Finite automata and algebraic extensions of function fields

Abstract

We give an automata-theoretic description of the algebraic closure of the rational function field 𝔽q(t) over a finite field 𝔽q, generalizing a result of Christol. The description occurs within the Hahn-Mal’cev-Neumann field of “generalized power series” over 𝔽q. In passing, we obtain a characterization of well-ordered sets of rational numbers whose base p expansions are generated by a finite automaton, and exhibit some techniques for computing in the algebraic closure; these include an adaptation to positive characteristic of Newton’s algorithm for finding local expansions of plane curves. We also conjecture a generalization of our results to several variables.