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Tropical linear spaces and tropical convexity [article]

Simon Hampe
2015 arXiv   pre-print
It is not difficult to see that each such space is tropically convex, i.e. closed under tropical linear combinations.  ...  In classical geometry, a linear space is a space that is closed under linear combinations.  ...  One might then be tempted to define a tropical linear space as a space that is closed under tropical linear combinations. This property is well-known under the name of tropical convexity.  ... 
arXiv:1505.02045v1 fatcat:astlrjo4urer7fjokzoz776rca

Tropical Linear Spaces and Tropical Convexity

Simon Hampe
2015 Electronic Journal of Combinatorics  
It is not difficult to see that each such space is tropically convex, i.e. closed under tropical linear combinations.  ...  In classical geometry, a linear space is a space that is closed under linear combinations.  ...  In other words, X is the quotient of a space closed under tropical linear combinations if and only if X is supported on a tropical linear space.  ... 
doi:10.37236/5271 fatcat:3aoatw4drfgflcgeol2mxxl4ha

Convexity in Tree Spaces

Bo Lin, Bernd Sturmfels, Xiaoxian Tang, Ruriko Yoshida
2017 SIAM Journal on Discrete Mathematics  
We study the geometry of metrics and convexity structures on the space of phylogenetic trees, which is here realized as the tropical linear space of all ultrametrics.  ...  Tropical convexity and the tropical metric behave better. They exhibit properties desirable for geometric statistics, such as geodesics of small depth.  ...  We thank Simon Hampe, Andrew Francis and Megan Owen for helpful conversations.  ... 
doi:10.1137/16m1079841 fatcat:timxs3tcg5c6pkmliy2sz2dpza

Affine Buildings and Tropical Convexity [article]

Michael Joswig, Bernd Sturmfels, Josephine Yu
2007 arXiv   pre-print
The notion of convexity in tropical geometry is closely related to notions of convexity in the theory of affine buildings.  ...  While the original inspiration was the work of Dress and Terhalle in phylogenetics, and of Faltings, Kapranov, Keel and Tevelev in algebraic geometry, our tropical algorithms will also be applicable to  ...  The tropical linear space L p is tropically convex, and it can be represented as a tropical lattice polytope as follows.  ... 
arXiv:0706.1918v1 fatcat:buwv765yk5erdmc6gkhjmdsd4u

Tropicalizing the positive semidefinite cone

Josephine Yu
2014 Proceedings of the American Mathematical Society  
We also show that the tropical PSD cone is the tropical convex hull of the set of symmetric matrices of tropical rank one and that every tropical PSD matrix can be factored as a tropical product of a matrix  ...  and its transpose.  ...  All matrices in the linear space is obtained from the all-zero matrix, which has tropical rank one, by tropically scaling rows and columns simultaneously, and tropical scaling does not change the tropical  ... 
doi:10.1090/s0002-9939-2014-12428-2 fatcat:pc4uk7pns5c57kmduby47ocjam

Tropical Principal Component Analysis and its Application to Phylogenetics [article]

Ruriko Yoshida and Leon Zhang and Xu Zhang
2017 arXiv   pre-print
In one approach, we study the Stiefel tropical linear space of fixed dimension closest to the data points in the tropical projective torus; in the other approach, we consider the tropical polytope with  ...  Here we define and analyze two analogues of principal component analysis in the setting of tropical geometry.  ...  The authors thank Bernd Sturmfels (UC Berkeley and MPI Leipzig) for many helpful conversations.  ... 
arXiv:1710.02682v2 fatcat:ib222ntxuzaendledj43ajb3la

Algorithms and Effectivity in Tropical Mathematics and Beyond (Dagstuhl Seminar 16482)

Stéphane Gaubert, Dima Grigoriev, Michael Joswig, Thorsten Theobald, Marc Herbstritt
2017 Dagstuhl Reports  
This report documents the Dagstuhl Seminar on Algorithms and Effectivity in Tropical Mathematics and Beyond, which took place from November 27 -December 02, 2016.  ...  Examples include: Tropical arithmetic (Min or Max -it's your choice!), tropical convex hull computations, tropical linear spaces, tropical polynomials and tropical hypersurfaces.  ...  As an application we obtain a dimension bound for the moduli space of those tropical linear spaces, the Dressian.  ... 
doi:10.4230/dagrep.6.11.168 dblp:journals/dagstuhl-reports/GaubertGJT16 fatcat:xrlmqx4cpjckhfayqbdyyy63py

