An algebraic geometric approach to Nivat's conjecture

Kari, Jarkko; Szabados, Michal

doi:10.1016/j.ic.2019.104481

Cited by 42 publications

(42 citation statements)

References 11 publications

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“…Kari and Szabados introduced in [13] an algebraic approach to study low complexity configurations. The present paper heavily relies on this technique.…”

Section: Algebraic Conceptsmentioning

confidence: 99%

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Decidability and Periodicity of Low Complexity Tilings

Kari

Moutot

2021

Theory Comput Syst

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In this paper we study colorings (or tilings) of the two-dimensional grid ${\mathbb {Z}}^{2}$ ℤ 2 . A coloring is said to be valid with respect to a set P of n × m rectangular patterns if all n × m sub-patterns of the coloring are in P. A coloring c is said to be of low complexity with respect to a rectangle if there exist $m,n\in \mathbb {N}$ m , n ∈ ℕ and a set P of n × m rectangular patterns such that c is valid with respect to P and |P|≤ nm. Open since it was stated in 1997, Nivat’s conjecture states that such a coloring is necessarily periodic. If Nivat’s conjecture is true, all valid colorings with respect to P such that |P|≤ mn must be periodic. We prove that there exists at least one periodic coloring among the valid ones. We use this result to investigate the tiling problem, also known as the domino problem, which is well known to be undecidable in its full generality. However, we show that it is decidable in the low-complexity setting. Then, we use our result to show that Nivat’s conjecture holds for uniformly recurrent configurations. These results also extend to other convex shapes in place of the rectangle. After that, we prove that the nm bound is multiplicatively optimal for the decidability of the domino problem, as for all ε > 0 it is undecidable to determine if there exists a valid coloring for a given $m,n\in \mathbb {N}$ m , n ∈ ℕ and set of rectangular patterns P of size n × m such that |P|≤ (1 + ε)nm. We prove a slightly better bound in the case where m = n, as well as constructing aperiodic SFTs of pretty low complexity. This paper is an extended version of a paper published in STACS 2020 (Kari and Moutot 12).

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“…Kari and Szabados introduced in [13] an algebraic approach to study low complexity configurations. The present paper heavily relies on this technique.…”

Section: Algebraic Conceptsmentioning

confidence: 99%

“…It was observed in [13] that if a configuration has low complexity with respect to some shape D then it is annihilated by some non-zero polynomial f = 0. If X ⊆ A Z 2 is a set of configurations, we denote by Ann(X) the set of Laurent polynomials that annihilate all elements of X.…”

Section: Algebraic Conceptsmentioning

confidence: 99%

Decidability and Periodicity of Low Complexity Tilings

Kari

Moutot

2021

Theory Comput Syst

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“…It should be noted that various authors ([Tij95], [Sze98], [KS20], [Bha16], [GT20]) had discovered the above lemma in one form or another.…”

Section: Preparatory Lemmasmentioning

confidence: 99%

A Periodicity Result for Tilings of $\mathbb Z^3$ by Clusters of Prime-Squared Cardinality

Khetan¹

2021

Preprint

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“…Kari and Szabados introduced in [9] an algebraic approach to study low complexity configurations. The present paper heavily relies on this technique.…”

Section: Algebraic Conceptsmentioning

confidence: 99%

Decidability and Periodicity of Low Complexity Tilings

Kari¹,

Moutot²

2019

Preprint

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We investigate the tiling problem, also known as the domino problem, that asks whether the two-dimensional grid Z 2 can be colored in a way that avoids a given finite collection of forbidden local patterns. The problem is well-known to be undecidable in its full generality. We consider the low complexity setup where the number of allowed local patterns is small. More precisely, suppose we are given at most nm legal rectangular patterns of size n × m, and we want to know whether there exists a coloring of Z 2 containing only legal n × m patterns. We prove that if such a coloring exists then also a periodic coloring exists. This further implies, using standard arguments, that in this setup there is an algorithm to determine if the given patterns admit at least one coloring of the grid. The results also extend to other convex shapes in place of the rectangle.

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An algebraic geometric approach to Nivat's conjecture

Cited by 42 publications

References 11 publications

Decidability and Periodicity of Low Complexity Tilings

Decidability and Periodicity of Low Complexity Tilings

A Periodicity Result for Tilings of $\mathbb Z^3$ by Clusters of Prime-Squared Cardinality

Decidability and Periodicity of Low Complexity Tilings

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