2016/03/14 by Boris Adamczewski, B Adamczewski, Adamczewski, B +5 · 5 citations
Mathematics · #Advanced Combinatorial Mathematics #Advanced Mathematical Identities #Algebraic Geometry and Number Theory #Combinatorics (math.CO) #FOS: Mathematics #Number Theory (math.NT) #math.CO #math.NT
paper · pdf · doi:10.48550/arxiv.1603.04187
arxiv created 2016/03/14 · openalex publication_date 2016/03/14 · arxiv updated 2016/03/15 · openalex created_date 2025/10/10 · openalex updated_date 2026/07/28
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".