Regular A-Determinants and A-Discriminants

  • Israel M. Gelfand
  • Mikhail M. Kapranov
  • Andrei V. Zelevinsky
Part of the Mathematics: Theory & Applications book series (MBC)


In the previous chapter we established some structural properties of the principal A-determinant E A . Now we shall apply this information to the A-discriminant Δ A . In the most important case when the toric variety X A is smooth, we have
$$ {E_A}(f) = \prod\limits_{\Gamma \subset Q} {{\Delta _{A \cap \Gamma }}} (f) $$
where the product is taken over all the faces of the polytope Q = Conv (A) (Theorem 1.2 Chapter 10). Since (in the case when X A is smooth) a similar equality holds for each E A⋂Г, we have a system of equalities relating the polynomials Δ A⋂Г and E A⋂Г that allows us to recover Δ A as as an alternating product of the E A⋂Г. Consequently, alternating sums and products will appear in the expressions for the Newton polytope and coefficients of Δ A .


Toric Variety Perverse Sheave Intersection Cohomology Constant Sheaf Real Algebraic Geometry 
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Copyright information

© Springer Science+Business Media New York 1994

Authors and Affiliations

  • Israel M. Gelfand
    • 1
  • Mikhail M. Kapranov
    • 2
  • Andrei V. Zelevinsky
    • 3
  1. 1.Department of MathematicsRutgers UniversityNew BrunswickUSA
  2. 2.Department of MathematicsNorthwestern UniversityEvanstonUSA
  3. 3.Department of MathematicsNortheastern UniversityBostonUSA

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