Convex bodies and algebraic equations on affine varieties

TL;DR

This paper introduces a new convex body construction associated with affine varieties and regular functions, linking geometric volume to solution counts of algebraic systems, and offers simplified proofs of key theorems in algebraic geometry and convex geometry.

Contribution

It generalizes Newton polytopes to affine varieties, connecting convex geometry with algebraic equations and providing new proofs of classical theorems.

Findings

Convex body volume correlates with solution counts of algebraic systems.

Provides simplified proofs of the Hodge Index Theorem and Alexandrov-Fenchel inequality.

Establishes a new bridge between algebraic geometry and convex geometry.

Abstract

Given an affine variety X and a finite dimensional vector space of regular functions L on X, we associate a convex body to (X, L) such that its volume is responsible for the number of solutions of a generic system of functions from L. This is a far reaching generalization of usual theory of Newton polytopes (which is concerned with toric varieties). As applications we give new, simple and transparent proofs of some well-known theorems in both algebraic geometry (e.g. Hodge Index Theorem) and convex geometry (e.g. Alexandrov-Fenchel inequality). Our main tools are classical Hilbert theory on degree of subvarieties of a projective space (in algebraic geometry) and Brunn-Minkowski inequality (in convex geometric).