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Advances in Geometry

Managing Editor: Grundhöfer, Theo / Joswig, Michael

Editorial Board: Bamberg, John / Bannai, Eiichi / Cavalieri, Renzo / Coskun, Izzet / Duzaar, Frank / Eberlein, Patrick / Gentili, Graziano / Henk, Martin / Kantor, William M. / Korchmaros, Gabor / Kreuzer, Alexander / Lagarias, Jeffrey C. / Leistner, Thomas / Löwen, Rainer / Ono, Kaoru / Ratcliffe, John G. / Scheiderer, Claus / Van Maldeghem, Hendrik / Weintraub, Steven H. / Weiss, Richard


IMPACT FACTOR 2018: 0.789

CiteScore 2018: 0.73

SCImago Journal Rank (SJR) 2018: 0.329
Source Normalized Impact per Paper (SNIP) 2018: 0.908

Mathematical Citation Quotient (MCQ) 2018: 0.53

Online
ISSN
1615-7168
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Volume 19, Issue 4

Issues

Criteria for strict monotonicity of the mixed volume of convex polytopes

Frédéric Bihan / Ivan Soprunov
Published Online: 2019-06-30 | DOI: https://doi.org/10.1515/advgeom-2018-0024

Abstract

Let P1, …, Pn and Q1, …, Qn be convex polytopes in ℝn with PiQi. It is well-known that the mixed volume is monotone: V(P1, …, Pn) ≤ V(Q1, …, Qn). We give two criteria for when this inequality is strict in terms of essential collections of faces as well as mixed polyhedral subdivisions. This geometric result allows us to characterize sparse polynomial systems with Newton polytopes P1, …, Pn whose number of isolated solutions equals the normalized volume of the convex hull of P1 ∪ … ∪ Pn. In addition, we obtain an analog of Cramer’s rule for sparse polynomial systems.

Keywords: Convex polytope; mixed volume; Newton polytope; sparse polynomial system; BKK bound

MSC 2010: Primary 52A39; 52B20; 14M25; Secondary 13P15

References

  • [1]

    D. N. Bernstein, The number of roots of a system of equations. Funct. Anal. Appl. 9 (1976), 183–185. MR0435072 Zbl 0328.32001Google Scholar

  • [2]

    B. Bertrand, F. Bihan, Intersection multiplicity numbers between tropical hypersurfaces. In: Algebraic and combinatorial aspects of tropical geometry, volume 589 ofContemp. Math., 1–19, Amer. Math. Soc. 2013. MR3088908 Zbl 1312.14139Google Scholar

  • [3]

    T. Chen, Unmixing the mixed volume computation. Preprint 2017, arXiv:1703.01684 [math.AG]Google Scholar

  • [4]

    D. A. Cox, J. Little, D. O’Shea, Using algebraic geometry. Springer 2005. MR2122859 Zbl 1079.13017Google Scholar

  • [5]

    A. Esterov, G. Gusev, Systems of equations with a single solution. J. Symbolic Comput. 68 (2015), 116–130. MR3283858 Zbl 1314.52011Web of ScienceCrossrefGoogle Scholar

  • [6]

    B. Huber, J. Rambau, F. Santos, The Cayley trick, lifting subdivisions and the Bohne-Dress theorem on zonotopal tilings. J. Eur. Math. Soc. 2 (2000), 179–198. MR1763304 Zbl 0988.52017CrossrefGoogle Scholar

  • [7]

    B. Huber, B. Sturmfels, A polyhedral method for solving sparse polynomial systems. Math. Comp. 64 (1995), 1541–1555. MR1297471 Zbl 0849.65030CrossrefGoogle Scholar

  • [8]

    A. G. Khovanskij, Newton polyhedra and the genus of complete intersections. Functional Anal. Appl. 12 (1978), 38–46. MR487230 Zbl 0406.14035CrossrefGoogle Scholar

  • [9]

    A. G. Kušnirenko, Newton polyhedra and Bezout’s theorem. (Russian) Funkcional. Anal. i Priložen. 10 (1976), 82–83. English translation: Functional Anal. Appl. 10 (1976), 233–235 (1977). MR0422272 Zbl 0328.32002Google Scholar

  • [10]

    J. M. Rojas, A convex geometric approach to counting the roots of a polynomial system. Theoret. Comput. Sci. 133 (1994), 105–140. MR1294429 Zbl 0812.65040CrossrefGoogle Scholar

  • [11]

    R. Schneider, Convex bodies: the Brunn–Minkowski theory, volume 151 of Encyclopedia of Mathematics and its Applications. Cambridge Univ. Press 2014. MR3155183 Zbl 1287.52001Google Scholar

  • [12]

    I. Soprunov, A. Zvavitch, Bezout inequality for mixed volumes.Int. Math. Res. Not. no. 23 (2016), 7230–7252. MR3632081Google Scholar

  • [13]

    B. Sturmfels, On the Newton polytope of the resultant. J. Algebraic Combin. 3 (1994), 207–236. MR1268576 Zbl 0798.05074CrossrefGoogle Scholar

About the article

Received: 2017-03-31

Revised: 2017-10-17

Revised: 2017-12-06

Published Online: 2019-06-30

Published in Print: 2019-10-25


Communicated by: M. Joswig


Citation Information: Advances in Geometry, Volume 19, Issue 4, Pages 527–540, ISSN (Online) 1615-7168, ISSN (Print) 1615-715X, DOI: https://doi.org/10.1515/advgeom-2018-0024.

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