Finite automata and algebraic extensions of function fields
Journal de théorie des nombres de Bordeaux, Volume 18 (2006) no. 2, pp. 379-420.

We give an automata-theoretic description of the algebraic closure of the rational function field 𝔽 q (t) over a finite field 𝔽 q , generalizing a result of Christol. The description occurs within the Hahn-Mal’cev-Neumann field of “generalized power series” over 𝔽 q . 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.

On donne une description, dans le langage des automates finis, de la clôture algébrique du corps des fonctions rationnelles 𝔽 q (t) sur un corps fini 𝔽 q . Cette description, qui généralise un résultat de Christol, emploie le corps de Hahn-Mal’cev-Neumann des “séries formelles généralisées” sur 𝔽 q . En passant, on obtient une caractérisation des ensembles bien ordonnés de nombres rationnels dont les représentations p-adiques sont générées par un automate fini, et on présente des techniques pour calculer dans la clôture algébrique ; ces techniques incluent une version en caractéristique non nulle de l’algorithme de Newton-Puiseux pour déterminer les développements locaux des courbes planes. On conjecture une généralisation de nos résultats au cas de plusieurs variables.

DOI: 10.5802/jtnb.551
Kedlaya, Kiran S. 1

1 Department of Mathematics Massachusetts Institute of Technology Cambridge, MA 02139, USA
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Kedlaya, Kiran S. Finite automata and algebraic extensions of function fields. Journal de théorie des nombres de Bordeaux, Volume 18 (2006) no. 2, pp. 379-420. doi : 10.5802/jtnb.551. http://www.numdam.org/articles/10.5802/jtnb.551/

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