys? In [1986] Korobov used recurrence relations to prove precise lower bounds for linear forms in the values of certain generalized hypergeometric functions at the point l/c, c E Z, c # 0. These estimates differ only by a constant factor from the upper bounds that are possible for these linear forms. Note that Korobov’s theorem was proved in 1981 and published only in 1986. In [1984] Galochkin used an effective analytic construction of approximating forms to generalize Korobov’s results and prove unimprovable bounds for the values of the functions p(z) in Theorem 2.17. We now give an exact formulation of these results.

132

Chapter 2. Effective Constructions

in Transcendental

Number Theory

Suppose that s E Z, s 2 2, Xi,. . . , A, are elements of the quadratic imaginary field II that are not equal to -1, -2, . . ., and a is a nonzero number such that a, aXi, . . . , aX, all belong to Zn. Let Z”

ti’(z) = g

v=o arn”y!(X1

where m = s + 1. Set uj = {ReXj}, a real number, and set j-1 -

A = min l<j<s --

+ 1)“. . . (X, + l)V ’

where

{ . } denotes the fractional

+ uj _ Cl + * *. + us S

s

part of

. >

In Theorem 2.19 there are also some positive numbers pl ,p2, n that depend on Xi,... , A, and are computed in [Galochkin 19841 in a way similar to A. For brevity we shall not give the explicit expressions for these numbers. Theofern 2.19. Suppose that b E 251,b # 0, ho,. . . , h, E 251are not all zero, max Ihi 1= H > 3, and

R = hs$(l/b)

+ h&(1/b)

+ ... + h,@)(l/b)

.

‘Then there exist effectively computable positive constants cl, ~2, ~3,q depending 012X1,..., X,, a and b, such that 1)

one has

2)

there exist infinitely many forms R for which

]RI > ci HmS(ln H)-‘(‘-“)

(ln In H)pl ;

IRI < c2HmS(ln H)-“(l-A)(lnln 3)

if

H)pl ;

R is a primitive form (see [Galochkin 19841for the definition) and if IRI > csH-‘(In

H)-S(‘-A)(Inln

then IRI > c4H-‘(In

H)P2 ,

H) rl- 41-A)

In comparison with [Galochkin 197Oa],the sequenceof approximating linear forms in the proof of Theorem 2.19 is explicitly constructed so as to be s times denser. For every N E N define T, t E Z by setting l‘.‘, s, and r is a contour that goes once around 0 and 1. These forms are linear combinations of the functions @c(z), +i (z), . . . , $*(z) with polynomial coefficients, and any m = s + 1 successive forms are linearly independent over C(z). It is from these forms at the point l/b that one obtains the required numerical approximating forms, which lead to a proof of part 1) of Theorem 2.19 and to an infinite sequence of forms satisfying the inequality in part 2). In [1990] Korobov used the Jacobi-Perron algorithm to find multidimensional continued fraction expansions for certain vectors connected to the values of generalized hypergeometric functions. These were multidimensional generalizations of certain classical continued fractions of Hurwitz, Euler, and Ramanujan. As a consequence Korobov was able to prove some unimprovable bounds for linear forms. Here is one of his results: Let s, q E N and set ‘$‘k

=g

v=.

((s

+

1)v

+Ik)!g(s+‘)y+”

k=0,1,...,

’

s.

Then there exists a positive constant cl such that the following inequality holds for any integer H > 3 and any ho, . . . , h, E Z not all zero, where maxihi] 5 H: --s(s+3)/(2s+2) Iho$o + . . . + h,&l

> c~H-~

In addition, there exists an infinite sequence of linear forms in the same numbers for which the opposite inequality holds with a different constant cq. In particular, for s = 3 one has the precise bound ]ho cosh(l/q)

+ hl cos(l/q) + h2 sinh(l/q)+hs

sin(l/q)]

>

> c~H-~

6.3. Ivaukov’s Construction. Generalizing Galochkin’s results, Ivankov [1986, 1991b] carried out a similar effective construction of simultaneous approximations for the values of arbitrary hypergeometric E-functions. Let cyi, . . . , a,. and pi,. . . , &, be rational numbers not equal to -1, -2,. . ., 0 5 T < m, and let F(z) be the generalized hypergeometric function with parameters 1, cri + 1,. . . , crp + 1 and pi + 1,. . . , pm + 1. If we denote a(x) = (57+ al) * *. (z + a,)

and

we have F(z)=gzvk@lg. v=o (The term for v = 0 is taken to be 1.)

b(z)

=

(2

+

Pl)

. . . (x

+

Pm)

,

134

Chapter

2. Effective

Constructions

in Transcendental

Number

Theory

Theorem 2.20. Suppose that none of the di$erences cxi - pj are rational integers, and WI,. . . , wt are pairwise distinct and nonzero elements of I. If bo, bij, 1 5 i 5 t, 1 5 j 5 m, are any nontrivial set of integers of II having absolute value at most H, H 2 3, then

b. + 2 2 bkjF(j-l) (wk) > ~10H-mt-Y4(~~~~~)-’ , kc1

j=l

where 74 is a constant depending only on the (Yi, bj, and wk. In the case T = 0 this result was proved in [Galochkin 19781. We now describe Ivankov’s construction in more detail. The functional approximating form R(z) needed to prove Theorem 2.20 is defined as follows:

(89) where r = mt(n + 1) + n,

k=l

in which the Bk(V) are polynomials in v of degree at most (n + 1)m - 1. In addition, R(z)

where PO(Z), the notation Xl@:>

=

Pkj

=

PO(z)

+

2

2

k=l

j=l

pkj(z)Fj(wkz)

,

(91)

(z) are polynomials in z of degree at most n. If we introduce

Xj(z)

1,

=

Cz + Pj-l>Xj-1

Cz>

for

j =2,...,m,

then the functions Fj(z) in (91) are given by the formulas Fj(z)

= Xj(G)F(z)

= eZyXj-l(u) v=o

k@l $$

.

(92)

It turns out that it is more convenient to construct functional linear forms in the Fj(z) than in the PlF(z). In order to explain where (91) comes from, we consider the following m(n+ 1) polynomials fj,(V) in the variable V: n-1 fjp(v)

=

xj(v-p)

n k=p

P--l 4~sk)

n k=O

b(v--k),

llj<m,

O 0 the number PC”)(a+ ky) is a polynomial with algebraic coefficients in the transcendental number In p, whereas the numbers f(“)(x)/ In” p are algebraic. 2.4. The Real Case. If cr, p, and 7 are real numbers, then Theorem 3.2 can be proved without using the theory of analytic functions. Such a proof was given in [Gel’fond 19621. Instead of the maximum principle one uses Lemma 3.1. Suppose that the real-valued function f(x) is continuous on the interval [a, b] along with its derivatives through the n-th order. If f (x) has n or more zeros on [a,b] ( counting multiplicities), then u I

n,

a<x 0 such that for anvtEAonehas IB(a, b) - (1 2 e-cdMT(lnT)M

,

where d = deg

n=l

c # 0, -1, -2,. . .) with rational parameters a, b, c. In [1988] Wolfart proved that, if one excludes certain exceptional triples a, b, c E Q and certain corresponding algebraic values of z, then F(u, b, c; z) is transcendental for all other a, b, c E Q and z E A. The exceptional 4-tuples (a, b,c,z) are described in [Wolfart 19881. It should be noted that these exceptional cases include not only the obvious algebraic hypergeometric functions, such as (1 - 2)-l/2

= F(-l/2,1,1;

z) ,

but also sometranscendental functions. For instance, F(1/12,5/12,1/2; z) and F(1/12,7/12,2/3; z ) are transcendental functions, but in [Beukers, Wolfart 19881it is shown that 12’ 12’ 2’ 1331

+ii

and

F(&;;~)=+.

In this connection see also [Flach 19891and [Joyce and Zucker 19911. In [1988] Wolfart takes advantage of the fact that the values of the hypergeometric function are related to the periods of abelian integrals, and uses a general theorem of Wiistholz [1984c] on algebraic groups. Wiistholz’s theorem generalizes such classical results of transcendental number theory as Lindemann’s theorem, and implies that certain classesof abelian integrals are transcendental. In this connection we should also cite papers by Bertrand [1983], Beukers [1991], Cohen [1993, to appear], Cohen and Wolfart [1993], Hirata-Kohno

$5. Bounds for Linear Forms with Two Logarithms

161

[1992], Holrapfel [1993], Laurent [1983], Masser [1977a], Masser and Wiistholz [1990-19951, Morita [1972], Shiga and Wolfart [1995], Wolfart [1981, 1983, 1984, 1985/1986, 19871,Wolfart and Wiistholz [1985], and Wiistholz [1986]. In [Masser 19761and [Coates and Lang 19761estimates are proved for linear forms in algebraic points of abelian functions. Masser’s paper also contains applications of his results to certain Diophantine problems. See [Bertrand 19871for a survey of the results in this active area of recent research and their application to Diophantine geometry.

$5. Bounds for Linear Forms with Two Logarithms 5.1. First Estimates for the Transcendence Measure of ab and lncu/ Ino. The method used to solve Hilbert’s seventh problem turned out to be suitable for obtaining quantitative results as well. We begin with the following equivalent problem (see (8) of the Introduction): estimate the difference between these numbers and algebraic numbers. The first two theorems of this type were proved by Gel’fond in [1935]. Theorem 3.19. Let a, b E A, aln a # 0, b 4 Q, n E N, E > 0. There exists an effective constant Ho = Ho(a, b, n, E) such that for any c E A with deg< 5 n and H(C) > Ho one has

lab - H-(‘“‘“H)6+E,

H = H(C) .

(19)

Theorem3.20. Leta,~~A,~~lncrln~#O,~=ln~/ln~,~~~,n~~, E > 0. There exists an effective constant HI = Hl(a, p, In CY,ln/3, n, E) such that for any C E A with degC 5 n and H(c) 2 HI one has

1~- e-(‘“H)4+c,

H = H(C) .

(20)

We give a sketch of the proof of Theorem 3.19. Suppose that 6 > 0, and for some C E A we have lab _ ,

(27)

Clearly, this first step of the proof is almost identical to the beginning of the proof of Gel’fond’s Theorem 3.2. The only difference is that, instead of a’, which we are not assuming to be algebraic, we use a number C E A that is a good approximation to ab. But now one must proceed in a different way than before. Unlike in the proof for Hilbert’s seventh problem, we do not have a large number of zeros of f(z) at integer points. Instead, we have the relation (27), which, when combined with (21)-(26), enables us to obtain the inequality f(S)

@,)

I

,

21 = (lnln H)2+a6 [

1.

(30)

55. Bounds

for Linear

Forms with

Two Logarithms

163

from (21)-(23), (25), (26), and (30) we now obtain the inequality

If we compare this inequality with the lower bound for nonzero cps,Zin (10) of Chapter 1, we see that cp8,Z - 0,

(s,z)

E Qn(so,a)

*

We are now able to obtain a stronger inequality than in (30), namely:

From this, using (29), the inequalities in (10) of Chapter 1, and (21), we arrive at the system of equations

p (0) = -g & c&l (k + lb)” = 0, k=O

s=O,l,...,

(Qo+ l)(q + 1) - 1 .

I=0

Since b is irrational, the determinant of this system is nonzero. Hence, we have a contradiction: the system of equations implies that all of the ck,l are zero, but this is not the case. The above method of proof has been applied many times by a number of authors to prove various theorems. New technical details have been developed in order to strengthen the inequalities, but the fundamental ideas have remained the same to the present day. 5.2. Refinement of the Inequalities (19) and (20) Using Gel’fond’s Second Method. By making technical improvements in his method, Gel’fond [1939] was able to prove a more precise inequality than (20):

I I

lna Inp - c > e-(l”w3+z

;

and in [1949c] (see also [Gel’fond 1960]) he proved the effective inequalities lab - e-,"3;nSC1)(n+lnH)In~+~(n+lnH+1)

I I

In Inp(Y _ c > ,-~Z(~+lnw2+c

.

In [1972] Cijsouw obtained the bound

I I lna 0-C

Be-

cnylnL)yl+lnn)-‘(l+lnL)=

>

L = L(C) .

It seemsthat the most precise bound that can be obtained by Gel’fond’s method has the form

164

Chapter 3. Hilbert’s

I I lncu ~np-C

Seventh Problem

cn2(nInn+lnH)2(1+lnn)-3

>e-

Unlike earlier papers, [Gel’fond 1949c] and later works contain bounds that explicitly take into account the degree of C. Stronger results that have been obtained using Baker’s method will be discussed in Chapter 4. 5.3. Bounds for Transcendence Measures. If we take into account the inequality (9) of the Introduction, we can use the bounds in the previous subsections to obtain similar estimates for the transcendence measures, namely:

Ip CabI I ’ e-cn2(n+ln and

-In (Y 1nP

H)

In2(n+ln

H+l)

ln-yn+l)

I( >I p

>

e-cn2(nInn+lnL)2(1+lnn)-S

,

where n = degP, H = H(P), L = L(P). We again emphasize that all of the constants in the inequalities in this section are effective. 5.4. Linear Forms with Two Logarithms. Using Lagrange’s formula, it is easy to obtain a trivial effective bound for the form

zln21-

ylnv,

z,y,u,w

E N.

Suppose that u” > vv. Then l~uZ-wV~(zlnu-ylnw) ]zlnZd - ylnv]

max eX; y In v<x<s In u

2 e-c(u~V)x,

X = max(z, y) .

This result can readily be extended to any algebraic numbers u and ‘u. Suppose that 1~~1> IVY], 21%# TJY,z,y E Z. We have IuZ -

vYl

= Iu51 le--slnu+ylnv

= WI

J

_ 11

yIIltJ--lIIlU etdt

0

5 1~~1lslnu - ylnvle (where we may, of course, assume that ]z lnzl - y lnv( 5 1). We use the inequality (10) of Chapter 1 to estimate the left side of the previous inequality. We obtain ]zIn21-

ylnv]

2 emCX,

c = c(u, v),

X = m=44, Ivl> .

(32)

The inequalities in $5.2 enable us to obtain much better bounds. A simple consequence of (31) is the bound

55. Bounds for Linear Forms with Two Logarithms

IG Ina0

- GlnP0I

QO,PO,ll,G n

=

> e

--cn*(nInn+lnL)z(l+ln7L)-3

E 4

7

CltlnaolnP0

#

JqCl/G)

degQ(Cl/L4,

165

=

L

(33)

0, .

Here c is an effective constant that depends only on In o. and In PO. 5.5. Generalizations to Non-archimedean Metrics. As early as [1935] Mahler proved a p-adic analogue of the Gel’fond-Schneider theorem. Theorem 3.21. Let CYand fi be p-adic algebraic integers, where

Then In, CY/In, ,l3 is either a rational number or a transcendental p-adic number. This theorem was then generalized to p-adic valuations. In [1940] Gel’fond proved the following quantitative result. Theorem 3.22. Suppose that p is a prime ideal of the algebraic number field K, Q, p E K, [al,, = l/?lP = 1, and the relation am - p” = 0 with m,n E Z holds only when m = n = 0. Then for any E > 0 Ord,, (a5 f BY) < ln3+E X X = m=(lzl,

Z,Y E z,

1~1)

2 X~(chB,p,e)

> 0.

If, in addition, pip with p a rational prime, ordpp = t, kp 2 ICI > t/(p - l), ord,,(o-1) = kl, andord,,(P-1) = kz, thenfor anyql,qz E Z with (ql,q2) = 1 one has

1nP I I -__

42

lno

ql

D

,

e-

ln3+c

q ,

Q=m4lqil,

lq21) L 40(lna,lnP,p,E)

> 0.

Multidimensional generalizations and refinements of Gel’fond’s inequalities have been obtained using Baker’s method. We will discuss this in Chapter 4. 5.6. Applications of Bounds on Linear Forms in Two Logarithms. These bounds have been used to answer many questions. We give some examples. 1) Let q(z) $0 be defined and completely multiplicative on Z, i.e., cp(mn) = cp(m)cp(n) for all m,n E Z. Suppose that cp(p) = 0 for all but finitely many primes p, and

166


for some constant

Seventh Problem

c and any N E N. Then cp(p) # 0 for at most one prime

p; and if there is such a prime po, we must have cp(po) = eia, a E IL!. This theorem

was proved by Linnik

and Chudakov

[1950].

2) An example in Gel’fond’s book [1960] has played a very important role in transcendental number theory and the theory of Diophantine equations. Let (Y = o(l) and /3 = p (l) be linearly independent integers of a real algebraic number field lK = K(l) of degree three, and let lKc2) and lKc3) be the conjugate fields. Consider the Diophantine equation Norm(aa:

+ /3y) = fi(&)z

+ /?(j)y) = 1 .

(34)

j=l

This is a special case of Thue’s equation (38) of Chapter 1. By Theorem 1.11, it has only finitely many solutions; however, that theorem does not give an effective bound on the number of solutions. Gel’fond gave a method for finding effective bounds for the solutions of (34), by reducing this problem to estimating linear forms in the logarithms of algebraic numbers. The value of Gel’fond’s method was not that he was able to find effective bounds for solutions of (34) (which had already been done before using methods of algebraic number theory; see, for example, [Delone 19221 and [Faddeev 1934]), but rather that he had the idea of reducing Diophantine problems to bounds for linear forms in the logarithms of algebraic numbers. We now sketch his solution to the above problem. Suppose that K c2) and lKc3) are complex fields. In that case, by the Dirichlet unit theorem, the units of lK are of the form flm, where I is a fundamental unit. This means that (34) is equivalent to the system of equations

a(l)x+ pq/ = (10)m &d2 +pwy = (9 m (w(3)2

+ p(34/

=

p39

m

for some m E Z, where It21 and 1c3) are the conjugates of I = 1(l) in Kc21 and lKt3). If we eliminate x and y from (35), we obtain the following equation in m:

(

a(3)p(2)

_ &)p(3)

) (qm

+

(&p(3) +

We may suppose 10~s that ]1t21 ] = m satisfying (36) n > 0. Note that

(,qw

- a(3q3u))

(qm

+

(36)

- &)p@) >(I(3)>m=0.

that 1(l) > 1. Since ]1(2)] = ]1(3)] and 1(1)1(2)1(3) = 1, it fol]1t3) ] = (]I(‘) ])-lj2. It is now clear that the positive integers can be effectively bounded from above. Next, let m = -n, (~(~)/3(‘) - (~(‘lp(~) # 0, since if (Y(~)//?(~) = &)//z?(~), then

55. Bounds

for Linear

Forms with

Two Logarithms

167

we would have c~(~l/p(~) E R. But a complex cubic field contains no irrational real numbers; hence, a (l) /p (l) E Q, which contradicts the linear independence of a and /3. Now from (36) we obtain the inequality

where c and ~0 are effective constants. We fix In A. From (37) we obtain the following inequality for some r E Z:

In 1952, when the Russian edition of [Gel’fond 19601appeared, no nontrivial effective lower bounds were known for the absolute value of a linear form in three logarithms of algebraic numbers. Hence, Gel’fond introduced another condition: that X be a root of 1, i.e., In X = hi, t E Q. Note that in any case ]A] = 1, since the numbers (Y(~)P(~) - a(1)@(2) and CX(~)/~(‘)- CY(~)~(~)are complex conjugates. From (38) with k = 2r - t we now obtain I(2) nlnm-kS where cl justified, below by I(2)/I(3),

<e

--c1n

,

where cs is an effective constant. This implies that we have the effective inequality /ml In+lkl _nc(oL, 0). This implies that the largest prime divisor of P, approaches infinity as n increases. Later, Schinzel [1967] found an explicit formula for no. It is clear from the definition of the Lehmer numbers that (Y and p are algebraic of degree 1, 2, or 4. Subsequently the numbers P,, were generalized using arbitrary algebraic numbers (Y and p, and a proper prime divisor was interpreted as a prime ideal. 6) In [1967] Schinzel refined Gel’fond’s inequalities and obtained explicit formulas for the constants. Here are some results that follow from his estimates. Theorem 3.24. Suppose that uo, ~1, P, Q E Z, PQ # 0, A = P2 - 4Q < 0, P2 # jQ for j = 1,2,3, uf - Puluo + Qui # 0, and un+l = Pu, - Qu+l for n >_2. If q is any prime divisor of Q/(P2, Q) and n > no, then


lunl > &(pZ.Q)“/‘exp Here no can be expressed explicitly Schinzel also obtained of u*.

