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hal-00698689, version 1

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

Boris Adamczewski (, ) 1, Jason P. Bell () 2

Inventiones Mathematicae 187 (2012) 343--393

Résumé : Let $K$ be a field of characteristic $p>0$ and let $f(t_1,\ldots ,t_d)$ be a power series in $d$ variables with coefficients in $K$ that is algebraic over the field of multivariate rational functions $K(t_1,\ldots ,t_d)$. 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 $(n_1,\ldots,n_d)\in \mathbb{N}^d$ for which the coefficient of $t_1^{n_1}\cdots t_d^{n_d}$ in $f(t_1,\ldots ,t_d)$ 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.

• 1 :  Institut Camille Jordan (ICJ)
• CNRS : UMR5208 – Université Claude Bernard - Lyon I – Ecole Centrale de Lyon – Institut National des Sciences Appliquées (INSA) - Lyon
• 2 :  Department of Mathematics
• Simon Fraser University
• Domaine : Mathématiques/Théorie des nombres

• hal-00698689, version 1
• oai:hal.archives-ouvertes.fr:hal-00698689
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• Soumis le : Jeudi 17 Mai 2012, 16:35:37
• Dernière modification le : Vendredi 18 Mai 2012, 08:53:23