Dépendance linéaire de fonctions arithmétiques et presque arithmétiques (Linear dependence of arithmetical and almost arithmetical functions)
type: Item
instance of: scholarly article
title: Dépendance linéaire de fonctions arithmétiques et presque arithmétiques (Linear dependence of arithmetical and almost arithmetical functions)
zbMATH Open document ID: 0577.10033
published in: Proceedings of the Japan Academy. Series A
publication date: 1985-01-01T00:00:00Z
review text: The author obtains by Schneider's method a very general transcendence theorem: Given an integer L, there exist two constants a and b with the following property. Let \(z_ n\) be a complex sequence and r(N) an upper bound for the \(| z_ n|\) (1\(\leq n\leq N)\). If \(f_ j(1\leq j\leq L)\) is an entire function such that \(f_ j(z_ n)\in {\mathbb{Z}}\) and log \(| f_ j(z)| \leq bN\) for \(| z| \leq a r(N)\), then the functions \(f_ j\) are \({\mathbb{Q}}\)-linearly dependent. This theorem contains for example Pólya's result (except for the value of the constant log 2): An entire function which maps \({\mathbb{N}}\) into \({\mathbb{Z}}\) and has a rate of growth smaller than exp(z log 2) is a polynomial. A similar proof gives a statement for functions such that \(f_ j(z_ n)\) is very near to an integer. This statement contains a particular case of a result by \textit{Ch. Pisot} [C. R. Acad. Sci., Paris 222, 1027-1028 (1946; Zbl 0060.215)]. Complete proofs can be found in ''Problèmes Diophantiens'' [Publ. Math. Univ. Paris VI, to appear].
Mathematics Subject Classification ID: 11J81
Mathematics Subject Classification ID: 30D99
zbMATH DE Number: 3922769
zbMATH Keywords: integral-valued entire functions
zbMATH Keywords: Schneider's method
zbMATH Keywords: general transcendence theorem
author: Noriko Hirata-Kohno
MaRDI profile type: Publication
full work available at URL: pjaa.61.274
OpenAlex ID: W1991765911
cites work: Q5583180
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DOI: 10.3792/PJAA.61.274
description: scientific article; zbMATH DE number 3922769