New inequalities for subspace arrangements

TL;DR

This paper introduces new inequalities for subspace arrangements that extend known conditions like Ingleton's inequality, providing a hierarchy of necessary criteria for the realizability of (poly)matroids, and discusses related open questions.

Contribution

The authors derive a new family of inequalities for subspace arrangements that generalize Ingleton's inequality to higher numbers of subspaces, advancing understanding of matroid realizability.

Findings

Recovered Ingleton's inequality at n=4

Established new inequalities for n>4

Discussed open problems on the cone of realizable polymatroids

Abstract

For each positive integer $n \geq 4$, we give an inequality satisfied by rank functions of arrangements of $n$ subspaces. When $n=4$ we recover Ingleton's inequality; for higher $n$ the inequalities are all new. These inequalities can be thought of as a hierarchy of necessary conditions for a (poly)matroid to be realizable. Some related open questions about the "cone of realizable polymatroids" are also presented.