Remarks on a paper by U. Zannier
type: Item
instance of: scholarly article
title: Remarks on a paper by U. Zannier
zbMATH Open document ID: 0969.39017
author: Takeshi Toshimitsu
published in: Aequationes Mathematicae
publication date: 2001-10-02T00:00:00Z
review text: The author considers some problems related to the results obtained by \textit{U. Zannier} [Aequationes Math. 55, No. 1-2, 15-43 (1998; Zbl 0894.39007)] for a Laurent series satisfying a functional equation of the form \[ f(x^m)= P\bigl(x,f(x) \bigr), \tag{1} \] where \(P(x,y)\in \mathbb{C}(x)[y]\). Zannier proved that if \(f(x)\) is not a rational function, then the set of positive integers \(m\) such that a relation (1) holds for some \(P=P_m\) consists precisely of the powers of a single integer. The author considers the case \[ f(x)= Q\bigl(x,f(x^m) \bigr). \tag{2} \] Assume that \(f(x)\) is a formal Laurent series with complex coefficients satisfying the functional equation (2) for an integer \(m\geq 2\) and \(Q(x,y)\in \mathbb{C}(x)[y]\). Let us denote by \(B_f\) the set of positive integers \(m\) such that the equation (2) holds true for some \(Q=Q_m\). Then the following result is proved. Theorem 1. Let \(f(x)\) be a Laurent series with complex coefficient, not representing a rational function, and let \(a,b\in B_f\). Then \(a,b\) are multiplicatively dependent. The paper contains also some generalization of \textit{K. Nishioka}'s theorem for Polya's functional equation [Arch. Math. 44, 330-335 (1985; Zbl 0568.12014)]. This result reads as follows. Theorem 2. Suppose that \(f(x)\in \mathbb{C} ([x])\) satisfies the functional equation \[ f(x^m)= \varphi\bigl[ x,f(x), \dots,f(x^{m-1}) \bigr], \tag{3} \] where \(\varphi(x,y_1, \dots,y_{m-1})\in \mathbb{C}(x,y_1, \dots, y_{m-1})\). If \(f(x)\) is algebraic over \(\mathbb{C}(x)\), then \(f(x)\in \mathbb{C}(x)\). Moreover, interesting examples of equations of type (3) are presented in the paper.
reviewed by: Stefan Czerwik
Mathematics Subject Classification ID: 39B32
Mathematics Subject Classification ID: 30B10
Mathematics Subject Classification ID: 30D05
zbMATH DE Number: 1538473
zbMATH Keywords: formal Laurent series
zbMATH Keywords: Polya's functional equation
MaRDI profile type: Publication
full work available at URL: s000100050144
OpenAlex ID: W1987626275
DOI: 10.1007/S000100050144
Recommended article: On a functional equation relating a Laurent series \(f(x)\) to \(f(x^m)\)
Recommended article: Sur les équation fonctionelles aux itérées. (On iterated functional equations)
Recommended article: Q4938207
Recommended article: Convergence of formal solutions of certain functional equations
Recommended article: Q4313413
Recommended article: Q4528783
Recommended article: Transcendental Solutions of a Class of Minimal Functional Equations
Recommended article: The Diophantine Equation f(x) = g(y)
Recommended article: Q4313419
Recommended article: Generalized sine equations. III
description: scientific article; zbMATH DE number 1538473