On the exceptional set of transcendental functions with integer coefficients in a prescribed set: the problems A and C of Mahler
type: Item
instance of: scholarly article
title: On the exceptional set of transcendental functions with integer coefficients in a prescribed set: the problems A and C of Mahler
zbMATH Open document ID: 1452.11086
author: Diego Marques
author: C. Moreira
published in: Journal of Number Theory
publication date: 2020-10-07T00:00:00Z
review text: A \textit{transcendental function} is a function \( f(x) \) such that the only complex polynomial \( P \) satisfying \( P(x, f(x)) =0 \), for all \( x \) in its domain, is the zero polynomial. Trigonometric functions, the exponential function, and their inverses are some of the examples of transcendental functions. Denote by \( \bar{\mathbb{Q}} \) the field of algebraic numbers. For a function \( f \) analytic in the complex domain \( \mathcal{D} \), define the exceptional set \( S_f \) of \( f \) as \(\displaystyle{S_f=\left\{ \alpha \in \bar{\mathbb{Q}}\cap \mathcal{D}: f(\alpha) \in \bar{\mathbb{Q}} \right\}}\). For example, the exceptional sets of the functions \( 2^{z} \) and \( e^{z\pi+1} \) are \( \mathbb{Q} \) and \( \emptyset \), respectively, as shown by the Gelfond-Schneider theorem and Baker's theorem. In the paper under review, the authors consider Problem A and Problem C in the book of \textit{K. Mahler} [Lectures of transcendental numbers (1976; Zbl 0332.10019)], who suggested three problems, which he named Problem A, B and C, on the arithmetic behaviour of transcendental functions. Problems B and C have been completely solved by the authors in [Math. Ann. 368, No. 3--4, 1059--1062 (2017; Zbl 1387.11056); Bull. Aust. Math. Soc. 98, No. 1, 60--63 (2018; Zbl 1422.11162)], but Problem A remains open in general. Recall that, as usual, \( \mathbb{Z}{\{z\}} \) denotes the set of the power series analytic in the unit ball \( B(0,1) \) and with integer coefficients. Problems A and C are stated as follows. \begin{itemize} \item[A.] Does there exist a transcendental function \( f\in \mathbb{Z}{\{z\}} \) with bounded coefficients and such that \( f(\bar{\mathbb{Q}}\cap B(0,1)) \subseteq \bar{\mathbb{Q}} \)? \item[C.] Does there exist for every choice of \( S \) (closed under complex conjugation and such that \( 0\in S \)) a transcendental entire function with rational coefficients for which \( S_f=S \)? \end{itemize} In this paper, the authors generalize the mai…
reviewed by: Mahadi Ddamulira
Mathematics Subject Classification ID: 11J81
Mathematics Subject Classification ID: 11J91
Mathematics Subject Classification ID: 30D20
Mathematics Subject Classification ID: 11N05
Mathematics Subject Classification ID: 30B10
zbMATH DE Number: 7257229
zbMATH Keywords: Mahler problem
zbMATH Keywords: transcendental function
MaRDI profile type: Publication
full work available at URL: j.jnt.2020.07.007
OpenAlex ID: W3048833467
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DOI: 10.1016/J.JNT.2020.07.007
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description: scientific article; zbMATH DE number 7257229