Tropical Linear Spaces

We define tropical analogues of the notions of linear space and Plücker coordinate 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 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. In addition we describe the relation of tropical linear spaces to linear spaces defined over power series fields and give many examples and counter-examples illustrating aspects of this relationship. We describe a family of particularly nice series-parallel linear spaces, which we term tree spaces, that realize the conjectured maximal f -vector and are constructed in a manner similar to the cyclic polytopes.