The University of Michigan Combinatorics Seminar


Abstract 

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 K^{n}, we define Trop X to be the closure of the image of v: X∩(K^{*})^{n}→Q^{n}. 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 K^{n}, 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 seriesparallel 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 fvector and conjecture that this fvector is maximal for all tropical linear spaces. 