Signs of the Leading Coefficients of the Resultant

TL;DR

This paper introduces an $ ext{F}_2$-valued analogue of mixed volume for lattice polytopes, which helps compute signs of leading coefficients of the resultant, revealing new algebraic and geometric insights.

Contribution

It defines a novel $ ext{F}_2$-valued mixed volume and applies it to determine signs of the resultant's leading coefficients, a problem previously solved only in special cases.

Findings

Defined a new $ ext{F}_2$-valued mixed volume concept.

Derived a closed-form expression for signs of the resultant's leading coefficients.

Showed the 2-mixed volume's relevance in algebraic geometry.

Abstract

We construct a certain $\F_2$-valued analogue of the mixed volume of lattice polytopes. This 2-mixed volume cannot be defined as a polarization of any kind of an additive measure, or characterized by any kind of its monotonicity properties, because neither of the two makes sense over $\F_2$. In this sense, the convex-geometric nature of the 2-mixed volume remains unclear. On the other hand, the 2-mixed volume seems to be no less natural and useful than the classical mixed volume -- in particular, it also plays an important role in algebraic geometry. As an illustration of this role, we obtain a closed-form expression in terms of the 2-mixed volume to compute the signs of the leading coefficients of the resultant, which were by now explicitly computed only for some special cases.