Could anyone give me a proof of Christol's theorem which states that a formal Laurent series is algebraic over the field of rational functions (over some finite field) iff it is automatic ?
2026-04-01 18:26:50.1775068010
Christol's Theorem
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The proofs of Christol’s theorem given in [1, 2, 3] all suffer from a gap. An erratum has be given by Joost Winter in 2015, see this link.
[1] J.-P. Allouche and J. O. Shallit, Automatic Sequences – Theory, Applications, Generalizations. Cambridge University Press, 2003.
[2] V. Berthé, Automatic sequences and transcendence. In N. Pytheas Fogg, editor, Substitutions in Dynamics, Combinatorics and Arithmetics, volume 1794 of Lecture Notes in Mathematics, pages 53–80. Springer, 2002.
[3] J. Berstel and C. Reutenauer, Noncommutative Rational Series with Applications. Cambridge University Press, 2011.