标题： 关于$E$-和$G$-函数的代数独立性，II：一个有效版本 显示英文标题

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

摘要： 设$K$是$\mathbb{Q}_p$的有限扩张，并且在$\mathbb{Q}_p$上完全分歧。集合$\mathcal{M}\mathcal{F}(K)$包含$1+zK[[z]]$中的幂级数，这些幂级数是$K(z)[d/dz]$中微分算子的解，带有强弗罗贝尼乌斯结构，并且在零点满足最大阶乘条件（MOM）。 结果表明，这个集合包含了一类有趣的$E$和$G$函数。 在本文中，我们提供了一个判定条件，用于确定属于$\mathcal{M}\mathcal{F}(K)$的元素在解析元域上的代数无关性。 作为该判定条件的一个示例，我们展示了某些$E$和$G$函数在解析元域上的代数无关性。 显示英文摘要

摘要： 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.