On the Algebraic Independence of $E$- and $G$-Functions, II: An Effective Version

Abstract

Let $K$ be a finite extension of $\mathbb{Q}_p$ that is totally ramified over $\mathbb{Q}_p$. The set $\mathcal{M}\mathcal{F}(K)$ consists of power series in $1+zK[[z]]$ that are solutions of differential operators in $K(z)[d/dz]$ equipped with strong Frobenius structure and satisfying maximal order multiplicty (MOM) condition at zero. It turns out that this set contains an interesting class of $E$- and $G$-functions. In this work, we provide a criterion for determining the algebraic independence, over the field of analytic elements, of elements belonging to $\mathcal{M}\mathcal{F}(K)$. As an illustration of this criterion, we show the algebraic independence of some $E$- and $G$-functions over the field of analytic elements.