Finite automata and algebraic extensions of function fields

Details Finite automata and algebraic extensions of function fields Finite automata and algebraic extensions of function fields

2004-10-17

arxiv.org/abs/math/0410375v2

Mathematics

Commutative Algebra

40 pages; expanded version of math.AC/0110089; v2: refereed version, includes minor edits

Scientific paper

We give an automata-theoretic description of the algebraic closure of the rational function field F_q(t) over a finite field, generalizing a result of Christol. The description takes place within the Hahn-Mal'cev-Neumann field of "generalized power series" over F_q. Our approach includes a characterization of well-ordered sets of rational numbers whose base p expansions are generated by a finite automaton, as well as 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.

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