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 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 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 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 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.

scholarly journals Tropical ideals

2018 ◽  
Vol 154 (3) ◽  
pp. 640-670 ◽  
Author(s):  
Diane Maclagan ◽  
Felipe Rincón
Keyword(s):  
Special Class ◽  
Tropical Geometry ◽  
Basic Structure ◽  
Chain Condition ◽  
Duke Math ◽  
Polyhedral Complex ◽  
Algebraic Foundation ◽  
Tropical Varieties ◽  
Associated Variety ◽  
Ascending Chain Condition

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 geometry. The class of tropical ideals strictly includes the tropicalizations of classical ideals, and allows us to define subschemes of tropical toric varieties, generalizing Giansiracusa and Giansiracusa [Equations of tropical varieties, Duke Math. J. 165 (2016), 3379–3433]. We investigate some of the basic structure of tropical ideals, and show that they satisfy many desirable properties that mimic the classical setup. In particular, every tropical ideal has an associated variety, which we prove is always a finite polyhedral complex. In addition we show that tropical ideals satisfy the ascending chain condition, even though they are typically not finitely generated, and also the weak Nullstellensatz.

scholarly journals Kernels in tropical geometry and a Jordan–Hölder theorem

2018 ◽  
Vol 17 (04) ◽  
pp. 1850066 ◽  
Author(s):  
Tal Perri ◽  
Louis H. Rowen
Keyword(s):  
Tropical Geometry ◽  
Rational Functions ◽  
Composition Series ◽  
Dimension Theory ◽  
Ordered Group ◽  
Additive Structure ◽  
Ordered Groups ◽  
Lattice Ordered Groups ◽  
Roots Of Polynomials ◽  
Tropical Varieties

When considering affine tropical geometry, one often works over the max-plus algebra (or its supertropical analog), which, lacking negation, is a semifield (respectively, [Formula: see text]-semifield) rather than a field. One needs to utilize congruences rather than ideals, leading to a considerably more complicated theory. In his dissertation, the first author exploited the multiplicative structure of an idempotent semifield, which is a lattice ordered group, in place of the additive structure, in order to apply the extensive theory of chains of homomorphisms of groups. Reworking his dissertation, starting with a semifield[Formula: see text][Formula: see text], we pass to the semifield[Formula: see text][Formula: see text] of fractions of the polynomial semiring[Formula: see text], for which there already exists a well developed theory of kernels, which are normal convex subgroups of [Formula: see text]; the parallel of the zero set now is the [Formula: see text]-set, the set of vectors on which a given rational function takes the value 1. These notions are refined in supertropical algebra to [Formula: see text]-kernels (Definition 4.1.4) and [Formula: see text]-sets, which take the place of tropical varieties viewed as sets of common ghost roots of polynomials. The [Formula: see text]-kernels corresponding to tropical hypersurfaces are the [Formula: see text]-sets of what we call “corner internal rational functions,” and we describe [Formula: see text]-kernels corresponding to “usual” tropical geometry as [Formula: see text]-kernels which are “corner-internal” and “regular.” This yields an explicit description of tropical affine varieties in terms of various classes of [Formula: see text]-kernels. The literature contains many tropical versions of Hilbert’s celebrated Nullstellensatz, which lies at the foundation of algebraic geometry. The approach in this paper is via a correspondence between [Formula: see text]-sets and a class of [Formula: see text]-kernels of the rational [Formula: see text]-semifield[Formula: see text] called polars, originating from the theory of lattice-ordered groups. When [Formula: see text] is the supertropical max-plus algebra of the reals, this correspondence becomes simpler and more applicable when restricted to principal [Formula: see text]-kernels, intersected with the [Formula: see text]-kernel generated by [Formula: see text]. For our main application, we develop algebraic notions such as composition series and convexity degree, leading to a dimension theory which is catenary, and a tropical version of the Jordan–Hölder theorem for the relevant class of [Formula: see text]-kernels.

Photoreceptor disk membrane substructures by means of the complementary freeze-fracture-replica methods

1978 ◽  
Vol 36 (2) ◽  
pp. 156-157
Author(s):  
A. Tonosaki ◽  
M. Yamasaki ◽  
H. Washioka ◽  
J. Mizoguchi
Keyword(s):  
Cell Membranes ◽  
Freeze Fracture ◽  
Purple Membrane ◽  
Remarkable Case ◽  
Single Face ◽  
Disk Membrane ◽  
Associated Proteins ◽  
Photoreceptor Membranes

A vertebrate disk membrane is composed of 40 % lipids and 60 % proteins. Its fracture faces have been classed into the plasmic (PF) and exoplasmic faces (EF), complementary with each other, like those of most other types of cell membranes. The hypothesis assuming the PF particles as representing membrane-associated proteins has been challenged by serious questions if they in fact emerge from the crystalline formation or decoration effects during freezing and shadowing processes. This problem seems to be yet unanswered, despite the remarkable case of the purple membrane of Halobacterium, partly because most observations have been made on the replicas from a single face of specimen, and partly because, in the case of photoreceptor membranes, the conformation of a rhodopsin and its relatives remains yet uncertain. The former defect seems to be partially fulfilled with complementary replica methods.

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