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.
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References
Beauquier, D., Nivat, M.: On translating one polyomino to tile the plane. In: Discrete & Computational Geometry 6 (1991)
Cyr, V., Kra, B.: Complexity of short rectangles and periodicity. In: (submitted) (2013). arXiv: 1307.0098 [math.DS]
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
Epifanio, C., Koskas, M., Mignosi, F.: On a conjecture on bidimensional words. In: Theor. Comput. Sci. 1–3(299) (2003)
Lagarias, J.C., Wang, Y.: Tiling the Line with Translates of One Tile. Inventiones Mathematicae 124, 341–365 (1996)
Lind, D., Marcus, B.: An Introduction to Symbolic Dynamics and Coding. Cambridge University Press (1995)
Morse, M., Hedlund, G.A.: Symbolic Dynamics. American Journal of Mathematics 60(4), 815–866 (1938)
Nivat, M.: Invited talk at ICALP, Bologna (1997)
Quas, A., Zamboni, L.Q.: Periodicity and local complexity. Theor. Comput. Sci. 319(1-3), 229–240 (2004)
Sander, J.W., Tijdeman, R.: The complexity of functions on lattices. Theor. Comput. Sci. 246(1-2), 195–225 (2000)
Sander, J.W., Tijdeman, R.: The rectangle complexity of func tions on two-dimensional lattices. Theor. Comput. Sci. 270(1-2), 857–863 (2002)
Szegedy, M.: Algorithms to tile the infinite grid with finite clusters. In: FOCS, pp. 137–147. IEEE Computer Society (1998)
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Kari, J., Szabados, M. (2015). An Algebraic Geometric Approach to Nivat’s Conjecture. In: Halldórsson, M., Iwama, K., Kobayashi, N., Speckmann, B. (eds) Automata, Languages, and Programming. ICALP 2015. Lecture Notes in Computer Science(), vol 9135. Springer, Berlin, Heidelberg. https://doi.org/10.1007/978-3-662-47666-6_22
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DOI: https://doi.org/10.1007/978-3-662-47666-6_22
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