Transcendence properties of the Artin-Hasse exponential modulo $p$

Abstract

Let $E_p(x)$ denote the Artin-Hasse exponential and let ${\overline{E}}_p(x)$ denote its reduction modulo $p$ in ${\mathbb{F}}_p[[x]]$. In this article we study transcendence properties of ${\overline{E}}_p(x)$ over ${\mathbb{F}}_p[x]$. We give two proofs that ${\overline{E}}_p(x)$ is transcendental, affirmatively answering a question of Thakur. We also prove algebraic independence results: i) for $f_1,\dots ,f_r\in x{\mathbb{F}}_p[x]$ satisfying certain linear independence properties, we show that the ${\overline{E}}_p(f_1),\dots ,{\overline{E}}_p(f_r)$ are algebraically independent over ${\mathbb{F}}_p[x]$ and ii) we determine the algebraic relations between ${\overline{E}}_p(cx)$, where $c\in {\mathbb{F}}_p^{\times }$. Our proof studies the higher derivatives of ${\overline{E}}_p(x)$ and makes use of iterative differential Galois theory.