Polarization Identities

In this paper we formulate and prove an equality expressing the discriminant of a sum of $$n$$ degree $$d$$ polynomials in one variable, where $$n \geqslant 2d - 1$$ , in terms of subsums of those polynomials. To that end we study the properties of polarizations of arbitrary degree of maps $$f:M \to N$$ of abelian groups, defined as certain generalizations of the classical polar form of a homogeneous polynomial in several variables. We show that the vanishing of $$(n + 1)$$ -st polarization of a map $$f$$ is equivalent to polylinearity of its $$n$$ th polarization. In case $$M = {R^s}$$ , $$N = R$$ and $$f$$ is given by a polynomial of degree $$n$$ we prove that its $$(n + 1)$$ -st polarization vanishes and simultaneously obtain a formula for its $$n$$ th polarization that recovers the classical polarization identity.