The University of Michigan Combinatorics Seminar
Fall 2004
October 29, 4:10-5:00, 3866 East Hall



Combinatorics of tropical linear spaces

David Speyer

University of California at Berkeley


Abstract

Let K denote the field of Laurent series with complex coefficients and rational exponents and let vK*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 vX∩(K*)nQn. 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.