  In tropical algebraic geometry, the solution sets of polynomial equations are
piecewise-linear. We introduce the tropical variety of a polynomial ideal, and
we identify it with a polyhedral subcomplex of the Grobner fan. The tropical
Grassmannian arises in this manner from the ideal of quadratic Plucker
relations. It is shown to parametrize all tropical linear spaces. Lines in
tropical projective space are trees, and their tropical Grassmannian G_{2,n}
equals the space of phylogenetic trees studied by Billera, Holmes and Vogtmann.
Higher Grassmannians offer a natural generalization of the space of trees.
Their facets correspond to binomial initial ideals of the Plucker ideal. The
tropical Grassmannian G_{3,6} is a simplicial complex glued from 1035
tetrahedra.
