  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.
