## 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.

### MSC:

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

Zbl 0789.65034
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### References:

 [1] D.N. Bernstein, “The number of roots of a system of equations,” Functional Anal. Appl.9 (1975), 183-185. · Zbl 0328.32001 [2] U. Betke, “Mixed volumes of polytopes,” Archiv der Mathematik58 (1992), 388-391. · Zbl 0766.52006 [3] L.J. Billera and B. Sturmfels, “Fiber polytopes,” Ann. Math.135 (1992), 527-549. · Zbl 0762.52003 [4] R.C. Buck, “Partitions of space,” Amer. Math. Monthly50 (1943), 541-544. · Zbl 0061.30609 [5] J. Canny and I. Emiris, “An efficient algorithm for the sparse mixed resultant,” Proc. AAECC, Puerto Rico, May 1993, Springer Lect. Notes in Comput. Science 263 (1993), pp. 89-104. · Zbl 0789.65034 [6] I.M. Gelfand, M.M. Kapranov, and A.V. Zelevinsky, “Discriminants of polynomials in several variables and triangulations of Newton polytopes,” Algebra i analiz (Leningrad Math. J.)2 (1990), 1-62. [7] I.M. Gelfand, M.M. Kapranov, and A.V. Zelevinsky, “Discriminants and Resultants,” Birkhäuser, Boston, 1994. · Zbl 0827.14036 [8] I.M. Gelfand, M.M. Kapranov, and A.V. Zelevinsky, “Generalized Euler integrals and A-hypergeometric functions,”Adv. Math.84 (1990), 255-271. · Zbl 0741.33011 [9] I.M. Gelfand, M.M. Kapranov, and A.V. Zelevinsky, “Newton polytopes of the classical resultant and discriminant,” Adv. Math.84 (1990), 237-254. · Zbl 0721.33002 [10] B. Huber and B. Sturmfels, “Homotopies preserving the Newton polytopes,” presented at the Workshop on Real Algebraic Geometry, MSI Cornell, August 1992. [11] M. Kapranov, B. Sturmfels, and A Zelevinsky, “Chow polytopes and general resultants,” Duke Math. J.67 (1992), 189-218. · Zbl 0780.14027 [12] Lee, C.; Gritzmann, P. (ed.); Sturmfels, B. (ed.), Regular triangulations of convex polytopes, No. 4, 443-456 (1991), Providence, RI · Zbl 0746.52015 [13] F.S. Macaulay, “Some formulae in elimination,” Proc. London Math. Soc.33 1, (1902), 3-27. · JFM 34.0195.01 [14] P. Pedersen and B. Sturmfels, “Product formulas for resultants and Chow forms,” Mathematische Zeitschrift, 214 (1993), 377-396. · Zbl 0792.13006 [15] B. Sturmfels, “Sparse elimination theory,” in Computational Algebraic Geometry and Commutative Algebra Proc. Cortona, D. Eisenbud and L. Robbiano, eds., Cambridge University Press, 1993, pp. 377-396.
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