A refinement of Christol’s theorem for algebraic power series

Abstract

A famous result of Christol gives that a power series $F(t)=\sum _{n\ge 0} f(n)t^n$ with coefficients in a finite field $\mathbb {F}_q$ of characteristic p is algebraic over the field of rational functions in t if and only if there is a finite-state automaton accepting the base-p digits of n as input and giving f(n) as output for every $n\ge 0$. An extension of Christol’s theorem, giving a complete description of the algebraic closure of $\mathbb {F}_q(t)$, was later given by Kedlaya. When one looks at the support of an algebraic power series, that is the set of n for which $f(n)\ne 0$, a well-known dichotomy for sets generated by finite-state automata shows that the support set is either sparse—with the number of $n\le x$ for which $f(n)\ne 0$ bounded by a polynomial in $\log (x)$—or it is reasonably large in the sense that the number of $n\le x$ with $f(n)\ne 0$ grows faster than $x^{\alpha }$ for some positive $\alpha$. The collection of algebraic power series with sparse supports forms a ring and we give a purely algebraic characterization of this ring in terms of Artin–Schreier extensions and we extend this to the context of Kedlaya’s work on generalized power series.