A refinement of Christol's theorem for algebraic power series

TL;DR

This paper refines Christol's theorem by characterizing algebraic power series over finite fields with sparse support sets using algebraic extensions, extending Kedlaya's work on generalized power series.

Contribution

It provides an algebraic characterization of algebraic power series with sparse support sets, connecting automata theory with Artin-Schreier extensions and extending Kedlaya's results.

Findings

Support sets of algebraic power series are either sparse or large.

Sparse support series form a ring characterized algebraically.

Extension of results to generalized 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 $f(n)\neq 0$, 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 $f(n)\neq 0$ bounded by a polynomial in $\log(x)$---or it is reasonably large in the sense that the number of $n\le x$ with $f(n)\neq 0$ 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.