Criteria of algebraic independence of complex numbers over a field of finite transcendence type
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).
scientific article; zbMATH DE number 4101310