# Tropical Linear Spaces

@article{Speyer2008TropicalLS, title={Tropical Linear Spaces}, author={David E Speyer}, journal={SIAM J. Discret. Math.}, year={2008}, volume={22}, pages={1527-1558} }

We define the tropical analogues of the notions of linear spaces and Plucker 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… Expand

#### 142 Citations

Tropical Linear Spaces and Tropical Convexity

- Mathematics, Computer Science
- Electron. J. Comb.
- 2015

It is shown that the converse is true: Each tropical variety that is also tropically convex is supported on the complex of a valuated matroid, and it is proved a tropical local-to-global principle: Any closed, connected, locally tropic convex set is tropically conveyed. Expand

Stiefel tropical linear spaces

- Computer Science, Mathematics
- J. Comb. Theory, Ser. A
- 2015

It is proved that the L ( A ) s obtained in this way are dual to certain matroid subdivisions of polytopes of transversal matroids, and their combinatorics to a canonically associated tropical hyperplane arrangement. Expand

Local Tropical Linear Spaces

- Mathematics, Computer Science
- Discret. Comput. Geom.
- 2013

The tropical linear space L can be expressed as the union of all its local tropical linear spaces, which are homeomorphic to Euclidean space, and it is proved that they are dual to mixed subdivisions of Minkowski sums of simplices. Expand

Tropical Intersection Products and Families of Tropical Curves

- Mathematics
- 2012

This thesis is devoted to furthering the tropical intersection theory as well as to applying the
developed theory to gain new insights about tropical moduli spaces.
We use piecewise polynomials… Expand

Moduli spaces of codimension-one subspaces in a linear variety and their tropicalization

- Mathematics
- 2020

We study the moduli space of d-dimensional linear subspaces contained in a fixed (d + 1)-dimensional linear variety X, and its tropicalization. We prove that these moduli spaces are linear subspaces… Expand

Intersection Theory on Tropical Toric Varieties and Compactifications of Tropical Parameter Spaces

- Mathematics
- 2011

We study toric varieties over the tropical semifield. We define tropical cycles inside these toric varieties and extend the stable intersection of tropical cycles in R^n to these toric varieties. In… Expand

Tropical ideals

- Mathematics
- Compositio Mathematica
- 2018

We introduce and study a special class of ideals, called tropical ideals, in the semiring of tropical polynomials, with the goal of developing a useful and solid algebraic foundation for tropical… Expand

Dissimilarity Vectors of Trees and Their Tropical Linear Spaces (Extended Abstract)

- Mathematics
- 2011

We study the combinatorics of weighted trees from the point of view of tropical algebraic geometry and tropical linear spaces. The set of dissimilarity vectors of weighted trees is contained in the… Expand

The Tropical Symplectic Grassmannian

- Mathematics
- International Mathematics Research Notices
- 2021

We launch the study of the tropicalization of the symplectic Grassmannian, that is, the space of all linear subspaces isotropic with respect to a fixed symplectic form. We formulate tropical… Expand

Obstructions to Approximating Tropical Curves in Surfaces Via Intersection Theory

- Mathematics
- Canadian Journal of Mathematics
- 2015

Abstract We provide some new local obstructions to approximating tropical curves in smooth tropical surfaces. These obstructions are based on a relation between tropical and complex intersection… Expand

#### References

SHOWING 1-10 OF 26 REFERENCES

The tropical Grassmannian

- Mathematics
- 2003

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… Expand

First steps in tropical geometry

- Mathematics
- 2003

Tropical algebraic geometry is the geometry of the tropical semiring $(\mathbb{R},\min,+)$. Its objects are polyhedral cell complexes which behave like complex algebraic varieties. We give an… Expand

Intersection theory on toric varieties

- Mathematics
- 1994

Abstract The operational Chow cohomology classes of a complete toric variety are identified with certain functions, called Minkowski weights, on the corresponding fan. The natural product of Chow… Expand

Enumerative Real Algebraic Geometry

- Mathematics, Computer Science
- Algorithmic and Quantitative Aspects of Real Algebraic Geometry in Mathematics and Computer Science
- 2001

Treating both sparse polynomial systems and enumerative geometry together in the context of Question 1.1 gives useful insight, which is the motivating question of enumerative real algebraic geometry. Expand

Tropical Mathematics

- Mathematics
- 2004

These are the notes for the Clay Mathematics Institute Senior Scholar Lecture which was delivered by Bernd Sturmfels in Park City, Utah, on July 22, 2004. The topic of this lecture is the ``tropical… Expand

Tropical Convexity

- Mathematics
- 2003

The notions of convexity and convex polytopes are introduced in the setting of tropical geometry. Combinatorial types of tropical polytopes are shown to be in bijection with regular triangulations of… Expand

Combinatorial Geometries, Convex Polyhedra, and Schubert Cells

- Mathematics
- 1987

This paper is a continuation of [GM] which was published in the same journal. We will explore a remarkable connection between the geometry of the Schubert cells in the Grassmann manifold, the theory… Expand

Grassmann-Plücker relations and matroids with coefficients

- Mathematics
- 1991

Abstract Matroids with coefficients in a fuzzy ring have been introduced in [A. W. M. Dress, Adv. in Math. 59 (1986), 97–123]. Based on this paper and [A. W. M. Dress and W. Wenzel, Adv. in Math. 77… Expand

The Bergman complex of a matroid and phylogenetic trees

- Computer Science, Mathematics
- J. Comb. Theory, Ser. B
- 2006

It is shown that the Bergman fan B˜(Kn) of the graphical matroid of the complete graph Kn is homeomorphic to the space of phylogenetic trees Tn × R, which leads to a proof that the link of the origin in Tn is homeomorphism to the order complex of the proper part of the partition lattice Πn. Expand

Lectures on Polytopes

- Mathematics
- 1994

Based on a graduate course given at the Technische Universitat, Berlin, these lectures present a wealth of material on the modern theory of convex polytopes. The clear and straightforward… Expand