Tropical Linear Spaces

  title={Tropical Linear Spaces},
  author={David E Speyer},
  journal={SIAM J. Discret. Math.},
  • D. Speyer
  • Published 21 October 2004
  • Mathematics, Computer Science
  • SIAM J. Discret. Math.
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… 
Tropical Linear Spaces and Tropical Convexity
  • Simon Hampe
  • 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.
Stiefel tropical linear spaces
Local Tropical Linear Spaces
  • Felipe Rincón
  • 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.
Tropical Intersection Products and Families of Tropical Curves
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
Moduli spaces of codimension-one subspaces in a linear variety and their tropicalization
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
Intersection Theory on Tropical Toric Varieties and Compactifications of Tropical Parameter Spaces
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
Tropical ideals
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
Dissimilarity Vectors of Trees and Their Tropical Linear Spaces (Extended Abstract)
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
The Tropical Symplectic Grassmannian
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
Obstructions to Approximating Tropical Curves in Surfaces Via Intersection Theory
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


The tropical Grassmannian
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
First steps in tropical geometry
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
Enumerative Real Algebraic Geometry
  • F. Sottile
  • 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.
Tropical Mathematics
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
Tropical Convexity
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
The Bergman complex of a matroid and phylogenetic trees
Lectures on Polytopes
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