Local Tropical Linear Spaces

Felipe Rincón
2013 Discrete & Computational Geometry  
We also study a certain class of tropical linear spaces that we call conical tropical linear spaces, and we give a simple proof that they satisfy Speyer's f-vector conjecture.  ...  In this paper we study general tropical linear spaces locally: For any basis B of the matroid underlying a tropical linear space L, we define the local tropical linear space L_B to be the subcomplex of  ...  Local Tropical Linear Spaces In this section we define local tropical linear spaces and prove that they are homeomorphic to Euclidean space.  ... 
doi:10.1007/s00454-013-9519-8 fatcat:n7m3jyznjjap7kkrm7z7fc64je

Isotropical linear spaces and valuated Delta-matroids

Felipe Rincón
2012 Journal of combinatorial theory. Series A  
In this paper we tropicalize this picture, and we develop a combinatorial theory of tropical Wick vectors and tropical linear spaces that are tropically isotropic.  ...  Our theory generalizes several results for tropical linear spaces and valuated matroids to the class of Coxeter matroids of type D.  ...  National Science Foundation (DMS-0456960 and DMS-0757207).  ... 
doi:10.1016/j.jcta.2011.08.001 fatcat:slzakngcfjckvg6qy2tprk62t4

Tropical medians by transportation [article]

Andrei Comăneci, Michael Joswig
2022 arXiv   pre-print
This method has several desirable properties; e.g., it is Pareto and co-Pareto on rooted triplets.  ...  Fermat-Weber points with respect to an asymmetric tropical distance function are studied. It turns out that they correspond to the optimal solutions of a transportation problem.  ...  The space of equidistant trees T n is a max-tropical linear space and thus max-tropically convex; see [16, Proposition 10.33].  ... 
arXiv:2205.00036v1 fatcat:fp7z4xtm35genj4lk3nokhcyue

Computing Convex Hulls in the Affine Building of SL_d [article]

Leon Zhang
2018 arXiv   pre-print
As a consequence, we bound the dimension of the tropical projective space needed to realize the convex hull as a tropical polytope.  ...  These convex hulls describe the relations among a finite collection of invertible matrices over K.  ...  . , v n ) be a d × n matrix of rank d over K and let L be its associated tropical linear space.  ... 
arXiv:1811.08884v1 fatcat:5kfj5lql2zc2vgk5kyxetpowhu

Ring of conditions for affine space [article]

Boris Kazarnovskii
2020 arXiv   pre-print
The construction of this ring is based on the definition of associated to EAS algebraic subvariety of some multidimensional torus and on the applying tropical algebraic geometry to this subvariety.  ...  We consider ES and EAS as an analogs of Laurent polynomial and of algebraic variety in complex torus (C∖0)^n.  ...  · s * [4] Let s ′ : V * → E * be an adjoint to s linear operator and ∆ ⊂ V * be a convex polyhedron.  ... 
arXiv:2007.09046v1 fatcat:gsxemxr5xfgcnbg4wgk5oy7u3y

Multiplicative structure of 2×2 tropical matrices

Marianne Johnson, Mark Kambites
2011 Linear Algebra and its Applications  
Using ideas from tropical geometry, we give a complete description of Green's relations and the idempotents and maximal subgroups of this semigroup. (M. Johnson), Mark.Kambites@manchester.ac.uk (M.  ...  We study the algebraic structure of the semigroup of all 2 × 2 tropical matrices under multiplication.  ...  Acknowledgments The authors thank the participants of the Manchester Tropical Mathematics Reading Group, and the organisers, speakers and participants of the First de Brún Workshop on Computational Algebra  ... 
doi:10.1016/j.laa.2009.12.030 fatcat:mhcg4joxaja7hiy3u6acrxoli4

Tropical Secant Varieties of Linear Spaces

Mike Develin
2005 Discrete & Computational Geometry  
In this paper, we investigate tropical secant varieties of ordinary linear spaces.  ...  sets of ordinary toric varieties; we show that their interesting parts are combinatorially isomorphic to a certain natural subcomplex of the complex of regular subdivisions of a corresponding point set, and  ...  In particular, theories of tropical convexity [4] , tropical polytopes [5] , tropical linear spaces [8] , tropical linear algebra [3] , tropical geometry [7] , and tropical algebraic geometry [6]  ... 
doi:10.1007/s00454-005-1182-2 fatcat:q5cdsf3kifhxdjfiv3nt6j6o3m

Tropical secant varieties of linear spaces [article]

Mike Develin
2004 arXiv   pre-print
In this paper, we investigate tropical secant varieties of ordinary linear spaces.  ...  sets of ordinary toric varieties; we show that their interesting parts are combinatorially isomorphic to a certain natural subcomplex of the complex of regular subdivisions of a corresponding point set, and  ...  In particular, theories of tropical convexity [4] , tropical polytopes [5] , tropical linear spaces [8] , tropical linear algebra [3] , tropical geometry [7] , and tropical algebraic geometry [6]  ... 
arXiv:math/0405115v1 fatcat:4ckvwpspfrhvtbqbsz2rz67koi
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