Skip to Main content Skip to Navigation
Journal articles

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

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".
Document type :
Journal articles
Complete list of metadatas

Cited literature [20 references]  Display  Hide  Download
Contributor : Boris Adamczewski <>
Submitted on : Monday, November 9, 2020 - 5:32:58 PM
Last modification on : Tuesday, November 17, 2020 - 2:59:46 PM


Files produced by the author(s)



Boris Adamczewski, Jason Bell, Eric Delaygue. Algebraic independence of G-functions and congruences "à la Lucas". Annales scientifiques de l'Ecole normale supérieure, 2019, 52 (3), pp.515-559. ⟨10.24033/asens.2392⟩. ⟨hal-02091793⟩



Record views


Files downloads