Algebra & Number Theory

Automatic sequences and curves over finite fields

Andrew Bridy

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Abstract

We prove that if y = n=0a(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+g1 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
Received: 20 June 2016
Revised: 20 November 2016
Accepted: 19 December 2016
First available in Project Euclid: 19 October 2017

Permanent link to this document
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

Subjects
Primary: 11B85: Automata sequences
Secondary: 11G20: Curves over finite and local fields [See also 14H25] 14H05: Algebraic functions; function fields [See also 11R58] 14H25: Arithmetic ground fields [See also 11Dxx, 11G05, 14Gxx]

Keywords
automatic sequences formal power series algebraic curves finite fields

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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