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

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.