Algebraic independence criteria.
type: Item
instance of: scholarly article
title: Algebraic independence criteria.
zbMATH Open document ID: 0615.10044
DOI: 10.1007/BF02699191
author: Patrice Philippon
published in: Publications Mathématiques
publication date: 1986-01-01T00:00:00Z
full work available at URL: http://www.numdam.org/item
full work available at URL: https://eudml.org/doc/104018
review text: A special case of the main theorem of this paper is the following. Let n be a positive integer. There exists \(c_ 0=c_ 0(n)>0\) with the following property. Let \(\eta >0\), and \(\theta =(\theta_ 1,...,\theta_ n)\in {\mathbb{C}}^ n\); assume that there exists a sequence \(I_ N=(P_{1,N},...,P_{m(N),N})\), \(N\geq N_ 0\) of ideals of \({\mathbb{Z}}[X_ 1,...,X_ n]\), whose set of zeros in the ball \(B(\theta, \exp (-3C_ 0N^{\eta}))\) of \({\mathbb{C}}^ n\) is finite, and such that, for each \(N\geq N_ 0\), \[ t(P_{i,N})\leq N\quad (1\leq i\leq m(N))\quad and\quad 0<\max \{| P_{i,N}(\theta)|; 1\leq i\leq m(N)\}\leq \exp (-C_ 0N^{\eta}). \] Then \(\eta <n+1.\) The case \(n=1\) is due to \textit{A. O. Gel'fond} [Transcendental and algebraic numbers (Moscou GITTL 1952; Zbl 0048.03303, and Dover-New York 1960; Zbl 0090.26103)]. In the case \(n\geq 1\), the conclusion \(\eta <2^ n\) was already known, thanks to Chudnovsky's semi-resultant [\textit{G. V. Chudnovsky}, Contributions to the theory of transcendental numbers (Math. Surv. Monogr.. 19) (1984; Zbl 0594.10024); \textit{E. Reyssat}, J. Reine Angew. Math. 329, 66-81 (1981; Zbl 0459.10023); the reviewer and \textit{Zhu Yaochen}, C. R. Acad. Sci., Paris, Sér. I 297, 229-232 (1983; Zbl 0531.10037)]. The conclusion \(\eta <n+1\) had already been derived by the author under the stronger hypothesis that the set of zeros of the homogeneous ideal associated with \(I_ N\) is of zero dimension in all of \({\mathbb{P}}_ n({\mathbb{C}})\) [the author, Pour une théorie de l'indépendance algébrique, Thèse d'Etat, Paris XI, 1983]. The new criterion has very interesting consequences. For instance the author quotes the following: if \(\alpha\) and \(\beta\) are algebraic numbers, with \(\alpha\neq 0\), log \(\alpha\neq 0\) and \(\beta\) of degree \(d\geq 2\) over \({\mathbb{Q}}\), then at least [d/2] of the numbers \(\alpha^{\beta},\alpha^{\beta^ 2},...,\alpha^{\beta^{d-1}}\) are algebraically independent. It should be noted that \text…
reviewed by: Michel Waldschmidt
Mathematics Subject Classification ID: 11J85
zbMATH DE Number: 3995793
zbMATH Keywords: algebraic independence criterion
zbMATH Keywords: transcendence
zbMATH Keywords: transcendence degree
zbMATH Keywords: diophantine approximation
zbMATH Keywords: exponential function
zbMATH Keywords: commutative algebraic groups
MaRDI profile type: Publication
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description: scientific article; zbMATH DE number 3995793