(iv-)

169

.

in terms of P, Q, q, uo, ~1.

an effective lower bound for the largest prime divisor

Theorem 3.25. Let D be an odd natural number not equal to 2” - 1 for k E N. Then 1) The Diophantine equation

x~+D=~~ has at most one solution with m > 80 and x > 0. 2) If p is any prime divisor of 40 + 1, then the Diophantine equation x2+D=pm has at most one solution with m > 1 + 6(10 + lnlnp)/ Theorem

3.26.

lnp .

Let CYbe a real quadratic irrationality,

and let g E N, g > 1.

Then

Il~dYl > c7+ exp(43 for n E N, where c is an effective constant.

For a E Z let P(a) denote the largest prime divisor of a. Theorem

3.27.

Suppose that f(x)

E Z[x], deg f (x) = 2, and f(x)

has sim-

ple roots. Then

wherek=2iff(

lim inf p(f (x>> > ~ z-boo lnlnx - 7 ’ x ). as irreducible and k = 1 otherwise.

7) Let R = Q(A) be a quadratic imaginary field, where -d E N is squarefree. It is well known that any cr E ZK can be uniquely written in the form a+bw,a,bEZ,where 4-i w = { i(l+

A),

ifdE2,3(mod4), if d E 1 (mod 4) .

If w’ is the conjugate of w (i.e., either -&

or (1 - &)/2),

DC1 w2 I I 1 w’

= (w - w’)2

is called the discriminant of K. It is easy to see that

then


170

D =

Seventh Problem

4d,

if df

2,3 (mod 4),

d,

ifdrl

(mod4).

Unlike Q, most quadratic imaginary fields do not have unique factorization into prime elements. One first has to decide what numbers play the role of primes in algebraic number fields. Let a,P E Zx. We say that cr and p are associates, and write cr - /3, if (Y = ,0(‘ f or some algebraic unit C. If p E & is not a unit, we say that p is irreducible in # if all elements of Zn< that divide p are either associates of p or units. In Q there are only the two units fl, so that k - m for k, m E Z simply means that lc = fm. The irreducible elements of Z are of the form p or -p, where p is a prime. We say that the field lK has unique factorization if, whenever 7rl

. * .7r,

=

p1

* * * pt

with irreducible ni and pj, we have s = t, and each ni can be paired with a in corresponding pi so that pi - pj. Besides Q, we have unique factorization such fields as Q(G) and Q(n). On the other hand, lK = Q(a) does not have unique factorization, as we see from the example 2.3=(1+-)(1-G), in which all four numbers are irreducible, and 2 and 3 are not associates of l&G. For a long time nine values of d were known for which the corresponding quadratic imaginary field has unique factorization, namely: -1, -2, -3, -7,-11, The corresponding

discriminants -4,-8,

-19, -43,-67,

-163.

(46)

-67, -163.

(41)

are

-3, -7,-l&

-19,-43,

It was Gauss who found these discriminants, although question that at first looks completely different. Let A,B,C

E Z,

D=B2-4ACO,

F(x, y) = Ax2 + Bxy + Cy2 . F(z, y) is called a positive definite binary quadratic form, and D = D(F) is called its discriminant. If @(x, y) is another binary quadratic form, we say that F(z, y) and G(x, y) are equivalent, and write F - I, if there exist a, b, c, d E Z, ad - bc = 1, such that @(x, y) = F(az

+ by, cz + dy) .

(We note that equivalence of quadratic forms can be defined in much greater generality; but we shall only need to consider positive definite binary quadratic forms.)


171

It is easy to check that D(F) = D(Q) for equivalent forms F(s, y) and @(z, y). Since F - @ and F - G imply that @ N G, we can divide all forms with a given discriminant D into equivalence classes. It can be shown that for fixed D there are only finitely many equivalence classes. Gauss wrote that the nine numbers (41) are all the discriminants he was able to find that have only one equivalence class of forms. These are called the discriminants of class number one. A basic concept of algebraic number theory is that of an ideal class in a field lK c A. We say that two ideals 2t and 93 of K belong to the same class if there exist a,@ E Zx, (YP # 0, such that (o)!2l = (/?)%, where (o) and (fi) denote the principal ideals generated by (Y and ,0. It turns out that K has unique factorization if and only if it has only one ideal class, i.e., if and only if all ideals of the field are principal. Now we need only observe that there is a one-to-one correspondence between ideal classes of a quadratic imaginary field of discriminant D and equivalence classes of positive definite binary quadratic forms of the same discriminant D. Thus, Gauss’ nine numbers (41) are also the discriminants of quadratic imaginary fields with unique factorization. Gauss conjectured that there are at most finitely many discriminants miss.ing from his list of discriminants of class number one. This conjecture attracted the attention of many mathematicians. Finally, in [1934] Heilbronn and Linfoot proved that there can be no more than one discriminant missing from the list (41), and that such a tenth discriminant DIO, if it exists, must be large. Increasingly large lower bounds were found for DUO: Dickson [1911] obtained IDlo > 1.5. 106; Lehmer [1933] proved that lDl,-,I > 5.10’; and Stark [1967] was able to show that ~DIO[ > exp(2.2. 107). In [1948] Gel’fond and Linnik proved that, if DIO exists, then the following inequality holds: ]zilnor

+22lncYs

+lncrs]

<e --Y&m.

Here ~1, (~2, and (~3 are fixed algebraic numbers independent over Q; x1,22 E %, with l~ll,l~2l

whose logarithms

(42) are linearly

< -Yom;

and y and 70 are effective constants. In 1948 no one had yet been able to prove a nontrivial and effective lower bound for the absolute value of a linear form with three logarithms of algebraic numbers; hence, (42) had to be kept in reserve. If it became possible to show that the absolute value of the linear form in (42) is bounded below by exp(-cp(]Dru])), where cp(z)/& + 0 as x + 00, then one would have an upper bound for [DIo~. It would then just be a matter of comparing this upper bound with the lower bounds given above, and, if they are incompatible, then we would know that DIO does not exist. This plan came to fruition twenty years later. We shall return to this question in Chapter 4.

172

Chapter

3. Hilbert’s

Seventh Problem

In [1952] Heegner published a proof that Die does not exist. But his proof contained a gap, and he had to withdraw his claim. The deficiency in Heegner’s proof was finally corrected by Stark in [1967, 19691.But actually the solution had been within reach of Gel’fond and Linnik. As Chudakov showed in [1969], the problem of showing that Dia does not exist can be reduced to finding a suitable bound for a linear form with two logarithms (this was also shown independently by Stark). Chudakov proved the inequality IHln(2 + &)

- 2?rmI

< 73.5e-“ml12,

HEZ.

(43)

After making the computations, it turned out that J&e] < 1042, and this bound is incompatible with Stark’s inequality l&e] > exp(2.2 . 107). 8) Gel’fond’s inequalities were used in [Babaev 1966] and [Segal 1939, 19401 to answer questions about the distribution of integer points on certain curves and about the distribution of integers that are divisible only by primes in a fixed finite set. 9) A final application is connected with coding theory. In [Bassalygo, Fel’dman, Leont’ev, Zinov’ev 19751the inequality (zln3 - yln2]

> e-210(‘nY)2,

Z,Y E N,

Y22,

is used to solve a problem related to error-correcting codes.

96. Generalization of Hilbert’s Seventh Problem to Liouville Numbers 6.1. Ricci’s Theorem. In [1935] Ricci showed that the number a in Theorem 3.1 can be replaced by any Liouville number. Theorem

3.28.

Let a,/? E A, (Y # 0, 1, p $! Q. If K. $ Q, and for some E > 0

the inequality K, I

has infinitely scendental.

!2

0 the inequality

n-P<e 9I I

-(ln

Suppose that for

q)a+c

has infinitely many solutions p E Z, q E N. Then the number (cr~)fl is transcendental.

$7. Numbers

Connected

with

the Exponential

Function

173

6.2. Later Results. In [1937] Franklin proved Theorem 3.30. Let IK be a fixed algebraic number field of finite degree. Suppose that a # O,l, b 4 Q, and

where &, &,6,, E IK are irrational and have conjugates that are uniformly bounded. Further suppose that A, E N and &A,, &A,, &A, E 23~. If la - EnI + lb - &I + lab - &I < e-Inb An for n = 1,2,. . ., then k 5 7.

In [1957] Schneider proved that the inequality la - aI + lb - /3[ + la” - yI < e-Ins+’ H,

&>O,

has only finitely many solutions (Y,p, y E A, p $ Q, H = max(a(a), H(P), H(y)), if the degrees are bounded. Schneider’s inequality was later refined by a series of authors, including Bundschuh, Cijsouw, Mignotte, Shmelev, Waldschmidt, and Wiistholz. We give a result from [Waldschmidt 1978b]. Theorem 3.31. Let a, b E Cc, aln a # 0. Then for any (Y,/3, y E A, ,6 $! Q, one has Ia - aI + lb - @I+ lab - yI > e--cn4(‘nH)3(‘n’nH)-2 , where H = max(H(a), constant.

H(P), H(y))

and c-c-

(1na,blna)>Oisane#ective

The condition that /I $?!Q cannot be removed, as one seesfrom the following theorem of Bijlsma [1977, 19781. Theorem 3.32. Given any K.> 0, there exist irrational such that the inequality la - al + lb - PI + lab - yl < e-In” H,

numbers a, b E (0,l)

H = m=(H(a),

H(P), H(y))

,

has infinitely many solutions (~,p,y E Q.

$7. Transcendence

Measure of Some Other Numbers Connected with the Exponential Function

7.1. Logarithms of Algebraic Numbers. In Chapter 2 we discussedthe irrationality measures and the transcendence of 7rand In Q for Q E A. The bounds in that chapter were obtained by methods that various authors developed using ideas of Hermite. We had good bounds in terms of H for ]e - e- cn(nln

n+lnL)(l+ln

7%)

7

(44)

where c is an effective constant. If we compare (44) with (7) and (9) of the Introduction, we see that for In L >> n Inn the only factor that might be “unnecessary” in the exponent of (44) is the increasing term (1 + Inn). The constant c has been computed; in any case, it is known that one can take c = 389. We note that stronger inequalities can be obtained under some additional conditions on C, for example, if C E Q(e2AilN), iV E N. Theorem 3.34. Szlppose that m 2 1, cri, . . . , Q, E A, and lnoi, . . . , lna, are fixed valzles of the logan’thm that are linearly independent over Q. There exists an eflective constant c = C(CQ,. . . , cr,; In cyl, . . . , In (Ye) > 0 such that for any

e-cn’+“~(nlnn+lnL)/(lnn+l)

)

k=l

where

nlnLi -+...+m

L = exp nk

=

d% > IC=l,...,m,

nm L( 0 such that

IP- Cl> e-c(ln

fIq(ln

In Hy

for any C E A, deg< 5 n, H(6) 5 H. Corollary 3.2. Let p 4 A, n E N, a E A, aln a # 0. Suppose that for any CQ> 0 the inequality -co(lnH)s(lnInH)”

IP - Cl < e

has infinitely many solutions C with degc 5 n, H(c) 5 H. Then p and up are algebraically independent.

176


$8. Transcendence

Seventh Problem

Measure of Numbers Elliptic Functions

Connected

with

In 53 we gave several theorems on transcendence of numbers connected with the Weierstrass function Q(Z). Bounds for the transcendence measure of such numbers have been obtained in many papers by such authors as Beukers, Brownawell, Chudnovsky, Hirata-Kohno, Kholyavka, Masser, Nagaev, Philippon, Reyssat, Tubbs, and Waldschmidt. We shall give some of these inequalities below. We shall use the notation of $3 and the notion of a multiplier of a lattice that was introduced in $3.2. 8.1. The Case of Algebraic Invariants. In this subsection we shall suppose that the invariants g2 and g3 of p(z) are algebraic. We further suppose that t,tL,Ez E A, n = degt, nj = degtj, H = H(C), Hj = H(&), L = L(C), Lj = L(&), j = 1,2. All of the c and cj will be effective positive constants. Here are some of the many inequalities that have been proved:

1) ]W - 51 > exp (-c(w)n2(n(lnn)3

+ (InHlnlnH)‘)),

2) ]wi - &I + ]WZ- 521> exp(-cvl\rln’iV),

w E R.

where

3) If & wi + &WZ # 0, then ISlwl +

Gw2l

>

exp(-c2vM3)

,

where M=lnN+vlnv+vlnln(N+l)+civ,

. 4) Let u be an algebraic point of p(z), i.e., p(u) E A. Then ]u-El

> exp(-csn4(lnH(lnlnH)2+n(lnn)3)(1+lnn)-4)

.

5) Withuasin4),

147~)- El > exp (- c4n4(ln H(lnln 6)

Iv - 51> exp (- csn2(In H(ln

H)’ + n3(lnn)‘)(1

+ Inn)-4)

In H)’ + n2 (ln n)“)) .

These inequalities are essentially due to Reyssat [198Oc].

.

58. Numbers

Connected

with

Elliptic

Functions

177

In [1990a, 1990b] Hirata-Kohno obtained a refinement of 3) and 4) and also a new inequality: 3’) ]&wi + &wz] > exp (-csv3(lnN 4’) Iu - (1 > exp (- crn2(lnH 7) I&w +

(274

+ lnv)(lnlnN

+ nlnn)(lnlnH

+ lnv)2).

+ lnn)2).

2 exp (- cs~2(lnN+lnv)(lnlniV+lnv)).

Because of the inequality (8) of the Introduction, one can obtain similar bounds for the transcendence measures of these numbers. 8.2. The Case of Algebraic Periods. From $3 we know that both wi and ws are transcendental whenever gs, gs E A. Hence, if wi, ws E A, then at least one of the numbers gs,gs is transcendental. Notice, by the way, that in that case one of the invariants might still be algebraic. For instance, if the lattice is constructed from the basis { 1, i}, then gs = 0 E A. We give two theorem of Kholyavka [1987].

E A, n = degQ(wi,ws,&,&),

Theorem 3.36. Suppose that wl,w2, exp(-cnNlnN)

Theorem 3.37. Suppose that &,<s,wi Ns = (ns +ns)

(

.

E A, n = degQ(&,&,wi),

and

2

elnL2

+ 5lnLs

+n+nlnn

>

.

Then

I92-

(21 + I93 - 531 > exp (-cn2NslnNo)

.

8.3. Values of p(z) at Non-algebraic Points. Bijlsma [1977] extended the results of s6.1 to p(z). Theorem 3.38. Let g(z) have algebraic invariants and period lattice R, let lK be the field of multipliers of p(z), and let a, b E Cc,a, ab $! 0. Then for any a,p, y E A with /3 4 K one has ]@(a) - (Y)+ l&ab) - PI + lb - y] > exp (-cn6(lnH)6(lnlnH)-5)

,

where n = deg Q(Q, P, r>,

H = madee, H(a), H(P), H(y))

Here c is an effective constant depending only on p(z).

.

178


Seventh Problem

Bijlsma also extended his Theorem 3.32 to p(z), and proved that the condition p 4 K cannot be removed in Theorem 3.38. In [Bundschuh 19711 bounds were found for the transcendence measure of roots of an equation of the form P(z, p(z)) = 0, where g(z) is the Weierstrass function.

51. Linear

Forms in the Logarithms

of Algebraic

Chapter 4 Multidimensional Generalization Seventh Problem $1. Linear


Numbers

179

of Hilbert’s

of Algebraic

Numbers

1.1. Preliminary Remarks. Let (~1,. . . , cr, E A, and let In ai,. . . , Ina, be fixed values of their logarithms. For the duration of this chapter we will use the notation A=bilnai+~~~+b,lna,,

bi,...,b,EZ;

Ao=bo+bllnalf...+b,lncr,, Al =bilnai

b0,

+*..+b,lna,,

(1)

h ,...,b,

bi,...,&

E A;

(2)

EA.

(3)

It is easy to prove a trivial lower bound for IAl. Let x+1...&-

0 < IAl < 1,

-1.

If L is the line segment connecting 0 and A in the complex plane, then

1x1= /”0 I

e’dz

5 IAl

rnLp leZl < e[Al .

I

The number X is the value at (crYf’, . . . , a%), &j = sgn bj, of the polynomial

Since A # 0, it follows that X = 0 is possible only when A = 2kri for nonzero k E Z, in which case \A( >_ 2n. So we may assume that X # 0. From (11) of Chapter 1 we obtain the inequality IAl 2 emCX,

x = ozym -- lbjl 3

where c = c(cri,..., a,) > 0 is an effective constant. (In the case m = 2 a similar estimate was obtained in 55.4 of Chapter 3.) The inequality (4) can be strengthened using any of the ThueRoth theorems. This is what Gel’fond did in [1948]. Theorem 4.1. Let E > 0. There exists a constant He = He(c,ai,. lncri,..., lna,) > 0 such that if A # 0, then IAl > emEx,

X = oFjym -- lbjl 2

HO

.

. . ,cY,,

(5)

The constant He is non-effective, since (5) comes from the non-effective Theorem 1.16. Gel’fond also obtained a padic analogue of Theorem 4.1.

180

Chapter

4. Generalization

of Hilbert’s

Seventh Problem

In the case m = 2, Gel’fond’s method was used to obtain bounds that

are stronger than (4) or (5), are effective,and apply to forms with algebraic coefficients. This was discussed in @5.4-5.5 of Chapter 3. 1.2. The

First

Effective

Theorems

in the General

Case. In [1966a]

Baker

proved the following fundamental result. Theorem 4.2. Suppose that m 2 2, cr1, . . . , cxy, E A, 6 > m + 1, n E N, and the numbers In (~1,. . . , In cr,, and rri are linearly independent over Q. There exists an eflective constant c = c(m, al,. . . , am, In (wl, . . . , lna,, K, n) > 0 such that for any nontrivial m-tuple PI,. . . ,p,,, E A, deg/?j 5 n, H(fij) 5 H, j = l,..., m, one has IAl 1> ce-lnffi H . (6)

When m = 2 this inequality is the same as (27) of Chapter 3. Baker gave two consequencesof his theorem that resolve the multidimensional analogue of Hilbert’s seventh problem. Theorem 4.3. Suppose that ~1,. . . , cx,,, are real, algebraic, and not equal to 0 or 1; and supposethat PI, . . . , &,, are also real and algebraic. If 1, pi,. . . , & are linearly independent over Q, then the number

is transcendental. Theorem4.4. Letcq ,..., cr, l A,crl...cr, #O. Ifthenumberslncxl,..., and rri are linearly independent over Q, then they are linearly indepenlnff,, dent over A.

When m = 1, Theorem 4.3 is the same as Theorem 3.3, and Theorem 4.4 is the same as Theorem 3.1, except that the conditions in Theorems 3.1 and 3.3 are less stringent. Soon after, Baker [1967b, 1967b] refined his theorems as follows. Theorem

4.5. Suppose that /3&l * * *,&

H = mm H(pj),

# 0, IE > m + 1, n = maxdeg&,

j = 0, 1, . . . ,m. Then IA01

wherec=c(m,cul,...,

o,,lncri

>

ce

-In=

H

>

,... , In a,, n, n) > 0 is an effective constant.

Theorem 4.6. Suppose that either In ~1,. . . , lncw, or else PI,. . . , & are linearly independent over Q. Let n = maxdeg& and H = max H(/?j). If K > m, then IAll > ce-I”” H ,

wherec=c(m,ai

,...,

(r,,lnal,...

, In o,, 6, n) > 0 is an effective constant.

31. Linear


of Algebraic

Numbers

181

Theorem 4.7. Let (~1,. . . ,CX, E A, cq *..cr, # 0. Then lnoi ,..., lno, are linearly independent over A if and only if they are linearly independent over Q.

