On a new criterion for isomorphism of Artinian Gorenstein algebras

To every Gorenstein algebra $A$ of finite vector space dimension greater than 1 over a field $\FF$ of characteristic zero, and a linear projection $\pi$ on its maximal ideal ${\mathfrak m}$ with range equal to the annihilator $\Ann({\mathfrak m})$ of ${\mathfrak m}$, one can associate a certain algebraic hypersurface $S_{\pi}\subset{\mathfrak m}$, which is the graph of a polynomial map $P_{\pi}:\ker\pi\ra\Ann({\mathfrak m})\simeq\FF$. Recently, in {\rm\cite{FIKK}}, {\rm\cite{FK}} the following surprising criterion was obtained: two 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 CR-geometric argument. In the present paper we give a short algebraic proof of this statement. We also compare the polynomials $P_{\pi}$ with Macaulay's inverse systems. Namely, we show that the restrictions of $P_{\pi}$ to certain subspaces of $\ker\pi$ are inverse systems for $A$.