Computing minimal Gorenstein covers

Abstract

We analyze and present an effective solution to the minimal Gorenstein cover problem: given a local Artin k-algebra $A=k[[x1,...xn]]/I$, compute an Artin Gorenstein $k$-algebra $G=k[[x1,...xn]]/J$ 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.