Ifcq

Theorem4.8. number

,...,

(wm,/30,P1,...,

,&EA,~~~~~%&#O,

thenthe

eoaap . . ..om m

is transcendental. At the 1970 International Congress of Mathematicians in Nice, Alan Baker was awarded the Fields Medal for his series of papers on bounds for linear forms in the logarithms of algebraic numbers and their applications to various problems of number theory. Theorem 4.5 also has consequences for the theory of integrals of rational functions. In his book [1949] Siegel noted that transcendental number theory had not at that time been able to determine the algebraic nature of the number

J

’ dx -= 0 2s+1

?ln2+A. 3 3&

Theorem 4.5 implies that this number is transcendental; and, in fact, one can prove a much more general fact (see [van der Poorten 19711). Let P(z), Q(z) E a[~], (P(z), Q(z)) = 1. Let ~1,. . . ,cr, be all of the distinct poles of the function P(z)/&(z), and let vi,. . . , vm be the corresponding residues. If the contour r in Ccis a closed path, or if it joins two points or extends to infinity, and if the integral

Jrp(“)dx Q(z) m SC

exists and is equal to a, then a E A if and only if

rkcl

“kdz=O. z--k

In particular, if deg P(z) < deg Q(z), then either a = 0 or else a $ A. In 1968 the exponent K in (7) was replaced by 1. Theorem 4.9. Let In al,. . . , lno, be any fixed values of the logarithms of the algebraic numbers al,. . . , crm, and let n E N. There exists an effective constant cs = cc,(ai ,..., om,lncri ,..., lna,,n) > 0 such that if -40 # 0,

deg~(al,...,cr,,po,...,P,)

L-co,

L

=

o~k~mL(Pk) --

.

(8)

When m = 2 and PO = 0, this theorem gives a better bound in terms of L than (29) and (32) of Chapter 3. On the other hand, the theorem does not

182

Chapter 4. Generalization

show how CXJdepends on 3. Theorem 1.9 shows us Lmco cannot be replaced power of L. The exponent because of its importance detail.

of Hilbert’s

n, whereas

Seventh Problem

this is done in (29) and (32) of Chapter

that the dependence on L in (8) has the right form: by a function that decreases more slowly than a c~ has been the subject of much research, especially in applications. In $1.4 we shall discuss this in more

1.3. Baker’s Method. Many authors have introduced various technical refinements - some of which are rather complicated - in order to strengthen Baker’s original bound (6). However, Baker’s basic idea that allowed him to improve on Gel’fond’s second method still lies at the heart of the proof of these results. We shall describe this basic idea without dwelling on the technical details, which can be found, for example, in [Baker 1975a], [Stolarsky 19741, [Waldschmidt 1974c, 1979b]. For simplicity, we shall assume that cq, . . . , cr,, pi,. . . , ,&-i E A, and shall sketch the proof of the following assertion: if

lna,

=/3ilncrl

+...+/3,-ilnom-r,

(9)

then lnai,... , In CX, are linearly dependent over Q. (Note that when m = 2 this is Theorem 3.3.) 1) Let q E N, n = degQ(m,. s = (4mq)2m+l, LqU,V)

={(s,z)

. . ,a,,

c = 2qsm,

x =qm, : Sl)...,

PI,. . . ,&-I),

Sm-l,ZENO,

s= (Sl)...)

sl+“‘+Sm-l

3,-l)

,L(dC))

and since L(&))

5 ~13, it follows -w

From this inequality,

fort=l,...,

I

.

(35) gives the bound C12pq

;

from (38) and Lemma 4.4 that I

C141Jw16

.

(35), (36), and (39) we obtain the bounds H

pq

Seventh Problem

I

5~16

+C17lnIMI

,

l/~e+3+~17~~~~~)

C2pq

0 and C23 = C23((Y) > 0 SUCh that

I I a - ;

> C22qC23--n

for any p E Z, q E N. Proof.

Let a,.?

+ . . . + a0 = a, fi

(z - &)

j=l

be the minimal

polynomial

of o = o(l), and set

f (x, y) = anxn + an-~xn--ly

+ . . * + aoyn = a, fJ (x - Gy) j=l

fhq)=M. Since f (z, y) is irreducible,

it follows

from (40) that

)

32. Applications

of Bounds on Linear Forms

193

We may suppose that ]CX- p/q1 < 1; then

Now

from which

the theorem

follows

immediately.

The inequality (40) and Theorem 4.19 were obtained in 1971 as corollaries of Theorem 4.18. However, the first (less precise) inequalities were found by Baker [1968b], who derived them from (29). The later refinements of (29) led to improvements on Baker’s original bounds for solutions of Thue’s equation and for ICY- p/ql. As noted before, the above constants cis, czi, ~22, and ~23 are effective. Various investigators - Baker, Gyijry, Kotov, Papp, Sprindzhuk, Stark, and others - have found explicit formulas for them. For details, we refer the reader to [GyGry 19801, [Sprindzhuk 19821, [Shorey and Tijdeman 19861, [Schmidt 19911, and [de Weger 19891. Here we shall only give two inequalities from [Baker and Stewart 19881. Let a E N, a # b3 for b E N. Let C be the smallest unit of Q(G) that is greater than 1. Set Cl =

C(501nw

7

c2 = 1012 In C .

Then 1)

ForanypEZandqENonehas

I I &G‘-

2)

g > cq+,

where

c = (34~1,

K. = 3 - ;

.

Ifz,yEZ,mEN,and X3

- ay3 = m ,

then max(]z],

Iv]) < cymcz

.

Unlike the similar inequalities in $33.5-3.6 of Chapter 1, which were proved using Pad& approximations, the bounds of Baker and Stewart were proved using estimates for linear forms in the logarithms of algebraic numbers. 2.4. The Thue-Mahler

Equation.

We already spoke of the equation

194


of Hilbert’s

Seventh Problem

in $10 of Chapter 1. The p-adic versions of effective bounds for the linear form A make it possible to obtain effective versions of Theorem 1.49. The first such results were due to Coates [1969, 1970a, 1970b]. Theorem 4.20. Suppose that f(x, y) E Z[x, y] is an irreducible form of degree n 2 3, N E Z, N # 0, P = {PI,. . . ,pS} is a fixed finite set of primes, and N = mM, m, M E Z, where all of the prime divisors of m are in P and CM, PI -..ps) = 1. l-f N,x, Y E Z, g.c.d.(x, y,n . . .pS) = 1, satisfy the equation f(x,y)

= N 9

then for any p > 0 one has max(]x],

]y]) < ce(‘“IMi)“, c = exp

v = %(s P Theorem 4.21. If x,y f (x, y) is greater than

(

(

n=n(s+l)+l+p,

2vaP2svn2 (1+

+ l)K’,

H(f F))

>

P = max(pr , . . . ,p,, 2) .

E Z, (x, y) = 1, then the greatest prime

1nlnX lO*n*(lnH(f)

divisor

of

l/4

+ 1) >

’

X = m@4 IYI) .

(41)

Coates obtained similar bounds for the equation y2 - x3 = k as a consequence of his p-adic version of Baker’s first estimate for A. The subsequent improvement of the bounds for A and their p-adic analogues led to refinements of Coates’ theorems. For details, see [Sprindzhuk 19821 and [Shorey and Tijdeman 19861. See also [Serre 19891. 2.5. Solutions in Special Sets. Let P be a fixed finite set of primes, and let S be the set of all rational integers whose prime divisors all belong to P (we also suppose that fl E S). We have already studied solutions of equations in S, for example, in the case of the equation x-y = c (see s4.2 of Chapter 1 and 55.6 of Chapter 2). There has been much work on solutions of equations or inequalities in rational numbers whose numerators and denominators (or only the denominators) have prime divisors that all belong to P. We encountered one such situation in 556.3 and 10.1 of Chapter 1. The notion of the set S can be extended to algebraic number fields, where instead of prime numbers one takes either irreducible elements of the field or else prime ideals. Many theorems have been proved about the solutions in S of various problems. One can find a great deal of information on this in the books [Schmidt 19911 and [Shorey, Tijdeman 19861.

$2. Applications


195

In some cases one can approach these problems using bounds for linear forms in the logarithms of algebraic numbers. We illustrate this using theorems from [Tijdeman 1973b] and [Shorey, Tijdeman 19861. Theorem 4.22. Let P be a finite set of primes, and let n1 < n2 < . . . be the sequence of natural numbers all of whose prime divisors belong to P. Then

nj+l

- nj 2 nj(lIITIj)-c,

nj 2 3,

where c = c(P) > 0 is an effective constant. Theorem 4.23. Let K be an algebraic number field of finite degree over Q, let P be a fixed finite set of prime ideals of IK, and let S be the set of elements of & whose prime divisors all lie in P. Suppose that CY~,. . . , Q~ are distinct elements of IK, m,r,rl ,..., rnEN,m>3, bEI& and f(x, z) = (x - a12)”

* * * (x - cYn,Zp .

If

at least two of the rj are equal to 1, and if x,y equation fb, z) = bym 9

E &,

t E S satisfy

the

where msy(min(ord,x,ord,,y))

5 7,

then there exist a unit n E K and an eflective constant such that the heights of qx and qz satisfy the bound H(v)

+ H(v)

c = c(K, P,

f, b, m,

T)

5 c*

2.6. Catalan’s Equation. In [1844] Catalan conjectured that, except for 8 and 9, there are no cases of consecutive integers that are perfect powers of natural numbers. In other words, the equation xm-yn=l, in the four variables

X,Y,m,n

> 1,

(42)

X, Y, m, n E N has only one solution X=3,

Y=2,

m=2,

n=3.

(43)

Before 1976 only partial results had been obtained. In [1850] Lebesgue proved that there are no solutions with n = 2. In [1921] Nagell showed that (43) is the only solution with n = 3, and there are no solutions with m = 3; and in [1965] Ko Chao showed that (43) is the only solution with m = 2. A confirmation of Catalan’s conjecture came within reach after Tijdeman [1976b] proved that the solutions of (42) are bounded, i.e., max(X,

Y, m, n) < C ,

(44

196

Chapter

4. Generalization

of Hilbert’s

Seventh Problem

where C is an effective constant. He obtained this result using bounds for linear forms in two or three logarithms. To prove Catalan’s conjecture it then remained to compute C and check the values of X, Y, m, n E N satisfying (44). The first numerical result was due to Langevin [1975/1976], who showed that X”

< exp exp exp exp 730

and

P(mn) < exp241 ,

where P(mn) denotes the largest prime divisor of mn. Without loss of generality one may assume that the exponents m = p and n = q are prime. In [1992] Mignotte proved that q < 1.31 x lol* .

p < 1.06 x 1026,

Various necessary conditions on p and q have been proved. The following criterion is due to Inkery [1964]. If p E 3 (mod 4), then either p*-’ E 1 (mod q”), or else the class number of Q(G) is not divisible by q. If p E 1 (mod 4), then either p*‘-l s 1 (mod q2), or else the class number of Q(e2”i/P) is not divisible by q. These conditions enable one to substantially reduce the work required to run through the 4-tuples that could be solutions to (42). However, it is not yet computationally possible to complete the proof of Catalan’s conjecture. More information about Catalan’s equation and its generalizations can be found in [Shorey and Tijdeman 19861. 2.7. Some Results Connected with Fermat’s Last Theorem. * We now describe some results that are related to Fermat’s conjecture that the equation

xn + yn = zn

(45)

has no solutions x, y, z E N for n E N, n 2 3. Proofs of the theorems below are given in [Shorey and Tijdeman 19861. Theorem 4.24. Let x,y, z be a solution of (45), where n > 2. If n is odd and z - y > 1, then z - y > 2n/n. Moreover, there exist effective constants cl and c2 such that l-cl(lnn)3/n.

IY-xl>z

7

I(z

-

y)

-

(y

-

x)I

>

zl-4w3/~

.

Theorem 4.25. Let B E N, and let S be the set oft E N all of whose prime divisors are 5 B. If the triple x,y,z is a solution to (45), and if at least one of the six numbers x,y,z,y - x, y + z,x + z (respectively, at least one of the first four of these numbers) lies in S when n is odd (resp. even), then n + x + y + z < cs, where cs = Q(B) is an effective constant.

Suppose that F(X,Y) is a quadratic form over Z, n > 2, and the triple x, y, z is a solution to (45). If at least one of the numbers F(x, y), Theorem

4.26.

* The results in this section have been superseded by [Wiles 19951.

32. Applications


197

F(x,z), F(y,z) belongs to the set S in Theorem 4.25, then x + y + z < ~4, where c4 = cq(n, B, F) is an eflective constant. 2.8. Some Other Theorem Q(m,...

Diophantine

Equations.

We shall give just a few examples.

4.27. Let m 2 3, crl, . . . , (Yrn,pE , am, p) . Then the equation

z.4, ai # aj for i #j, K =

(X - a1Y) * * * (X - amY) = ji has only finitely many solutions x,y E Zn3, degK = n = 122-aant, a E Z, a # 0, Ial, lY!EJ 5 A, i = 2,..., t, M E Z. Then any solution (xl,. . . , xt) E Zt of the equation

198

Chapter

4. Generalization

of Hilbert’s

Seventh Problem

a Norm(zl + ~2~x2+ . +f + QCQ) = M

(48)

satisfies the bound lyzt -where R

is

1~1 < exp(cR(R + ln(AIMI))

ln2(R + e)) ,

the regulator of IK and c = c(n) is an effective

constant.

Theorem 4.30. Let (~2,. . . , at E A, a E Z, a # 0, (al, lTiSJ 5 A, I& = Q, l& = JK+l(ai), [l& : l&-l] 2 3, i = 2,. . . , t. Then any solution (xl,. . . , xi) E Zt of (48) satisfies the bound

l’=zt -- I4 < < exp QIq3n2-9n+6

1q$n~-$n+3

(

+ ln(AIMI))

(

(ln ID~)3nS-gnz+6’+1) ,

where D is the discriminant of the field Q(a2,. . . , at) and co = co(n) is an effective constant. In [Gybry 19801 similar results are obtained for some special cases of the generalized Thue-Mahler equation Norm(zr + zpop + . . . + ztat)

= m$

. . .pf’ ,

where m E Z, {PI,... ,p8} is a fixed set of primes, and we are interested in solutions (21,. . . , zt, 21,. . . , 2,) E ZFs. Here is another example from [Shorey and Stewart 19831. Theorem t

4.31.

Let a, b,c,d E Z, acd(b2 - 4ac) # 0. Ifx,y,

t E Z, 1x1> 1,

> 1, and ax2t + bxty + cy2 = d ,

then

I4 + IYI + t I c , where c = c(a, b, c, d) is an effective constant. We cannot drop the assumption that t > 1 in Theorem 4.31, since when = 1 there may be values of d for which the equation has infinitely many solutions (see the discussion of Pell’s equation in $2.2 of Chapter 1). In $5.1 of Chapter 1 we gave Siegel’s theorem on integer points on algebraic curves of genus 1. This was a non-effective theorem. In [1970] Baker and Coates proved an effective version of this theorem. t

Theorem 4.32. Suppose that F(X, Y) E Z[X, Y] is irreducible of degree n and height H, and suppose that the curve F(X, Y) = 0 is of genus 1. Then any solution x, y E Z of the Diophantine equation

F(X,Y) satisfies the bound

= 0

$2. Applications max(]z],


]y]) 5 expexpexp

(

(2H)‘““‘o)

199

.

2.9. The abc-Conjecture. For a, b, c E N let G(a, b, c) denote the squarefree part of abc, i.e., G=G(a,b,c) = n p. plabc,p a prime Masser and Oesterle conjectured: If a + b = c with (a, b, c) = 1, then for any E > 0 there exists 70 = TO(E) such that c < yoG’+& . (49) A proof of this conjecture would bring us closer to solving several problems in number theory. For example, let a = xn, b = yn, c = P, n, x, y, .z E N, n 2 3. If abc-conjecture is true, then any solution x, y, I of the Fermat equation xn+yn=zn would satisfy the bound c = Z” 5 -yo(~)G1+E . Let E = 0.25. Since G= n

p 5 xyz < z3 ,

PbW

it follows that zn

5

-y~(1/4)z3+"~75

)

and so Z

n-3.75 < 7)(1/4) .

This would mean that Fermat’s Last Theorem holds for n > no, and for every n E N, 4 < n 5 no, one would obtain an upper bound on the value of z in a solution. This bound would be effective if 7s is effective. It should be noted that for the purposes of Fermat’s Last Theorem it suffices to have c < ylG’rather than (49). In [1991] Stewart and Yu Kun-rui proved Theorem 4.33. There exists an effective constant 7 > 0 such that c < exp G2j3 +y/lnlnG (

>

.

2.10. The Class Number of Imaginary Quadratic Fields. In 55.6 of Chapter 3 we mentioned that bounds for linear forms in three logarithms of algebraic

200

Chapter

4. Generalization

of Hilbert’s

Seventh Problem

numbers can be used to prove that there is no tenth imaginary quadratic field of class number one. In [1969] Bundschuh and Hock implemented the idea of Gel’fond and Linnik [1948], and used Baker’s inequality (6) to prove that Die does not exist. Now suppose that the field Q(a), w here d E N is not a perfect square, has class number 2. It is known that in that case 0 < ]&lnai

+/3zlnaz

+Pslnas]

< e-O.l&,

where cri and ~2 are the fundamental units of certain real quadratic fields, CQ= -1, pi, P2 E Z, and /3s = 64a/31. Here Ipi] and I/32] are bounded by d”, where c is an effective constant. After strengthening Theorem 4.17, Baker and Stark were able to obtain an effective upper bound for d. Their calculations showed that d = 427 is the largest discriminant of a field of class number 2. For details, see [Baker 1971a, 1971b, 1975a], [Baker and Stark 19711, [Stark 1971, 19721. The paper [Baker and Schinzel 19711is also related to these questions. Using completely different techniques (from the theory of elliptic curves and L-series), Goldfeld [1976] and Gross and Zagier [1986] proved an effective bound for the discriminants of imaginary quadratic fields of any given class number. 2.11. Applications in Algebraic Number Theory. Let (Y E A, degcr = n > 1, and let o(l) = o,. . . , ocn) be the conjugates of (Y. The disctiminant of (Y is the number D(a) = n (Q(i) - (y(j))2 . l 0 such that

m M 2 clIn D(lnln

0)”

*

2.12. Recursive Sequences. By a recursive sequence we mean a sequence {un} that satisfies an equation of the form Un+k

n=O,l,...

= uk-lZLn+k-l+~~~+~lUn+l+~OZln,

.

(51)

Such a sequence is uniquely determined by its initial conditions, i.e., by the Ic-tuple ~0, . . . ,u~-~. Suppose that a~,. . . , ok-1 are complex numbers, and define the characteristic polynomial of the sequence to be P(Z) = Zk - ~k-lzk--l

- “* - a0

= fi(Z-Oj)Uj,

ei # eh

for i # h.

j=l

Then the general solution of (51) has the form un = 2

Rj(n)ej”

,

(52)

j=l

where the Rj (z) are arbitrary polynomials of degree Vj - 1 with complex coefficients. A particular solution is determined by initial conditions, which enable one to solve for the coefficients of the polynomials Rj(z). If as,. . . , a&i, uc, . . . , u&i are algebraic, then so are all of the u,. It is frequently possible to prove certain properties of the sequence {un} using bounds for the differences between terms in the sequence, i.e., bounds

202

Chapter

4. Generalization

of Hilbert’s

Seventh Problem

for the differences between exponential expressions as in (5) of Chapter 1. This sometimes reduces to estimating linear forms in logarithms. We have already examined similar questions in connection with bounds for linear forms in two logarithms (e.g., Theorem 3.24). We now give some results that can be proved using bounds for linear forms in several logarithms. Suppose that k = 2, i.e., un+2 = uun+i + bun, where a, b,ue,ui E Z, ~ue~+~ui~>0,a2+4b#0.Then U n-

-A#+Bpn,

A=

‘LLOP p-a

Ul

w ’

B=

p-cl!

uoa ’

where Q and /? are the roots of the polynomial .z2 - az - b. Suppose that I4 2 PI, AB # 0, and (Y/P is not a root of unity. Theorem 4.37. There exist effective constants c and co depending only on a, b, UO,UI such that

Iu, - 21,) 1 Iapqm

+ 2)-cl++2),

mfn,

M=max(m,n)>c0.

