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

TL;DR

This paper proves that the set of zero coefficients in algebraic power series over fields of positive characteristic is p-automatic, extending classical and recent theorems, with applications to Diophantine equations and the Mordell–Lang theorem.

Contribution

It generalizes Derksen's analogue of the Skolem-Mahler-Lech theorem and Christol's theorem by showing zero coefficient sets are p-automatic, with implications for Diophantine problems.

Findings

Zero coefficient sets are p-automatic in algebraic power series.

The result extends classical theorems to multivariate cases.

Applications include effective results for S-unit equations and Mordell–Lang over 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 $(n_1,...,n_d)\in \mathbb{N}^d$ 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.