A tropical morphism related to the hyperplane arrangement of the complete bipartite graph

TL;DR

This paper explores a piecewise linear map related to tropical geometry and hyperplane arrangements, revealing its structure and subdivisions, with implications for statistical models and tropical algebraic varieties.

Contribution

It characterizes the map's domains and images, linking them to hyperplane arrangements and providing detailed polyhedral subdivisions with enumerative properties.

Findings

The map's domains correspond to regions of a hyperplane arrangement.

The image admits two polyhedral subdivisions, one refining the other.

The finer subdivision has a maximum number of cells related to combinatorial counts.

Abstract

We undertake a combinatorial study of the piecewise linear map g : R^{2m+2n} --> R^{mn} which assigns to the four vectors a, A in R^m and b, B in R^n the m by n matrix given by g_{ij} = min (a_i + b_j, A_i+B_j). This map arises naturally in Pachter and Sturmfels's work on the tropical geometry of statistical models. The image of g has been a subject of recent interest; it is the positive part of the tropical algebraic variety which parameterizes n-tuples of points on a tropical line in m-space. The domains of linearity of g are the regions of the real hyperplane arrangement A_{m,n}, corresponding to the complete bipartite graph K_{m,n}. We explain how the images of (some of) the regions provide two polyhedral subdivisions of the image of g, one of which is a refinement of the other. The finer subdivision is particularly nice enumeratively: it has 2 {m \choose 2} {n \choose 2} r_{m-2,n-2} maximum-dimensional cells, where r_{m-2,n-2} is the number of regions of the arrangement A_{m-2,n-2}.