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 describe discriminants in a general context, and focus on exploiting the sparseness of polynomials via the theory of Newton polytopes and sparse (or toric) elimination. We concentrate on bivariate polynomials and establish an original formula that relates the 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 formula of the discriminant as well as to establish multiplicativity formulas arising when one polynomial can be factored.