Automatic Sequences and Curves over Finite Fields

TL;DR

This paper establishes a new upper bound on the automaton complexity needed to generate algebraic power series over finite fields, linking automata theory with algebraic geometry through Cartier operators.

Contribution

It introduces a novel bound on automaton states for algebraic power series, improving previous estimates by connecting automata with algebraic geometry techniques.

Findings

Automaton with at most q^{h+d+g-1} states generates the sequence.

Connects automata theory with algebraic geometry via Cartier operators.

Provides a significant improvement over earlier bounds.

Abstract

We prove that if $y=\sum_{n=0}^\infty{\bf a}(n)x^n\in\mathbb{F}_q[[x]]$ is an algebraic power series of degree $d$, height $h$, and genus $g$, then the sequence ${\bf a}$ is generated by an automaton with at most $q^{h+d+g-1}$ states, up to a vanishingly small error term. This is a significant improvement on previously known bounds. Our approach follows an idea of David Speyer to connect automata theory with algebraic geometry by representing the transitions in an automaton as twisted Cartier operators on the differentials of a curve.