A tropical morphism related to the hyperplane arrangement of the complete bipartite graph
| dc.creator | Ardila, Federico | |
| dc.date | 2004-04-16 | |
| dc.date.accessioned | 2026-07-07T05:07:28Z | |
| dc.date.available | 2026-07-07T05:07:28Z | |
| dc.description | 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}. | |
| dc.description | 18 pages, 5 figures | |
| dc.identifier | https://arxiv.org/abs/math/0404287 | |
| dc.identifier | http://arxiv.org/abs/math/0404287 | |
| dc.identifier.uri | http://salesiana.dossiersoluciones.com/handle/123456789/70872 | |
| dc.subject | Combinatorics | |
| dc.subject | 52B99 (Primary), 05A15 (Secondary) | |
| dc.title | A tropical morphism related to the hyperplane arrangement of the complete bipartite graph | |
| dc.type | text |