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


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.


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

Mathematics Subject Classification (2000)

13P15 14M25 14T05 52B20 



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.


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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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