An Algebraic Geometric Approach to Nivat’s Conjecture

Conference paper
Part of the Lecture Notes in Computer Science book series (LNCS, volume 9135)

Abstract

We study multidimensional configurations (infinite words) and subshifts of low pattern complexity using tools of algebraic geometry. We express the configuration as a multivariate formal power series over integers and investigate the setup when there is a non-trivial annihilating polynomial: a non-zero polynomial whose formal product with the power series is zero. Such annihilator exists, for example, if the number of distinct patterns of some finite shape D in the configuration is at most the size |D| of the shape. This is our low pattern complexity assumption. We prove that the configuration must be a sum of periodic configurations over integers, possibly with unbounded values. As a specific application of the method we obtain an asymptotic version of the well-known Nivat’s conjecture: we prove that any two-dimensional, non-periodic configuration can satisfy the low pattern complexity assumption with respect to only finitely many distinct rectangular shapes D.

Preview

Unable to display preview. Download preview PDF.

Unable to display preview. Download preview PDF.

References

  1. [BN91]
    Beauquier, D., Nivat, M.: On translating one polyomino to tile the plane. In: Discrete & Computational Geometry 6 (1991)Google Scholar
  2. [CK13a]
    Cyr, V., Kra, B.: Complexity of short rectangles and periodicity. In: (submitted) (2013). arXiv: 1307.0098 [math.DS]
  3. [CK13b]
    Cyr, V., Kra, B.: Nonexpansive \(\mathbb{Z}\)2-subdynamics and Nivat’s conjecture. Trans. Amer. Math. Soc. (2013). http://dx.doi.org/10.1090/S0002-9947-2015-06391-0
  4. [EKM03]
    Epifanio, C., Koskas, M., Mignosi, F.: On a conjecture on bidimensional words. In: Theor. Comput. Sci. 1–3(299) (2003)Google Scholar
  5. [LW96]
    Lagarias, J.C., Wang, Y.: Tiling the Line with Translates of One Tile. Inventiones Mathematicae 124, 341–365 (1996)MATHMathSciNetCrossRefGoogle Scholar
  6. [LM95]
    Lind, D., Marcus, B.: An Introduction to Symbolic Dynamics and Coding. Cambridge University Press (1995)Google Scholar
  7. [MH38]
    Morse, M., Hedlund, G.A.: Symbolic Dynamics. American Journal of Mathematics 60(4), 815–866 (1938)MathSciNetCrossRefGoogle Scholar
  8. [Niv97]
    Nivat, M.: Invited talk at ICALP, Bologna (1997)Google Scholar
  9. [QZ04]
    Quas, A., Zamboni, L.Q.: Periodicity and local complexity. Theor. Comput. Sci. 319(1-3), 229–240 (2004)MATHMathSciNetCrossRefGoogle Scholar
  10. [ST00]
    Sander, J.W., Tijdeman, R.: The complexity of functions on lattices. Theor. Comput. Sci. 246(1-2), 195–225 (2000)MATHMathSciNetCrossRefGoogle Scholar
  11. [ST02]
    Sander, J.W., Tijdeman, R.: The rectangle complexity of func tions on two-dimensional lattices. Theor. Comput. Sci. 270(1-2), 857–863 (2002)MATHMathSciNetCrossRefGoogle Scholar
  12. [Sze98]
    Szegedy, M.: Algorithms to tile the infinite grid with finite clusters. In: FOCS, pp. 137–147. IEEE Computer Society (1998)Google Scholar

Copyright information

© Springer-Verlag Berlin Heidelberg 2015

Authors and Affiliations

  1. 1.Department of Mathematics and StatisticsUniversity of TurkuTurkuFinland

Personalised recommendations