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

We undertake a combinatorial study of the piecewise linear map g : R → R which assigns to the four vectors a,A in R and b, B in R the m×n matrix given by gij = min(ai + bj, Ai +Bj). 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 arrangementAm,n, corresponding to the complete bipartite graphKm,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 2 )( n 2 ) rm−2,n−2 maximum-dimensional cells, where rm−2,n−2 is the number of regions of the arrangement Am−2,n−2.