# Algebraic independence of $G$-functions and congruences "à la Lucas".

3 CTN - Combinatoire, théorie des nombres
ICJ - Institut Camille Jordan [Villeurbanne]
Abstract : We develop a new method for proving algebraic independence of $G$-functions. Our approach rests on the following observation: $G$-functions do not always come with a single linear differential equation, but also sometimes with an infinite family of linear difference equations associated with the Frobenius that are obtained by reduction modulo prime ideals. When these linear difference equations have order one, the coefficients of the $G$-function satisfy congruences reminiscent of a classical theorem of Lucas on binomial coefficients. We use this to derive a Kolchin-like algebraic independence criterion. We show the relevance of this criterion by proving, using p-adic tools, that many classical families of $G$-functions turn out to satisfy congruences "à la Lucas".
Keywords :
Type de document :
Pré-publication, Document de travail
2016
Domaine :

Littérature citée [40 références]

https://hal.archives-ouvertes.fr/hal-01287140
Contributeur : Eric Delaygue <>
Soumis le : vendredi 11 mars 2016 - 18:33:04
Dernière modification le : jeudi 8 février 2018 - 11:10:35
Document(s) archivé(s) le : lundi 13 juin 2016 - 09:37:35

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ABD_09mars2016.pdf
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• HAL Id : hal-01287140, version 1
• ARXIV : 1603.04187

### Citation

B Adamczewski, Jason P. Bell, E Delaygue. Algebraic independence of $G$-functions and congruences "à la Lucas".. 2016. 〈hal-01287140〉

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