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- W2133822347 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}." @default.
- W2133822347 created "2016-06-24" @default.
- W2133822347 creator A5040287199 @default.
- W2133822347 date "2004-04-16" @default.
- W2133822347 modified "2023-09-27" @default.
- W2133822347 title "A tropical morphism related to the hyperplane arrangement of the complete bipartite graph" @default.
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- W2133822347 hasPublicationYear "2004" @default.
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