Let M E Z, M # 0. We let P(M) denote the largest prime divisor of M, and let Q(M) denote the squarefree part of M (i.e., the product of all primes dividing M). Let v = dega (thus, Y = 1 or 2). Theorem 4.38. There exist effective constants cl, ~2, cg, and ~4, depending on a, b, uo, ~1, such that

P(u,)

2 Cl (+Jl’++l)

)

n 2

~2 ,

Q(4

2

(~3%)

,

n2

c4

and exp

.

Now suppose that k 1 2, and the sequence {un} is given by algebraic initial conditions and the relation (51) with algebraic coefficients. In (52) suppose that Pll 2 18212 **- L l&l and p11 = . . . =

l&l

>

l&,lI

*

Theorem 4.39. Suppose that r = 3, 101I > 1, and at least one of the numbers e1/e2, e1/e3, e2/e 3 is not a root of unity. There exist efective constants cg, cg, ~7, and cfj, depending only on the coeficients in (51) and the initial conditions, such that lu,l 2 14 in exp(-c5 ln2 n), nlcs;

Iu, - urnI > ~el~~exp(-c71n2nln(m

+ 2)),

n>m,

n>c8.

$2. Applications

of Bounds

on Linear

Forms

203

Theorems 4.37 and 4.38 are due to Shorey, and Theorem 4.39 was proved by Mignotte, Tijdeman, and Shorey. Proofs are given in Chapters 3 and 4 of [Shorey and Tijdeman 19861, where one can also find discussions of work on this question by Beukers, GySry, van der Poorten, Stewart, and others. Stewart’s survey [1985] should also be mentioned. 2.13. Prime Divisors of Successive Natural Numbers. For n E N the numbers 2n - 1, 2n + 1, and 2n are pairwise relatively prime. Hence, if n > 1, each of these numbers has its own prime divisors. For n E N let g = g(n) denote the greatest natural number such that there exist distinct primes pi, . . . ,p, with ptl(n + t), t = l,..., g. Several papers have been written giving lower bounds for g(n). Using estimates for linear forms in the logarithms of algebraic numbers, Ramachandra, Shorey, and Tijdeman [1975] proved that g>c

( > Inn lnlnn

3

n>3,

’

where c > 0 is an effective absolute constant. In (19761 the same authors proved that if k E N satisfies the inequality k 5 exp(ce&%)

,

where ~0 is an effective absolute constant, then there are at least k distinct prime divisors of the number (n + 1) . . . (n + k). Several theorems have been proven about the prime divisors of the product (n + 1) . . . (n + k) and the prime divisors of a product of successiveterms in a general arithmetic progression, i.e., (a + b)(a + 2b) . . . (a + kb), a, b E N. Once again we refer the reader to [Shorey and Tijdeman 19861for more information about this. 2.14. Dirichlet Series. We now give one of the results proved in [Baker, Birch, and Wirsing 19731. Suppose that q E N, (q,cp(q)) = 1, x(n) is a character modulo q, and

Wx)=C,,--

O” x(4

n=l

If none of the characters x is the principal character, then the numbers L( 1, x) are linearly independent over A. This paper also proves more general results about the series

f(n)7 L(s) =cO”y$n=l

.-

where f(n) is a periodic function that takes algebraic values on Z. In particular, it is pointed out that in certain casesone can use Theorem 4.9 to obtain the inequality

204


where H is the maximum period of f(n)).

of Hilbert’s

Seventh Problem

height of the numbers

$3. Elliptic

f(l),

. . . , f(q)

(here q is the

Functions

3.1. The Theorems of Baker and Coates. Just as the solution of Hilbert’s seventh problem stimulated work on the properties of elliptic functions, the work of Baker similarly led to new theorems in this area. We shall use the notation introduced in 53 of Chapter 3. In this section we shall assume that the @functions have algebraic invariants. Theorem 4.40. Let q and v’ be the qua&periods corresponding to the periods w and w’ of the Weierstrass g-functions ~(2) and p1 (z), and let (Y, /3, y, 6 E A. Then the number X=aw+pw’+yq+6q’ is either zero or transcendental. Theorem 4.41. Let CEO,(~1, (~2 E A, QO # 0, H = max H((Yo, CQ,CXZ), n = deg Q(~o, al, 4. Then Ia0 + cylwl + where K is an absolute constant. Theorem

constant

a2w2I

> ce-'"x

H ,

and c = c(gg, gs, WI, ws) > 0 e’s an effective

4.42. Let a E A, (Y # 0. Then Ip(a)I

where K. is an absolute constant.

constant

< Gel”= H(a) , and ~0 = Q(g2, gs, ~1, ~2) is an effective

Proofs of these theorems of Baker and related results can be found in [Baker 1969c, 1970a, 1970b, 1975a]. Many of these theorems were later refined and generalized. For example, in [1971a] Coates proved that one can add the term ET, E E A, to X in Theorem 4.40. .3.2. Masser’s Theorems. We say that (Y E Cc is an algebraic point of the Weierstrass pfunction if p(o) E A or if (Y belongs to the period lattice R of p(z). If Q(Z) h as algebraic invariants (this is the case we are considering in this section), then it follows from the addition theorem ((8) of Chapter 3) that for r E Q and p $ Q either both &?) and p(rp) are algebraic, or else both are transcendental. If w E 0 and w/2 $ R, then @‘(w/2) = 0, and so it follows from (7) of Chapter 3 that @(w/2) E A.

$3. Elliptic

205

Functions

The first bounds for linear forms in an arbitrary number of algebraic points appeared in [Masser 19751.Let r be the ratio of the two fundamental periods of P(Z), and let K = Q(7) if r E A and K = Q if 7 is transcendental. Theorem 4.43. Let ul, . . . , urn be algebraic points of p(z) that are linearly independent over K, and let c > 0. Then ~1,. . . , u, are linearly independent over A, and the following inequality holds for any pi,. . . , Pm E A, max ]oj ] > 0, degpj 5 n, H(Pj) 2 H, j = 1,. . . , m: l/3lul + . . . + ,¨ > cemHE, wherec=c(g2,g3,u1,...

(53)

, u,, c, m) > 0 is an effective constant.

Masser also proved that if one drops the condition that ~1,. . . , u, be linearly independent over K, then the linear form in (53) is either zero or transcendental (see Appendix 3 of [Masser 19751). In [1977b] Masser proved the following relations between the A-dimensions of certain vector spaces: dim{w,uz,w,m,~)

= 2dim{w,a}

+ 1

(the right side is clearly equal to 3 if Q(Z) h as complex multiplication, otherwise), dim{Lw,w,w,r/2,~)

= dim{w,ww,m,~}

dim{w,m,w,m) dim{l,tii,Wz,qi,n2}

and 5

+ 1,

= 2dim{w,a), = 2dim{wi,wa}

+ 1.

3.3. Further Results. The next theorem, which strengthens Theorem 4.43, is due to Anderson [1977]. Theorem 4.44. Suppose that p(z) has complex multiplication field K, ul,...,u, are algebraic points of g(z) that are linearly dent over !K, n E N, and K. > m + 1. There exists an effective u,) > 0 such that the following inequality c = c(~,n,g2,g3,u1,--., & E A, degpj 5 n, H(/?j) 5 H, H 2 3, j = any Po,Pl,..., IIXiXl~jl > 0: po

+

PlUl

+

*. . +

Pm u m I>ce

- In H(ln

In H)=

over the indepenconstant holds for 1,. . . , m,

(54)

In [1990b] Hirata-Kono improves and generalizes this result. She proves that, under certain conditions, if uj is an algebraic point of pj(z), then (54) holds with K.= m + 1. She also shows how the bound depends on the heights of pj(uj) and the degree of the field generated by the /3j and pj(Uj). Note that Theorem 4.44 implies, in particular, that the numbers 1 and uj are linearly independent over A. In the complex multiplication case this qualitative result was proved earlier by Masser [1976]. In [Bertrand and Masser

206


of Hilbert’s

Seventh Problem

198Oa] a different method (see [Schneider 19411) is used to prove an analogous theorem in the non-complex multiplication case. These results are elliptic analogues of Baker’s Theorems 4.4 and 4.7. 4.45. Suppose that p(z) does not have complex multiplication. Let be algebraic points of p(z) that are linearly independent over Q.. Then 1,2~~,...,2d~ are linearly independent over A. Theorem

w,**.,%z

Nagaev’s paper [1977] is also concerned with these questions. 3.4. Wiistholz’s Theorems. The results given below appeared in [Wiistholz 1984a]. Let c(z) be the Weierstrass C-function corresponding to P(Z), let w be a nonzero period, and let n = q(w) = C(z + w) - C(z) be the corresponding quasiperiod. Let 2~1,. . . , urn be algebraic points of p(z) that are not in the period lattice, and set X(uj, w) = wC(uj) - VZL~.Note that the function UC(Z) nz is periodic with period w. Theorem

4.46.

If

x = crlX(U1)w) + **. + wn~(%, w>+ Pw + Porl# 0 , whereal,...

,(~~,/I,/30 E A, then X is transcendental.

An important consequence of this theorem is that the periods of elliptic integrals are transcendental. Theorem 4.47. Let R(x, y) E A(x, y), and let y be a closed contour on the Riemann surface of the curve y2 = 4x2 - g2x - gs, where g2,gs E A. Then

JYR(x, Y&

is either zero or a transcendental number. This result follows from Theorem 4.46 because the integral in Theorem 4.47 can be written as a linear combination with algebraic coefficients of numbers of the form X(ui, w), . . . , X(u,,,, w), w, 71,where uj are algebraic points of p(z). In [Laurent 198Oa] the last theorem was proved under the additional assumption that all of the residues of the differential form in the integral are rational. Laurent also proved some other special casesof the theorem. The next theorem answers the question: Under what conditions can the linear form X in Theorem 4.46 vanish? 4.46. Let T be the number of elements in a maximal subset of . . , urn} that is linearly independent over Q. Then there are exactly r + 3 numbers in the set { 1, Xi, . . . , A,,, , w, 7) that are linearly independent over Q. Theorem

{w,.

54. Generalizations

of the Theorems in $1 to Liouville Numbers

$4. Generalizations of the Theorems to Liouville Numbers

207

in $1

4.1. Walliser’s Theorems. The results in Theorems 4.4 and 4.6 remain valid if the exponents have “good” approximations by algebraic numbers. We have already encountered such generalizations in $6 of Chapter 3. The next two theorems were proved by Walliser [1973a]. Theorem 4.49. Let al,. . . , am be distinct algebraic numbers not equal to 0 or 1. Suppose that the numbers 1, @I,. . . , P,,, are linearly independent over Q, n E N, IS > m + 1, and for arbitrarily large H the inequality

has solutions the number

m+ 1, and for arbitran’ly large H the inequality

has solutions the number

0, n E N, and the logarithms ofal,..., a, are linearly independent over Q. Further suppose that there exist a strictly increasing sequence of natural numbers Aj for which the ratios In In Aj+l / In In Aj are bounded from above and an infinite sequence of vectors (orj, . . . , om,j) E A.- with H(ai,j) 5 Aj and degaij 5 n that satisfy the inequality Ial -

'Yl,jl

+

. . . +

la, -

CXm,j

1
~-(lnH)“‘-*+~

.

4.1. If

lal-~l;I+...+la,-~,I<X-

(In H)“2-“+c

,

then (21 In al + . . . + xm ln urnI > x-(‘“w2-n+c

.

This theorem was proved in [Wolfart and Wiistholz 19851. In [1979] Wiistholz obtained a lower bound for the following sum when al,. . . , a,, bl,. . . , b, are Liouville numbers: lal-all+...+lu,-a,l+Ibl-pll+... -.a+ lb, -/3,,J + ~u;~Yz~ whereal ,...,

a,,&

,...,

&,+ycA.

---/I,

51. E-Functions

209

Chapter 5 of Analytic Functions That Satisfy Differential Equations

Values

Linear

51. E-Functions 1.1. Siegel’s Results. In [1929/1930] Siegel studied the transcendence erties of the values at algebraic points of the function KA(Z)

= 1+ 2 (-lY z 272 n=l n!(X + 1). *. (A + n) 0 5 ’ x # -1, -2,.

and its derivative K:(z) for rational function Jx(z) as follows: Jxb) and it satisfies

prop-

=

the differential

.. )

A. This function

1 r(x+l)

z k(z) 05

is related to the Bessel

;

equation

2x + 1 y” + --yy’+y=o. Before Siegel’s work, Legendre, Hurwitz, Stridsberg, and Maier had studied the values of functions closely connected with KA(z) (see 54 of the Introduction to [Shidlovskii 1989a]). Siegel credited Maier’s work with stimulating him to develop a method that would make it possible to obtain much more general results. In particular, Siegel was able to prove the following theorems. Theorem 5.1. If X is a rational number not equal to half of an odd number, and if [ is a nonzero algebraic number, then the numbers Kx(c) and Ki( are algebraically

independent

m 2

i=l,...,

m,

whose squares

j=l,...,

are dis-

n,

over Q.

Theorem 5.3. Let < be a nonzero algebraic number. For any s 2 1 and H 2 1 and for any polynomial P E Z[xl, x2], P $0, such that

210

Chapter

5. Functions

That

Satisfy Linear

deg P < s,

Differential

Equations

H(P) 5 H ,

one has IP(Jo(O, J;(t))1 L ~~~~~~~~~~ , where h is the degree oft

and LTis a constant depending only on [ and s.

Theorems 5.2 and 5.3 are generalizations of Theorem 5.1. From the relations K-r/2(~) = cosz, Kl_,,,(z) = - sinz, and Kx-1 = KA + &Ki 1 K’ X-l = -zT; = Kx ,

,

Kx = -aK’ .3 Ki = &KM 1

X-l

7

(1)

4x=K, + F

X-l

1

it follows that for any n E Z the functions Kn+1,2(z) and K;+,,,(z) are algebraically dependent over C(z), and their values at a nonzero point 5 E A are algebraically dependent over Q. From (1) it also follows that if Xr - X2 E Z, then the functions Kxl, Kil, Kx2, Ki2 are linearly dependent over C(z), and their values at a nonzero point E.E A are algebraically dependent over Q. One also has the relation (see 57 of Chapter 9 of [Shidlovskii 1989a]) KA(z)KI_,(z)

- K~(z)K-~(z)

- FKA(z)Kmx(z)

+ $

= 0,

Kx,(

h,

b2,.

. . ) briZ

=

2

(4

n=O

(bl),

* * * (4z * * * (b&

Zn ’

where the symbol (a), is defined by setting (ajo = 1,

(cx)~ = a(a + 1). . . (0 + n - 1)

for

n= 1,2,...

,

satisfies the differential equation

j&5 i=l

+ bi - 1) - z b(6 + ai)

y = (br - 1) . . . (b, - 1) ,

i=l

where 6 = z& (see [Luke 19691,formula (5) of Chapter V, $5.7.1). When 1 is added or subtracted to the parameters, the hypergeometric function behaves as follows (formulas (9)-(10) of $5.2.2 in Chapter V of [Luke 19691):

212

Chapter 5. Functions That Satisfy Linear Differential

Equations

One has similar identities when one of the ok is decreased by 1 or one of the bk is increased by 1. If t = m - 1 > 0, the function

is an entire function of order l/t. In transcendental number theory it is customary to make the change of variables z I+ (z/t)t, so that the resulting function has order 1 and type 1. In particular,

The function ‘+lFm satisfies

the differential fi(6

equation + tui)

+ t(bi - 1)) - z’ f&5

y = tm(bI - 1) +. . (b, - 1) .

i=l

i=l

In [1929/1930] Siegel proved that if ui, bj E Q, and if the bj are not equal to negative integers, then ‘+lFm

l,Ui,...,Q bI,b2,...,b,’ (

is an E-function; and, in fact, in requirements E-function one can replace O(nEn) by O(P) c that depends on the ui and bj. A function l,w

‘+lFrn

,...,a1

( bl,b2,...,b,’

z t (0t 2) and 3) of the definition of an for a sufficiently large constant

z

t

0t

with these conditions on the ui and bj is called a hypergeometric E-function. In [1949] Siegel conjectured that any E-function that satisfies a linear diflerential equation with coeficients in C(z) can be expressed as a polynomia! with algebraic coeficients in z and a finite number of hypergeometric E-functions

51. E-Functions

213

or functions obtained from them by a change of variables of the form z I-+ az for cr E A. This conjecture still has not been either proved or disproved, even in the case of an Efunction that satisfies a first order non-homogeneous linear differential equation. In [1981] Galochkin proved the following criterion for a hypergeometric function to be an Efunction: the function ‘+lFm with parameters

ai, bj E Cc satisfying

ai, bj # 0, -1, -2,. . . ,

the conditions l

had been

1.3. Siegel’s General Theorem. In [1949] Siegel proved the following general theorem on algebraic independence of the values of Efunctions. Theorem 5.4. Suppose that the E-functions fl(z), . . . , fm(z) form a solution of the system of homogeneouslinear differential equations Y; = 2

Qki(Z)Yi,

k=1,2

,...,

m,

(3)

i=l

where Qk:i(z) E C(z). Further supposethat for any N E N the (“LN) products of powers f,“l(z>***fkm(z,,

O c1 PI ci=lm IJw’>l Hi H~*e*Hrn ’ where A = detllaijll and cr = &I maxr~j~m Iwjl. Suppose that for all N sufficiently large we are able to construct of linearly independent linear forms (8) with aij E Z such that

IL&ql = o@v2-“),

ma laij I L N, &j

Further suppose that the numbers i.e., there exists a linear form @)

= blzl

WI,. . . , w,

lFjym -- PA = H,

l &N1-TIAl

.

(13)

km--r+1

Since A is a nonzero algebraic integer, we have IAl 1 m account that [al < cgN’, and hence IAl > ~siV-‘(~-i), we obtain the inequality 1 - v > 1 - rh, i.e., T>K

-(‘+l). Taking into from (11) and (13)

h’

In the proof of the next lemma (Lemma 16 of Chapter 3 of [Shidlovskii 1989a]) we construct the linear forms (8) satisfying (11) that are needed to prove Theorem 5.5. Lemma 5.1. Suppose that the E-functions fi(z), . . . , fm(z) form a solution of the system of differential equations (3) and are linearly independent over C!(z). Let CYbe any algebraic number not equal to 0 or a singular point of the system (3), and let K be an algebraic number field that contains cr and the Taylor coeficients of the E-functions fi(z), . . . , fm(z), [K : Q] = h. Then for any E, 0 < E < 1, and for all n E N suficiently large there exist linearly independent linear forms (8) with aij E & such that maxl=)=O i,j

IJuf1(a),

. * * 7fm(a))l

( n (l+a)n), = 0 (n-(m-l-E)n)

i,j=l,..., ,

m, i = 1,. . . ,m .

(15)

(16)

Let T be the cardinality of a maximal subset of {fi(a), . . . , fm(a)} that is linearly independent over IK. It follows from Lemma 5.1 and (14) that

52. The Siegel-Shidlovskii m r ’ h(l+&) Since E is arbitrary, Corollary

5.1.

Method

217

.

we have the following Under the conditions in Lemma 5.1, r>m/h.

