  In this paper we consider homaloidal polynomial functions $f$ such that their
multiplicative Legendre transform $f_*$, defined as in
\cite[Section3.2]{MR1890194}, is again polynomial. Following Dolgachev
\cite{MR1786486}, we call such polynomials EKP-homaloidal. We prove that every
EKP-homaloidal polynomial function of degree three is a relative invariant of a
symmetric prehomogeneous vector space. This provides a complete proof of
\cite[Theorem 3.10, p.~39]{MR1890194}. With respect to the original argument of
Etingof, Kazhdan and Polischuk our argument focuses more on prehomogeneous
vector spaces and, in principle, it may suggest a way to attack the more
general problem raised in \cite[Section 3.4]{MR1890194} of classification of
EKP-homaloidal polynomials of arbitrary degree.
