scholarly journals Pure dimension and projectivity of tropical polytopes

2016 ◽  
Vol 303 ◽  
pp. 1236-1263 ◽  
Author(s):  
Zur Izhakian ◽  
Marianne Johnson ◽  
Mark Kambites
Keyword(s):  
Pure Dimension ◽  
Tropical Polytopes

Higher Order Tangents to Analytic Varieties along Curves

2003 ◽  
Vol 55 (1) ◽  
pp. 64-90 ◽  
Author(s):  
Rüdiger W. Braun ◽  
Reinhold Meise ◽  
B. A. Taylor
Keyword(s):  
Differential Operators ◽  
Necessary Conditions ◽  
Forthcoming Paper ◽  
Puiseux Series ◽  
Open Set ◽  
Partial Differential Operators ◽  
Real Analytic Functions ◽  
Pure Dimension ◽  
Analytic Varieties ◽  
Real Analytic

AbstractLet V be an analytic variety in some open set in which contains the origin and which is purely k-dimensional. For a curve γ in , defined by a convergent Puiseux series and satisfying γ(0) = 0, and d ≥ 1, define Vt := t−d(V − (t)). Then the currents defined by Vt converge to a limit current Tγ,d[V] as t tends to zero. Tγ,d[V] is either zero or its support is an algebraic variety of pure dimension k in . Properties of such limit currents and examples are presented. These results will be applied in a forthcoming paper to derive necessary conditions for varieties satisfying the local Phragmén-Lindelöf condition that was used by Hörmander to characterize the constant coefficient partial differential operators which act surjectively on the space of all real analytic functions on .

scholarly journals Approximating the volume of tropical polytopes is difficult

2019 ◽  
Vol 29 (02) ◽  
pp. 357-389 ◽  
Author(s):  
Stéphane Gaubert ◽  
Marie MacCaig
Keyword(s):  
Approximation Algorithm ◽  
Integer Points ◽  
Dimension Formula ◽  
Parallel Body ◽  
Tropical Polytopes

We investigate the complexity of counting the number of integer points in tropical polytopes, and the complexity of calculating their volume. We study the tropical analogue of the outer parallel body and establish bounds for its volume. We deduce that there is no approximation algorithm of factor [Formula: see text] for the volume of a tropical polytope given by [Formula: see text] for the volume of a tropical polytope given by [Formula: see text] vertices in a space of dimension [Formula: see text], unless P[Formula: see text]NP. Neither is there such an approximation algorithm for counting the number of integer points in tropical polytopes described by vertices. It follows that approximating these values for tropical polytopes is more difficult than for classical polytopes. Our proofs use a reduction from the problem of calculating the tropical rank.

scholarly journals Triangulations of $\Delta_{n-1} \times \Delta_{d-1}$ and Tropical Oriented Matroids

2011 ◽  
Vol DMTCS Proceedings vol. AO,... (Proceedings) ◽  
Author(s):  
Suho Oh ◽  
Hwanchul Yoo
Keyword(s):  
Oriented Matroids ◽  
Oriented Matroid ◽  
Nous Proposons ◽  
Combinatorial Objects ◽  
New Class ◽  
Nous Montrons ◽  
International Audience ◽  
Tropical Polytopes

International audience Develin and Sturmfels showed that regular triangulations of $\Delta_{n-1} \times \Delta_{d-1}$ can be thought of as tropical polytopes. Tropical oriented matroids were defined by Ardila and Develin, and were conjectured to be in bijection with all subdivisions of $\Delta_{n-1} \times \Delta_{d-1}$. In this paper, we show that any triangulation of $\Delta_{n-1} \times \Delta_{d-1}$ encodes a tropical oriented matroid. We also suggest a new class of combinatorial objects that may describe all subdivisions of a bigger class of polytopes. Develin et Sturmfels ont montré que les triangulations de $\Delta_{n-1} \times \Delta_{d-1}$ peuvent être considérées comme des polytopes tropicaux. Les matroïdes orientés tropicaux ont été définis par Ardila et Develin, et ils ont été conjecturés être en bijection avec les subdivisions de $\Delta_{n-1} \times \Delta_{d-1}$. Dans cet article, nous montrons que toute triangulation de $\Delta_{n-1} \times \Delta_{d-1}$ encode un matroïde orienté tropical. De plus, nous proposons une nouvelle classe d'objets combinatoires qui peuvent décrire toutes les subdivisions d'une plus grande classe de polytopes.

