Abstract
A valuation of a matroid M with bases B is a function ν : B → R that satisfies a quantitative version of the symmetric base exchange axiom. Matroid valuations are otherwise known as tropical linear spaces. Each valuation induces a decomposition of the matroid basis polyhedron into matroid basis polyhedra of other matroids on the same ground set.
The Dressian of M, defined as the set of valuations, is the disjoint union of finitely many open polyhedral cells. Each cell is the collection of all valuations that induce the same matroid basis decomposition. We show that if M is a matroid of rank r ≥ 3 on a ground set with n elements, then the dimension of the Dressian and the number of cells can be bounded from above. We also provide a lower bound for uniform matroids.