Properties of the coordinate ring of a convex polyomino

TL;DR

This paper classifies convex polyominoes with Gorenstein coordinate rings and provides formulas for their regularity and multiplicity, advancing understanding of their algebraic properties.

Contribution

It offers a complete classification of convex polyominoes with Gorenstein coordinate rings and introduces recursive formulas for regularity and multiplicity of stack polyominoes.

Findings

Classification of convex polyominoes with Gorenstein coordinate rings

Formulas for Castelnuovo-Mumford regularity of stack polyominoes

Recursive computation method for the multiplicity of stack polyominoes

Abstract

We classify all convex polyomino whose coordinate rings are Gorenstein. We also compute the Castelnuovo-Mumford regularity of the coordinate ring of any stack polyomino in terms of the smallest interval which contains its vertices. We give a recursive formula for computing the multiplicity of a stack polyomino.