Iterated discriminants

It is shown that the discriminant of the discriminant of a multivariate polynomial has the same irreducible factors as the product of seven polynomials each of which is defined as the GCD of the generators of an elimination ideal. Under relatively mild conditions of genericity, three of these polynomials are irreducible and generate the corresponding elimination ideals, while the other four are equal to one. Moreover the irreducible factors of two of these polynomials have multiplicity at least two in the iterated discriminant and the irreducible factors of two others of the seven polynomials have multiplicity at least three.