Maximal Unipotent Monodromy, congruences "à la Lucas" and Algebraic independence (2103.15192v2)

Abstract: Let $f(z)$ be in $1+z\mathbb{Q}[[z]]$ and $\mathcal{S}$ be an infinite set of prime numbers such that, for all $p\in\mathcal{S}$, we can reduce $f(z)$ modulo $p$. We let $f(z){\mid p}$ denote the reduction of $f(z)$ modulo $p$. Generally, when $f(z)$ is D-finite, $f(z){\mid p}$ is algebraic over $\mathbb{F}p(z)$. It turns out that if $f(z)$ is a solution of a polynomial of the form $X-A_p(z)X^{p^l}$, we can use this type of equations to obtain results of transcendence and algebraic independence over $\mathbb{Q}(z)$. In the present paper, we look for conditions on the differential operators annihilating $f(z)$ to guarantee the existence of these particular equations. Suppose that $f(z)$ is solution of a differential operator $\mathcal{H}\in\mathbb{Q}(z)[d/dz]$ having a strong Frobenius structure for all $p\in\mathcal{S}$ and we also suppose that $f(z)$ annihilates a Fuchsian differential operator $\mathcal{D}\in\mathbb{Q}(z)[d/dz]$ such that zero is a regular singular point of $\mathcal{D}$ and the exponents of $\mathcal{D}$ at zero are equal to zero. Our main result states that, for almost every prime $p\in\mathcal{S}$, $f(z){\mid p}$ is solution of a polynomial of the form $X-A_p(z)X^{p^l}$, where $A_p(z)$ is a rational function with coefficients in $\mathbb{F}_p$ of height less than or equal to $Cp^{2l}$ with $C$ a positive constant that does not depend on $p$. We also study the algebraic independence of these power series over $\mathbb{Q}(z)$.