How to Draw Tropical Planes

@article{Herrmann2009HowTD,
  title={How to Draw Tropical Planes},
  author={Sven Herrmann and Anders Nedergaard Jensen and Michael Joswig and Bernd Sturmfels},
  journal={Electron. J. Comb.},
  year={2009},
  volume={16}
}
The tropical Grassmannian parameterizes tropicalizations of linear spaces, while the Dressian parameterizes all planes in $\TP^{n-1}$. We study these parameter spaces and we compute them explicitly for $n \leq 7$. Planes are identified with matroid subdivisions and with arrangements of trees. These representations are used to draw pictures. 
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References

SHOWING 1-10 OF 40 REFERENCES
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 anExpand
The homology of tropical varieties
Given a closed subvarietyX of an algebraic torusT, the associated tropical variety is a polyhedral fan in the space of 1-parameter subgroups of the torus which describes the behaviour of theExpand
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 polyhedralExpand
Tropical Linear Spaces
  • D. Speyer
  • Mathematics, Computer Science
  • SIAM J. Discret. Math.
  • 2008
TLDR
The main result is that all constructible tropical linear spaces have the same $f-vector and are “series-parallel”, and it is conjectured that this $f$-vector is maximal for all Tropical linear spaces, with equality precisely for the series-par parallel tropicallinear spaces. Expand
The Cayley Trick and Triangulations of Products of Simplices
We use the Cayley Trick to study polyhedral subdivisions of the product of two simplices. For arbitrary (fixed) l ≥ 2, we show that the numbers of regular and non-regular triangulations ofl × � kExpand
Tropical hyperplane arrangements and oriented matroids
We study the combinatorial properties of a tropical hyperplane arrangement. We define tropical oriented matroids, and prove that they share many of the properties of ordinary oriented matroids. WeExpand
Introduction to Tropical Geometry (notes from the IMPA lectures in Summer 2007)
These condensed notes treat some basic notions in Tropical Geometry (varieties, cycles, modifications, equivalence). These topics are to be extended, illustrated and included to the upcoming bookExpand
Matroid polytopes, nested sets and Bergman fans
The tropical variety defined by linear equations with constant coefficients is the Bergman fan of the corresponding matroid. Building on a self-contained introduction to matroid polytopes, we presentExpand
GEOMETRY OF CHOW QUOTIENTS OF GRASSMANNIANS
We consider Kapranov’s Chow quotient compactification of the moduli space of ordered n-tuples of hyperplanes in P r−1 in linear general position. For r = 2 ,t his is canonically identified with theExpand
Combinatorial Geometries, Convex Polyhedra, and Schubert Cells
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 theoryExpand
...
1
2
3
4
...