A Criterion for Isomorphism of Artinian Gorenstein Algebras

Let $A$ be an Artinian Gorenstein algebra over an infinite field $k$ with either $\hbox{char}(k)=0$ or $\hbox{char}(k)>\nu$, where $\nu$ is the socle degree of $A$. To every such algebra and a linear projection $\pi$ on its maximal ideal ${\mathfrak m}$ with range equal to the socle $\hbox {Soc}(A)$ of $A$, one can associate a certain algebraic hypersurface $S_{\pi}\subset{\mathfrak m}$, which is the graph of a polynomial map $P_{\pi}:\hbox{ker}\,\pi\to \hbox{Soc}(A)\simeq k$. Recently, the author and his collaborators have obtained the following surprising criterion: two Artinian Gorenstein algebras $A$, $\tilde A$ are isomorphic if and only if any two hypersurfaces $S_{\pi}$ and $S_{\tilde\pi}$ arising from $A$ and $\tilde A$, 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 $P_{\pi}$ and Macaulay inverse systems.