scholarly journals Tropical Linear Spaces and Tropical Convexity

10.37236/5271 ◽  
2015 ◽  
Vol 22 (4) ◽  
Author(s):  
Simon Hampe
Keyword(s):  
Convex Set ◽  
Tropical Geometry ◽  
Tropical Variety ◽  
Linear Combinations ◽  
Linear Spaces ◽  
Classical Geometry ◽  
Tropical Convexity ◽  
Valuated Matroid ◽  
Tropical Varieties ◽  
Global Principle

In classical geometry, a linear space is a space that is closed under linear combinations. In tropical geometry, it has long been a consensus that tropical varieties defined by valuated matroids are the tropical analogue of linear spaces. It is not difficult to see that each such space is tropically convex, i.e. closed under tropical linear combinations. However, we will also show that the converse is true: Each tropical variety that is also tropically convex is supported on the complex of a valuated matroid. We also prove a tropical local-to-global principle: Any closed, connected, locally tropically convex set is tropically convex.

scholarly journals COMPLETIONS, REVERSALS, AND DUALITY FOR TROPICAL VARIETIES

2011 ◽  
Vol 10 (06) ◽  
pp. 1141-1163
Author(s):  
ZUR IZHAKIAN ◽  
LOUIS ROWEN
Keyword(s):  
Algebraic Structure ◽  
Tropical Geometry ◽  
Tropical Variety ◽  
Single Face ◽  
Polyhedral Complexes ◽  
Tropical Varieties

The object of this paper is to present two algebraic results with straightforward proofs, which have interesting consequences in tropical geometry. We start with an identity for polynomials over the max-plus algebra, which shows that any polynomial divides a product of binomials. Interpreted in tropical geometry, any tropical variety W can be completed to a union of tropical primitives, i.e. single-face polyhedral complexes. In certain situations, a tropical variety W has a "reversal" variety, which together with W already yields the union of primitives; this phenomenon is explained in terms of a map defined on the algebraic structure, and yields a duality on tropical hypersurfaces.

scholarly journals Tropical ideals do not realise all Bergman fans

2021 ◽  
Vol 8 (3) ◽  
Author(s):  
Jan Draisma ◽  
Felipe Rincón
Keyword(s):  
Tropical Geometry ◽  
Tensor Products ◽  
Tropical Variety ◽  
Polyhedral Complex ◽  
Direct Sum ◽  
Weight One ◽  
Polyhedral Complexes

AbstractEvery tropical ideal in the sense of Maclagan–Rincón has an associated tropical variety, a finite polyhedral complex equipped with positive integral weights on its maximal cells. This leads to the realisability question, ubiquitous in tropical geometry, of which weighted polyhedral complexes arise in this manner. Using work of Las Vergnas on the non-existence of tensor products of matroids, we prove that there is no tropical ideal whose variety is the Bergman fan of the direct sum of the Vámos matroid and the uniform matroid of rank two on three elements and in which all maximal cones have weight one.

scholarly journals Tropical geometry and the motivic nearby fiber

2011 ◽  
Vol 148 (1) ◽  
pp. 269-294 ◽  
Author(s):  
Eric Katz ◽  
Alan Stapledon
Keyword(s):  
Euler Characteristic ◽  
Hodge Structure ◽  
Tropical Geometry ◽  
Mixed Hodge Structure ◽  
General Fiber ◽  
Algebraic Torus ◽  
Tropical Varieties

AbstractWe construct motivic invariants of a subvariety of an algebraic torus from its tropicalization and initial degenerations. More specifically, we introduce an invariant of a compactification of such a variety called the ‘tropical motivic nearby fiber’. This invariant specializes in the schön case to the Hodge–Deligne polynomial of the limit mixed Hodge structure of a corresponding degeneration. We give purely combinatorial expressions for this Hodge–Deligne polynomial in the cases of schön hypersurfaces and matroidal tropical varieties. We also deduce a formula for the Euler characteristic of a general fiber of the degeneration.

scholarly journals On Rational Equivalence in Tropical Geometry

2016 ◽  
Vol 68 (2) ◽  
pp. 241-257 ◽  
Author(s):  
Lars Allermann ◽  
Simon Hampe ◽  
Johannes Rau
Keyword(s):  
Tropical Geometry ◽  
Chow Groups ◽  
Independent Interest ◽  
Main Step ◽  
Rational Equivalence ◽  
Basic Properties ◽  
Boundary Points ◽  
Intersection Ring ◽  
Tropical Varieties

AbstractThis article discusses the concept of rational equivalence in tropical geometry (and replaces an older, imperfect version). We give the basic definitions in the context of tropical varieties without boundary points and prove some basic properties. We then compute the “bounded” Chow groups of Rn by showing that they are isomorphic to the group of fan cycles. The main step in the proof is of independent interest. We show that every tropical cycle in Rn is a sum of (translated) fan cycles. This also proves that the intersection ring of tropical cycles is generated in codimension 1 (by hypersurfaces).

scholarly journals Tropical Chow Hypersurfaces

2017 ◽  
Vol 2019 (14) ◽  
pp. 4302-4324
Author(s):  
Paolo Tripoli
Keyword(s):  
Projective Variety ◽  
Classical Case ◽  
Geometric Description ◽  
Tropical Variety ◽  
Linear Spaces ◽  
Chow Variety

