9 - Resultants and Discriminants

Summary

This chapter is devoted to the applications of our methods to generalized discriminants and resultants. Here the nature of the applications is twofold. First, the geometric method is useful for knowing in which cases the discriminant type variety has the expected dimension, i.e., it has codimension in the ambient space.

In such cases one can look at the twisted complexes F(ℒ)• and try to classify those that have only two nonzero terms. The point is that in such case the matrix giving a differential of F(ℒ)• gives a determinantal expression of the defining variety, i.e. the resultant or discriminant in question. We call such complexes F(ℒ)• the determinantal complexes.

In section 9.1 we give the basic definitions regarding resultants. In section 9.2 we apply our methods to classify (following [WZ]) the determinantal complexes for the resultants of sets of multihomogeneous polynomials.

In section 9.3 we give the basic definitions regarding discriminants. Finally in section 9.4 we consider the case of discriminants of multilinear forms — so-called hyperdeterminants. We give several examples of determinantal complexes, but we also show that for the multilinear forms of general format the determinantal complexes do not exist.

We conclude that section with the analysis of formats of three dimensional matrices for which the hyperdeterminantal variety has codimension bigger that one.