Non-Generic Tropical Hyperplane Arrangements and the Secondary polytope of $\Delta_{n-1} \times \Delta_{d-1}$

TL;DR

This paper explores non-generic tropical hyperplane arrangements, revealing their connection to subdivisions of product of simplices and providing initial insights beyond the generic case.

Contribution

It extends the understanding of tropical hyperplane arrangements to non-generic cases, linking them to subdivisions of the product of simplices, which was not previously addressed.

Findings

Non-generic arrangements encode subdivision information.

Preliminary results on non-generic hyperplanes.

Connections to secondary polytopes of product of simplices.

Abstract

Ardila and Develin's paper on tropical oriented hyperplane arrangements and tropical oriented matroids defines tropical oriented matroids and conjectures a bijection between them and triangulations of products of simplices $\Delta_{n-1} \times \Delta_{d-1}$. Oh and Yoo recently confirmed this conjecture; however, neither group addressed the case of hyperplanes that are not in generic position. These non-generic arrangements do not correspond to tropical oriented matroids, but they encode information about subdivisions of $\Delta_{n-1} \times \Delta_{d-1}$. This note considers the non-generic case and presents some preliminary results in the area.