Abstract Given a projective variety $X\subset\mathbb{P}^n$ of codimension $k+1$, the Chow hypersurface $Z_X$ is the hypersurface of the Grassmannian $\operatorname{Gr}(k, n)$ parametrizing projective linear spaces that intersect $X$. We introduce the tropical Chow hypersurface $\operatorname{Trop}(Z_X)$. This object only depends on the tropical variety $\operatorname{Trop}(X)$ and we provide an explicit way to obtain $\operatorname{Trop}(Z_X)$ from $\operatorname{Trop}(X)$. We also give a geometric description of $\operatorname{Trop}(Z_X)$. We conjecture that, as in the classical case, $\operatorname{Trop}(X)$ can be reconstructed from $\operatorname{Trop}(Z_X)$ and prove it for the case when $X$ is a curve in $\mathbb{P}^3$. This suggests that tropical Chow hypersurfaces could be the key to construct a tropical Chow variety.

scholarly journals The Relationship between Two Kinds of Generalized Convex Set-Valued Maps in Real Ordered Linear Spaces

2013 ◽  
Vol 2013 ◽  
pp. 1-4
Author(s):  
Zhi-Ang Zhou
Keyword(s):  
Convex Set ◽  
Locally Convex Spaces ◽  
Linear Spaces ◽  
Algebraic Interior ◽  
Vector Closure ◽  
Ordered Linear Spaces ◽  
The Relationship ◽  
Locally Convex ◽  
Convex Spaces

A new notion of the ic-cone convexlike set-valued map characterized by the algebraic interior and the vector closure is introduced in real ordered linear spaces. The relationship between the ic-cone convexlike set-valued map and the nearly cone subconvexlike set-valued map is established. The results in this paper generalize some known results in the literature from locally convex spaces to linear spaces.

scholarly journals PUISEUX POWER SERIES SOLUTIONS FOR SYSTEMS OF EQUATIONS

2010 ◽  
Vol 21 (11) ◽  
pp. 1439-1459 ◽  
Author(s):  
FUENSANTA AROCA ◽  
GIOVANNA ILARDI ◽  
LUCÍA LÓPEZ DE MEDRANO
Keyword(s):  
Algebraic Curves ◽  
Newton Polygon ◽  
Series Solutions ◽  
Systems Of Equations ◽  
Algebraic Equations ◽  
Puiseux Series ◽  
Tropical Variety ◽  
Power Series Solutions ◽  
Tropical Varieties ◽  
The Ideal

We give an algorithm to compute term-by-term multivariate Puiseux series expansions of series arising as local parametrizations of zeroes of systems of algebraic equations at singular points. The algorithm is an extension of Newton's method for plane algebraic curves, replacing the Newton polygon by the tropical variety of the ideal generated by the system. As a corollary we deduce a property of tropical varieties of quasi-ordinary singularities.

scholarly journals Characterisation of normed linear spaces with Mazur's intersection property

1978 ◽  
Vol 18 (1) ◽  
pp. 105-123 ◽  
Author(s):  
J.R. Giles ◽  
D.A. Gregory ◽  
Brailey Sims
Keyword(s):  
Euclidean Space ◽  
Convex Set ◽  
Intersection Property ◽  
Linear Spaces ◽  
Normed Linear Spaces ◽  
Strongly Exposed Points ◽  
Denting Points ◽  
Mazur's Intersection Property ◽  
Closed Convex Set ◽  
Space Property

Normed linear spaces possessing the euclidean space property that every bounded closed convex set is an intersection of closed balls, are characterised as those with dual ball having weak* denting points norm dense in the unit sphere. A characterisation of Banach spaces whose duals have a corresponding intersection property is established. The question of the density of the strongly exposed points of the ball is examined for spaces with such properties.

scholarly journals Tropical geometry and Newton–Okounkov cones for Grassmannian of planes from compactifications

2020 ◽  
pp. 1-33
Author(s):  
Christopher Manon ◽  
Jihyeon Jessie Yang
Keyword(s):  
Tropical Geometry ◽  
Flag Varieties ◽  
Tropical Variety ◽  
Affine Cone ◽  
Toric Degenerations

Abstract We construct a family of compactifications of the affine cone of the Grassmannian variety of $2$ -planes. We show that both the tropical variety of the Plücker ideal and familiar valuations associated to the construction of Newton–Okounkov bodies for the Grassmannian variety can be recovered from these compactifications. In this way, we unite various perspectives for constructing toric degenerations of flag varieties.

scholarly journals Phylogenetic trees and the tropical geometry of flag varieties

2012 ◽  
Vol DMTCS Proceedings vol. AR,... (Proceedings) ◽  
Author(s):  
Christopher Manon
Keyword(s):  
Phylogenetic Trees ◽  
Tropical Geometry ◽  
Type A ◽  
Hyperplane Arrangements ◽  
Nous Allons ◽  
Flag Varieties ◽  
Dynkin Diagrams ◽  
The Family ◽  
International Audience ◽  
Tropical Varieties

International audience We will discuss some recent theorems relating the space of weighted phylogenetic trees to the tropical varieties of each flag variety of type A. We will also discuss the tropicalizations of the functions corresponding to semi-standard tableaux, in particular we relate them to familiar functions from phylogenetics. We close with some remarks on the generalization of these results to the tropical geometry of arbitrary flag varieties. This involves the family of Bergman complexes derived from the hyperplane arrangements associated to simple Dynkin diagrams. Nous allons discuter de quelques théorèmes récents concernant l'espace des arbres phylogénétiques aux variétés Tropicales de chaque variété de drapeaux de type A. Nous allons également discuter des tropicalisations des fonctions correspondant à tableaux semi-standard, en particulier, nous les rapporter à des fonctions familières de la phylogénétique. Nous terminerons avec quelques remarques sur la généralisation de ces résultats à la géométrie tropicale de variétés de drapeaux arbitraires. Il s'agit de la famille de complexes Bergman provenant des arrangements d'hyperplans associés à des diagrammes de Dynkin simples.

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