Regular A-Determinants and A-Discriminants

Abstract

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 .