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 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. © 2011 Universitat de Barcelona.