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Mathematische Zeitschrift

, Volume 274, Issue 3–4, pp 761–778 | Cite as

Mixed discriminants

  • Eduardo Cattani
  • María Angélica Cueto
  • Alicia DickensteinEmail author
  • Sandra Di Rocco
  • Bernd Sturmfels
Article

Abstract

The mixed discriminant of \(n\) Laurent polynomials in \(n\) variables is the irreducible polynomial in the coefficients which vanishes whenever two of the roots coincide. The Cayley trick expresses the mixed discriminant as an \(A\)-discriminant. We show that the degree of the mixed discriminant is a piecewise linear function in the Plücker coordinates of a mixed Grassmannian. An explicit degree formula is given for the case of plane curves.

Keywords

A-discriminant Degree Multiple root Cayley polytope  Tropical discriminant Matroid strata Mixed Grassmannian 

Mathematics Subject Classification (2000)

13P15 14M25 14T05 52B20 

Notes

Acknowledgments

MAC was supported by an AXA Mittag-Leffler postdoctoral fellowship (Sweden) and an NSF postdoctoral fellowship DMS-1103857 (USA). AD was supported by UBACYT 20020100100242, CONICET PIP 112-200801-00483 and ANPCyT 2008-0902 (Argentina). SDR was partially supported by VR Grant NT:2010-5563 (Sweden). BS was supported by NSF Grants DMS-0757207 and DMS-0968882 (USA). This project started at the Institut Mittag-Leffler during the Spring 2011 program on “Algebraic Geometry with a View Towards Applications”. We thank IML for its wonderful hospitality.

References

  1. 1.
    Ardila, F., Klivans, C.J.: The Bergman complex of a matroid and phylogenetic trees. J. Combin. Theory Ser. B 96(1), 38–49 (2006)MathSciNetzbMATHCrossRefGoogle Scholar
  2. 2.
    Benoist, O.: Degrés d’homogénéité de l’ensemble des intersections complètes singulières. Annales de l’Institut Fourier 62(3), 1189–1214 (2012)Google Scholar
  3. 3.
    Bernstein, D.N.: The number of roots of a system of equations. Funkcional Anal. i Prilož. 9(3), 1–4 (1975)Google Scholar
  4. 4.
    Casagrande, C., Di Rocco, S.: Projective \(\mathbb{Q}\)-factorial toric varieties covered by lines. Commun. Contemp. Math. 10(3), 363–389 (2008)MathSciNetzbMATHCrossRefGoogle Scholar
  5. 5.
    Cattani, E., Dickenstein, A., Sturmfels, B.: Rational hypergeometric functions. Compositio Math. 128(2), 217–239 (2001)MathSciNetzbMATHCrossRefGoogle Scholar
  6. 6.
    Curran, R., Cattani, E.: Restriction of \(A\)-discriminants and dual defect toric varieties. J. Symb. Comput. 42(1–2), 115–135 (2007)MathSciNetzbMATHCrossRefGoogle Scholar
  7. 7.
    Dickenstein, A., Feichtner, E.M., Sturmfels, B.: Tropical discriminants. J. Am. Math. Soc. 20(4), 1111–1133 (2007)Google Scholar
  8. 8.
    Dickenstein, A., Sturmfels, B.: Elimination theory in codimension 2. J. Symb. Comput. 34(2), 119–135 (2002)MathSciNetzbMATHCrossRefGoogle Scholar
  9. 9.
    Di Rocco, S.: Projective duality of toric manifolds and defect polytopes. Proc. London Math. Soc. (3), 93(1), 85–104 (2006)Google Scholar
  10. 10.
    Esterov, A.: Newton polyhedra of discriminants of projections. Discret. Comput. Geom. 44(1), 96–148 (2010)MathSciNetzbMATHCrossRefGoogle Scholar
  11. 11.
    Feichtner, E.M., Sturmfels, B.: Matroid polytopes, nested sets and Bergman fans. Port. Math. (N.S.). 62(4), 437–468 (2005)Google Scholar
  12. 12.
    Gel\(^{\prime }\)fand, I.M., Kapranov, M.M., Zelevinsky, A.V.: Discriminants, resultants, and multidimensional determinants. In: Mathematics: Theory & Applications. Birkhäuser Boston Inc., Boston (1994)Google Scholar
  13. 13.
    Katz, N.: Pinceaux de Lefschetz: théoréme d’existence. Exposé XVII in Groupes de Monodromie en Géométrie Algébrique (SGA 7 II). In: Lecturer Notes in Mathematics, vol. 340, pp. 212–253. Springer, Berlin (1973)Google Scholar
  14. 14.
    Matsui, Y., Takeuchi, K.: A geometric degree formula for \(A\)-discriminants and Euler obstructions of toric varieties. Adv. Math. 226(2), 2040–2064 (2011)Google Scholar
  15. 15.
    Nie, J.: Discriminants and non-negative polynomials. J. Symb. Comput. 47, 167–191 (2012)zbMATHCrossRefGoogle Scholar
  16. 16.
    Rincón, E.F.: Computing tropical linear spaces. J. Symb. Comput. (2012, in press)Google Scholar
  17. 17.
    Salmon, G.: A treatise on the higher plane curves: intended as a sequel to “A treatise on conic sections”, 3rd edn. Hodges and Smith, Dublin (1852)Google Scholar
  18. 18.
    Shub, M., Smale, S.: Complexity of Bézout’s theorem. I. Geometric aspects. J. Am. Math. Soc. 6(2), 459–501 (1993)MathSciNetzbMATHGoogle Scholar

Copyright information

© Springer-Verlag Berlin Heidelberg 2012

Authors and Affiliations

  • Eduardo Cattani
    • 1
  • María Angélica Cueto
    • 2
  • Alicia Dickenstein
    • 3
    Email author
  • Sandra Di Rocco
    • 4
  • Bernd Sturmfels
    • 5
  1. 1.Department of Mathematics and StatisticsUniversity of Massachusetts AmherstAmherstUSA
  2. 2.FB12 Institut für MathematikGoethe-Universität FrankfurtFrankfurt am MainGermany
  3. 3.Departamento de MatemáticaFCEN, Universidad de Buenos Aires and IMAS, CONICET, Ciudad UniversitariaBuenos AiresArgentina
  4. 4.KTH, MathematicsStockholmSweden
  5. 5.Department of MathematicsUniversity of CaliforniaBerkeleyUSA

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