Mixed Discriminants

TL;DR

This paper explores the properties of mixed discriminants of Laurent polynomials, linking their degree to geometric structures like the mixed Grassmannian and providing explicit formulas for specific cases.

Contribution

It establishes a connection between mixed discriminants and A-discriminants via the Cayley trick and derives a piecewise linear degree formula in Plucker coordinates.

Findings

Degree of mixed discriminant is piecewise linear in Plucker coordinates.

Explicit degree formula provided for plane curves.

Shows the mixed discriminant vanishes when roots coincide.

Abstract

The mixed discriminant of n Laurent polynomials in n variables is the irreducible polynomial in the coefficients which vanishes whenever two of the roots coincide. The Cayley trick expresses the mixed discriminant as an A-discriminant. We show that the degree of the mixed discriminant is a piecewise linear function in the Plucker coordinates of a mixed Grassmannian. An explicit degree formula is given for the case of plane curves.