A characterization of some families of Cohen--Macaulay, Gorenstein and/or Buchsbaum rings

TL;DR

This paper develops algorithms to determine key algebraic properties like Cohen--Macaulayness, Gorenstein, and Buchsbaum for specific semigroup rings derived from dilated convex polyhedra in three-dimensional space.

Contribution

It introduces algorithmic methods to verify these properties in families of semigroup rings constructed from convex polyhedra dilations, expanding computational tools in algebraic geometry.

Findings

Identified families of semigroup rings with Cohen--Macaulay, Gorenstein, or Buchsbaum properties.

Provided explicit algorithms for property verification.

Demonstrated the applicability of methods to rings from dilated convex polyhedra.

Abstract

We provide algorithmic methods to check the Cohen--Macaulayness, Buchsbaumness and/or Gorensteiness of some families of semigroup rings that are constructed from the dilation of bounded convex polyhedrons of $\R^3_{\geq}$. Some families of semigroup rings are given satifying these properties.