We now show how Theorem 5.5 follows from this corollary. Suppose that the hypothesis of Theorem 5.5 is satisfied, and the functions (5) are algebraically independent over C(z). Then for any N E N the products (4) are linearly independent over C(z). One easily verifies that these products form a solution of a system of homogeneous linear differential equations whose singular points are the same as those of the system (6). Since E-functions form a ring, all of the products in (4) are E-functions. If we apply Corollary 5.1 to them, we can conclude that among the numbers f~1(4--~f~m(4,

O V, and we have r > m/(1 + E) in Lemma 5.1, i.e. (since E is arbitrary) r > m. For K an arbitrary algebraic number field, the conjecture has been proved under the additional assumption that

218


Equations

tr deg,WI(Z), . . . , fm(z>> = tr degq,)W(z), . . . , fm(z>) (see Theorem 9 of $8, Chapter 4 in [Shidlovskii 1989a]). In [Nesterenko and Shidlovskii 19961 it is proved that Conjecture 5.1 holds for all (Y E A outside of a finite set. Finally, note that Conjecture 5.1 trivially implies Theorem 5.5. .2.2. Construction of a Complete Set of Linear Forms. A method for constructing a complete set of linearly independent linear forms satisfying (15) and (16) was suggested to Siegel by Thue’s proof of his theorem on approximating algebraic numbers by rational numbers. First, to obtain one such form, one constructs a functional linear form R(z)

=

Pl

(Z>fl(Z)

+

** * +

~m(z)fm(z)

,

(19)

Pi(z) E Z,[z], that has a zero at z = 0 of high order (ord R) compared to the degrees and magnitudes of the coefficients of the Pi(z). More precisely, for any E, 0 < B < 1, and for all n sufficiently large, there exist polynomials Pi(Z)

=

2bijZ’y

bij

E &I

i=l,...,m,

j=O

such that i=l,...,

1 = 0 (72’1+E)n) )

ma;trbij

m, j=O,l,...,

ordR 2 m(n + 1) - [en] - 1 .

n,

(20) (21)

The condition (21) means that the m(n + 1) unknown coefficients bij must satisfy m(n + 1) - [en] - 1 linear relations. It remains to prove that this system of homogeneous linear equations has a nontrivial solution satisfying (20). The function R(z) is an E-function. Hence, its Taylor coefficients approach zero very rapidly, and from (20) it immediately follows that for any E > 0 one has pqcr)I = 0 (n+--l--c)n) . Thus, we have constructed one linear form b(fl(Q),--*

,fm(a))

=

R(a)

=

s(a>fl(a)

+

**-

+Pm(~)fm(~)

in the numbers fi (a), . . . , fm(a) with algebraic coefficients that is rather small in absolute value. In order to construct a complete set of such forms, one first constructs a complete set of linearly independent functional linear forms h(z)

=

Pkl(Z)fl(Z)

+

***

+

Pkm(Z)fm(Z),

k=l,...,m,

(22)

that vanish to a high order at z = 0. If we differentiate both sidesof (19), then replace fi (z) , . . . , fA(z) by the expressions on the right in (3) with fi(Z) in place of yi, and finally multiply the result by a common denominator t(z) of

$2. The Siegel-Shidlovskii Method

219

the coefficients in (3)) then we obtain another linear form in fi (z), . . . , fm(z) with polynomial coefficients. The degrees of the coefficients do not increase much, and the multiplicity of zero at a = 0 of the new linear form is at most 1 less than that of R(z). This procedure can be repeated several times. In this way one constructs the linear forms (22): Rk(z) =

RI(Z) = R(z),

t(Z)-

d&--l dz

k=2,3,...

’

.

We now assume that the forms (22) turn out to satisfy the condition A(Z)

=

det

iipkiIli,k=l,...,

m

#

0 .

(23)

, Rm(a) will serve as the required set of linear If A(a) # 0, then RI(~),... forms in Lemma 5.1. If, on the other hand, A(a) = 0, it turns out that the polynomial A(Z) does not have a high order of zero at z = cr. Using bounds on the degrees of the &(z), one easily shows that degA(z)

5 mn + cl ,

where cl does not depend on n. Using Cramer’s formulas, it is also easy to prove that ord A(z) > mn - [En] - cz , and hence A(z) = zmn--jen]-cczA1(~) , where deg Al(z)

5 in + cs. But then there exists an integer 7, OO,

1

where for brevity

we let S denote the set of real number parameters S = , A,}. Here we assumethat m > 12 0, m 2 2, and A, = 1. We say that S is an admissible set of parameters if it satisfies at least one of the following conditions: bl,..

a) b)

.,pLI,h,--.

Xj - pi $! Z for 1 5 i 5 1, 1 5 j 5 m, and all of the sums Xj + pi for 1 5 i 5 j 5 m are distinct modulo Z; 1 = 0, m is odd or equal to 2, and the set {Xi,. . . , A,} modulo Z is not a union of arithmetic progressions {A, X + i, . . . , X + y} of fixed length d, where dim, d > 1.

WesaythattwosetsofparametersS={C11,...,~I,X1,...,Xm}andS’= {pi,. . . ,p;,,Xl,,. . . ,A;,} are similar if a) b)

1’ = 1, m’ = m; there exist ~1,X E lR and r E (0, 1) such that with a suitable indexing of the parameters one has pi E /.L+ (-l)‘pi A> z X + (-l)‘Xj

(mod Z), (mod Z),

i = l,...,l, j=l,...,m.

The next theorem is proved in [Beukers, Brownawell, and Heckman 19881. Theorem 5.18. Let S1, . . . , Sk be admissible sets of rational parameters (not necessarily all distinct), and let &, . . . , tk be nonzero algebraic numbers such that if Si and Sj, i # j, are similar sets of length 1,m, then *l.

Then for any X E Q, X # 0, -1, -2,. . ., and for any r E N the numbers e”,wx,l(a), . . . , WA,,.(~) are algebraically independent over Q. Theorem 5.20 implies, in particular, that for any nonzero Q!E A the numbers e”’ and Jo* eet dt are algebraically independent, as are the numbers e” and JoaP-l emtdt. Theorem

5.21.

Let

Then for any nonzero (Y E A and any r E N the r(r + 1)/2 numbers tip cay> 3

O 0 one has

Ibo+ h Jo(t) + where

the constant

b2

J;(S)1 > cH-~-‘,

H = maxi&l

,

c > 0 depends only on E and < E Q.

5.1. Bounds for Linear Forms in the Values of E-Functions. We first consider the general situation. We have an m-tuple WI,. . . ,w,, and we want to find a lower bound for Il(W)j, where

lyjym -- PA = H,

H>l,

that depends on the parameter H. In order to show how Siegel’s idea works, we recall the method of proving linear independence that was given in 52. As in $2, we suppose that for some infinite and rather dense sequenceof natural numbers N we have been able to construct a set of linearly independent linear forms Li = Li(l)

= ?aijZjf

i=l,...,m,

(34)

j=l

oij E Z, for which ma laij I 5 N, i,j

ILi(GT)l = o(N~-“),

l c~H-~N~-~

- N-l

2

ILi(c)l

2 c~H-~N~+ If we now choose N large enough so that

- o(N1-“)

> (36)

i=2

.

$5. Forms and Polynomials in the Values of EFunctions

o(N’-*) < &H-‘N’-”

)

235

(37)

then from (36) we obtain the bound

Clearly, the smaller we choose N satisfying (37), the better the bound we obtain for ]1(a) 1.The optimal choice of N depends on H, and so the expression on the right in (38) ultimately is a function of H. This gives the required estimate. Once Shidlovskii improved and perfected Siegel’s method for studying the values of E-functions, it became possible to prove quantitative results in the general case as well. In the situation in Lemma 5.1, if we make the additional assumption that lK = Q, we obtain a set of forms Lo with aij E Z and max ]aij ] < c27~(‘+‘)~, id ILi(f~(a),

. . . , fm(a))I

>

or H < C4n(l--E(“--l))~ .

(41)

In addition, if c is small enough so that 1 - .s(m - 1) > 0, and if n is the smallest natural number satisfying (41), then N w c5H(‘+“)/(1-“(“-1))

as

H+CO,

and (38) takes the form I@)[ > c,H-(“-l)(l+E)/(l-E(m-l))

.

(42)

By choosing E smaller and smaller, we can make the exponent of H on the right in (42) arbitrarily close to the best possible value 1 - m. A similar argument can be used in the casewhen K is a quadratic imaginary field. In [1967b] Shidlovskii proved the following general theorem. Theorem 5.26. Suppose that the E-functions fl(~),...,fm(Z),

m21,

236


Equations

have power series coeficients in a quadratic imaginary field K, are linearly independent over C(Z), and form a solution of the system of differential equations Y; = 2

Qdz)~i,

k=1,2

,...,

m,

i=l

where Qki(z) E Cc(z). Let cx E IK be nonzero and not equal to a pole of any of the Qki(Z). Then for any E > 0 there exists a constant cr such that for bj E &, 1 5 j 5 m, satisfying

one has lhfi(a)

+ . . . + bmfm(a)l > c~H’-~-~

.

Of course, it is not yet possible to obtain such an estimate for an arbitrary algebraic number field K, since we cannot even prove linear independence of the fi(a) in that case. For hypergeometric E-functions the bound O(nEn) in parts 2) and 3) of the definition of an E-function can be replaced by O(P), where c is a sufficiently large constant depending on the ai and bj. If we have E-functions that satisfy this stronger condition, the estimates (39) and (40) can be refined. The term o(N2-“) in (35) becomes a more rapidly decreasing function, and this leads to an improved bound for the linear form. More precisely;the following theorem was proved in [Shidlovskii 1967a] (see also [Shidlovskii 19791). Theorem 5.27. Suppose that the conditions in Theorem 5.26 are fulfilled, and the E-functions satisfy the above stronger version of the definition. Then there exists a constant cs such that for any bj E Z, 1 5 j 5 m, satisfying

one has

lhfi(~) + -.++

b,f,(cr)I

> csH 1 - n--f~d(lnlnH)-“*

.

Here and later, the letter y with various subscripts denotes positive constants that depend only on (Y, K, and the constants in the bounds for the power series coefficients of the fj(z) in parts 2) and 3) of the modified definition of an E-function. For some functions one is able to construct a system of linear forms L,(Z) with even better simultaneous approximations and thereby obtain more precise bounds. This is usually done using an explicit construction of approximating forms. For example, if 5 is a nonzero point in a quadratic imaginary field I[, we have the following bound for linear forms in the values of the Bessel function, which improves upon the result announced by Siegel: for bi E i%n

55. Forms and Polynomials Ibo

+hJo(t)

+

baJ;(t)(

>

cgH-

in the Values of E-Functions 2

-

-&lInH)-'

,

237

H = max ]bi] > 0 .

In certain casesit is possible in this manner to obtain best possible estimates; such results are discussed in 54 of Chapter 2. 5.2. Bounds for the Algebraic Independence Measure. The bounds for linear forms in the last section can be used to obtain bounds for the algebraic independence measure of values of E-functions at algebraic points. General results of this type were first obtained in [Shidlovskii 1967133.In fact, any polynomial P(fi(o), . . . , fm(o)), where P E Z[zi,. . . ,z,], may be regarded as a linear form in the products of powers fi(a)“l . . . fm(cr)km, kl + . . . + k, 5 d = deg P. Since the set of E-functions form a ring, it follows that the products h(z) “‘.-.f,(~)~~,

kl+...+k,

Id

(43)

are Efunctions, and they satisfy a system of linear differential equations with coefficients in C(z). Thus, Theorem 5.26 can be applied to the functions (43), so as to obtain a bound for the algebraic independence measure of , fm(cr). A similar result can be proved in the more general situation fl(~y>,... when the fi(z) have power series coefficients in a quadratic imaginary field. Theorem 5.28. Suppose that the E-functions

.*,fnz(z),

h(z),.

ml17

have power series coeficients in a quadratic imaginary field K, are algebraically independent over C(z), and form a solution of the system of differential equations Y; = &o(z)

+ 2

Qki(z)Yi,

k=1,2

,...,

m,

i=l

where Qki(z) E C(z). Further suppose that (Y E K is not equal to 0 or a pole of any of the Qki(Z). Let d be a natural number. Then for any E > 0 there exists a constant u depending on E, d, (Y, and the functions fi(z)p such that for any polynomial P E Z[xl, . . . ,x,1 with P $0 and deg P 5 d one has

IP(fl (a>, . . . , f m (a))1 2 cH(P)‘-~-~

.

We note that this lower bound differs from the known upper bounds only by the E in the exponent of H(P). There is also an analogue of Theorem 5.27 for polynomials (see Chapter 13 of [Shidlovskii 1989a]). It is also possible to prove bounds for the algebraic independence measure of the values of E-functions in the case when the Taylor coefficients or (Ylie in an arbitrary algebraic number field of finite degree over Q. As early as 1929, Siegel proved the first such result for the function JO(Z) and its derivative

238


Equations

(see Theorem 5.3). After Shidlovskii proved his First Fundamental Theorem (Theorem 5.5), it became possible to obtain a quantitative generalization of Theorem 5.3. This was done in [Lang 19621. Theorem

5.29.

Suppose that the E-functions h(z),

* * * 7 fm(z>,

m21,

have power series coeficients in an algebraic number field K of finite degree, are algebraically independent over C(z), and form a solution of the system of diaerential equations Y; =

Qko(z) + 2 Qki(z)~i,

k=1,2

,...,

m,

(44

i=l

where Qki(z) E C(z). Suppose that CI E IK is not equal to 0 or a pole of any of the Qki(z). Then for any P E Z[zl, . . . ,zm], P $0, one has IWl

(a>,

* * * 7 fm(a))l

L

cfwrbd”

7

where d = deg P, and b and c are positive constants, where b depends only on h = [IK : Q] and m, and c depends on m, cr, d, and the functions fi(z). In [Lang 19621this theorem was proved only in the case of a homogeneous system of differential equations. However, the proof in the general case does not require any important changes. In [1968] Galochkin obtained a refinement of Theorem 5.29 with the constant b computed explicitly. If the system of differential equations can be split up into r disjoint subsystems of orders ml,. . . , m,, then b = 2(2r)mmy1 . . .rnrr hm+i ml!. . em,! In addition, the term d” in the exponent of H(P) can be replaced by where dk is the total degree of P in the set of variables corre6;“’ ...drr, sponding to the functions fi(z) that form a solution to the k-th subsystem of differential equations. In particular, under the conditions of Theorem 5.29 we have b = 2”+‘mm(m!)-’ hm+‘. We briefly describe the basic ideas in the proof of Theorem 5.29. Suppose that we are given a set of linearly independent linear forms ii(z) = bilzl + . . . + bimzm,

bij E Z,

i=l,...,s,

andapointSj= (w~,...,w~) c(Cm, and we want to obtain a lower bound for in terms of the parameter H. Further suppose that for some m=lo.

Then, repeating the argument in 55.1 that led to (36) with appropriate modifications (see also §2.1), we obtain

H-’

l’=iy. I&(~)1 L cl (HW m--a)-h - N-l -2 cl (HsiVm-e)-h

2

IL&q

- o(iV-“)

2 (45)

i=s+1

,

from which we obtain an estimate for maxi 0. It is not hard to see that n can be chosen in such a way that

for some ~1 > 0 depending on E, and then from (45) we can obtain the following bound: , mu Ih(fi(a), . . . , fm(a))l > cZH1--h-=+ (46) l 0 and for some cs > 0 depending on Q, m, 6, and the functions fi(z). This bound is the same as (42) if s = h = 1. To prove Theorem 5.29 one applies the above considerations to the set of functions h(z) kl--f&)km,

k~+...+k,

5 (l+X)d,

(47)

where d = deg P and X is a large integer constant depending only on h = [IK : Q] and m. Then in the earlier argument the number of functions is replaced by the number

(

(1 + X)d+ m = (1 + X)m m, d” +0(&-l) m >

All of the polynomials Xlkl “‘2, km.P(sl

,...,

x,),

kl+.~.+k,<Xd,

may be regarded as linear forms with integer coefficients in the products of powers xf1 . . . xk) kl +.a. + k, < (1 + X)d , and one easily checks that these forms are linearly independent. The number s of linear forms is equal to

240

Chapter

5. Functions

That

Satisfy Linear

= $P

Differential

+ O(P-l)

Equations

.

If we substitute fi(o) in place of zi, these linear forms become numbers whose absolute values do not differ very much from ]P(fi(o), . . . , fm(a))I. It is easy to verify that the bound (46), when applied in this situation, gives the inequality in Theorem 5.29. The one drawback in the proof of Theorem 5.29 is that one does not obtain an effective expression for the constant c in terms of the degree d of the polynomial. For this reason the estimate in the theorem cannot be applied to polynomials with bounded coefficients and increasing degree d. The noneffectiveness of c is a consequence of the non-effectiveness of another constant - the o-, in the functional part of the Siegel-Shidlovskii method (see Lemma 5.2). The point is that, when proving Theorem 5.29, we apply Lemma 5.2 to the products of powers (47), and we want the lemma to give us a bound for ordR(z, fi(z), . . . , fm(z)), where R(xo,xI,. . . , x,) is a polynomial with complex coefficients. Clearly, in this situation the constant ~0 in Lemma 5.2 will depend on the system of equations that the functions (47) satisfy, and hence on d. If we impose some restrictions on the system of differential equations satisfied by the functions (47) (f or instance, the Siegel normality condition that we discussed in §2.3), then we can explicitly compute ~0, and hence the’ constant c in Theorem 5.29. However, this has not been done in the general, unrestricted case. In [1972] Nesterenko proved a generalization of Lemma 5.2 and used it to determine how c depends on d. The details appeared in [Nesterenko 19771, where it was proved that under the conditions of Theorem 5.29 one can take b = 4”h”(mh2

+ h + 1)

and

c = exp(- exp(rd2* In(d + 1))) .

Here it was assumed that each of the functions satisfies the modified definition of an E-function (see above). The constant r in the expression for c depends only on certain properties of the differential Galois group of the system (44) in Theorem 5.29. In many casesit can be explicitly determined - for example, if every nonzero solution of the corresponding homogeneous system m

YL = c

Qki(Z)Yi,

k= 1,2 ,...,

m,

i=l

consists of functions that are algebraically independent over C(z) (see [Brownawell 1985]), or if every solution of this system that has nonzero components is made up of functions that are algebraically independent over Cc(z) (see [Nesterenko 1988b]). Shidlovskii proved a homogeneous analogue of Theorem 5.29, in which he essentially finds bounds for the algebraic independence measure of the ratios , m. He also obtained bounds for polynomials in the fi(Q)/fl((y), i = %... values of E-functions that are connected by algebraic relations over (c(z). Detailed proofs of these and related results can be found in [Shidlovskii 19811.

$6. Bounds for Linear Forms that Depend on Each Coefficient

241

As examples of the consequences of these theorems, one obtains a bound for the transcendence measure of tana for nonzero cx E A, and also a bound for the algebraic independence measure of sin (~1, . . . , sin cr, for algebraic numberscri,...,o, that are linearly independent over Q. Shidlovskii also proved results when one has several sets of E-functions, each of which is a solution of a corresponding system of differential equations of the type (44). These and other results in this area are described in his book [Shidlovskii 1989a]. We also mention some quantitative results on the zeros of E-functions. In [1970b] Galochkin studied the arithmetic properties of the roots < of equations of the form P(z, fr(z), . . . , fm(z)) = 0, where P is an irreducible polynomial with algebraic coefficients and the fi(z) are algebraically independent E-functions that form a solution of the system (6). Shidlovskii’s fundamental theorem implies that c is transcendental. Galochkin obtained a bound for the transcendence measure of such numbers [ under the assumption that the height of the polynomial in the estimate is greater than some bound that depends (non-effectively) on its degree. This result was subsequently refined by VZininen [1977], Ivankov [1985], and Galochkin [1992].

