Higher cosystoles of matroids

TL;DR

This paper introduces a new matroid invariant called the three-cosystole, establishes an optimal upper bound for regular matroids of rank up to six, and explores its properties under matroid extensions.

Contribution

It defines the three-cosystole invariant, proves an optimal upper bound for it in certain regular matroids, and analyzes its behavior under matroid extensions.

Findings

Established an optimal upper bound for the three-cosystole in regular matroids of rank at most six.

Showed that the three-cosystole is increasing under matroid extensions.

Estimated the invariant for maximal simple regular matroids of rank at most six.

Abstract

We define a matroid invariant called the three-cosystole that is related to higher notions of cogirth for weighted matroids, and we prove an optimal upper bound for it in the class of regular matroids of rank at most six. To accomplish this, we show that it is increasing under matroid extensions and then estimate it for each of the maximal simple regular matroids of rank at most six.