Plane Mixed Discriminants and Toric Jacobians

Abstract

Polynomial algebra offers a standard approach to handle several problems in geometric modeling. A key tool is the discriminant of a univariate polynomial, or of a well-constrained system of polynomial equations, which expresses the existence of a multiple root. We concentrate on bivariate polynomials and establish an original formula that relates the mixed discriminant of two bivariate Laurent polynomials with fixed support, with the sparse resultant of these polynomials and their toric Jacobian. This allows us to obtain a new proof for the bidegree of the mixed discriminant as well as to establish multipicativity formulas arising when one polynomial can be factored.