Vol. 11, No. 3, 2017

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Automatic sequences and curves over finite fields

Andrew Bridy

Vol. 11 (2017), No. 3, 685–712
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.

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Keywords
automatic sequences, formal power series, algebraic curves, finite fields
Mathematical Subject Classification 2010
Primary: 11B85
Secondary: 11G20, 14H05, 14H25
Milestones
Received: 20 June 2016
Revised: 20 November 2016
Accepted: 19 December 2016
Published: 6 May 2017
Authors
Andrew Bridy
Department of Mathematics
Texas A&M University
Department of Mathematics
Mailstop 3368
College Station, TX 77843
United States