Criteria of algebraic independence of complex numbers over a field of finite transcendence type
type: Item
instance of: scholarly article
title: Criteria of algebraic independence of complex numbers over a field of finite transcendence type
zbMATH Open document ID: 0673.10030
author: Yaochen Zhu
published in: Chinese Science Bulletin
publication date: 1989-01-01T00:00:00Z
review text: The author extends his criterion for algebraic independence [C. R. Math. Acad. Sci., Soc. R. Can. 6, 297-302 (1984; Zbl 0558.10031) and Sci. Sin., Ser. A 31, No.8, 897-907 (1988; Zbl 0662.10025)] by replacing the base field \({\mathbb{Q}}\) by a field of finite transcendence type (according to Lang's definition). The main result is a generalization in several variables of a theorem of \textit{W. D. Brownawell} [Trans. Am. Math. Soc. 210, 1-26 (1975; Zbl 0312.10022)]. The method of proof goes back to \textit{G. V. Chudnovsky}'s work [Contributions to the theory of transcendental numbers (1985; Zbl 0594.10024)]. Reviewer's remark. A more elaborate and more powerful method has been introduced by Nesterenko and developed by \textit{P. Philippon} [Publ. Math. Inst. Hautes Etud. Sci. 64, 5-52 (1987; Zbl 0615.10044)]. However, in comparison with Philippon's result, an interesting feature of the present paper is that the diophantine hypothesis here is somewhat weaker \((\theta_ 1\) is fixed), which is useful in some applications (e.g. algebraic independence of values of one elliptic function).
reviewed by: Michel Waldschmidt
Mathematics Subject Classification ID: 11J85
Mathematics Subject Classification ID: 11J81
zbMATH DE Number: 4101310
zbMATH Keywords: algebraic independence
zbMATH Keywords: field of finite transcendence type
zbMATH Keywords: several variables
MaRDI profile type: Publication
description: scientific article; zbMATH DE number 4101310