{
  "id": "math/0404287",
  "version": "v1",
  "published": "2004-04-16T02:31:45.000Z",
  "updated": "2004-04-16T02:31:45.000Z",
  "title": "A tropical morphism related to the hyperplane arrangement of the complete bipartite graph",
  "authors": [
    "Federico Ardila"
  ],
  "comment": "18 pages, 5 figures",
  "categories": [
    "math.CO"
  ],
  "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}.",
  "revisions": [
    {
      "version": "v1",
      "updated": "2004-04-16T02:31:45.000Z"
    }
  ],
  "analyses": {
    "subjects": [
      "52B99",
      "05A15"
    ],
    "keywords": [
      "complete bipartite graph",
      "tropical morphism",
      "real hyperplane arrangement",
      "map arises",
      "sturmfelss work"
    ],
    "note": {
      "typesetting": "TeX",
      "pages": 18,
      "language": "en",
      "license": "arXiv",
      "status": "editable",
      "adsabs": "2004math......4287A"
    }
  }
}