Tropical Linear Spaces

Abstract

We define the tropical analogues of the notions of linear spaces and Plücker coordinates and study their combinatorics. We introduce tropical analogues of intersection and dualization and define a tropical linear space built by repeated dualization and transverse intersection to be constructible. Our main result is that all constructible tropical linear spaces have the same f-vector and are “series-parallel”. We conjecture that this f-vector is maximal for all tropical linear spaces, with equality precisely for the series-parallel tropical linear spaces. We present many partial results towards this conjecture.