HAL : hal-00698689, version 1

arXiv : 1205.4091

DOI : 10.1007/s00222-011-0337-4

Fiche détaillée

Récupérer au format BibTeX - EndNote - TEI - RefWorks -

Inventiones Mathematicae 187 (2012) 343--393

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

Boris Adamczewski 1, Jason P. Bell 2

(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

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