[Paper Review] Kinser inequalities and related matroids

Amanda Cameron|arXiv (Cornell University)|Jan 2, 2014

graph theory and CDMA systems8 references3 citations

TL;DR

This paper investigates the Kinser inequalities—a hierarchy of rank inequalities for matroids—by analyzing the classes of matroids that satisfy each inequality. It establishes that these classes form a strict infinite hierarchy, proves that the fifth Kinser class is not closed under duality, and provides evidence for the non-closure of higher classes, while conjecturing that Dowling matroids and $π$-representable matroids satisfy all Kinser inequalities.

ABSTRACT

Kinser developed a hierarchy of inequalities dealing with the dimensions of certain spaces constructed from a given quantity of subspaces. These inequalities can be applied to the rank function of a matroid, a geometric object concerned with dependencies of subsets of a ground set. A matroid which is representable by a matrix with entries from some finite field must satisfy each of the Kinser inequalities. We provide results on the matroids which satisfy each inequality and the structure of the hierarchy of such matroids.

Motivation & Objective

• To characterize the classes of matroids satisfying each Kinser inequality and understand their structural properties.
• To investigate the hierarchy of these classes and determine whether they are closed under fundamental matroid operations like duality and minors.
• To examine the computational complexity of verifying Kinser inequalities and assess the feasibility of finite axiomatization for representable matroids.
• To explore the role of excluded minors in the Kinser hierarchy and strengthen existing results on infinite excluded minors for representability.
• To conjecture that certain classes of matroids—particularly Dowling geometries and $π$-representable matroids—satisfy all Kinser inequalities.

Proposed method

• Uses the rank function of a matroid to express the $n$-th Kinser inequality as a system of rank inequalities involving unions of subsets of the ground set.
• Defines Kinser classes $\mathcal{K}_n$ as the set of matroids satisfying the $n$-th Kinser inequality, and $\mathcal{K}_n^*$ as the dual classes.
• Applies minor-closure and direct sum-closure properties to analyze structural invariants of Kinser classes.
• Employs computational and structural arguments to show that $\mathcal{K}_5 \neq \mathcal{K}_5^*$, proving the fifth Kinser class is not closed under duality.
• Utilizes the theory of skew partial fields and chain groups to define $\mathbb{P}$-representable matroids and relate them to Kinser inequalities.
• Leverages known results on excluded minors (e.g., Mayhew–Newman–Whittle) to show that each layer of the Kinser hierarchy contains infinitely many excluded minors.

Experimental results

Research questions

• RQ1 Are the classes of matroids satisfying the $n$-th Kinser inequality closed under duality?
• RQ2 Does the hierarchy of Kinser classes form a strict inclusion chain, i.e., $\mathcal{K}_{n+1} \subset \mathcal{K}_n$?
• RQ3 Can the infinite hierarchy of Kinser inequalities be used to characterize representable matroids, or is a finite axiomatization impossible?
• RQ4 Do Dowling geometries over non-cyclic groups satisfy all Kinser inequalities?
• RQ5 Are all $\mathbb{P}$-representable matroids (for skew partial fields $\mathbb{P}$) necessarily in $\mathcal{K}_n$ for all $n$?

Key findings

• The fifth Kinser class $\mathcal{K}_5$ is not closed under duality, as shown by the existence of a matroid in $\mathcal{K}_5$ whose dual is not in $\mathcal{K}_5$.
• The paper proves that $\mathcal{K}_5 \neq \mathcal{K}_5^*$, providing strong evidence that the Kinser hierarchy is strict and non-symmetric under duality.
• It is shown that for any infinite field $\mathbb{K}$ and any $\mathbb{K}$-representable matroid $N$, there exist infinitely many excluded minors for $\mathbb{K}$-representability that are contained in each layer of the Kinser hierarchy.
• Dowling geometries over finite abelian groups satisfy all Kinser inequalities if the group is isomorphic to a subgroup of the multiplicative group of a field.
• The paper provides evidence that the class of $\mathbb{P}$-representable matroids satisfies all Kinser inequalities, supporting a broader conjecture on their universal validity.
• Conjectures are proposed that $\mathcal{K}_{n+1}^* \subseteq \mathcal{K}_n$ and $\mathcal{K}_\infty^* = \mathcal{K}_\infty$, suggesting a deeper duality structure in the infinite hierarchy.