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Discriminants of polynomials in several variables. (English. Russian original) Zbl 0719.15003
Funct. Anal. Appl. 24, No. 1, 1-4 (1990); translation from Funkts. Anal. Prilozh. 24, No. 1, 1-4 (1990).
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.

15A15 Determinants, permanents, traces, other special matrix functions
15A54 Matrices over function rings in one or more variables
12E05 Polynomials in general fields (irreducibility, etc.)
14M20 Rational and unirational varieties
52A20 Convex sets in \(n\) dimensions (including convex hypersurfaces)
Full Text: DOI
[1] A. Cayley, Memoir on hyperdeterminants. Collected Papers,1, No. 13/14, 80-112 (1889). · doi:10.1017/CBO9780511703676.014
[2] I. M. Gel’fand, A. V. Zelevinskii, and M. M. Kapranov, ”A-discriminants and Cayley ? Koszul complexes,” Dokl. Akad. Nauk SSSR,307, No. 6, 1307-1310 (1989).
[3] I. M. Gel’fand, A. V. Zelevinskii, and M. M. Kapranov, ”Newton polyhedra of principal A-determinants,” Dokl. Akad. Nauk SSSR,308, No. 1, 20-23 (1989).
[4] I. M. Gel’fand, A. V. Zelevinskii, and M. M. Kapranov, ”Projectively dual manifolds and hyperdeterminants,” Dokl. Akad. Nauk SSSR,305, No. 6, 1294-1298 (1989).
[5] I. M. Gel’fand, A. V. Zelevinskii, and M. M. Kapranov, ”Hypergeometric functions and toroidal manifolds,” Funkts. Anal. Prilozhen.,23, No. 2, 12-26 (1989). · Zbl 0737.35116 · doi:10.1007/BF01078569
[6] T. V. Alekseevskaya, I. M. Gel’fand, and A. V. Zelevinskii, ”The location of real hypersurfaces and the decomposition function connected with it,” Dokl. Akad. Nauk SSSR,297, No. 6, 1289-1293 (1989).
[7] B. L. Van den Waerden, Algebra [Russian translation], Nauka, Moscow (1978).
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