This paper is a continuation of a series of papers by the authors. In this one they give without proofs a geometrical investigation of the set of monomials occurring in the discriminant \(\Delta(f)\) of a homogeneous multivariate polynomial f of degree d. The monomials correspond to points in \(R^ N\) where \(N=\binom{n+d-1}{d}\). A polyhedron \(M\subset R^ N\) is defined to be the convex hull of all such monomials with nonzero coefficients.
A complete description is given of the vertices of M and the corresponding coefficients. It turns out that the vertices of M correspond to the triangularization of an (n-1)-dim simplex, the Newton polyhedron of a general polynomial of degree d. In particular, for \(n=2\), i.e., in the classic case of the discriminant of a polynomial in one variable, the polyhedron M is a “skewed” cube. Actually, the paper contains the solution of a more general problem which includes hyperdeterminants of multivariate matrices and resultants of systems of polynomials in several variables. The proofs of the results in this paper and in two of the referenced papers are to be given in a future publication.