Primality of polyomino ideals by quadratic Gr\"obner basis

TL;DR

This paper characterizes when polyomino ideals have a quadratic Gr"obner basis and proves that such ideals are prime, leading to the discovery of new infinite families of prime polyominoes.

Contribution

It provides a necessary and sufficient combinatorial condition for polyomino ideals to have a quadratic Gr"obner basis and establishes their primality, introducing new prime polyomino families.

Findings

Characterization of polyomino ideals with quadratic Gr"obner basis

Proof that such ideals are prime when the basis condition holds

Identification of two new infinite families of prime polyominoes

Abstract

In this work, we provide a necessary and sufficient condition on a polyomino ideal for having the set of inner 2-minors as degree reverse lexicographic Gr\"obner basis, due to combinatorial properties of the polyomino itself. Moreover, we prove that when the latter holds the ideal coincides with the lattice ideal associated to the polyomino, that is the ideal is prime. As an application, we describe two new infinite families of prime polyominoes.