Algebra & Number Theory

Automatic sequences and curves over finite fields

Andrew Bridy

Full-text: Access denied (no subscription detected)

However, an active subscription may be available with MSP at

We're sorry, but we are unable to provide you with the full text of this article because we are not able to identify you as a subscriber. If you have a personal subscription to this journal, then please login. If you are already logged in, then you may need to update your profile to register your subscription. Read more about accessing full-text


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

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

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

Digital Object Identifier

Mathematical Reviews number (MathSciNet)

Zentralblatt MATH identifier

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]

automatic sequences formal power series algebraic curves finite fields


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.

Export citation


  • B. Adamczewski and J. P. Bell, “On vanishing coefficients of algebraic power series over fields of positive characteristic”, Invent. Math. 187:2 (2012), 343–393.
  • B. Adamczewski and J. P. Bell, “Diagonalization and rationalization of algebraic Laurent series”, Ann. Sci. Éc. Norm. Supér. $(4)$ 46:6 (2013), 963–1004.
  • J.-P. Allouche and J. Shallit, Automatic sequences, Cambridge University Press, 2003.
  • J. Berstel and C. Reutenauer, Rational series and their languages, EATCS Monographs on Theoretical Computer Science 12, Springer, 1988.
  • J. A. Brzozowski, “Canonical regular expressions and minimal state graphs for definite events”, pp. 529–561 in Proc. Sympos. Math. Theory of Automata (New York, 1962), Polytechnic Press, Brooklyn, NY, 1963.
  • P. Cartier, “Une nouvelle opération sur les formes différentielles”, C. R. Acad. Sci. Paris 244 (1957), 426–428.
  • G. Christol, “Ensembles presque periodiques $k$-reconnaissables”, Theoret. Comput. Sci. 9:1 (1979), 141–145.
  • G. Christol, T. Kamae, M. Mendès France, and G. Rauzy, “Suites algébriques, automates et substitutions”, Bull. Soc. Math. France 108:4 (1980), 401–419.
  • H. Derksen, “A Skolem–Mahler–Lech theorem in positive characteristic and finite automata”, Invent. Math. 168:1 (2007), 175–224.
  • S. Eilenberg, Automata, languages, and machines, vol. A, Pure and Applied Mathematics 58, Academic Press, New York, 1974.
  • G. Eisenstein, “Über eine allgemeine Eigenschaft der Reihen-Entwicklungen aller algebraischen Funktionen”, Preuss. Akad. d. Wissensch. Berlin (1852), 441–443. reprinted as pp. 765–767 in his Mathematische Werke, vol. II, Chelsea, New York, 1975.
  • J. Fresnel, M. Koskas, and B. de Mathan, “Automata and transcendence in positive characteristic”, J. Number Theory 80:1 (2000), 1–24.
  • T. Harase, “Algebraic elements in formal power series rings”, Israel J. Math. 63:3 (1988), 281–288.
  • T. Harase, “Algebraic elements in formal power series rings, II”, Israel J. Math. 67:1 (1989), 62–66.
  • R. Hartshorne, Algebraic geometry, Graduate Texts in Mathematics 52, Springer, 1977.
  • R. Lidl and H. Niederreiter, Introduction to finite fields and their applications, 1st ed., Cambridge University Press, 1994.
  • J.-P. Massias, J.-L. Nicolas, and G. Robin, “Effective bounds for the maximal order of an element in the symmetric group”, Math. Comp. 53:188 (1989), 665–678.
  • A. Salomaa and M. Soittola, Automata-theoretic aspects of formal power series, Springer, 1978.
  • W. M. Schmidt, “Eisenstein's theorem on power series expansions of algebraic functions”, Acta Arith. 56:2 (1990), 161–179.
  • J.-P. Serre, “Sur la topologie des variétés algébriques en caractéristique $p$”, pp. 24–53 in Symposium internacional de topologí a algebraica (Mexico City, 1956), Universidad Nacional Autónoma de México and UNESCO, Mexico City, 1958.
  • J. Shallit, A second course in formal languages and automata theory, Cambridge University Press, 2009.
  • J. H. Silverman, The arithmetic of elliptic curves, 2nd ed., Graduate Texts in Mathematics 106, Springer, 2009.
  • D. Speyer, “Christol's theorem and the Cartier operator”, 2010, hook \posturlhook.
  • H. Stichtenoth, Algebraic function fields and codes, 2nd ed., Graduate Texts in Mathematics 254, Springer, 2009.