A note on Christol's theorem

TL;DR

This paper presents an elementary approach to Christol's theorem, providing bounds on automaton size for algebraic power series over finite fields using diagonals of bivariate rational functions, simplifying previous algebraic geometry methods.

Contribution

It introduces a simpler, elementary proof for bounds in Christol's theorem, replacing complex algebraic geometry techniques with rational function diagonals.

Findings

Bounds on automaton size are comparable to previous geometric methods.

Elementary approach simplifies understanding of algebraic power series.

Provides a new perspective on automata characterization of algebraic series.

Abstract

Christol's theorem characterises algebraic power series over finite fields in terms of finite automata. In a recent article, Bridy develops a new proof of Christol's theorem by Speyer, to obtain a tight quantitative version, that is, to bound the size of the corresponding automaton in terms of the height and degree of the power series, as well as the genus of the curve associated with the minimal polynomial of the power series. Speyer's proof, and Bridy's development, both take place in the setting of algebraic geometry, in particular by considering K\"ahler differentials of the function field of the curve. In this note we show how an elementary approach, based on diagonals of bivariate rational functions, provides essentially the same bounds.