Plane mixed discriminants and toric jacobians

TL;DR

This paper introduces a novel formula connecting mixed discriminants of bivariate Laurent polynomials with their sparse resultants and toric Jacobians, providing new insights into their algebraic properties and factorizations.

Contribution

It establishes an original formula relating mixed discriminants, sparse resultants, and toric Jacobians for bivariate Laurent polynomials, offering new proofs and factorization formulas.

Findings

Derived a new formula linking mixed discriminants and sparse resultants.

Provided a new proof for the bidegree of the mixed discriminant.

Established multipicativity formulas for factored polynomials.

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.