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On homaloidal polynomial functions of degree 3 and prehomogeneous vector spaces

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.