Combinatorics of tropical linear spaces

Let K denote the field of Laurent series with complex coefficients and rational exponents and let v: K^*→Q be the map that takes a power series to the exponent of its lowest degree term. If X is any subvariety of Kⁿ, we define Trop X to be the closure of the image of v: X∩(K^*)ⁿ→Qⁿ. Trop X is a polyhedral complex, whose geometry is hoped to reflect the geometry of X∩(K^*)ⁿ.

In this talk, we discuss the case where X is a linear subspace of Kⁿ, so X∩(K^*)ⁿ is a hyperplane arrangement. Then the study of Trop X becomes very combinatorial and can be described in terms of polyhedral decompositions of the hypersimplex arising from certain collections of matroids -- so that it is possible to understand this theory without thinking about power series at all. We identify a special class of tropical linear spaces, the series-parallel linear spaces, which is closed under dualization and intersection and hence contains all easily constructible linear spaces. We prove that all members of this class have the same f-vector and conjecture that this f-vector is maximal for all tropical linear spaces.