An Algebraic Geometric Approach to Multidimensional Words

Kari, Jarkko; Szabados, Michal

doi:10.1007/978-3-319-23021-4_3

Cited by 5 publications

(4 citation statements)

References 9 publications

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“…However, Ord(c) = 2 case is special in the sense that c is then a sum of periodic configurations, that is, finitary power series. In general, any configuration with a non-trivial annihilator is a sum of periodic power series [9], but already when Ord(c) = 3 these power series may be necessarily non-finitary [8]. It seems then that proving Nivat's conjecture for configurations of order three would reflect the general case better than the order two case.…”

Section: Discussionmentioning

confidence: 99%

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Low-Complexity Tilings of the Plane

Kari

2019

Descriptional Complexity of Formal Systems

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A two-dimensional configuration is a coloring of the infinite grid Z 2 with finitely many colors. For a finite subset D of Z 2 , the D-patterns of a configuration are the colored patterns of shape D that appear in the configuration. The number of distinct D-patterns of a configuration is a natural measure of its complexity. A configuration is considered having low complexity with respect to shape D if the number of distinct D-patterns is at most |D|, the size of the shape. This extended abstract is a short review of an algebraic method to study periodicity of such low complexity configurations.

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Section: Discussionmentioning

confidence: 99%

“…We first proposed this approach in [9,10] to study Nivat's conjecture. It led to a number of subsequent results [6,7,8,14]. In this presentation we review the main results without proofs -the given references can be consulted for more details.…”

Section: Introductionmentioning

confidence: 99%

Low-Complexity Tilings of the Plane

Kari

2019

Descriptional Complexity of Formal Systems

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“…Then c = c 3 − c 1 − c 2 is a finitary integral configuration (over alphabet {0, 1}), annihilated by the polynomial (X (1,0) − 1)(X (0,1) − 1)(X (1,−1) − 1), but it cannot be expressed as a sum of finitary periodic configurations as proved in [KS15a]. Figure 3 illustrates the setup for α being the golden ratio.…”

Section: By Induction Hypothesis We Havementioning

confidence: 99%

An Algebraic Geometric Approach to Nivat’s Conjecture

Kari

Szabados

2015

Automata, Languages, and Programming

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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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“…This elegant and apparently simple conjecture has been formulated by M. Nivat in 1997 during an invited talk at ICALP and has lead to various approaches and numerous results. See for instance [CK15, KM18,KS15] for further references and examples of the latests developments on this conjecture.…”

Section: Introductionmentioning

confidence: 99%

Balancedness and coboundaries in symbolic systems

Berthé

Cecchi

2019

Theoretical Computer Science

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This paper studies balancedness for infinite words and subshifts, both for letters and factors. Balancedness is a measure of disorder that amounts to strong convergence properties for frequencies. It measures the difference between the numbers of occurrences of a given word in factors of the same length. We focus on two families of words, namely dendric words and words generated by substitutions. The family of dendric words includes Sturmian and Arnoux-Rauzy words, as well as codings of regular interval exchanges. We prove that dendric words are balanced on letters if and only if they are balanced on words. In the substitutive case, we stress the role played by the existence of coboundaries taking rational values and show simple criteria when frequencies take rational values for exhibiting imbalancedness.

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An Algebraic Geometric Approach to Multidimensional Words

Cited by 5 publications

References 9 publications

Low-Complexity Tilings of the Plane

Low-Complexity Tilings of the Plane

An Algebraic Geometric Approach to Nivat’s Conjecture

Balancedness and coboundaries in symbolic systems

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