Singular strata of cuspidal type for the classical discriminant

We consider an algebraic equation with variable complex coefficients. For the reduced discriminant set of such an equation we obtain parametrizations of the singular strata corresponding to the existence of roots of multiplicity at least $j$. These parametrizations are the restrictions of the Horn-Kapranov parametrization of the whole discriminant set to a chain of nested linear subspaces of the projective space. It is proved that such strata can be transformed into reduced $A$-discriminant sets by monomial transformations.
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