Decision Problems for Linear Recurrence Sequences

  • Joël Ouaknine
  • James Worrell
Part of the Lecture Notes in Computer Science book series (LNCS, volume 7550)


Linear recurrence sequences permeate a vast number of areas of mathematics and computer science. In this paper, we survey the state of the art concerning certain fundamental decision problems for linear recurrence sequences, namely the Skolem Problem (does the sequence have a zero?), the Positivity Problem (is the sequence always positive?), and the Ultimate Positivity Problem (is the sequence ultimately always positive?).


Decision Problem Recurrence Relation Algebraic Number Galois Theory Fibonacci Sequence 
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  1. 1.
    Berstel, J., Mignotte, M.: Deux propriétés désirables des suites récurrentes linéaires. Bull. Soc. Math. 104 (1976)Google Scholar
  2. 2.
    Blondel, V.D., Portier, N.: The presence of a zero in an integer linear recurrent sequence is NP-hard to decide. Linear Algebra and Its Applications (2002)Google Scholar
  3. 3.
    Cohen, H.: A Course in Computational Algebraic Number Theory. Springer (1993)Google Scholar
  4. 4.
    Everest, G., van der Poorten, A., Shparlinski, I., Ward, T.: Recurrence Sequences. American Mathematical Society (2003)Google Scholar
  5. 5.
    Halava, V., Harju, T., Hirvensalo, M.: Positivity of second order linear recurrent sequences. Discrete Applied Mathematics 154(3) (2006)Google Scholar
  6. 6.
    Halava, V., Harju, T., Hirvensalo, M., Karhumäki, J.: Skolem’s problem — on the border between decidability and undecidability. Technical Report 683, Turku Centre for Computer Science (2005)Google Scholar
  7. 7.
    Laohakosol, V., Tangsupphathawat, P.: Positivity of third order linear recurrence sequences. Discrete Applied Mathematics 157(15) (2009)Google Scholar
  8. 8.
    Lech, C.: A note on recurring series. Ark. Mat. 2 (1953)Google Scholar
  9. 9.
    Lipton, R.J.: Mathematical embarrassments. Blog entry (December 2009),
  10. 10.
    Litow, B.: A decision method for the rational sequence problem. Electronic Colloquium on Computational Complexity (ECCC) 4(55) (1997)Google Scholar
  11. 11.
    Mahler, K.: Eine arithmetische Eigenschaft der Taylor Koeffizienten rationaler Funktionen. Proc. Akad. Wet. 38 (1935)Google Scholar
  12. 12.
    Mahler, K.: On the Taylor coefficients of rational functions. Proc. Cambridge Philos. Soc. 52 (1956)Google Scholar
  13. 13.
    Mignotte, M., Shorey, T.N., Tijdeman, R.: The distance between terms of an algebraic recurrence sequence. Journal für die Reine und Angewandte Mathematik 349 (1984)Google Scholar
  14. 14.
    Salomaa, A.: Growth functions of Lindenmayer systems: Some new approaches. In: Lindenmayer, A., Rozenberg, G. (eds.) Automata, Languages, Development. North-Holland (1976)Google Scholar
  15. 15.
    Skolem, T.: Ein Verfahren zur Behandlung gewisser exponentialer Gleichungen. In: Comptes Rendus du Congrès des Mathématiciens Scandinaves (1934)Google Scholar
  16. 16.
    Soittola, M.: On D0L synthesis problem. In: Lindenmayer, A., Rozenberg, G. (eds.) Automata, Languages, Development. North-Holland (1976)Google Scholar
  17. 17.
    Tao, T.: Open question: effective Skolem-Mahler-Lech theorem. Blog entry (May 2007),
  18. 18.
    Vereshchagin, N.K.: The problem of appearance of a zero in a linear recurrence sequence. Mat. Zametki 38(2) (1985) (in Russian)Google Scholar
  19. 19.
    Waldschmidt, M.: Diophantine approximation on linear algebraic groups. Springer (2000)Google Scholar

Copyright information

© Springer-Verlag Berlin Heidelberg 2012

Authors and Affiliations

  • Joël Ouaknine
    • 1
  • James Worrell
    • 1
  1. 1.Department of Computer ScienceOxford UniversityUK

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