$6. Bounds for Linear Forms that Depend on Each Coefficient In certain situations - for example, when estimating the deviation of a uniformly distributed sequence - one needs bounds on linear forms in several numbers that depend on all of the coefficients. From metric considerations (see Theorem 13 of [Sprindzhuk 19791) it follows that for any E > 0 and for almost all 3 = (&,... , S,) E ll? in the sense of Lebesgue measure, there exists a constant c = ~(6, E) > 0 such that for any nonzero vector h = (he,. . . , h,) E Zs+’ one has Iho + hl?91 + . . . + h,6,1 > c fr

ii-‘(ln(I

+ h))-“-”

,

(49)

i=l

where & = max(1, Ihi]) and h = maxrci,...,flnl(Z),...,fml(Z),...,fmn,(Z)

are, together with 1, linearly independent over C(z); and suppose that they form a solution of the system of differential equations Y:j = Qijo(z) + 2 Qijs(Z)Yis, &?=I

i=l

,...,m,

j=l

,.-.,ni,

with Qijs(z) E C(z). Further suppose that K is a quadratic imaginary field, and cr E R is not equal to 0 or a pole of any of the Qijs(z). Then there exists

$6. Bounds for Linear Forms that Depend on Each Coefficient a constant c > 0 depending only on cr and the functions fij(z), j = l,..., ni, such that for any hij E Z not all zero, one has

245

i = 1,. . . , m,

-1

h-“(h) where hi = ma.xl<j property. For any nonzero HI,. . . , H,,, E Z with H = > ~0 one has either Hlfl(r)

+ . . . + H, fm(r)

= 0 ,

or else IHlfl(r)

+..s+

Hmfm(r)l

> IHI...H,I-~H~-~.

It should be noted that what Chudnovsky actually proves is not this result, but rather a weaker assertion in which several additional conditions are imposed on the functions fl(z), . . . , fm(z) in the course of the proof. The proof is carried out under the assumption that the so-called conjugate differential equations to the ones in the theorem satisfy the Siegel normality condition. It is further assumed that r is not a singular point of any of the systems of differential equations. The result has not yet been proved in the version given above. We also note that the conditions imposed on the fi(z) make it impossible to include the derivatives of any of these functions in the set of functions. In the special case fl(z) = 1, fa(z) = KA(z) (see §l), Titenko [1987] was able to refine Chudnovsky’s argument and prove the following result.

246


Equations

Theorem 5.32. Let X E Q, X # O,-l,-2,.. ., and X # (2k + 1)/2 for any k E Z; and let a be a nonzero rational number. Then for any p, q E Z, IqI > go > 0, one has lPKx(~) where go and y are positive A substantial

+ 91 >

Iql- 1 - -#nln

constants

generalization

1q1)-“3

7

that depend only on X and cr.

of this theorem was obtained

$7. G-Functions

in [Zudilin

19951.

and Their Values

7.1. G-F’unctions. The Siegel-Shidlovskii method can be applied to the values of the generalized hypergeometric functions al,...,al

O”

IF, h,.

“,

bm

”

=nTo

(al>,..-(ad,

(bl)n...(bm)nn!Zn

in the case 1 = m + 1, i.e., when the power series has finite radius of convergence. As before, we assume that ai, bj E Q. In [1929] Siegel defined the class of G-functions, which includes the generalized hypergeometric functions with 1 = m + 1, and discussed the possibility of studying the values of G-functions at algebraic points. Here is the exact definition. Definition

5.4. An analytic

function

f(z) = 5 c,zn n=O is said to be a G-function 1) 2)

if the following

are fulfilled:

all of the coefficients cn lie in an algebraic number field lK of finite degree over Q; for some constant C > 1 and any n 2 1 we have l-z-l

3)

conditions

SC”,

where m denotes the maximum modulus of the conjugates of cr; and there exists a sequence of natural numbers ql,q2, . . ., where qn 5 C”, such that for all n for O<j = Y(X)?

fm(x)

= I”

ydx .

As mentioned before, these functions are G-functions; and they are linearly independent over e(z). In addition, they form a solution of a certain system of linear differential equations with coefficients in Cc(x). The claim that the numbers f?(t), 0 5 j 5 m, are linearly independent over a certain field means, in particular, that s,’ y dx does not belong to that field. This is what Siegel stated. It is also easy to see that the first result of Siegel mentioned above follows from the linear independence over Q of the values of the G-functions I,

Z~l ( 112$‘2;z)

,

and

zFi ( 1’2f’2;z)

at a point x = p/q that satisfies the given conditions. The first general theorems on the values of G-functions were proved in [Nurmagomedov 19711. Among other results, he showed the following. Suppose that fi(z),... , fm(z) are G-functions that are linearly independent over c(z) and satisfy the system of linear differential equations Y;

= 5

Qki(z)~i,

k=

1,2 ,...,

m,

(56)

i=l

where Qki E Cc(z). Let (Y E A and b, H E IV, where b is greater than an explicit constant bo that depends only on H, cr, and the functions fi(z). Further suppose that a/b is not equal to a singular point of the system of differential equations. Then the numbers fi(cr/b), . . . , fm(a/b) are not connected to one another by a linear equation with rational integer coefficients of absolute value not exceeding H. Under similar conditions he proved a lower bound for the

$7. G-Functions

and Their

Values

249

absolute value of a linear form in these numbers; and he obtained analogous results for polynomials. In [Chirskii and Nurmagomedov 1973a, 1973b] and [Chirskii 1973b] the authors applied these results of Nurmagomedov to functions zFi (y ;z) with rational parameters, their derivatives, and elliptic integrals in Legendre form. A serious drawback of these results is that the lower bound be for the denominators of o/b depends on the parameter H, i.e., on the size of the coefficients of the linear form. In other words, for a fixed point a/b one is able to prove only that there are no relations with coefficients of bounded size among the numbers fi(cr/b), . . . , fm(o/b). This does not, however, imply that the numbers are linearly independent. Moreover, in [1978a, 197813, 1978c] Chirskii showed that equally strong results can be proved for a broader class of functions that includes both Efunctions and G-functions. In particular, such results hold for hypergeometric functions with irrational algebraic parameters, and also for a set of functions consisting of eaiz andln(l-pja)with~li,pjEA,lLiIm,lIjLn. In order to understand the reason for this deficiency in the results, we briefly recall the argument. First, in the Siegel-Shidlovskii method one constructs a functional linear form R(z)

= fi(Z)fl(Z)

+***+

P*f*(z)

,

where Pi(z) E Z,[z], that has a zero at z = 0 of sufficiently high order compared to the degrees of the Pi(z) and the size of their coefficients. One then constructs a sequence of functional linear forms Rk(Z)

= Pkl(Z)fl(Z)

+-..+Pkm(Z)fm(Z),

k>l,

with Pki(z) E C(z), that have high order zeros at z = 0. The operation of producing new linear forms can be conveniently interpreted as an action of the differential operator

on the linear form R = Pl(z)zl

+ -.. + Pm(z)z,

.

Then RI(Z) = R,

%+1(Z) = (t(z))“D”R(Z),

k= 1,2,...

.

Here t(z) is a common denominator for the coefficients of the system of differential equations (56). We have somewhat modified the method of constructing the forms Rk that was used in our study of E-functions. Next, one can prove that there are m linearly independent numerical linear forms among the forms

250

Chapter 5. Functions That Satisfy Linear Differential hl

(ah

+

. . . +

&I

(Q>Gn

9

Equations

l 1, such that for all n > 1 and for any m-tuple of polynomials PI(Z), . . . , P,(z) E &[z] the coefficients of the linear forms ~(t(z))“D”R(Z),

k= 1,2 ,..., n,

are polynomials in ZK[Z], and one has Ian] 5 cn for some constant c 2 1. In [1974] Galochkin proved that products of powers of ln(1 + Q~z), cri E A, have the G-property, as do products of powers of (1 + Q~z)“’ for algebraic CQ and rational Vi. In particular, he proved the following theorem. Theorem 5.33. Suppose that H E IR, q, d E N, and the functions f~ (z), . . . . . . , fm(z) have the G-property and are not related to one another by any nontrivial algebraic equation of degree at most d with coeficients in Cc(z). Further suppose that P E Z[xl, . . . ,x,1, P f 0, is a polynomial of degree at

97. G-J?unctionsand Their Values

251

most d whose coeficients have absolute value at most H. Then there exist positive constants A = A(K, fi, d), X = X(K, fi, d), and /I = p(lK, fi, d, q) such that for q > A one has > q-XH-p

.

The constants A and p, which depend upon the parameters in a rather complicated way, are given explicitly in [Galochkin 19741. In particular, Galochkin proves results for ln(1 + al/q), . . . , ln(1 + a,/q) for pairwise distinct oi,...om E A. For example, if q > e7g5, then ln(l-~)ln(l+~) is irrational. If d = 1, then Theorem 5.33 gives us linear independence of 1, fiWq>, . . . , fmWq) over Q. In [1975] Galochkin proved a result analogous to Theorem 5.30 - i.e., a bound on linear forms that takes into account the rates of growth of all of the coefficients - for the values of any set of G-functions that have Taylor coefficients in a quadratic imaginary field K, are (along with 1) linearly independent over e(z), and satisfy the G-property. Here each of the G-functions must be a solution of a first order linear differential equation with coefficients in Cc(z) (see IS). Galochkin applied this result to the values of ln(1 - criz) and products of powers of (1 + (Y~z)~‘. We note that in [1972] Fel’dman had proved such a result for the functions

which also satisfy first order differential equations. In [1981] Matala-aho and V&%-&en proved that the function

F(z)= r’2f’2;z) 2F1

and its derivative have the G-property. Using this, they obtained lower bounds for polynomials in F(cr/q) and F’(a/q) with integer coefficients, provided that cr and q satisfy certain conditions. In particular, they showed that at least one of the numbers FCp/q) or F’(p/q) satisfies the assertion announced by Siegel with a constant C in place of 10. In [1983] Beukers, Matala-aho, and VBananen proved that the functions 2F1

and

2F,’

have the G-property for any (Y,p, y E Q with y # 0, -1, -2,. . .. In the same paper they proved a theorem similar to Theorem 5.31 for the values of

252

Chapter 5. Functions That Satisfy Linear Differential Equations

where cri, /3i, yi E Q satisfy certain natural conditions, at a point p/q E Q, where q > qo = q,-,(p). This result implies, in particular, that these values are linearly independent over Q. 7.3. Arithmetic Type. In [1981] Bombieri proposed another condition that can take the place of the G-property and similarly make it possible to use the Siegel-Shidlovskii method for G-functions. Let lK be an algebraic number field of finite degree over Q such that the coefficients Qij (z) in the system (56) lie in K(z). Bombieri’s approach is based on studying the properties of the solutions of (56) over all of the completions of the field K. For each absolute value 1 IV on lK we let K, denote the completion of lK with respect to v, and we let R, denote the completion of the algebraic closure of KU. We also use the notation 1 Iv for the extension of the absolute value to 0,. If we fix a point < E fl,,, we can consider the solutions of the system of differential equations (56) in the ring of formal power series in z - < with coefficients in a,. If Iz - 0 by setting log+(z) = In max(1, x), and Vo is the set consisting of all non-archimedean valuations on K. If the inequality (57) holds for the system (56), then the system (more precisely, the differential operator corresponding to the system) is called a Fuchsian operator of atithmetic type. In Andre’s book [1989] the term G-operator is also used. The condition (57), which replaced the condition on canceling factorials, enabled Bombieri to apply the Siegel-Shidlovskii method and prove several general theorems on the values of G-functions. Here we shall only give two consequencesof these theorems that concern algebraic functions and polylogarithms. Theorem 5.34. Suppose that K is an algebraic number field, d = [IK : Q], and u(z) is an algebraic function of one variable that is regular in a neighborhood of 0 and satisfies the relation P(z,u) = 0, where P is a polynomial with rational coeficients. Further suppose that s: u(t) dt is not an algebraic function. Then there exist two constants cl and ~2, depending only on u(z), with the following property. If < E K, c # 0, h(J) > cl, and

$7. G-Functions

and Their

253

Values

ICI < exp(--c2dJin h(J) lnln h(C)) , where ) 1 is the usual absolute value in C, then the complex numbers,’ u(t) dt does not lie in K. (Here and later h(J) denotes the so-called Yogarithmic height” of the algebraic number [.) This theorem is a refinement of the result on the values of abelisn integrals that was announced by Siegel. Condition (57) can easily be verified using the Eisenstein criterion. Theorem 5.35. Suppose that K is an algebraic number field, v is a valuation of K, and [ E II6 satisfies ] exp (c3(m)62m+1 ln(6 + 1)) ,

151v< exp( -~4(m)61~(2m)(lnh(~))1-1~(2m)(lnlnh(J))1~(2m)

> are fulfilled, then the numbers Ll(J), . . . , L,(J), regarded as elements of K,,, do not satisfy any polynomial equation of degree6 with coeficients in K. 7.4. Global Relations. Bombieri’s approach relates the local and global properties of the solutions of differential equations. In particular, in [1981] he investigated the question of which G-functions have “global relations.” Definition 5.6. Suppose that fi(z), 1 5 i 5 m, are formal power serieswith coefficients in the field IK, c E K, and Q E lK[xi, . . . ,x,1. A relation

Q(fi(C), . . . 7fm(C)) = 0 is said to be global if it holds in K,, for every place v of lK for which I2;

and

1 1512< -4 ’

254


Equations

For instance, for t = 8/9 the series f(t) converges in lR and Q2. Note that f(8/9) converges to +3 in lR and to -3 in Qz. The relation f(c) - 3 = 0 is not global, since it does not hold in Q2. If we take c = -3, then f(t) converges only in Qs, where it converges to -l/2; hence f(c) + l/2 = 0 is a global relation for t = -3. In [1981] Bombieri proves the following general result about global relations among the values of G-functions. Suppose that our system of differential equations is Fuchtype, i.e., the property (57) holds; and suppose that (fl(ZL...> fm(z)> is a solution consisting of G-functions that are linearly independent over A(z). Then the points E E A where there are global linear dependence relations among the numbers fi( 0 such that for any xi, x2, x3 E Z with X = max(]xr 1,1x21,1x3I) suficiently large one has --yx

lxlbl + x2b2 + x3b31 > e

In x

(2)

Then the field L2 has transcendence degree at least 2. Theorem 6.2. Suppose that p = 3, q = 2, the numbers al = 1, a2, as are linearly independent over Q, and the numbers bl # 0 and b2 satisfy the following condition: there exists a constant y > 0 such that for any x1,x2 E Z with X = max(]xr 1,1x21) suficiently large one has

I

x1 +x2-

bz I bl

> emTX21nX .

Then the field L3 has transcendence degree at least 2. Applying Theorem 6.1 to Gel’fond’s third conjecture with ai = /Ii-l and bi = pi-l In a, i = 1,2,3, we seethat if d 2 3, then at least two of the numbers afl, &, & cup* are algebraically independent over Q. Here the condition (2) holds by Liouville’s theorem, since p is algebraic. In particular, this means that tr deg Q ( cyp,a@, . . . , oPdel) 2 2 for d>3. (4)

262

Chapter

6. Values of Functions

That

Have an Addition

Law

When d = 3, one obtains the following result, for which Gel’fond had published a direct proof somewhat earlier (see [1949b]). Theorem 6.3. Let cr be an algebraic number not equal to 0 or 1, and let p be a cubic irrationality. Then

are algebraically independent over Q. As Gel’fond pointed out in [1949d], Theorems 6.1 and 6.2 imply that each of the following sets contains at least two numbers that are algebraically independent over Q (here v E Q, v # 0, and a E A, a # 0,l): 1) e, eeY, ee2”, eeSY; 2) e, ueY , ae2”, a”“, a’*” ; 3) In a, aln” a, &12” a, &IS” 0.

1.2. Bound for the Transcendence Measure. Gel’fond’s method of proving algebraic independence was based on a new technique for estimating the transcendence measure. As an indication of the importance of such estimates, Gel’fond gave a proof of the following theorem along with the proofs of Theorems 6.1 and 6.2 in [1949d]. Theorem 6.4. Let a, b, (Y,p E A, a # 0, 1, and supposethat b and In (.y/ In fi are irrational. Further suppose that P(x) E Z[x], deg P = p > 0, H(P) = H 2 1. Then for any fixed E > 0 and H > Ho(e) one has -&(P+ln

IW)l and

fJ)(qP+ln

>e

w)2+r

I( >I p

$

>

e-P2(P+m2+=

.

In these inequalities for the first time p = deg P and H = H(P) can vary independently of one another. In particular, the bounds hold for fixed H and increasing p (compare with Theorems 3.19 and 3.20). We briefly describe the idea behind Gel’fond’s estimate for the transcendence measure. Let P(x),Q(x) E Z[x] be two polynomials with no roots in common. Then there exists a nonzero integer R and two polyomials A(x),B(x) E Z[x] such that P(x)A(x)

+ Q(x)B(x)

= R.

(5)

Here R is the resultant of P(x) and Q(z); and the coefficients of A(x) and B(x) can be expressed in terms of the coefficients of P(x) and Q(x). Using these expressions, one derives bounds on the degrees and coefficients of A(x) and B(x) in terms of the degrees and coefficients of P(x) and Q(x). From

$1. Gel’fond’s Method and Results

263

these bounds (and the fact that ]R] > 1) we obtain the following lemma as a consequence of (5). Lemma 6.1. Suppose that w E C!, and the polynomials P(z), Q(z) E Z[z] have no roots in common. Then

m4W411 IQ64 L 03+ ~)-lII~II-qIIQII-P , where p = deg P, q = deg Q, lIPI is the length of the vector of coeficients of P, and ~~Q~~ is the length of the vector of coeficients of Q. This lemma can be used to obtain a bound for the transcendence messure of w, i.e., a lower bound for IQ(w)] in terms of degQ and H(Q), as follows. Suppose that for given Q(z) we have somehow managed to construct a polynomial P(z) whose a) b) c)

roots are distinct from the roots of Q(z); degree deg P and height H(P) are not very different from the degree and height of Q(z); value at w is fairly small, namely, IP( 5 IQ(w)l.

Then the inequality in Lemma 6.1 implies the bound

IQ(w)I 2 (P+ d-lIIPII-qIIQII-P 7 which, by property b), can be expressed in terms of just q = deg Q and H(Q). It is interesting to note that the analytic construction of P(z) that, Gel’fond used in his proof of Theorem 6.4 was similar to the construction of the auxiliary function in the solution to Hilbert’s seventh problem. lJe obtained a polynomial P(z) whose coefficients are not in Z, but rather in the ring of integers Zk of a certain algebraic extension of Q. Then Gel’fond used a lemma analogous to Lemma 6.1 for P(z), Q(z) E &[z]. The above version of Lemma 6.1, which is an improvement of Gel’fond’s original lemma in the case k = Q, is taken from [Brownawell 19741. Another important consideration is that in the analytic construction of P(x) one cannot be assured that condition a) holds. For this reason one imposes an additional restriction on Q(z): Q( x ) must be an irreducible polynomial. Condition a) can then be replaced by the weaker condition a’) P(x) is not divisible by Q(z). Since one can ensure that condition a’) holds in the analytic construction of P(z), in this way one can prove the lower bound for IQ(w)1 for irreducible polynomials Q(z). To obtain a bound for arbitrary Q(x), one usesthe following lemma of Gel’fond. Lemma 6.2. Let &I (21,. . . , xm), . . . , Qs(zl,. . . , x,) be arbitrary polynomials incC[zi,... ,x,1. Let ni, i = 1,. . . ,m, denote the degree of

Qh,.

. . ,4

= &1(~1,...,xm>...Q~(x1,...,

x,)

264

Chapter 6. Values of Functions That Have an Addition Law

in the variable xi, and set n = nl + . . . + ns. Then H(Q)

2 ewn H(Qd

. . . H(Qs)

.

