A refinement of Christol’s theorem for algebraic power series

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