The pure dimension of rings and modules

1982 ◽  
Vol 37 (5) ◽  
pp. 170-171
Author(s):  
S N Fedin
Keyword(s):  
Pure Dimension

scholarly journals Comparison of Poisson structures on moduli spaces

2021 ◽  
Author(s):  
Indranil Biswas ◽  
Francesco Bottacin ◽  
Tomás L. Gómez
Keyword(s):  
Moduli Space ◽  
Moduli Spaces ◽  
Poisson Structures ◽  
Projective Curve ◽  
Smooth Projective Curve ◽  
Algebraic Vector ◽  
Pure Dimension ◽  
Stable Sheaves ◽  
Fixed Rank ◽  
Dimension One

AbstractLet X be a complex irreducible smooth projective curve, and let $${{\mathbb {L}}}$$ L be an algebraic line bundle on X with a nonzero section $$\sigma _0$$ σ 0 . Let $${\mathcal {M}}$$ M denote the moduli space of stable Hitchin pairs $$(E,\, \theta )$$ ( E , θ ) , where E is an algebraic vector bundle on X of fixed rank r and degree $$\delta $$ δ , and $$\theta \, \in \, H^0(X,\, {\mathcal {E}nd}(E)\otimes K_X\otimes {{\mathbb {L}}})$$ θ ∈ H 0 ( X , E n d ( E ) ⊗ K X ⊗ L ) . Associating to every stable Hitchin pair its spectral data, an isomorphism of $${\mathcal {M}}$$ M with a moduli space $${\mathcal {P}}$$ P of stable sheaves of pure dimension one on the total space of $$K_X\otimes {{\mathbb {L}}}$$ K X ⊗ L is obtained. Both the moduli spaces $${\mathcal {P}}$$ P and $${\mathcal {M}}$$ M are equipped with algebraic Poisson structures, which are constructed using $$\sigma _0$$ σ 0 . Here we prove that the above isomorphism between $${\mathcal {P}}$$ P and $${\mathcal {M}}$$ M preserves the Poisson structures.

scholarly journals Strong disk property for domains in open Riemann surfaces

Filomat ◽  
2016 ◽  
Vol 30 (7) ◽  
pp. 1711-1716
Author(s):  
Makoto Abe ◽  
Gou Nakamura
Keyword(s):  
Riemann Surface ◽  
Riemann Surfaces ◽  
Approximation Property ◽  
Stein Space ◽  
Open Riemann Surface ◽  
Holomorphic Approximation ◽  
Open Set ◽  
Pure Dimension

We study the relation between the holomorphic approximation property and the strong disk property for an open set of an open Riemann surface or a Stein space of pure dimension 1.

Absolute Purity

1975 ◽  
Vol 27 (1) ◽  
pp. 6-10 ◽  
Author(s):  
David J. Fieldhouse
Keyword(s):  
Finitely Generated ◽  
Pure Dimension

Throughout this paper we use the Bourbaki [1] conventions for rings and modules: all rings are associative but not necessarily commutative and have a 1; all modules are unital.Our purpose is to extend and simplify some recent results of Maddox [7], Megibben [8], Enochs [3], and the author [5] on absolutely pure modules by introducing several new dimensions, and using the absolutely pure dimension introduced by the author in [6], This completes some work on character modules and dimension in [5] and [6].An A -module will be called an FFR-module if and only if it has a resolution by finitely generated free A -modules.

scholarly journals Tropical Polytopes and Cellular Resolutions

2007 ◽  
Vol 16 (3) ◽  
pp. 277-291 ◽  
Author(s):  
Mike Develin ◽  
Josephine Yu
Keyword(s):  
Cellular Resolutions ◽  
Tropical Polytopes

scholarly journals Convexity of tropical polytopes

2015 ◽  
Vol 485 ◽  
pp. 531-544 ◽  
Author(s):  
Marianne Johnson ◽  
Mark Kambites
Keyword(s):  
Tropical Polytopes

scholarly journals Defining integer-valued functions in rings of continuous definable functions over a topological field

2020 ◽  
Vol 20 (03) ◽  
pp. 2050014
Author(s):  
Luck Darnière ◽  
Marcus Tressl
Keyword(s):  
Definable Set ◽  
Ordered Field ◽  
Dimension Formula ◽  
Valued Field ◽  
Pure Dimension ◽  
Isolated Points ◽  
Topological Field ◽  
Structure Formula ◽  
Definable Functions ◽  
Ring Of Functions

Let [Formula: see text] be an expansion of either an ordered field [Formula: see text], or a valued field [Formula: see text]. Given a definable set [Formula: see text] let [Formula: see text] be the ring of continuous definable functions from [Formula: see text] to [Formula: see text]. Under very mild assumptions on the geometry of [Formula: see text] and on the structure [Formula: see text], in particular when [Formula: see text] is [Formula: see text]-minimal or [Formula: see text]-minimal, or an expansion of a local field, we prove that the ring of integers [Formula: see text] is interpretable in [Formula: see text]. If [Formula: see text] is [Formula: see text]-minimal and [Formula: see text] is definably connected of pure dimension [Formula: see text], then [Formula: see text] defines the subring [Formula: see text]. If [Formula: see text] is [Formula: see text]-minimal and [Formula: see text] has no isolated points, then there is a discrete ring [Formula: see text] contained in [Formula: see text] and naturally isomorphic to [Formula: see text], such that the ring of functions [Formula: see text] which take values in [Formula: see text] is definable in [Formula: see text].

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