Advertisement

Discrete & Computational Geometry

, Volume 44, Issue 1, pp 96–148 | Cite as

Newton Polyhedra of Discriminants of Projections

  • A. Esterov
Article

Abstract

For a system of polynomial equations, whose coefficients depend on parameters, the Newton polyhedron of its discriminant is computed in terms of the Newton polyhedra of the coefficients. This leads to an explicit formula (involving Euler obstructions of toric varieties) in the unmixed case, suggests certain open questions in general, and generalizes a number of similar known results (Gelfand et al. in Discriminants, resultants, and multidimensional determinants. Birkhäuser, Boston, 1994; Sturmfels in J. Algebraic Comb. 32(2):207–236, 1994; McDonald in Discrete Comput. Geom. 27:501–529, 2002; Gonzalez-Perez in Can. J. Math. 52(2):348-368, 2000; Esterov and Khovanskii in Funct. Anal. Math. 2(1), 2008).

Keywords

Discriminant Newton polyhedron Mixed fiber polyhedron Mixed volume Elimination theory Euler obstruction Dual defectiveness Cayley trick 

References

  1. 1.
    Bernstein, D.N.: The number of roots of a system of equations. Funct. Anal. Appl. 9(3), 183–185 (1975) MATHCrossRefGoogle Scholar
  2. 2.
    Billera, L.J., Sturmfels, B.: Fiber polytopes. Ann. Math. (2) 135(3), 527–549 (1992) CrossRefMathSciNetGoogle Scholar
  3. 3.
    Cattani, E., Dickenstein, A., Sturmfels, B.: Rational hypergeometric functions. Compos. Math. 128, 217–240 (2001) MATHCrossRefMathSciNetGoogle Scholar
  4. 4.
    Curran, R., Cattani, E.: Restriction of A-discriminants and dual defect toric varieties. J. Symb. Comput. 42, 115–135 (2007) MATHCrossRefMathSciNetGoogle Scholar
  5. 5.
    Di Rocco, S.: Projective duality of toric manifolds and defect polytopes. Proc. Lond. Math. Soc. (3) 93(1), 85–104 (2006) MATHCrossRefGoogle Scholar
  6. 6.
    Dickenstein, A., Feichtner, E.M., Sturmfels, B.: Tropical discriminants. J. Am. Math. Soc. 20, 1111–1133 (2007) MATHCrossRefMathSciNetGoogle Scholar
  7. 7.
    Ernström, L.: A Plücker formula for singular projective varieties. Commun. Algebra 25, 2897–2901 (1997) MATHCrossRefGoogle Scholar
  8. 8.
    Esterov, A.: Indices of 1-forms, resultants, and Newton polyhedra. Russ. Math. Surv. 60(2), 352–353 (2005) MATHCrossRefMathSciNetGoogle Scholar
  9. 9.
    Esterov, A.: Indices of 1-forms, intersection indices, and Newton polyhedra. Sb. Math. 197(7), 1085–1108 (2006) MATHCrossRefMathSciNetGoogle Scholar
  10. 10.
    Esterov, A.: Determinantal singularities and Newton polytopes. Proc. Steklov Inst. 259, 16–34 (2007) MATHCrossRefMathSciNetGoogle Scholar
  11. 11.
    Esterov, A.: On the existence of mixed fiber bodies. Mosc. Math. J. 8(3), 433–442 (2008) MATHMathSciNetGoogle Scholar
  12. 12.
    Esterov, A.: Determinantal singularities and Newton polyhedra. arXiv:0906.5097
  13. 13.
    Esterov, A., Khovanskii, A.G.: Elimination theory and Newton polytopes. Funct. Anal. Math. 2(1) (2008) Google Scholar
  14. 14.
    Gelfand, I.M., Kapranov, M.M., Zelevinsky, A.V.: Discriminants, Resultants, and Multidimensional Determinants. Birkhäuser, Boston (1994) MATHCrossRefGoogle Scholar
  15. 15.
    Gonzalez-Perez, P.D.: Singularites quasi-ordinaires toriques et polyedre de Newton du discriminant. Can. J. Math. 52(2), 348–368 (2000) MATHMathSciNetGoogle Scholar
  16. 16.
    Hall, M.: Combinatorial Theory, 2nd edn. Wiley, New York (1986) MATHGoogle Scholar
  17. 17.
    Karpenkov, O.: Completely empty pyramids on integer lattices and two-dimensional faces of multidimensional continued fractions. Monatsh. Math. 152(3), 217–249 (2007) MATHCrossRefMathSciNetGoogle Scholar
  18. 18.
    Kazarnovskii, B.: c-fans and Newton polyhedra of algebraic varieties. Izv. RAN. Ser. Mat. 67(3), 23–44 (2003) MathSciNetGoogle Scholar
  19. 19.
    Khovanskii, A.G.: Newton polyhedra and the genus of complete intersections. Funct. Anal. Appl. 12, 38–46 (1978) MathSciNetGoogle Scholar
  20. 20.
    Matsui, Y., Takeuchi, K.: A geometric degree formula for A-discriminants and Euler obstructions of toric varieties. arXiv:0807.3163
  21. 21.
    Matsui, Y., Takeuchi, K.: Milnor fibers over singular toric varieties and nearby cycle sheaves. arXiv:0809.3148
  22. 22.
    McDonald, J.: Fractional power series solutions for systems of equations. Discrete Comput. Geom. 27, 501–529 (2002) MATHMathSciNetGoogle Scholar
  23. 23.
    McMullen, P.: Mixed fibre polytopes. Discrete Comput. Geom. 32, 521–532 (2004) MATHCrossRefMathSciNetGoogle Scholar
  24. 24.
    Oka, M.: Principal zeta-function of non-degenerate complete intersection singularity. J. Fac. Sci. Univ. Tokyo 37, 11–32 (1990) MATHMathSciNetGoogle Scholar
  25. 25.
    Sturmfels, B.: On the Newton polytope of the resultant. J. Algebraic Comb. 3(2), 207–236 (1994) MATHCrossRefMathSciNetGoogle Scholar
  26. 26.
    Sturmfels, B., Tevelev, J.: Elimination theory for tropical varieties. Math. Res. Lett. 15, 543–562 (2008) MATHMathSciNetGoogle Scholar
  27. 27.
    Sturmfels, B., Yu, J.: Tropical implicitization and mixed fiber polytopes. In: Software for Algebraic Geometry. The IMA Volumes in Mathematics and its Applications, vol. 148, pp. 111–131. Springer, New York (2008) CrossRefGoogle Scholar
  28. 28.
    Tevelev, E.: Compactifications of subvarieties of tori. Am. J. Math. 129(4), 1087–1104 (2007) MATHCrossRefMathSciNetGoogle Scholar
  29. 29.
    Varchenko, A.N.: Zeta-function of monodromy and Newton’s diagram. Invent. Math. 37, 253–262 (1976) MATHCrossRefMathSciNetGoogle Scholar

Copyright information

© Springer Science+Business Media, LLC 2010

Authors and Affiliations

  1. 1.Laboratoire J.-A. DieudonneUniversite de Nice—Sophia Antipolis Parc ValroseNice Cedex 02France

Personalised recommendations