Finite automata and algebraic extensions of function fields

Kiran S. Kedlaya[1]

  • [1] Department of Mathematics Massachusetts Institute of Technology Cambridge, MA 02139, USA

Journal de Théorie des Nombres de Bordeaux (2006)

  • Volume: 18, Issue: 2, page 379-420
  • ISSN: 1246-7405

Abstract

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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.

How to cite

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Kedlaya, Kiran S.. "Finite automata and algebraic extensions of function fields." Journal de Théorie des Nombres de Bordeaux 18.2 (2006): 379-420. <http://eudml.org/doc/249608>.

@article{Kedlaya2006,
abstract = {We give an automata-theoretic description of the algebraic closure of the rational function field $\mathbb\{F\}_q(t)$ over a finite field $\mathbb\{F\}_q$, generalizing a result of Christol. The description occurs within the Hahn-Mal’cev-Neumann field of “generalized power series” over $\mathbb\{F\}_q$. 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.},
affiliation = {Department of Mathematics Massachusetts Institute of Technology Cambridge, MA 02139, USA},
author = {Kedlaya, Kiran S.},
journal = {Journal de Théorie des Nombres de Bordeaux},
language = {eng},
number = {2},
pages = {379-420},
publisher = {Université Bordeaux 1},
title = {Finite automata and algebraic extensions of function fields},
url = {http://eudml.org/doc/249608},
volume = {18},
year = {2006},
}

TY - JOUR
AU - Kedlaya, Kiran S.
TI - Finite automata and algebraic extensions of function fields
JO - Journal de Théorie des Nombres de Bordeaux
PY - 2006
PB - Université Bordeaux 1
VL - 18
IS - 2
SP - 379
EP - 420
AB - We give an automata-theoretic description of the algebraic closure of the rational function field $\mathbb{F}_q(t)$ over a finite field $\mathbb{F}_q$, generalizing a result of Christol. The description occurs within the Hahn-Mal’cev-Neumann field of “generalized power series” over $\mathbb{F}_q$. 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.
LA - eng
UR - http://eudml.org/doc/249608
ER -

References

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  17. O. Salon, Suites automatiques à multi-indices (with an appendix by J. Shallit). Sem. Théorie Nombres Bordeaux 4 (1986–1987), 1–27. Zbl0653.10049
  18. J.-P. Serre, Local Fields (translated by M.J. Greenberg). Springer-Verlag, 1979. Zbl0423.12016MR554237
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