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 $x^{\hat{I}}\pm$ for some positive $\hat{I}\pm$. 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.

21 pages; statement of main theorem updated slightly