On the Newton polytope of the resultant. (English) Zbl 0798.05074

The reviewed paper continues the researches of Gelfand, Kapranov, Zelevinsky and others on the study of Newton polytopes of resultants and discriminants. Let \({\mathcal A}_ 0,{\mathcal A}_ 1,\dots,{\mathcal A}_ n\subset{\mathbf Z}^ n\) be subsets which jointly span the affine lattice \({\mathbf Z}^ n\), and \(\text{card}({\mathcal A}_ i)= m_ i\). Then \(\mathcal R\) is the unique (up to scaling) irreducible polynomial in \(m= m_ 0+ m_ 1+\cdots+ m_ n\) variables \(c_{i,{\mathbf a}}\), which vanishes whenever the Laurent polynomials \[ f_ i(x_ 1,\dots,x_ n)=\sum_{{\mathbf {a}}\in{\mathcal A}_ i}c_{i,{\mathbf{a}}} {\mathbf{x}}^{\mathbf a}\quad (i=0,1,\dots,n) \] have a common zero in \(({\mathbf C}^*)^ n\). The Newton polytope \({\mathcal N}({\mathcal R})\) is the convex hull in \({\mathbf R}^ m\) of the exponent vectors of all monomials appearing with nonzero coefficient in \(\mathcal R\). The main result is a combinatorial construction of the Newton polytope \({\mathcal N}({\mathcal R})\) of the sparse mixed resultant \(\mathcal R\). Moreover, the paper is organized as follows. In Section 1 the author collects some basics, including the precise definition of the sparse mixed resultant, and a dimension formula for the variety of solvable systems. Section 2 deals with the monomials corresponding to vertices of \({\mathcal N}({\mathcal R})\), which are called the extreme monomials. The author presents a combinatorial construction for the extreme monomials of \(\mathcal R\) using mixed polyhedral decomposition of \(Q\). In Section 3 he generalizes the Canny-Emiris formula by showing that for each extreme monomial \(m\) of \(\mathcal R\) there exists a determinant as in the paper by J. Canny and I. Emiris [An efficient algorithm for the sparse mixed resultant, Lect. Notes Comput. Sci. 263, 89-104 (1993; Zbl 0789.65034)], for which \(m\) appears as a factor of the main diagonal product. In Section 4 the author proves that all faces of resultant polytopes are Minkowski sums of resultant polytopes, expresses each initial form \(\text{init}_ \omega({\mathcal R})\) of the sparse mixed resultant as a product of resultants corresponding to subsets of the \({\mathcal A}_ i\) and for each extreme monomial \(\mathcal R\) determines the exact coefficient, which is either \(-1\) or \(+1\). In Section 5 he examines the relationship between the sparse mixed resultants and the \(\mathcal A\)- discriminants. In Section 6 combinatorial properties of resultant polytopes are explored. The author characterizes the edges of \({\mathcal N}({\mathcal R})\) in terms of mixed circuits, and uses this to show that the resultant polytope has the same dimension as the fiber polytope from the Section 5, namely \(\dim({\mathcal N}({\mathcal R}))= m- 2n-1\). Moreover, he characterizes all resultant polytopes of dimensions 2 and 3.


05E99 Algebraic combinatorics
14M25 Toric varieties, Newton polyhedra, Okounkov bodies


Zbl 0789.65034
Full Text: DOI


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