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Matroid polytopes, nested sets and Bergman fans. (English) Zbl 1092.52006
This paper is concerned with the tropical variety defined by a system of linear equations with constant coefficients. It turns out, that this variety is the Bergman fan of an associated matroid. After discussing the relation between Bergman fan and nested set complexes of (arbitrary) lattices, the authors refine a result due to Ardila-Klivans in relation with triangulations of Bergman complexes. A relation of combinatorial results and algebraic geometry (in terms of complements of arrangements of hyperplanes in the complex space) is also described.

MSC:
52B20Lattice polytopes (convex geometry)
05B35Matroids, geometric lattices (combinatorics)
14D99Families, fibrations
52B40Matroids (convex geometry)
52C35Arrangements of points, flats, hyperplanes