On vanishing coefficients of algebraic power series over fields of positive characteristic.

Let K be a field of characteristic p>0 and let f( t,..., t) be a power series in d variables with coefficients in K that is algebraic over the field of multivariate rational functions K( t,..., t). 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,..., n)∈ℕ for which the coefficient of $t_{1}^{n_{1}}\cdots t_{d}^{n_{d}}$ in f( t,..., t) 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. [ABSTRACT FROM AUTHOR]