Intersection theory of polymatroids

TL;DR

This paper extends algebraic geometric techniques to polymatroids via augmented Chow rings, revealing that intersection numbers are governed by the Hall--Rado condition, a novel insight even for matroids.

Contribution

It introduces augmented Chow rings for polymatroids and establishes the Hall--Rado condition as key to intersection theory, generalizing matroid techniques.

Findings

Intersection numbers are determined by the Hall--Rado condition.

Augmented Chow rings provide a new framework for polymatroid intersection theory.

The Hall--Rado condition is a novel criterion applicable to matroids and polymatroids.

Abstract

Polymatroids are combinatorial abstractions of subspace arrangements in the same way that matroids are combinatorial abstractions of hyperplane arrangements. By introducing augmented Chow rings of polymatroids, modeled after augmented wonderful varieties of subspace arrangements, we generalize several algebro-geometric techniques developed in recent years to study matroids. We show that intersection numbers in the augmented Chow ring of a polymatroid are determined by a matching property known as the Hall--Rado condition, which is new even in the case of matroids.