\(G\)-functions and Hilbert’s irreducibility theorem. (\(G\)-fonctions et théorème d’irréductibilité de Hilbert.) (English) Zbl 0565.12012

Dèbes, Pierre

Acta Arith. 47, 371-402 (1986).

In a famous article where he successfully investigated the diophantine approximation properties of values of \(E\)-functions, C. L. Siegel also defined \(G\)-functions and announced some results which could be obtained by his techniques. It is only fifty years later that E. Bombieri gave the detailed proofs as well as a general statement on the arithmetical nature of values of \(G\)-functions. A similar statement is proved here. The original idea of the paper is to apply the Gel’fond’s method which avoids the complications of the Siegel’s one.
Algebraic functions are typical G-functions. The study of that case leads to some improvements of the results of P. Bundschuh, T. Schneider and V. G. Sprindzhuk on Hilbert’s irreducibility theorem. Following Bombieri, as it is explained here, those last results have actually algebraic origins, namely the quadraticity of the height on abelian varieties. Finally, new corollaries are given, and in particular a new version of Hilbert’s irreducibility theorem which asserts that every Hilbert subset of a number field contains a geometric progression \((ab^ m)_{m\geq 1}\), for “a lot of” rational numbers \(b\).

Reviewer: Pierre Dèbes

MSC:

12E25	Hilbertian fields; Hilbert’s irreducibility theorem
11J91	Transcendence theory of other special functions
11G50	Heights
11R09	Polynomials (irreducibility, etc.)
11R58	Arithmetic theory of algebraic function fields
14G25	Global ground fields in algebraic geometry

Keywords:

diophantine approximation; G-functions; algebraic functions; Hilbert’s irreducibility theorem; height on abelian varieties