##
**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.

Reviewer: N.I.Osetinski (Moskva)

### Keywords:

elimination theory; mixed subdivision; Newton polytopes; resultants; Laurent polynomials; sparse mixed resultant; extreme monomials; mixed polyhedral decomposition; mixed circuits; fiber polytope### Citations:

Zbl 0789.65034
Full Text:
DOI

### 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 <Emphasis Type=”Italic“>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. |

This reference list is based on information provided by the publisher or from digital mathematics libraries. Its items are heuristically matched to zbMATH identifiers and may contain data conversion errors. In some cases that data have been complemented/enhanced by data from zbMATH Open. This attempts to reflect the references listed in the original paper as accurately as possible without claiming completeness or a perfect matching.