Plane mixed discriminants and toric Jacobians

Abstract

Polynomial algebra offers a standard approach to handle severalproblems in geometric modeling. A key tool is the discriminant of aunivariate polynomial, or of a well-constrained system of polynomialequations, which expresses the existence of a multiple root. We describediscriminants in a general context, and focus on exploiting the sparsenessof polynomials via the theory of Newton polytopes and sparse (or toric)elimination. We concentrate on bivariate polynomials and establish anoriginal formula that relates the discriminant of two bivariate Laurentpolynomials with fixed support, with the sparse resultant of thesepolynomials and their toric Jacobian. This allows us to obtain a newproof for the bidegree formula of the discriminant as well as to establishmultiplicativity formulas arising when one polynomial can be factored.