Multivariate Generalization of Christol’s Theorem

Abstract: Christol's theorem (1979), which sets ground for many interactions between theoretical computer science and number theory, characterizes the coefficients of a formal power series over a finite field of positive characteristic $p>0$ that satisfy an algebraic equation to be the sequences that can be generated by finite automata, that is, a finite-state machine takes the base-$p$ expansion of $n$ for each coefficient and gives the coefficient itself as output. Namely, a formal power series $\sum_{n\ge 0} f(n) t^n$ over $\mathbb{F}_p$ is algebraic over $\mathbb{F}_p (t)$ if and only if $f(n)$ is a $p$-automatic sequence. However, this characterization does not give the full algebraic closure of $\mathbb{F}_p (t)$. Later it was shown by Kedlaya (2006) that a description of the complete algebraic closure of $\mathbb{F}_p (t)$ can be given in terms of $p$-quasi-automatic generalized (Laurent) series. In fact, the algebraic closure of $\mathbb{F}_p (t)$ is precisely generalized Laurent series that are $p$-quasi-automatic. We will characterize elements in the algebraic closure of function fields over a field of positive characteristic via finite automata in the multivariate setting, extending Kedlaya's results. In particular, our aim is to give a description of the full algebraic closure for multivariate fraction fields of positive characteristic.

algebraic geometrynumber theory

Audience: researchers in the discipline

SFU NT-AG seminar

Series comments: The Number Theory and Algebraic Geometry (NT-AG) seminar is a research seminar dedicated to topics related to number theory and algebraic geometry hosted by the NT-AG group (Nils Bruin, Imin Chen, Stephen Choi, Katrina Honigs, Nathan Ilten, Marni Mishna).

We acknowledge the support of PIMS, NSERC, and SFU.

For Fall 2025, the organizers are Katrina Honigs and Peter McDonald.

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