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

Abstract

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 )∈ℕd for which the coefficient of \(t_{1}^{n_{1}}\cdots t_{d}^{n_{d}}\) 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.