A refinement of the Kushnirenko-Bernstein estimate

TL;DR

This paper improves the Kushnirenko-Bernstein estimate by introducing refined combinatorial invariants and a generalized mixed volume called the mixed integral, using new techniques from relative toric geometry.

Contribution

It presents a refined bound on the number of polynomial roots by developing new invariants and a generalized mixed volume concept, advancing the understanding of polynomial systems in toric geometry.

Findings

Introduction of mixed integral of concave functions

Enhanced bounds on polynomial roots in toric systems

Application of relative toric geometry techniques

Abstract

A theorem of Kushnirenko and Bernstein shows that the number of isolated roots of a system of polynomials in a torus is bounded above by the mixed volume of the Newton polytopes of the given polynomials, and this upper bound is generically exact. We improve on this result by introducing refined combinatorial invariants of polynomials and a generalization of the mixed volume of convex bodies: the mixed integral of concave functions. The proof is based on new techniques and results from relative toric geometry.