On Homaloidal Polynomials

Let P be the projective space over a field of characteristic zero. If F is a homogeneous polynomial we say that F is homaloidal if the polar map ∂F defined by the partial derivatives of F is a birational selfmap of P. Although the problem of determining homaloidal polynomials has a classical flavour the theme only recently was raised in an algebro-geometric context by Dolgachev ([Do]), following suggestions stemming from the theory of prehomogeneous varieties: the relative invariants of prehomogeneous spaces are in fact homaloidal polynomials ([KiSa],[EKP]). Dolgachev classifies homaloidal polynomials in P (see also [Di]) and characterizes homaloidal polynomials in P which are products of linear forms as products of four independent linear forms. Dolgachev also raises the question if it is true that a non square free product of linear forms is homaloidal if and only the product of its factors with multiplicity one is. This question has been given a positive answer in a specific case (see [KS]) and in full generality ([DiPa]) in a topological context. We will give an algebraic proof of the following: