An axiomatization of Nesterenko's method and applications on Mahler functions
type: Item
instance of: scholarly article
title: An axiomatization of Nesterenko's method and applications on Mahler functions
zbMATH Open document ID: 0813.11043
author: Thomas Töpfer
published in: Journal of Number Theory
publication date: 1995-06-07T00:00:00Z
review text: The author proves a new criterion for algebraic independence of complex numbers, using Nesterenko's method based on Cayley-Chow forms [\textit{Yu. V. Nesterenko}, Math. USSR, Izv. 11, 239-270 (1977); translation from Izv. Akad. Nauk SSSR, Ser. Mat. 41, 253-284 (1977; Zbl 0354.10026)]. The main difference with Philippon's earlier criterion [\textit{P. Philippon}, Publ. Math., Inst. Hautes Etud. Sci. 64, 5-52 (1986; Zbl 0615.10044)] is that here, only a finite sequence of polynomials is required [see also \textit{E. M. Jabbouri}, Approximations diophantiennes et nombres transcendants, C.-R. Colloq., Luminy/Fr. 1990, de Gruyter, 195- 202 (1992; Zbl 0777.11024)]. Several applications are provided. The first one is a generalization to several variables of Nishioka's results on algebraic independence of values of Mahler's functions [\textit{K. Nishioka}, J. Reine Angew. Math. 407, 202-219 (1990; Zbl 0694.10035)]. The second one yields algebraic independence for values at Liouville points of functions satisfying functional equations. The third one, also connected with Mahler's method, is an extension of a result of \textit{M. Amou} [Acta Arith. 59, 71-82 (1991; Zbl 0735.11031)]. Finally the author applies his criterion to the usual exponential function, but his result is weaker than the one of \textit{G. Diaz} [J. Number Theory 31, 1-23 (1989; Zbl 0661.10047)].
Mathematics Subject Classification ID: 11J85
Mathematics Subject Classification ID: 11J91
zbMATH DE Number: 653304
zbMATH Keywords: algebraic independence of complex numbers
zbMATH Keywords: Nesterenko's method
zbMATH Keywords: Cayley- Chow forms
zbMATH Keywords: values of Mahler's functions
zbMATH Keywords: values at Liouville points
zbMATH Keywords: functions satisfying functional equations
zbMATH Keywords: Mahler's method
zbMATH Keywords: exponential function
reviewed by: Michel Waldschmidt
MaRDI profile type: Publication
full work available at URL: jnth.1994.1077
OpenAlex ID: W2072909547
DOI: 10.1006/JNTH.1994.1077
description: scientific article; zbMATH DE number 653304