A refinement of Christol's theorem for algebraic power series

Abstract

A famous result of Christol gives that a power series F (t) = n≥0 f (n)t n with coefficients in a finite field Fq 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 ≥ 0. An extension of Christol's theorem, giving a complete description of the algebraic closure of Fq(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) = 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 ≤ x for which f (n) = 0 bounded by a polynomial in log(x)-or it is reasonably large in the sense that the number of n ≤ x with f (n) = 0 grows faster than x α for some positive α. 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.

Automata and finite order elements in the Nottingham group

“…This contradiction shows the result. Proposition 11.1.2 (Albayrak-Bell [4], special case). Let σ ∈ N (F 2 ) denote a power series that is algebraic over F 2 (t).…”