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Half a century of transcendence. (Un demi-siècle de transcendance.) (French) Zbl 0977.11030

Pier, Jean-Paul (ed.), Development of mathematics 1950-2000. Basel: Birkhäuser. 1121-1186 (2000).
This paper gives a very detailed survey of Transcendental Number Theory between 1950 and 2000. It is written in a very lively and lucid style and its 66 pages contain a lot of information. As it is almost impossible to give a complete review of it, I shall only present a list of the main topics treated in this survey.
The different sections are: 1) Irrationality and rational approximations. 2) Around Mahler’s method. 3) The Gelfond-Schneider’s method: transcendence and algebraic independence. 4) The method of Siegel-Shidlovskiĭ: \(E\)-functions and \(G\)-functions, hypergeometric functions. 5) The Gelfond-Baker’s method. 6) Zero lemmas, 7) Transcendence in function fields.
The paper ends with a list of references concerning books, surveys and proceedings of conferences.
The first chapter on irrationality and rational approximations begins with Liouville’s theorem on the approximation of algebraic numbers. It contains a section on irrationality measures (examples: \(\pi\), \(\zeta(3)\), \(e^\pi\), …). Lehmer’s problem is studied in a second section. The next sections present the Thue-Siegel theorem culminating with Roth’s theorem and W. Schmidt’s subspace theorem. The last two sections of this chapter are devoted to Padé approximations and to the irrationality of certain numbers related to the Fibonacci sequence.
The chapter on Mahler’s method begins with a section on arithmetical entire functions. The second section presents functional equations as studied first by Mahler. The next section indicates the link of these functional equations with fractals and dynamical systems. The last section deals with modular functions, a beautiful example of this theory is: \(\pi\), \(e^\pi\) and \(\Gamma(1/4)\) are algebraically independent (Yu. Nesterenko, 1996).
The third chapter begins with a survey on algebraic independence. The second section is “algebraic groups and the four exponential conjecture”, for example the following problem remains unsolved: does there exist an irrational real number \(x\) such that \(2^x\) and \(3^x\) are both rational integers? In the next section the case of functions of several variables is studied. This chapter ends with a short section on the classification of transcendental numbers.
The \(E\)-functions and \(G\)-functions which are the theme of the fourth chapter were introduced by Siegel in 1929. The results on \(E\)-functions are presented in the first section and those on \(G\)-functions in the next one. For example, they lead to transcendence results for hypergeometric functions. The last section is devoted to Hilbert’s irreducibility theorem.
The first section of the chapter on the Gelfond-Baker method gives a survey on lower bounds for linear forms in logarithms, including the effective refinement of Liouville’s theorem and presenting some conjectures (“\(abc\)” and the Lang-Waldschmidt conjecture). The second section deals with Diophantine equations, more precisely the consequences of Baker’s method: bounds for the solutions of certain Diophantine equations and the possibility to solve some of them completely, which is for me the greatest achievement of this theory. This section is very detailed and contains many examples. The third section is “algebraic groups and Diophantine geometry” and presents numerous deep works which began in the 80’s.
Chapter six is about zero lemmas, maybe the deepest question in this theory where to get a proof we almost always construct some number \(\gamma\) which is small for analytic reasons and which cannot be too small because of arithmetical reasons (Liouville estimate)…if it is nonzero, and very often the main problem is to prove (by some “zero-lemma”) that \(\gamma\not=0\). This subject is also related to effective versions of Hilbert Nullstellensatz.
The last chapter is a survey on transcendence in function fields. The field of Laurent series \(K((T^{-1}))\) has numerous similarities with the field of real numbers, this is the reason for this study. The case of nonzero characteristic leads to many problems. This theory has also links with automata theory.
To conclude I translate (freely) the conclusion of the author: “Very important progress has been made during the last fifty years, but the amount of open problems shows that this theory is far from being completely mature. Many questions, very easy to state, remain unsolved. But, the present methods are rich of varied and deep applications. I am sure that the future of this theory will be beautiful”.
For the entire collection see [Zbl 0947.00008].

MSC:

11Jxx Diophantine approximation, transcendental number theory
11-02 Research exposition (monographs, survey articles) pertaining to number theory
11-03 History of number theory
11Dxx Diophantine equations
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