  Let $K$ be a field of characteristic $p>0$ and let $f(t_1,...,t_d)$ be a
power series in $d$ variables with coefficients in $K$ that is algebraic over
the field of multivariate rational functions $K(t_1,...,t_d)$. We prove a
generalization of both Derksen's recent analogue of the Skolem-Mahler-Lech
theorem in positive characteristic and a classical theorem of Christol, by
showing that the set of indices $(n_1,...,n_d)\in \mathbb{N}^d$ for which the
coefficient of $t_1^{n_1}...t_d^{n_d}$ in $f(t_1,...,t_d)$ is zero is a
$p$-automatic set. Applying this result to multivariate rational functions
leads to interesting effective results concerning some Diophantine equations
related to $S$-unit equations and more generally to the Mordell--Lang Theorem
over fields of positive characteristic.