Lemma 6.2 is used repeatedly in the proof of Theorem 6.4. We shall describe only one of the circumstances when it is used. If a lower bound has already been proved for the value at w of irreducible polynomials, then one can apply the lemma to obtain a bound for IQ(w)] f or arbitrary Q(x). To do this, one writes Q(x) as a product of irreducible factors, for each of which one has a lower bound for the value at w. This leads to a lower bound for IQ(w) 1, but one that is expressed in terms of the degrees and heights of the irreducible factors of Q(x). If we now use Lemma 6.2 with m = 1, along with the fact that each irreducible factor has degree 5 deg Q(z), we can express our estimate in terms of the degree and height of Q(x). This gives the required result. Finally, we note that lemmas similar to Lemmas 6.1 and 6.2 (with m = 1) were first proved and the above method of bounding polynomials was first used in [Koksma and Popken 19321 to obtain a result on the transcendence measure of en. In 1948, Gel’fond was the first to apply this argument to prove algebraic independence of numbers. In what follows we shall give an outline of the analytic construction of the polynomials P(x) satisfying a’), b), and c).. 1.3. Gel’fond’s “Algebraic Independence Criterion” and the Plan of Proof of Theorem 6.3. We now give a brief description of Gel’fond’s technique for proving Theorem 6.3. We shall make use of some technical simplifications and refinements that were found later. The next lemma, which concerns dense sequences of polynomials, provides the algebraic basis for the proof of Theorem 6.3. The lemma is sometimes called Gel’fond’s “algebraic independence criterion” for reasons that will soon become clear. Lemma 6.3. Suppose that w E Cc, and a(N) zs . a real-valued function that is defined for all N E N suficiently large, is monotonic increasing, approaches infinity as N + 00, and has the property that a(N + 1)/c(N) is bounded. Further suppose that there exists a sequence of polynomials PN(x) E Z[x], PN $0, such that t(PN>

5 a(N),

where c is a suficiently large constant. &T(W) = 0 for all N suficiently large.

IpN(w)l

< e-

co’(N)

,

Then w is an algebraic

In Gel’fond’s version of this lemma, in that increases monotonically to +oo. The proved in [Lang 19651. The proof uses the A simple consequence of Lemma 6.3, Gei’fond, was essentially what was used

number,

and

place of c one has a function O(N) above form of the lemma was first ideas that were described in $1.2. although not stated explicitly by in his proof of Theorem 6.3. It is

51. Gel’fond’s Method and Results because of this corollary that Lemma 6.3 is sometimes gebraic independence criterion.”

265 called Gel’fond’s

“al-

Corollary 6.1. Suppose that ~1, ~2, . . . , wm E Cc, m 2 2, and the function cl(N) satisfies the same conditions that o(N) did in Lemma 6.3. Further suppose that there exists a sequence of polynomials QN(x~ , x2,. . . , x,) E ,x,1 that satisfy the following conditions for any constant cl > 0 q7a,x2,... and any suficiently large N E N:

t(QN) I

m(N),

0
2

for

p=q=3;

(6)

trdegL2

2 2

for

p=4,

q=2;

(7)

trdegLs

2 2

for

p=3,

q=2.

(8)

All of the results proved using these ideas can be combined in the following theorem. Theorem 6.5. Let al,. . . , aP and bl, . . . , b, be two sets of complex numbers that are linearly independent over Q. Then: I) 2) 3)

if pq 2 2(p + q), then tr deg L1 2 2; if pq > p + 2q, then tr deg L2 > 2; ifpq >p+q, thentrdegLs 2 2.

Part 1) of Theorem 6.5 was proved in [Shmelev 19711with an additional technical condition similar to (2) ( a condition that automatically holds if

268

Chapter 6. Values of Functions That Have an Addition

Law

the ai and bj are real). An unconditional proof using Lemma 6.5 was given independently by Waldschmidt [1971] and Brownawell [1972]. Part 2) of the theorem follows from Tijdeman’s inequalities (6) and (7), and it was also published by Waldschmidt [1971], who had independently proved a result similar to Lemma 6.4. Part 3) follows from Tijdeman’s inequality (8) and was also proved in [Waldschmidt 19711. For a short proof of this result without using any lemma similar to Lemma 6.4, see Theorem 3 of [Shmelev 19711. Different proofs of Lemma 6.3, as well as refinements and modifications of the lemma, can be found in many places. For a list of references, see [WaldSchmidt 1974c]. Brownawell [1975] and Waldschmidt [1973a] independently proved a generalization of Lemma 6.3 in which deg PN and In II are bounded by different functions 6(N) and o(N). They used this result to prove the following theorem. Theorem 6.6. Suppose that al, a2 and bl, bz are two pairs of complex numbers each of which is linearly independent over Q. If

are algebraic numbers, then at least two of the six numbers al,

a2, bl,

b2, ealbz, ew&z

are algebraically independent. If we choose al = b2 = 1 and a2 = bl = er for some nonzero r E Q, then this theorem implies that either ee’ or ee2’ must be transcendental. In the case r = 1 this proves a conjecture of Schneider (see p. 138 of [Schneider 19571). Another consequence of the theorem is the following: if e”* E A, then e and T are algebraically independent. To seethat, take al = bl = ni and a2 = b:! = 1. The above method has also been used by Chudnovsky to prove bounds for transcendence measures of various numbers. 1.5. Fields of Finite Transcendence Type. The discreteness of Z has been very important in proving the results in this section. The analogous fact for algebraic numbers is Liouville’s theorem giving a lower bound for the modulus of a nonzero algebraic integer in terms of the degree and height of the number. In [1966] Lang proposed an inductive procedure for bounding the transcendence degree of finitely generated fields. His method is based on the notion of.a field of finite transcendence type. Let K = Q(wr , . . . , wq, C) be a finitely generated field extension of Q, where wi , . . . , wq are algebraically independent over Q and C is algebraic of degree v over Q(wi , . . . , We). Any cy E K can be written in the form

Cl= where Pi E Z[sl,...,

Pa@) + Pz(iq

x~],z=

+ .** + P”(qc”-l PO(4

(wl ,...,

up).

,

$1. Gel’fond’s

Method

and Results

269

Definition 6.1. By the size of o, denoted s(a), we mean the smallest B such that there exists a representation (9) with t(Pi) 5 B for i = 0, 1, . . . , V. Following Lang, for r 2 2 we shall say that IK is a field of transcendence type at most r if K has a set of generators ~1,. . . , wq, < relative to which every nonzero Q E lK satisfies the inequality

I4 2 exp C---c4~)~)

(10)

for some constant c that depends only on WI,. . . , wp, C. In that case we write T(K) 5 7. In [Waldschmidt 1974c] it is proved (see Lemma 4.2.23) that bounds on the transcendence type of a field do not depend on the choice of generators. If we have a sufficiently good bound on the algebraic independence measure for some transcendence basis of a field, then we obtain an upper bound for the transcendence type of the field. It is easy to verify that Gel’fond’s Theorem 6.4 implies that, under the conditions in the theorem, 7- (Q(ab)> I4

and

r(Q($))

14+e

for any E > 0. And from Theorem 3.33, which gives a bound for the transcendence measure of X, we obtain r(Q(r)) 5 2 + E for any E > 0. In [Cijsouw 1972, 19741one can find bounds on the transcendence measure of other numbers, from which one obtains upper bounds for the transcendence type of the corresponding fields. Using the Dirichlet pigeonhole principle, it is not hard to prove that the transcendence type of a field K cannot be less than 1 + tr deg@. The inequality (10) can be used in place of Liouville’s theorem to prove that numbers are transcendental or algebraically independent over the field K. If one is able to show that there are at least m of the numbers 51,. . . , J,, that are algebraically independent over K, then this means that there are at least m+qofthenumberswi ,..., wp,Ji ,..., & that are algebraically independent over Q. As an illustration of what can be obtained in this direction, we give the following result (see Part 4 of [Waldschmidt 1974c]). Theorem 6.7. Let T > 1 be a real number, and let K be a finitely generated subfield of @ having transcendence type at most r. Suppose that al, . . . , aP and b are sets of complex numbers that are both linearly independent over tri-y

* Pl>dP+d

9

then at least one of the numbers eaibj, 1 5 i 5 p, 1 2 j 5 q, is transcendental over IK.

In the special case when lK = Q(wi) with wi transcendental, this theorem is proved in Chapter 5 of [Lang 19661. Similar results for the numbers

270


Law

{ai, ea;bj}, 1 I i 5 p, 1 5 j 5 q, for (pq +p)/(p

+ q) 2 r and for the numbers {Ui, }, 1 I i 5 p, 1 I j 5 q, for (pq + p + q)/(p + q) 2 7 were announced in [Brownawell1972] and [Waldschmidt 19711. Proofs can be found in [Waldschmidt 1973b]. There are more general results of this type in [Shmelev 1975a]. .One also has analogues of Theorem 6.5 that hold for fields of finite transcendence type. These theorems can be proved by carrying over Lemmas 6.1-6.3 to such fields (see [Shmelev 19741 and [Brownawell 19751). The next theorem is taken from [Brownawell 19751. bj,eci*bj

Theorem 6.8. Suppose that r(K) 5 7, and both al,. . . ,ap and br,. . . , b, are linearly independent over Q. Then: I) ifpq/(p + q) > 27, then tr deg,lK(e”‘bj) 2 2; 2) if (pq + p)/(p + q) > 27, then tr deg,lK(oi, eaibj) 2 2; 31 if(pq+p+q)/(p+q) 2 2r, thentrdegK~(ai,bj,eaibj)

2 2.

The first example of a field of transcendence degree 2 having finite transcendence type was found in [1982a] by Chudnovsky, who proved the following: if the Weierstrass function p(z) has algebraic invariants g2 and 93, complex multiplication, and period lattice R, and if w E 0\20 and q is the correspond‘ing quasiperiod, then for any E > 0 the transcendence type of Q(x/w, v/w) is at most 3 + E. We shall give an estimate for the algebraic independence measure of r/w and q/w in $5 (see Theorem 6.38). In this case the above bound on the transcendence type differs only by E from the best possible value of 3. The first (and apparently the only) example of classes of numbers that generate fields of transcendence degree greater than 2 and finite transcendence type were given in [Becker 1991a] and [Nishioka 1991a]. Those authors proved good bounds depending on H(P) and deg P for the algebraic independence measure of the values at algebraic points of functions that satisfy the so-called Mahler functional equations. The transcendence and algebraic independence of the values of such functions were first studied by Mahler in [1929, 1930a, 1930b]. Here is one of the results that follow from Theorem 2 of [Nishioka 1991a]: Let d E N, d 2 2, and let p(z) be defined by the series

cp(.z)= 2 Zd” . n=O Then for any Q E A, 0 < 1~11< 1, and for any E > 0 one has -r (QM~,Q~~~>, This upper bound lower bound. The uses ideas that will Finally, we note the transcendence

. . . , dad-‘I)

L d + 1+ E.

on the transcendence type is rather close to our earlier method of proof in [Becker 1991a] and [Nishioka 1991a] be described later (see s3.3). that for almost all w E Iw in the sense of Lebesgue measure, type of Q( w ) is less than or equal to 2 (see [Nesterenko

32. Successive

Elimination

of Variables

271

1974a]). An analogous result holds for almost all complex numbers w (in the senseof Lebesgue measure in C). For m 2 2 the transcendence type of fields of the form Q(wi, . . . ,w,)ClRis~m+2foralmostall(wi,...,w,)EBm (see [Nesterenko 1974a]). It has been conjectured that the transcendence type ,w,) c lR is 5 m+l for almost all (wi,...,w,) E R”. Here OfQ(Wl,... “almost all” is meant in the senseof Lebesgue measure in I+?. In [1990, 19941 Amoroso proved that for almost all points in Cc” the transcendence type of the corresponding field is 5 m + 1. The proof usesideas that will be discussed in 53.

$2. Successive Elimination

of Variables

In this section we describe improvements of Gel’fond’s method that under certain conditions enable one to prove that some fields generated by values of the exponential function have transcendence degree at least 3, 4, etc. The techniques used all involve repeated application of results similar to Lemma 6.1. 2.1. SmaIl Bounds on the Transcendence Degree. In [1972a] Shmelev published the first result giving three algebraically independent numbers among the values of the exponential function at transcendental points. Namely, he was interested in finding a lower bound for the transcendence degree of the field where Q E A, a # 0, 1, and p is an algebraic number of degree d. Shmelev proved that then tr deg K 2 3 . d 1 19, if (11) His argument follows the plan of proof of Theorem 6.4. The role of Z is played by the ring Z [&I, and Liouville’s theorem is replaced by the bound in Theorem 6.4 for the transcendence measure of op. Thus, his proof is an example of an argument using the transcendence type, as described at the end of the last section. In [1972] Brownawell announced several results on algebraic independence of values of the exponential function. In [1975] he published a proof of the following result, which strengthens Shmelev’s theorem:

if

d > 15,

then

tr deg lK 13 .

(12)

A further improvement on (11) and (12) was given by Chudnovsky [1974a]: if

d 2 7,

then

trdeglKz3.

(13)

Then in [1974/1975] Waldschmidt refined Chudnovsky’s argument and proved that


272

if

h e improved

and in [1975/1976]

then

d 2 31,

if

trdegK24;

this result as follows:

d 2 23,

then

trdegK

2 4.

Other expository treatments of this approach can be found in [Waldschmidt 1977a] and [Brownawell 1979b]. In this connection we should also mention [Chudnovsky 1977a, 1977b], [B rownawell1975/1976,1979a], [Brownawell and Waldschmidt 19771, and [Mignotte 1974/1975]. 2.2. An Inductive Procedure. The identity (5), which lies at the heart of the proof of Lemma 6.1, holds for polynomials in one variable over an arbitrary ring. This means that it can be applied to polynomials P, Q E , z,], regarded as polynomials in 21 with coefficients in the ring Z[z1,22,... x,]. Then the resultant, which we denote R = R(x2,. . . ,x,) = qx2,..., Res,, (P, Q), is a polynomial in Z[xa, . . . , xm]. By analogy with Lemma 6.1, one has the following inequalities for any m-tuple of complex numbers Wl,...,W,Z t(R) . . , w,)l

IR(w2,.

clt(WQ)

5

7

5 eczt(P)t(Q)max(lP(wl,w2,.

. . ,wm)],

(14)

IQh,~2,-~-,an)I)

1

where cl > 0 and c2 > 0 depend only on m, wi, . . . , w,. Of course, there is a condition that must be satisfied in order for P and Q to have nonzero resultant - a condition that is also sufficient for (14) to hold. Namely, the polynomials P(Xl,X2,*.. ,x,> and Q(a,m . . . ,x,) cannot have any common divisors in xm] in which the variable xi appears. qn,x2, f.. , We now consider a hypothetical procedure for bounding the transcendence degree of a field K. Let wi, ws, . . . , w, E Cc,m 2 1, be a transcendence basis for K, so that m = tr deg K. Suppose that we have somehow been able to construct a set of polynomials PI,. . . , P, E iZ[xl, x2,. . . , z,,J, s > m, such that t(P,> I T,

. , w,)l

IPi(W,W2,~~

“‘, s, there exist two that have no common root:. Applying (14) m - 1 times, we find that t (P/“-“) IP!“-“(w,)l 2

2 CATTY-‘, 5 exp (-U

+ CATTY-‘)

i = m, . . . , s , i = m,. . . ,s .

,

Here and later, the letter c with various subscripts denotes positive constants that depend only on m, WI,. . . , wm. Finally, if we apply Lemma 6.1 to the two polynomials PJ”-l) (z,) that have no common roots, we obtain the inequality z exp (-U

+ CATTY) > exp (-csT~~)

,

from which it follows that U 5 CATTY. This inequality gives a lower bound for m = tr deg K. When bounding the transcendence degrees of the fields Li, it should be noted that Gel’fond’s construction, which was used by Tijdeman, Shmelev, Waldschmidt, Brownawell, and Chudnovsky, enables one to obtain a sequence of polynomials P~(z1, . . . , z,) with parameters T M NP+Q, T M Np+q,

U

T M Nf’+q,

U x NPqInN w

NP9+P(ln u

N)P/(P+‘d x

NPQ+P+Q

in the case of LI , in the case of L2 ,

(15)

in the case of L3

(see Theorems 7.1.6, 7.2.8, and 7.3.4 of [Waldschmidt 1974c]). In the case of LI, for example, this line of argument should give the bound m 2 l+ [log2 (iz)] . However, in general this set of polynomials does not have the properties needed to eliminate the variables. It turns out that it is a complicated matter to ensure that the conditions hold for all of the steps in the elimination of variables. The results in $2.1 come from an implementation of these ideas when the number m of variables is small. The first attempt to carry out this approach in the general case (see Theorem 6.10 below) was due to Chudnovsky [1974d, 1974e]. Although he was not able to give a complete proof of the results he announced (he published a proof only in the “nondegenerate case,” i.e., when one essentially assumes that all of the steps go through in the elimination of variables), nevertheless the preprints [Chudnovsky 1974d, 1974e] played an important role as a stimulus that inspired many researchers to look for a complete proof of these and similar results based on Chudnovsky’s ideas. The difficulties encountered in carrying out this plan of proof caused people to pay special attention to the algebraic problems connected with elimination theory, and to look for conditions on the original set of polynomials Pi E

274


Law

1 5 i 5 s, that are as simple as possible and yet ensure . . , z,], that all of the successive steps in the elimination can be realized. With this purpose in mind, attempts were made to generalize Lemma 6.5 (Gel’fond’s algebraic independence criterion) using the techniques described above. The direct generalization of Corollary 6.1 of Lemma 6.3, with (61(N))& rather than ((TV) 2 in the exponent of e in the hypothesis and with the phrase “there are at least K of the numbers” in the conclusion, turns out to be false (see Theorem 14 in Chapter 5 of [Cassels 19571 for a counterexample). The first attempt at a correct generalization of Lemma 6.3 was made in [Dvornichich 19781, but that result was never applied to prove algebraic independence of numbers. Subsequently, several generalizations of Lemma 6.3 appeared that were suitable for proving algebraic independence. The first was due to Reyssat [1981]. The paper [Waldschmidt and Zhu Yao Chen 19831 should also be mentioned. We shall state a result that can be derived from [Nesterenko 1983b] and is a slight generalization of the theorem in [Reyssat 19811. q21,22,.

Theorem 6.9. Suppose that ~1, . . . , w,,, E Cc; (~1(N) and 02(N) are nonnegative functions of a natural number argument such that ol(N + 1) 5 sol(N) for some a 2 1; K > 1; and the sequence of polynomials PN E Z[xl, . . . , xm], ‘N > No, satisfies the conditions

-N”ol(N)

< In ~PN(w~, . . . ,wm)l

5 -N”czz(N)

.

Then there exists a constant c, depending only on WI,. . . , w, and K, such that a2(Nj2

1

cm(N)

all N 2 No, then there are at least 1 + [log, K] of the numbers WI, . . . , w, that are algebraically independent over Q.

for

In [1981] Philippon used a padic version of the criterion in [Reyssat 19811 to prove that if certain bounds hold for linear forms in the p-adic numbers al,..., ap and bl,. ..,bg, then PC7 tr deg L1 2 log, p+q

Pq+P tr deg L2 2 log, P+q

’

Pq+P+q P+q * In particular, if (Y # 0,l is algebraic and /3 is an algebraic number of degree d 2 2, then there are at least log,(d + 2) - 1 of the padic numbers tr deg LI 2 log,

lna,

c#,

#,

. . . , QIBd-’

that are algebraically independent over Q.

92. Successive

Elimination

of Variables

275

There are some difficulties in applying Reyssat’s criterion with Gel’fond’s construction, as was done, for example, by Chudnovsky when he proved (13). In [1981] Philippon proposed an analytic method involving functions of several complex variables (“false” variables) to construct the polynomials PN E %[xl,... , zm] that are needed to apply the criterion. To prove the bound on the transcendence degree of Ns, Philippon constructs the following auxiliary function F(Z) of d complex variables f = zd) (we have slightly modified Philippon’s construction to make it (G,..., better suited to the complex case): (16) _Here4_hesummat@ is over all vectors Ji = (pi,. . . , pd), 0 5 pi 5 No, and x = (A,,... ,Ad), , Xi,), 0 5 Xij < N, with integer components Ai = (kl,... and with parameters N and NO, where No depends on N in some explicit way. The vector Z in (16) has the form h = (al,. . . , up). The coefficients C(x, Ti> are chosen to be rational integers, and it is this that distinguishes Philippon’s construction from Gel’fond’s; in particular, Philippon can use the trivial lower bound maxIF ]C(x, Ti>] 1 1. Recall that in Gel’fond’s construction the coefficients were’ chosen to be polynomials in the numbers being studied with integer coefficients, and it was not easy to find a lower bound for the maximum coefficient. The .coefficients C(x, JX)E Z are chosen so that the function F(Z) has zeros of multiplicity at least K at the points hibi + . . . + h,b, for 0 5 hj < H, where H and K are parameters that are chosen as a function of N. F(Z) can be represented as an exponential polynomial in zd whose coefficients are functions of a similar type in the variables zi , . . . , z&i. This makes it possible to use analytic results for functions of one variable (the Hermite interpolation formula, and Tijdeman’s lemma giving an upper bound for the coefficients of an exponential polynomial) along with induction on the number of variables. If the number d of variables is chosen greater than a certain bound that depends only on p and q, then it turns out that there exist parameters Lo, H, and K for which one can construct polynomials satisfying the conditions in Reyssat’s criterion with upper bound only a little worse than (15).

