On forced periodicity of perfect colorings

TL;DR

This paper investigates the forced periodicity of two-dimensional perfect colorings using algebraic methods, providing new proofs and conditions for periodicity on various grids and exploring configurations with low abelian complexity.

Contribution

It introduces an algebraic framework for analyzing perfect colorings, establishes new sufficient conditions for forced periodicity, and extends results to different grid types and low-complexity configurations.

Findings

Any perfect coloring can have a non-trivial annihilator polynomial.

Provides new proofs for forced periodicity on square and triangular grids.

Establishes a new result on forced periodicity in the king grid.

Abstract

We study forced periodicity of two-dimensional configurations under certain constraints and use an algebraic approach to multidimensional symbolic dynamics in which $d$-dimensional configurations and finite patterns are presented as formal power series and Laurent polynomials, respectively, in $d$ variables. We consider perfect colorings that are configurations such that the number of points of a given color in the neighborhood of any point depends only on the color of the point for some fixed relative neighborhood, and we show that by choosing the alphabet suitably any perfect coloring has a non-trivial annihilator, that is, there exists a Laurent polynomial whose formal product with the power series presenting the perfect coloring is zero. Using known results we obtain a sufficient condition for forced periodicity of two-dimensional perfect colorings. As corollaries of this result we get simple new proofs for known results of forced periodicity on the square and the triangular grids. Moreover, we obtain a new result concerning forced periodicity of perfect colorings in the king grid. We also consider perfect colorings of a particularly simple type: configurations that have low abelian complexity with respect to some shape, and we generalize a result that gives a sufficient condition for such configurations to be necessarily periodic. Also, some algorithmic aspects are considered.