Quantitative results on algebraic independence for algebraic groups
type: Item
instance of: scholarly article
title: Quantitative results on algebraic independence for algebraic groups
zbMATH Open document ID: 0758.11032
DOI: 10.1016/0022-314X(92)90021-G
author: Mohammed Ably
published in: Journal of Number Theory
publication date: 1993-01-16T00:00:00Z
review text: The results of algebraic independence, due to Gel'fond, Chudnovsky and others, concerning the values \(\exp(x_iy_j)\) of the exponential function (for complex numbers \(x_1,\ldots x_n\) and \(y_1,\ldots,y_m\)) have been improved by \textit{G. Diaz} [ J. Number Theory 31, 1--23 (1989; Zbl 0661.10047)]. Here, the author extends these theorems of Diaz on large transcendence degree to one parameter subgroups of commutative algebraic groups. At the same time he provides quantitative versions (measures of algebraic independence). His main theorem contains a lot of corollaries; see for instance the author's note reviewed below [C. R. Acad. Sci., Paris, Sér. I 313, No. 10, 653--655 (1991; Zbl 0758.11033)]. One of his main tools is the use of Chow forms introduced by Nesterenko and developed by Philippon (see also \textit{E. M. Jabbouri} [Sur un critère pour l'indépendance algébrique de P. Philippon, Approximations diophantiennes et nombres transcendants, Luminy 1990, 195--202 (1992; Zbl 0777.11024)]).
reviewed by: Michel Waldschmidt
Mathematics Subject Classification ID: 11J85
Mathematics Subject Classification ID: 11J82
zbMATH DE Number: 93289
zbMATH Keywords: large transcendence degree
zbMATH Keywords: one-parameter subgroups of commutative algebraic groups
zbMATH Keywords: measures of algebraic independence
zbMATH Keywords: Chow forms
MaRDI profile type: Publication
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description: scientific article; zbMATH DE number 93289