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 dept