Polar Cremona transformations

In this paper we address the following question arising from the work of P. Etingof, D. Kazhdan and A. Polishchuk (math.AG/0003009): given a homogeneous complex polynomial, when the rational map defined by its partials is of degree 1? We answer this question for reduced polynomials in three variables and also for reduced polynomials in four variables which factor into the product of linear polynomials.