Vol. 11, No. 3, 2017

Download this article
Download this article For screen
For printing
Recent Issues

Volume 14
Issue 8, 2001–2294
Issue 7, 1669–1999
Issue 6, 1331–1667
Issue 5, 1055–1329
Issue 4, 815–1054
Issue 3, 545–813
Issue 2, 275–544
Issue 1, 1–274

Volume 13, 10 issues

Volume 12, 10 issues

Volume 11, 10 issues

Volume 10, 10 issues

Volume 9, 10 issues

Volume 8, 10 issues

Volume 7, 10 issues

Volume 6, 8 issues

Volume 5, 8 issues

Volume 4, 8 issues

Volume 3, 8 issues

Volume 2, 8 issues

Volume 1, 4 issues

The Journal
About the Journal
Editorial Board
Subscriptions
Editors' Interests
Submission Guidelines
Submission Form
Editorial Login
Ethics Statement
ISSN: 1944-7833 (e-only)
ISSN: 1937-0652 (print)
Author Index
To Appear
 
Other MSP Journals
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.

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