We know that the Bergman complex of a variety (as defined in Chapter 9 of Sturmfels' book on ``Solving systems of polynomial equations''), i.e. the tropical variety, is a polyhedral complex. For linear subspaces, the paper of Ardila and Klivans shows that this Bergman complex has a very nice combinatorial structure: a subdivision of it is the order complex of the lattice of flats of the associated matroid, a very well understood combinatorial object. (As a corollary we get the topology, etc.)
Question: Can we find a similar combinatorial description for other classes of Bergman complexes?
(contributed by Federico Ardila)
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