In [1984] Endell published a brief outline of a direct proof of the following theorem, which had been announced earlier by Chudnovsky (see [Chudnovsky 1974d, 1974e, 1977, 1982c]). Theorem 6.10. Let al,. . . , up and bl, . . . , b, be complex numbers for some constant y > 0

and

lwbl + . . * + vqb,l > exp(-y]??])

such that

276

for

anyE= Then

Chapter

(Ul,...


,up) E zp a7dv=

That

Have an Addition

Law

(WI,... ) wq) E zq WJith IEl > 0, Iv’1 > 0.

1) ifpq/O,+q) 2 2”, thentrdegLi >n+l; 21 if(pq++)/(p+q) >2n, thentrdegL2 >n+l; 3) if(pq+p+q)/(p+q) >2”, thentrdegLs >n+l. The proof of the algebraic independence criterion in [Nesterenko 1983c] was based on ideas that will be described in the next section. Using the same ideas, in [Nesterenko 1983a] it was proved that trdeg~(cuP,aprr,...,apd-I)

2 [log,(d+l)],

(17)

where Q # 0,l is an algebraic number, and /3 is an algebraic number of degree d 2 3. It is easy to see that this result is a special case of part 2) of Theorem 6.10.

§3. Applications

of General

Elimination

Theory

3.1. Definitions and Basic Facts. The proofs in [Nesterenko 1983a, 1983cj of the algebraic independence criterion and the inequality (17) were based on the idea of working with ideals of the ring Z[ze, . . . , z,] in the same way as with polynomials. The basic definitions and facts needed for this section can be found in [Zariski and Samuel 19601. Here we shall only give a few of the definitions. We first define the rank of an ideal. (In [Zariski and Samuel 19601this is called the height, but we prefer the term “rank,” because “height” has a very different meaning in transcendental number theory.) Definition 6.2. The rank h(p) of a prime ideal p c Z[ze, . . . , z,] is the maximum length of an increasing chain of prime ideals that are strictly contained in p. If J is an arbitrary ideal, then its rank h(3) is defined as the minimal rank of the prime ideals containing 3. Note that principal ideals have rank 1. Recall that any ideal 3 c Z[se, . . . . . . 7z,] has finitely many associated prime ideals pi, . . . , pB (these are analogues of the prime divisors of an integer). Definition 6.3. An ideal J c Z[zs, . . . , zrm] is said to be unmixed if all of its associated prime ideals pj have the same rank h(3). In the case of an unmixed ideal 3 the set of associated prime ideals is uniquely determined. In [Nesterenko 1983a, 1983c] certain numerical invariants deg3 and H(3) - analogous to the degree and height of a polynomial P E Z[ze, . . . , zm] were associated to any unmixed ideal J c Z[ss, . . . , z,] that is homogeneous (i.e., generated by homogeneous polynomials). For any CJ = (we,. . . , w,) E

$3. Applications

of General

Elimination

Theory

277

@m+1 \ {0}, the magnitude of the ideal at Z, denoted ]J(ij)], was also defined;

this is analogous to the absolute value of P(Z) for a polynomial P. The precise definitions of deg J, H(J), and ]J(Ts)] are based on the elimination theoretic notion of the u-resultant or Chow form of the ideal J. In the case of homogeneous prime ideals of the ring Ic[zc, . . . ,zm], where Ic is a field, this object was introduced in [Chow and van der Waerden 19371,where it was called the “zugeordnete Form.” It is the resultant of a set of basis polynomials for 3 and a certain set of linear forms with indeterminate coefficients. The properties of the Chow form are explained in detail in [Hodge and Pedoe 19471, where it is called the “Cayley form.” The name “Chow form” goes back to 1972, when the Russian edition of [Shafarevich 19941 was published. It should be noted that the non-homogeneous analogue of the Chow form had appeared earlier in articles by Hentzelt [1923] and Noether [1923], where it was called the “Elementarteilerform.” Its properties were subsequently studied by Krull [1948], who referred to the ‘LGrundpolynom.” We also note van der Waerden’s paper [1958]. In these classical papers the Chow form was studied from an algebrageometric point of view. The behavior of the coefficients, which is what one has to understand in order to apply it in transcendental number theory, was not examined, although essentially all of the necessary tools for this had been developed. The invariants deg J, H(3), and ID@)] of an unmixed homogeneous ideal J that were introduced in [Nesterenko 1983a, 1983c] have properties similar to those of the degree and height of a polynomial and the value of a polynomial at a point. If J = (P) C Z[ZO,. . . ,z,] is a principal ideal, then one has (see Proposition 1 of [Nesterenko 1984b]): deg J = deg P,

In H(J) 5 In H(P) + 2m2 deg P ,

We now consider the following problem: Given a point z = (us,. . . , w,) E Cm+‘, find a lower bound for 13(w)I in terms of H(3) and deg3. In the case of a principal ideal J = (P) it follows from (18) that a bound for ]J(ss)] leads to a lower bound for IP( in terms of H(P) and deg P. Results analogous to Lemmas 6.1-6.3 hold for these invariants of ideals of Z[zs, . . . , zm]. The numbers deg 3, In H(3), and In ]YJ(z)] behave “almost linearly” when the ideal is factored into an intersection of primary ideals. This makes it possible to reduce the problem of estimating ]J(~s)] for an arbitrary unmixed ideal 3 to that of estimating In@)] for a prime ideal p. An analogue of Lemma 6.1 enables one to construct an inductive argument (with induction on the rank of 3) establishing a bound for ]?(Ts)]. Given a prime ideal p and a polynomial Q E Z[zc, . . . , z,] \p, this lemma gives an ideal 3 with h(3) > h(p) such that H(3), deg3, and ]3@)] can be bounded from above in terms of the corresponding invariants of p and Q. In essence,the lemma replaces the

278


procedure of elimination of variables. The basic idea of [Nesterenko 1983a, 1983c] was to use induction on the rank of the ideals in place of the elimination of variables. In [1984, 1985, 19861 Philippon made a major advance in the theory that enabled him to strengthen all of the bounds on the transcendence degrees of the number fields, by replacing logarithmic bounds with linear ones (see Theorem 6.12 below). Philippon refined the analogue of Lemma 6.1 by introducing a new invariant (the projective distance p from the point i5 to the variety of zeros of the ideal) and finding a connection between 13(Z)] and the distance p. In [Philippon 1984, 1985, 19861 the invariants of ideals are determined for the polynomial ring k[zc, . . . , z,] over a finite extension k of Q; in particular, he uses the Weil height of a point in projective space to determine H(J). All of the results are proved for non-archimedean as well as archimedean absolute values. In what follows, when we discuss Philippon’s results, we shall consider only the archimedean case, and shall assume that the ring of coefficients is Z. We shall use the notation introduced above. At present there are three approaches to estimating the transcendence degree of number fields that are different in appearance, but are all based on the ideas described above. The first uses Philippon’s algebraic independence criterion (see [Philippon 1984, 1985, 1986]) and resembles Gel’fond’s proof of Theorems 6.1-6.3. The second approach, which is similar to the proof of Theorem 6.4, is based on direct estimates for ID(Z)] (see [Nesterenko 198413, 1985a, 19891). The third method makes use of a multidimensional generalization of Lemma 6.1 that is due to Brownawell [1987a, 19881. We shall examine all of these approaches, focusing our attention on how they obtain bounds for the transcendence degrees of the fields Li. 3.2. Philippon’s Criterion. In [1984, 19861 Philippon published a general theorem that included all of the known algebraic independence criteria as special cases and led to substantially improved bounds on transcendence degrees. Theorem 6.11. Let Ts = (WI,. . . , w,) E Cm, and let 6 be the prime ideal OfZ[Xl,... , x~] consisting of polynomials that vanish at GT. Let m - k, where 0 5 k 5 m, denote the codimension of this ideal. Suppose that CT,6, R, and S are increasing functions defined on N that for N E N suficiently large take values 1 1. Further suppose that CT+ 6 approaches +co as N + +co; the function S

(u + 6)6” is an increasing S(N)k+2

function;

-> c(a(N

and for all N E N suficiently

+ 1) + 6(N + l))S(N

+ 1)” (S(N)“+l

large one has + R(N

+ l)“+‘)

,

where c 2 1 is a constant depending only on m and 6. Under these conditions there does not exist a sequence of ideals 3~7 C Z[xl, . . . , x,] that have a

$3. Applications

of General

Elimination

Theory

finite number of zeros in the ball of radius exp(-R(N)) generated by polynomials QiN’, . . . , Q$$ for which degQiN) 2 5 6(N) >

279

centered at 15 and are

’ 1nH ~Q~‘)(LT), QtN’ 5 o(N), exp(-,S’(ii.l’*” ( >

0 < l 0 is a constant that is suficiently large compared to m. There does not exist a sequence of homogeneous ideals 3~ = (P~,N, . . . , Pm(~),~) C Z[xl, . . . , xm], N 2 No, that have a finite number of zeros in the ball of radius exp(-3ciNq) ‘centered at ij and satisfy the inequalities t&N)

0

0 and X 1 X(E), and for all vectors iG= (IQ,.. . , I+) E Zp and I= (ll,. . . ,l,) E Zq with Ihi 5 X and lliI 5 X:

lklal + . *. Then:

+ kpapl1 exp(--XE),

lllbl + . . . + l,b,l 2 exp(-X’)

. (20)

280


1)

tr degQLl

> 5

- 1;

2) 3)

tr degqLz tr deg&3

2 s 2 5.

- 1;

In [1986] Philippon gives a brief proof of this theorem (using the example of part 3) to show how it follows from his criterion). He uses the analytic construction with “false variables” that was described in [Philippon 19811. In [1984a, 19861 Waldschmidt uses Philippon’s criterion to obtain a large number of results on the transcendence degree of fields generated by values of the exponential function and elliptic functions. In [1984a] Waldschmidt proves Theorem 6.12 as a consequence of a general theorem on algebraic groups. Another proof of Theorem 6.12, which instead of Philippon’s criterion uses a direct inductive bound on the ideals, is described in [Nesterenko 1985a], where part 2) is treated in detail (the method of proof will be discussed in $3.3 below). Finally, in [1987a] Brownawell proposes a third way to prove Theorem 6.12. Like Philippon [1981], he uses a construction with “false variables.” Brownawell investigates both the exponential and elliptic functions, and gives a detailed proof of the elliptic analogue of part 1) of Theorem 6.12. See $3.4 for more details on Brownawell’s work. In [1987] Diaz published a sketch of a proof of a further strengthening of the first inequality in Theorem 6.12, and he announced a result that strengthened the second inequality as well. In [Diaz 19891 he gave detailed proofs and also improved the technical conditions (20). We now state these theorems, which at present give the best bounds known for the transcendence degrees of LI and Lz. Theorem

4

6.13.

Let al,. . . , ap and bl, . . . , b, be complex numbers for which

there exists a constant X (bl , . . . , b,) > 0 such that the-following inequality holds for any X 2 X(bl, . . . , b,) and for all vectors 1 = (ll,. . . ,I,) E Zq with llil 5 X: ; lllbl +. . . + l,b,l 2 exp -Xpq/(2P+q) ( >

b)

there exists a constant X(al, . . . , aP) > 0 such that the following inequality holds for any X > X(al, . . . , ap) and for all vectors x = (ICI,. . . , kp) E Zp with llcil 2 x: Ihal + ... + kpapl 1 exp (-min

(XlnX,XPq/(P+2q)))

.

Then the following bound holds for pq > p-k q: tr degq LI > Theorem

6.14.

[ 1 P9 P+9

.

Let al,. . . , ap and bl , . . . , b, be complex numbers for which

$3. Applications

a)

of General Elimination

Theory

there exists a constant X(bl, . . . , b,) > 0 such that the following inequality holds for any X 2 X(bl,. . . , b,) and for all vectors i = (11,. . . , Zq)E I@ with lZiI 5 X: lllbl + ... + lqbql L exp -xP(Q+W(2P+d (

b)

281

; >

there exists a constant X(al, . . . , ap) > 0 such that the following inequality holds for any X 2 X(al,. . . , up) and for all vectors z = (ICI,. . . , kp) E ZP with lkil 5 X: lklal +*..+kpapl

>exp (-min(XlnX,Xp(q+‘)/(p+2q+1)))

.

Then the following bound holds for pq + p > 2: tr degQ L2 2

[ 1 Pq+P P+9

.

In particular, Theorem 6.14 implies a bound for the transcendence degree in Gel’fond’s conjecture that is the best that is currently known. Corollary 6.3. Let cr # 0,l be an algebraic number, and let /3 be an algebraic number of degree d > 2. Then tr deg&

(aP, CY~‘,. . . , oOOd-’)+gq.

To derive the corollary one takes p = q = d, ai = pi-‘, and bi = pi-l In (Y, i = l,..., d, in Theorem 6.14. In [1991] Ably proved an analogue of Corollary 6.3 in which Q! E A is replaced by a transcendental number that in some sensehas “good approximations” by rational numbers (see also [Amou 1991a]). Namely, he proved: If y is a transcendental number that has ‘good approximations” by rational numbers, and if /3 is an algebraic number of degree d 1 2, then tr deg&

(y, yO, yB2,. . . , yfidml )qg].

The improvement of the bound in Theorem 6.12 was achieved by changing the analytic construction of the polynomials needed in Philippon’s criterion. Diaz applied an idea of Chudnovsky (see p. 60-61 of [Chudnovsky 1982a]) in this situation. The result was that the auxiliary analytic functions were now functions of one variable, and their coefficients were polynomials in the numbers being studied, as they had been in Gel’fond’s original papers. We give a brief description of Diaz’s construction as it applies to Theorem 6.13. We set m = pq and w = (ealbl,. . . , e@q) E Cc”. We further set y = . , Ypq), and denote wll,..

282

Chapter


That

Have an Addition

Law

Nd(L) = {(II, e.. 7Id) E Zd ; 0 5 Zi < Ly i = 1,. . . , d} * We let N denote a large natural number, and Xi, As,. . . denote positive constants that depend only on p, q, ai, and bj. In the first part of the proof Siegel’s lemma is used to construct a nontrivial set of polynomials

q EmT>

i = (ll,. ..,b> ENQ(L), L = ‘3’+lNp

7

such that the degree of 5 in each variable is less than 2P+‘NP+Q, the height Of Pi satisfies lnH(Pi) 5 Xi In N , and the following identity holds for all E = (Ici , . . . , lc,) E lVp(M) , M = NP: QI(y)

=

c e(v) SEW(L)

fr zf(y,j)ki” i=l

s 0.

(21)

j=l

For any vector r = (~11, . . . , rpq) E ;Zpq with rij 2 0 we denote

and Q#)

=

c iEN’

P&P)

fr

ti(Y.j)‘“i’j

i=l

j=l

.

(22)

We set R(N) = 8qNPq In N and let B be the ball of radius p = exp(-R(N)) centered at Z. We fix a point t E B and consider the set of all vectors 5; = (Tll, * . . , rpq) E Zpq, rij 2 0, for which there exists a polynomial Pi,?(Y) with PI?(~) # 0. We choose a vector in this set that has the smallest sum of its coordinates, and call it i?(t). The set of points f E B is infinite, but the set 93 = {F(if) ; f E B} is finite. We now choose MI = X2M with As E N sufficiently large, and consider the finite set of polynomials !3X = {Qz,JT)

; % E NP(M&

T E !R} .

(23)

It is to this set of polynomials that we apply Philippon’s criterion. It can be checked that deg Qx.,~ 5 AsNpfq,

In H(Qk,?) 5 AsNP+q,

Qz,, E ?JJZ7

and so one can take o(N) = 6(N) = XsNP+Q in Philippon’s criterion.

$3. Applications

To find an upper

of General

bound for IQ@)]

point $ = ( Qrn--d (clTm) e- C2Trn

.

In [1991] Ably published a series of results giving lower bounds for the approximation measure of a set of values of abelian functions (and, in particular,, exponential or elliptic functions) by fields of large transcendence degree. His method of proof was based on the ideas in s3.2.

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344

Index

Index Algebraic - dependence, - independence, - integer, 74 - number, 11 - unit, 74 Approximation - Diophantine, - of algebraic - rational, 23, -- of e, 112 - simultaneous, Bombieri’s

17 17

78 numbers, 26, 78

15, 110

78, 79

Theorem,

Confluent Function, Conjugate, 22 Continued Fraction, - and e”, 112 - for functions, 95 - irregular, 95 Convergent, 13, 24

51 97 13, 24, 78

Degree - of a polynomial, 22 - of an algebraic number, 22 Diophantine - approximation, 78 - equation, 15, 29 Dirichlet - pigeonhole principle, 40 - unit theorem, 74 Dual Basis, 71 Dyson’s - inequality, 49 - lemma, 50 Effectiveness, 33, 34, 36, 45, 78 Euclidean Algorithm, 13 Functional Approximation, - of Hermite-Pad& 98

93

Gauss - class number 1 problem, 16 - hypergeometric function, 95, 112 Gel’fond’s - method, 82 - theorem, 48 Geometry of Numbers, 70

Height - of a polynomial, 22 - of an algebraic number, Hermite’s - identity, 84, 92 - method, 80, 83 HermitePade Functional tion, 98 Hilbert’s - seventh problem, 100 Hypergeometric - function, 95, 112

22

Approxima-

Index of a Polynomial, 56 Irrationality - exponent, 106 - measure, 17 -- of rr, 106 - of ir, 13, 113 - of Jz, 11 - of e, 78 Irregular Continued Fraction, Jacobi Identity,

95

97

Length - of a polynomial, 22 - of an algebraic number, 22 Lindemann’s Theorem, 88 Lindemann-Weierstrass Theorem, 103 Linear Form, 39, 64 Liouville’s - numbers, 26, 27 - theorem, 26, 44 Logarithm of Algebraic Number - transcendence measure, 104 Mahler ‘s - theorem, 54, 76 Measure - of algebraic independence, 18 - of irrationality, 17 -- of rr, 106 - of linear independence, 17 - of relative transcendence, 18 - of transcendence, 18 -- of ?r, 104 -- of e, 99 Minimal Polynomial, 22

88,

Index Minkowski’s Theorem, 66 Module - full, 74 - in a number field, 74 - non-degenerate, 75 - similar, 75 Non-archimedean Metrics, 40, 76 Norm, 22 Norm Form Equation, 73 Pade Approximation, 95 - of first kind, 98 - of second kind, 98 Partial Quotient, 24 Pell’s Equation, 30 Pigeonhole Principle, 40 Quadratic

Irrationality,

Rational - approximation, -- of e, 112 - subspace, 68 Roth’s - lemma, 57 - theorem, 55 Schmidt’s

23, 26

25

345

- approximation theorem, - finiteness theorem, 75 - subspace theorem, 68 Schneider’s - theorem, 54 Siegel’s - method, 53, 82 - theorem, 45, 53 Squaring the Circle, 13 Successive Minima, 71 Thue’s - equation, 28, 29 - method, 31, 41, 45, 53 - theorem, 42 Transcendence - measure, 18 -- of 7r, 104 -- of e, 99 -- relative, 18 - of r, 13 - of e, 83 - of e*, 82, 100 Transcendental - number, 11, 14 Transfer Theorem, 70 Waring

Problem,

16

60, 64, 65

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