On the Exceptional Set of Transcendental Entire Functions in Several Variables

In this paper, among other things, we prove that any subset of ${\bar{\mathbb{Q}}}^m$ (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.