  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 $A$.
