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arXiv · 1605.05929

An Algebraic Geometric Approach to Nivat's Conjecture

Jarkko Kari·Michal Szabados
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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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Jarkko Kari, Michal Szabados. 2016-05-19. An Algebraic Geometric Approach to Nivat's Conjecture. https://arxiv.org/abs/1605.05929

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