On the generalization of Christol's theorem to domains of finite type over finite fields

Abstract

We generalize Christol's theorem---which characterizes algebraic power series over a finite field in terms of automatic coefficient sequences---to completions of domains of finite type over a finite field~$\mathbb{F}_{q}$. Let $A$ be such a domain of dimension~$d$ and $\mathfrak{a}$ an ideal of depth~$d$. In the $\mathfrak{a}$-adic completion $\hat{A}$, choose an $\hat{A}$-regular sequence $\boldsymbol{x}=\{x_{1},\dots,x_{d}\}\subset\hat{\mathfrak{a}}$ and a set of representatives $E$ of $\hat{A}/(\boldsymbol{x})$ that is an $\mathbb{F}_{q}$-vector subspace consisting of algebraic elements. Every element $y\in\hat{A}$ then admits a unique expansion $y=\sum_{u\in\mathbb{N}^{d}}a_{u}x^{u}$ with coefficients $a_{u}\in E$. We prove that $y$ is algebraic over $A$ if and only if the coefficient sequence $\{a_{u}\}_{u\in\mathbb{N}^{d}}$ is $q$-automatic. The proof relies on a careful study of henselizations, which are shown to coincide with the algebraic closures inside the completions under certain conditions. As an application, we obtain a complete characterization of the automaticity of $β$-expansions for formal Laurent series over $\mathbb{F}_{q}$, answering questions posed by Hbaib--Mkaouar and Scheicher--Sirvent.