Iterated discriminants
In this paper the authors prove that the discriminant of the discriminant of a multivariate polynomial has the same irreducible factors of the product of seven polynomials. These seven polynomials are obtained as the GCD of the generators of an elimination ideal. This result is similar to the one by \textit{L. Busé} and \textit{B. Mourrain} [Math. Comput. 78, No. 265, 345--386 (2009; Zbl 1200.13042)] for bivariate polynomials. In this paper the authors study the factors of the discriminant of the discriminant of a multivariate polynomial using an elimination ideal instead of resultants; in this way they can avoid degree hypotheses, however they do not get complete information on the multiplicity of the factors. The main result is obtained studying the order at a point of a polynomial and its discriminant, the singularities of the hypersurface defined by a discriminant and the use of the notion of ``generic point'' of an irreducible variety in the sense of Weil. As an application, the authors directly compute the factors of an iterated discriminant in a problem of quantifier elimination.
singularities of discriminant hypersurface
scientific article; zbMATH DE number 5573847