The coordinate ring of a simple polyomino

TL;DR

This paper proves that simple polyominoes have balanced properties, leading to their coordinate rings being normal Cohen-Macaulay domains, thus connecting geometric simplicity with algebraic properties.

Contribution

It establishes the equivalence between simplicity and balancedness in polyominoes and characterizes their coordinate rings as normal Cohen-Macaulay domains.

Findings

Simple polyominoes are balanced if and only if they are simple.

Coordinate rings of simple polyominoes are normal Cohen-Macaulay domains.

Provides a characterization linking geometric and algebraic properties of polyominoes.

Abstract

In this paper it is shown that a polyomino is balanced if and only if it is simple. As a consequence one obtains that the coordinate ring of a simple polyomino is a normal Cohen-Macaulay domain.