Tropical Linear Spaces
Abstract
We define tropical analogues of the notions of linear space and Plucker 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 fvector and are ``seriesparallel''. We conjecture that this fvector is maximal for all tropical linear spaces with equality precisely for the seriesparallel tropical linear spaces. We present many partial results towards this conjecture. In addition we relate tropical linear spaces to linear spaces defined over power series fields and give many examples and counterexamples illustrating aspects of this relationship. We describe a family of particularly nice seriesparallel linear spaces, which we term tree spaces, that realize the conjectured maximal fvector and are constructed in a manner similar to the cyclic polytopes.
 Publication:

arXiv Mathematics eprints
 Pub Date:
 October 2004
 arXiv:
 arXiv:math/0410455
 Bibcode:
 2004math.....10455S
 Keywords:

 Combinatorics;
 Algebraic Geometry
 EPrint:
 40 pages, 5 figures