## Algebra & Number Theory

### Automatic sequences and curves over finite fields

Andrew Bridy

#### Abstract

We prove that if $y = ∑ n=0∞$a$(n)xn ∈ Fq[[x]]$ is an algebraic power series of degree $d$, height $h$, and genus $g$, then the sequence a is generated by an automaton with at most $qh+d+g−1$ 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.

#### Article information

Source
Algebra Number Theory, Volume 11, Number 3 (2017), 685-712.

Dates
Revised: 20 November 2016
Accepted: 19 December 2016
First available in Project Euclid: 19 October 2017

https://projecteuclid.org/euclid.ant/1508431777

Digital Object Identifier
doi:10.2140/ant.2017.11.685

Mathematical Reviews number (MathSciNet)
MR3649365

Zentralblatt MATH identifier
06722479

#### Citation

Bridy, Andrew. Automatic sequences and curves over finite fields. Algebra Number Theory 11 (2017), no. 3, 685--712. doi:10.2140/ant.2017.11.685. https://projecteuclid.org/euclid.ant/1508431777

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