Skip to main content
SpringerLink
Log in

Mixed discriminants

  • Published:
Mathematische Zeitschrift Aims and scope Submit manuscript

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.

This is a preview of subscription content, log in via an institution to check access.

Access this article

Log in via an institution

Price excludes VAT (USA)
Tax calculation will be finalised during checkout.

Instant access to the full article PDF.

Institutional subscriptions

Fig. 1

Similar content being viewed by others

References

  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)

    Article  MathSciNet  MATH  Google Scholar 

  2. Benoist, O.: Degrés d’homogénéité de l’ensemble des intersections complètes singulières. Annales de l’Institut Fourier 62(3), 1189–1214 (2012)

  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. Casagrande, C., Di Rocco, S.: Projective \(\mathbb{Q}\)-factorial toric varieties covered by lines. Commun. Contemp. Math. 10(3), 363–389 (2008)

    Article  MathSciNet  MATH  Google Scholar 

  5. Cattani, E., Dickenstein, A., Sturmfels, B.: Rational hypergeometric functions. Compositio Math. 128(2), 217–239 (2001)

    Article  MathSciNet  MATH  Google Scholar 

  6. Curran, R., Cattani, E.: Restriction of \(A\)-discriminants and dual defect toric varieties. J. Symb. Comput. 42(1–2), 115–135 (2007)

    Article  MathSciNet  MATH  Google Scholar 

  7. Dickenstein, A., Feichtner, E.M., Sturmfels, B.: Tropical discriminants. J. Am. Math. Soc. 20(4), 1111–1133 (2007)

    Google Scholar 

  8. Dickenstein, A., Sturmfels, B.: Elimination theory in codimension 2. J. Symb. Comput. 34(2), 119–135 (2002)

    Article  MathSciNet  MATH  Google Scholar 

  9. Di Rocco, S.: Projective duality of toric manifolds and defect polytopes. Proc. London Math. Soc. (3), 93(1), 85–104 (2006)

  10. Esterov, A.: Newton polyhedra of discriminants of projections. Discret. Comput. Geom. 44(1), 96–148 (2010)

    Article  MathSciNet  MATH  Google Scholar 

  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. 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)

  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)

  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. Nie, J.: Discriminants and non-negative polynomials. J. Symb. Comput. 47, 167–191 (2012)

    Article  MATH  Google Scholar 

  16. Rincón, E.F.: Computing tropical linear spaces. J. Symb. Comput. (2012, in press)

  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)

  18. Shub, M., Smale, S.: Complexity of Bézout’s theorem. I. Geometric aspects. J. Am. Math. Soc. 6(2), 459–501 (1993)

    MathSciNet  MATH  Google Scholar 

Download references

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.

Author information

Authors and Affiliations

  1. Department of Mathematics and Statistics, University of Massachusetts Amherst, Amherst, MA, 01003, USA

    Eduardo Cattani

  2. FB12 Institut für Mathematik, Goethe-Universität Frankfurt, Robert-Mayer-Str. 6-8, 60325 , Frankfurt am Main, Germany

    María Angélica Cueto

  3. Departamento de Matemática, FCEN, Universidad de Buenos Aires and IMAS, CONICET, Ciudad Universitaria, Pab, I - C1428EGA, Buenos Aires, Argentina

    Alicia Dickenstein

  4. KTH, Mathematics, S-10044 , Stockholm, Sweden

    Sandra Di Rocco

  5. Department of Mathematics, University of California, Berkeley, CA, 94720, USA

    Bernd Sturmfels

Authors
  1. Eduardo Cattani
    View author publications

    You can also search for this author in PubMed Google Scholar

  2. María Angélica Cueto
    View author publications

    You can also search for this author in PubMed Google Scholar

  3. Alicia Dickenstein
    View author publications

    You can also search for this author in PubMed Google Scholar

  4. Sandra Di Rocco
    View author publications

    You can also search for this author in PubMed Google Scholar

  5. Bernd Sturmfels
    View author publications

    You can also search for this author in PubMed Google Scholar

Corresponding author

Correspondence to Alicia Dickenstein.

Additional information

Dedicated to the memory of our friend Mikael Passare (1959–2011).

Rights and permissions

Reprints and permissions

About this article

Cite this article

Cattani, E., Cueto, M.A., Dickenstein, A. et al. Mixed discriminants. Math. Z. 274, 761–778 (2013). https://doi.org/10.1007/s00209-012-1095-8

Download citation

  • Received:

  • Accepted:

  • Published:

  • Issue Date:

  • DOI: https://doi.org/10.1007/s00209-012-1095-8

Keywords

Mathematics Subject Classification (2000)

Search

Navigation