On simple matroids with a unique minimal tropical basis

TL;DR

This paper characterizes simple matroids with a unique minimal tropical basis by analyzing circuits and cocircuits, providing a solution to a previously posed problem in the field.

Contribution

It offers a new characterization of simple matroids with a unique minimal tropical basis using circuit orthogonality and cocircuit determination.

Findings

Circuits determine cocircuits via orthogonality in simple matroids.

Closed circuits must be included in any tropical basis.

Identifies conditions for simple matroids to have a unique minimal tropical basis.

Abstract

In this note we characterize tropical bases as sets of circuits that by orthogonality determine the set of cocircuits of a simple matroid. Furthermore, we show that any circuit, which itself is closed, must be contained in any tropical basis. This yields a characterization of simple matroids which have a unique minimal tropical basis, giving a solution for a problem posted in the Matroid Union Blog.