Abstract
In this paper we consider homaloidal polynomial functions f such that their multiplicative Legendre transform f *, defined as in Etingof et al. (Sel. Math. (N.S.) 8(1):27–66, 2002), Section 3.2 is again polynomial. Following Dolgachev (Michigan Math. J. 48:191–202, 2000), 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 Etingof et al. (Sel. Math. (N.S.) 8(1):27–66, 2002), Theorem 3.10, p. 39. Our argument may suggest a way to attack the more general problem raised in Etingof et al. (Sel. Math. (N.S.) 8(1):27–66, 2002), Section 3.4 of EKP-homaloidal polynomials of arbitrary degree.
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