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Contributions to the theory of transcendental numbers. (”Portions of this Volume were translated from the Russian by G. A. Kandall”). (English) Zbl 0594.10024

Mathematical Surveys and Monographs, No. 19. Providence, R.I.: American Mathematical Society (AMS). XI, 450 p. $ 80.00 (1984).
This volume consists of a set of papers on the theory of transcendental numbers and diophantine approximations, written by the author since 1970. Most of the material of this volume is the translation into English of papers written by the author in Russian in 1970 and in the period from 1974 till 1977. The papers preserved principally their original form. Only changes allowing the interested reader to trace recent achievements in this field are made. Some modernization of the former material is done in order to connect different directions of researches. In the reviewer’s opinion it would be necessary to include recent results of Yu. V. Nesterenko and P. Philippon for the completeness of the survey of the latest achievements in the theory of transcendental numbers. When writing this volume they were probably known to the author. Sometimes the assertions are formulated in a stronger form than proved. For example, it refers to the Appendix where the talk was about extremality of certain manifolds. The reviewer has been acquainted with the initial variant of the corresponding author’s manuscript in Russian; there is the same gap in it.
Now let us come to the essence of the problem. In the preface the extended text of the author’s speech at the 1978 International Mathematical Congress in Helsinki is given. It is an analysis of the progress in transcendental number theory with particular emphasis on transcendence, algebraic independence and measures of algebraic independence of numbers connected with exponential and elliptic functions.
Chapter I deals with the proofs of the author’s 1974 results [Preprints IM-74-8/9, Kiev 1974; for further expositions and summaries see Zbl 0337.10023, Zbl 0345.10020] on the Schanuel conjecture, the essence of which is that the number of algebraically independent elements in the set of complex numbers of the form \(\{e^{\alpha_ i\beta_ j}\}\), \(\{\alpha_ i,e^{\alpha_ i\beta_ j}\}\), \(\{\alpha_ i,\beta_ j,e^{\alpha_ i\beta_ j}\}\) (1\(\leq i\leq N\), \(1\leq j\leq M)\) grows logarithmically with min(N,M). At present the author has improved his results. However, the methods of Chapter 1 are of interest and they are further developed in the following chapters of the monograph. The main method of Chapter 1 is based on the Gel’fond-Schneider method.
In Chapter 2 Baker’s analytic method concerning the problem of algebraic independence is considered. The results proved in Chapter 2 were first published by the author in 1974. In particular, it is proved that there are at least two algebraically independent numbers among \(\pi\), \(\pi^{\sqrt{D}}\), \(e^{i\pi \sqrt{D}}.\)
The main point of Chapter 3 are the author’s two papers on ”colored sequences” published in 1975 but unfortunately they did not reach wide circulation among the readers. Chapters 3 and 4 deal with the criterion of algebraic independence of \(n\geq 2\) numbers. The well-known Gel’fond criterion states that it is impossible for a transcendental number \(\theta\) \((n=1)\) to have a ”dense” sequence of algebraic approximations. Though a straightforward generalization of this criterion to \(n>1\) is impossible, various results that are important for applications are presented in Chapters 3 and 4. Chapter 3 deals mainly with the case when the algebraic independence of several numbers depends on the measure of transcendence of each of them. Chapter 4 presents an attractive and simple criterion of algebraic independence of two numbers in terms of a ”dense” system of polynomials in two variables with rational integer coefficients, assuming small values at these numbers. This criterion and its n-dimensional generalization are the best possible, and constitute the necessary algebraic part in the proof of the algebraic independence of various numbers connected with exponential, elliptic and Abelian functions.
Chapter 5 is devoted to diophantine approximations. The main method of this chapter is Siegel’s method and its development in the famous papers of Mahler, Baker and the author. In this chapter are the results on diophantine approximations of numbers generated by values of E-functions at rational points, on the bound for the number of solutions of Thue’s equation, on the bounds of the measures of diophantine approximations of arbitrary numbers from the fields generated over \({\mathbb{Q}}\) by values of exponential and other E-functions, satisfying linear differential equations over \({\mathbb{Q}}(x)\), at rational points.
Chapter 6 is an expanded version of the author’s paper on diophantine sets published in Russian in 1970.
Chapter 7 considers transcendence and algebraic independence of numbers connected with elliptic and exponential functions. The results of algebraic independence of \(\pi\) /\(\omega\), \(\eta\) /\(\omega\) for a period \(\omega\) and quasi-period \(\eta =2\phi (\omega /2)\) of an elliptic curve defined over \({\bar {\mathbb{Q}}}\); algebraic independence of \(\pi\) and \(\omega\), and of \(\pi\) /\(\omega\) and \(e^{i\pi \sqrt{D}}\) for a period \(\omega\) of an elliptic curve with a complex multiplication by \(\sqrt{D}\) are given. In this chapter properties of L-functions of elliptic curves and the Birch-Swinnerton-Dyer conjecture used to bound heights of integer points on elliptic curves are analyzed. He considers analogs of the Lindemann-Weierstrass theorem for arbitrary systems of functions satisfying differential equations and the laws of addition.
In Chapter 8 the author points out the bound on the measure of the algebraic independence of \(\pi\) /\(\omega\) and \(\eta\) /\(\omega\), which is close to the best possible. Another important feature of this result is in its dependence of height of “\(H^{-c}\)” form. This result implies, e.g. an “\(H^{-c}\)” form of the measure of diophantine approximations for numbers such as \(\Gamma\) (1/3), \(\Gamma\) (1/4) etc.
Chapter 9 presents the results on a generalization of the Straus- Schneider theorem on the number of complex z such that a given meromorphic function f assumes, together with all its derivatives, rational integer values at z.
As a matter of fact the appendix is a paper on the extremality of certain multidimensional manifolds prepared by A. I. Vinogradov and the author in 1976.
Summarizing the reviewer whishes to point out that this monograph contains urgent material in modern interpretation. It is a good addition to the known books on the theory of transcendental numbers.
Reviewer: S.Kotov

MSC:

11J81 Transcendence (general theory)
11-02 Research exposition (monographs, survey articles) pertaining to number theory
11J85 Algebraic independence; Gel’fond’s method
11J04 Homogeneous approximation to one number
11J17 Approximation by numbers from a fixed field
11J68 Approximation to algebraic numbers
11D61 Exponential Diophantine equations
11K60 Diophantine approximation in probabilistic number theory