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.


52B20 Lattice polytopes in convex geometry (including relations with commutative algebra and algebraic geometry)
05B35 Combinatorial aspects of matroids and geometric lattices
14D99 Families, fibrations in algebraic geometry
52B40 Matroids in convex geometry (realizations in the context of convex polytopes, convexity in combinatorial structures, etc.)
52C35 Arrangements of points, flats, hyperplanes (aspects of discrete geometry)
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