A criterion for isomorphism of artinian gorenstein algebras

Let A be an Artinian Gorenstein algebra over an infinite field k of characteristic either 0 or greater than the socle degree of A. To every such algebra and a linear projection π on its maximal ideal m with range equal to the socle Soc(A) of A, one can associate a certain algebraic hypersurface Sπ ⊂ m, which is the graph of a polynomial map Pπ : ker π → Soc(A) ≃ k. Recently, the following surprising criterion has been obtained: two Artinian Gorenstein algebras A, Ã are isomorphic if and only if any two hypersurfaces Sπ and Sπ arising from A and Ã, respectively, are affinely equivalent. The proof is indirect and relies on a geometric argument. In the present paper, we give a short algebraic proof of this statement. We also discuss a connection, established elsewhere, between the polynomials Pπ and Macaulay inverse systems.