Algebraic equations and convex bodies

TL;DR

This paper reviews recent generalizations of the Bernstein-Kushnirenko theorem, connecting algebraic equations on varieties with convex geometry, and explores their applications in algebraic and convex geometry.

Contribution

It presents a broad generalization of the Bernstein-Kushnirenko theorem to systems on any quasi-projective variety, expanding its scope and applications.

Findings

Generalization of Bernstein-Kushnirenko theorem to quasi-projective varieties

Connections established between algebraic equations and convex bodies

Applications demonstrated in algebraic and convex geometry

Abstract

The well-known Bernstein-Kushnirenko theorem from the theory of Newton polyhedra relates algebraic geometry and the theory of mixed volumes. Recently the authors have found a far-reaching generalization of this theorem to generic systems of algebraic equations on any quasi-projective variety. In the present note we review these results and their applications to algebraic geometry and convex geometry.