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

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 $\widehat{A}$, choose an $\widehat{A}$-regular sequence $\mathit{x}=\{x_1,\dots ,x_d\}\subset \widehat{\mathfrak{a}}$ and a set of representatives $E$ of $\widehat{A}\mathrm{/}(\mathit{x})$ that is an ${\mathbb{F}}_q$-vector subspace consisting of algebraic elements. Every element $y\in \widehat{A}$ 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 $\beta$-expansions for formal Laurent series over ${\mathbb{F}}_q$, answering questions posed by Hbaib–Mkaouar and Scheicher–Sirvent.