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

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.