Algebraic function solutions of a certain class of functional equations
type: Item
instance of: scholarly article
title: Algebraic function solutions of a certain class of functional equations
zbMATH Open document ID: 0568.12014
DOI: 10.1007/BF01235775
author: Keiji Nishioka
published in: Archiv der Mathematik
publication date: 1985-01-01T00:00:00Z
review text: The following is a special case of the result proved in the paper. Suppose \(f(z_ 1,...,z_ n)\) in \({\mathbb C}((z_ 1,...,z_ n))\) satisfies the functional equation \[ f(z_ 1^{p_ 1},...,z_ n^{p_ n})=F(z_ 1,...,z_ n,f(z_ 1,...,z_ n)), \] where \(F\) is a rational function. If \(f\) is algebraic over \({\mathbb C}(z_ 1,...,z_ n)\), then it is in fact in \({\mathbb C}(z_ 1,...,z_ n)\). Theorems of this type are of interest in connection with Mahler's method in transcendence theory. The theorem is proved by a nice application of some differential algebra. For background comment and other results of this type obtained by less elegant methods, see the articles by the reviewer and \textit{A. J. van der Poorten}, pp. 211--226 (Zbl 0378.10020) and \textit{K. K. Kubota}, pp. 227-- 229 (Zbl 0378.10021) in ''Transcendence theory: advances and applications'', Proc. Conf., Cambridge 1976 (1977; Zbl 0357.00010).
reviewed by: John H. Loxton
Mathematics Subject Classification ID: 11J81
Mathematics Subject Classification ID: 11J91
Mathematics Subject Classification ID: 12H05
Mathematics Subject Classification ID: 39B99
zbMATH DE Number: 3906608
zbMATH Keywords: algebraic function solutions
zbMATH Keywords: formal power series
zbMATH Keywords: functional equation
zbMATH Keywords: rational function
zbMATH Keywords: Mahler's method
zbMATH Keywords: transcendence theory
zbMATH Keywords: differential algebra
MaRDI profile type: Publication
cites work: Q5810622
cites work: Arithmetic properties of certain functions in several variables
cites work: A class of hypertranscendental functions
cites work: First order algebraic differential equations. A differential algebraic approach
cites work: Q3967605
description: scientific article; zbMATH DE number 3906608