689 articles – 286 Notices  [english version]
 HAL : hal-00698689, version 1
 arXiv : 1205.4091
 Inventiones Mathematicae 187 (2012) 343--393
 On vanishing coefficients of algebraic power series over fields of positive characteristic
 (2012)
 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
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 hal-00698689, version 1 http://hal.archives-ouvertes.fr/hal-00698689 oai:hal.archives-ouvertes.fr:hal-00698689 Contributeur : Boris Adamczewski <> Soumis le : Jeudi 17 Mai 2012, 16:35:37 Dernière modification le : Vendredi 18 Mai 2012, 08:53:23