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. (Joint work with Jason Bell).