Algebraic Properties of the Coordinate Ring of a Convex Polyomino

Claudia Andrei

doi:10.37236/7728

Abstract

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

Highlights

A polyomino P is a finite connected set of adjacent cells in the cartesian plane N2
In the proposition we show that the bipartite graph GP associated with P is 2-connected
Since T X, there exists x ∈ X \ T with NY (x) NY (T ), by Lemma 17. It follows that we find y ∈ NY (x) \ NY (T ) ⊂ Y \ NY (T )

Summary

Introduction

A polyomino P is a finite connected set of adjacent cells in the cartesian plane N2. A cell in N2 is a unitary square. There it was shown that if P is a convex polyomino, K[P] is a normal Cohen-Macaulay domain The first main result of this paper appears, where we classify all convex polyominoes whose coordinate rings are Gorenstein (Theorem 21). For this classification, we use a result due to Ohsugi and Hibi ([8]) who classified all 2-connected bipartite graphs whose edge rings are Gorenstein.

Preliminaries

Gorenstein convex polyominoes

Gorenstein stack polyominoes