# Computing minimal Gorenstein covers

## TL;DR

This paper introduces an effective algorithm for computing minimal Gorenstein covers of local Artin algebras using Macaulay's inverse systems, with practical experimentation demonstrating its efficiency.

## Contribution

It presents a novel algorithm and characterizations for computing minimal Gorenstein covers, improving upon existing methods for low Gorenstein colength.

## Key findings

- Algorithm successfully computes minimal Gorenstein covers in practice.
- New characterizations facilitate the computation process.
- Experimental results show the method's practical efficiency.

## Abstract

We analyze and present an effective solution to the minimal Gorenstein cover problem: given a local Artin k-algebra $A = k[[x 1 ,. .. x n ]]/I$, compute an Artin Gorenstein $k$-algebra $G = k[[x 1 ,. .. x n ]]/J$ such that $\ell(G)--\ell(A)$ is minimal. We approach the problem by using Macaulay's inverse systems and a modification of the integration method for inverse systems to compute Gorenstein covers. We propose new characterizations of the minimal Gorenstein cover and present a new algorithm for the effective computation of the variety of all minimal Gorenstein covers of A for low Gorenstein colength. Experimentation illustrates the practical behavior of the method.