An Algebraic Geometric Approach to Nivat's Conjecture

TL;DR

This paper applies algebraic geometry to analyze low pattern complexity in multidimensional configurations, proving they are sums of periodic configurations and providing an asymptotic version of Nivat's conjecture.

Contribution

It introduces an algebraic geometric framework for studying Nivat's conjecture and proves that non-periodic configurations cannot have low pattern complexity for infinitely many shapes.

Findings

Configurations with low pattern complexity are sums of periodic configurations.

Non-periodic configurations satisfy low pattern complexity only for finitely many shapes.

An asymptotic version of Nivat's conjecture is established.

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 $D$.