On the Algebraic Independence of $E$- and $G$-Functions, I: A $p$-adic Criterion

TL;DR

This paper establishes a p-adic criterion linking algebraic dependence of certain $E$- and $G$-functions to the existence of a multiplicative relation among them, under specific differential and ramification conditions.

Contribution

It provides a new p-adic criterion for algebraic dependence of $E$- and $G$-functions satisfying Frobenius and MOM conditions, extending previous algebraic independence results.

Findings

Algebraic dependence characterized by multiplicative relations.

Criterion applies to functions with strong Frobenius structures.

Results facilitate studying algebraic independence over $E_p$.

Abstract

Let $K$ be a finite extension of $\mathbb{Q}_p$, and let $f_1(z),\ldots, f_m(z) \in K[[z]]$ such that, for every $1 \leq i \leq m$, $f_i(z)$ is a solution of a differential operator $\mathcal{L}_i \in E_p[d/dz]$, where $E_p$ is the field of analytic elements. Suppose that $K$ is totally ramified over $\mathbb{Q}_p$, and that for every $1 \leq i \leq m$, the operator $\mathcal{L}_i$ has a strong Frobenius structure and satisfies the maximal order multiplicity (MOM) condition at zero. Then, we show that $f_1(z),\ldots, f_m(z)$ are algebraically dependent over $E_p$ if and only if there exist integers $a_1,\ldots, a_m$, not all zero, such that $f_1(z)^{a_1} \cdots f_m(z)^{a_m}\in E_p$. The main consequence of this result is that it provides a tool to study the algebraic independence of a broad class of $G$-functions and certain $E$-functions over the field of analytic elements.