Algebraic independence of $G$-functions and congruences "\`a la Lucas"

TL;DR

This paper introduces a novel method for establishing the algebraic independence of $G$-functions by leveraging congruences similar to Lucas' theorem and associated difference equations, supported by p-adic techniques.

Contribution

The paper presents a new criterion for algebraic independence of $G$-functions based on Lucas-like congruences and difference equations, expanding the toolkit for such proofs.

Findings

Many classical $G$-functions satisfy Lucas-like congruences.

The algebraic independence criterion is effective for functions with order-one difference equations.

p-adic methods confirm the relevance of the criterion for various $G$-functions.

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 "\`a la Lucas".