  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.
