  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.
