On Vanishing Coefficients of Algebraic Power Series over Fields of Positive Characteristic

— Let K be a field of characteristic p > 0 and let f(t1, . . . , td) be a power series in d variables with coefficients in K that is algebraic over the field of multivariate rational functions K(t1, . . . , td). 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 (n1, . . . , nd) ∈ N d for which the coefficient of t1 1 · · · t nd d in f(t1, . . . , td) 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.