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2023•arXiv (Cornell University)Open access

On the Exceptional Set of Transcendental Entire Functions in Several Variables

Diego Alves, Jean Lelis, Diego Marques, Pavel Trojovský

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Abstract

In this paper, among other things, we prove that any subset of $\overline{\mathbb{Q}}^m$ (closed under complex conjugation and which contains the origin) is the exceptional set of uncountable many transcendental entire functions over $\mathbb{C}^m$ with rational coefficients. This result solves a several variables version of a question posed by Mahler for transcendental entire functions.

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What this paper is about

In this paper, among other things, we prove that any subset of $\overline{\mathbb{Q}}^m$ (closed under complex conjugation and which contains the origin) is the exceptional set of uncountable many transcendental entire functions over $\mathbb{C}^m$ with rational coefficients. This result solves a several variables version of a question posed by Mahler for transcendental entire functions.

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Available abstract

In this paper, among other things, we prove that any subset of $\overline{\mathbb{Q}}^m$ (closed under complex conjugation and which contains the origin) is the exceptional set of uncountable many transcendental entire functions over $\mathbb{C}^m$ with rational coefficients. This result solves a several variables version of a question posed by Mahler for transcendental entire functions.

Key concepts: Uncountable set, Transcendental number, Transcendental function, Entire function, Mathematics, Set (abstract data type), Transcendental equation, Pure mathematics

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