On the Exceptional Sets of Transcendental Analytic Functions in Several Variables with Integer Coefficients

TL;DR

This paper extends previous results to show that in multiple variables, any subset of algebraic points within the unit polydisc, closed under conjugation and containing zero, can be realized as the exceptional set of uncountably many transcendental analytic functions with integer coefficients.

Contribution

It generalizes the characterization of exceptional sets from one variable to several variables for transcendental functions with integer coefficients.

Findings

Any subset of algebraic points in the polydisc closed under conjugation can be an exceptional set.

Uncountably many transcendental functions with integer coefficients realize these sets as their exceptional sets.

The result broadens the understanding of the structure of exceptional sets in multivariable complex analysis.

Abstract

In 2020, Marques and Moreira proved that every subset of $\overline{\mathbb{Q}} \cap B(0,1)$, which is closed under complex conjugation and contains $0$, is the exceptional set of uncountably many transcendental analytic functions with integer coefficients. In this paper, we extend this result to transcendental analytic functions in several variables. In particular, we show that every subset of $\overline{\mathbb{Q}}^m$ contained in the unit polydisc $\Delta(0;1)$, closed under complex conjugation and containing the zero vector, is the exceptional set of uncountably many transcendental analytic functions in several variables with integer coefficients.