Tropical hyperplane arrangements and oriented matroids

TL;DR

This paper explores the combinatorial structure of tropical hyperplane arrangements, introducing tropical oriented matroids, and establishes their connection to subdivisions of products of simplices, suggesting a bijective correspondence.

Contribution

It defines tropical oriented matroids and proves their properties are similar to classical oriented matroids, linking them to geometric subdivisions.

Findings

Tropical oriented matroids share properties with classical oriented matroids.

A tropical oriented matroid determines a subdivision of a product of two simplices.

Conjecture: this correspondence is a bijection.

Abstract

We study the combinatorial properties of a tropical hyperplane arrangement. We define tropical oriented matroids, and prove that they share many of the properties of ordinary oriented matroids. We show that a tropical oriented matroid determines a subdivision of a product of two simplices, and conjecture that this correspondence is a bijection.