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

Abstract

Let $K$ be a finite extension of ${\mathbb{Q}}_p$, and let $f_1(z),\dots ,f_m(z)\in K[[z]]$ such that, for every $1\le i\le m$, $f_i(z)$ 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 $1\le i\le m$, the operator ${\mathcal{L}}_i$ has a strong Frobenius structure and satisfies the maximal order multiplicity (MOM) condition at zero. Then, we show that $f_1(z),\dots ,f_m(z)$ are algebraically dependent over $E_p$ if and only if there exist integers $a_1,\dots ,a_m$, 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.