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## Section: New Results

### Computing minimal Gorenstein covers

Participant : Bernard Mourrain.

In [7], we analyze and present an effective solution to the minimal Gorenstein cover problem: given a local Artin $k$–algebra $A=k\left[\left[{x}_{1},...,{x}_{n}\right]\right]/I$, compute an Artin Gorenstein $k$–algebra $G=k\left[\left[{x}_{1},...,{x}_{n}\right]\right]/I$ such that $\ell \left(G\right)-\ell \left(A\right)$ 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.

This is a joint work with Juan Elias and Roser Homs (Dep. de Matematiques i Informatica, Universitat de Barcelona).