The Tropical Grassmannian

TL;DR

The paper introduces the tropical Grassmannian as a polyhedral complex arising from polynomial ideals in tropical algebraic geometry, linking it to phylogenetic trees and providing explicit combinatorial structures for higher Grassmannians.

Contribution

It defines the tropical Grassmannian via tropical algebraic geometry, connecting it to phylogenetic trees and describing its combinatorial structure for specific cases.

Findings

Tropical Grassmannian parametrizes tropical linear spaces.

G_{2,n} equals the space of phylogenetic trees.

G_{3,6} is a simplicial complex with 1035 tetrahedra.

Abstract

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.