# Minimal tropical basis for Bergman fan of matroid

## TL;DR

This paper proves that every simple binary matroid has a unique minimal tropical basis for its Bergman fan, confirming a conjecture for regular matroids.

## Contribution

It establishes that all simple binary matroids possess a unique minimal tropical basis, extending the understanding of tropical bases in matroid theory.

## Key findings

- Every simple binary matroid has a unique minimal tropical basis.
- Confirmed the conjecture for regular matroids as a special case.
- Extended previous results verified for graphic, cographic, and R_{10} matroids.

## Abstract

The Bergman fan of a matroid is the intersection of tropical hyperplanes defined by the circuits. A tropical basis is a subset of the circuits set that defines the Bergman fan. Yu and Yuster posed a question whether every simple regular matroid has a unique minimal tropical basis of its Bergman fan, and verified it for graphic, cographic matroids and $R_{10}$. We show every simple binary matroid has a unique minimal tropical basis. Since the regular matroid is binary, we positively answered the question.