On Gr\"obner bases and Cohen-Macaulay property of closed path polyominoes

TL;DR

This paper introduces specific monomial orders for closed path polyominoes and proves that their associated ideals have reduced Gr"obner bases, leading to Cohen-Macaulay properties of their coordinate rings.

Contribution

It establishes new monomial orders ensuring Gr"obner basis properties and links geometric configurations to algebraic properties like Cohen-Macaulayness.

Findings

Generators form reduced Gr"obner basis under new monomial orders

Closed path polyominoes without zig-zag walks have Cohen-Macaulay coordinate rings

Prime ideals correspond to certain geometric configurations

Abstract

In this paper we introduce some monomial orders for the class of closed path polyominoes and we prove that the set of the generators of the polyomino ideal attached to a closed path forms the reduced Gr\"obner basis with respect to these monomial orders. It is known that the polyomino ideal attached to a closed path containing an L-configuration or a ladder of at least three steps, equivalently having no zig-zag walks, is prime. As a consequence, we obtain that the coordinate ring of a closed path having no zig-zag walks is a normal Cohen-Macaulay domain.