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Article Dans Une Revue Collectanea Mathematica Année : 2013

On homaloidal polynomial functions of degree 3 and prehomogeneous vector spaces

Pierre-Emmanuel Chaput

Résumé

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.

Dates et versions

hal-01256636 , version 1 (15-01-2016)

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Pierre-Emmanuel Chaput. On homaloidal polynomial functions of degree 3 and prehomogeneous vector spaces. Collectanea Mathematica, 2013, 64 (1), pp.135-140. ⟨10.1007/s13348-011-0046-8⟩. ⟨hal-01256636⟩
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