Questions tagged [tropical-geometry]
For questions about tropical geometry, piece-wise linear functions with integer slopes, tropical degenerations and applications of tropical geometry, max-plus algebra, and tropical semifields.
94 questions
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Is there a Galois theory for $\mathbb R_{\geq 0}$?
The broadest version of my question is the following:
Where can I find algebrogeometric abstract nonsense that handles "rings" and "fields" like $\mathbb R_{\geq 0}$ in which ...
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2
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0
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holomorphic curves in almost toric fibration and their relation to tropical curves
My goal is to get better understanding how the projection of holomorphic curves converge to tropical disks.
We are given an almost toric fibration $X\rightarrow B$ with special Lagrangian fibers with ...
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What can we say about tropical maps $\mathbb{P}^1 \to A$ for an Abelian variety $A$?
It's well known that maps $\mathbb{P}^1_\mathbb{C} \to A$ are constant for any Abelian variety $A$ (in fact, for any complex torus).
Is there any similar statement in the tropical case? Naively, the ...
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1
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Is there a tropical geometric proof for counting genus g curves in any n dimensional projective space?
Consider the following question: Let $X$ be a compact complex manifold
and $\beta \in H_2(X, \mathbb{Z})$ a fixed homology class. Let
$\mu_1, \mu_2, \ldots, \mu_k$ denote certain generic ...
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1
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Are schemes which agree on open set and its complement equal? - w/ applications to initial ideals/tropical basis
I appreciate the comments so far and am modifying based on something closer to the problem I'm interested in. I started out with something far too general.
This is probably easy, but I have been ...
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3
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1
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Looking for interesting, natural models of this algebraic theory in which $x^\dagger$ is not always the multiplicative inverse of $x$
It is easy to think up interesting, natural models of the algebraic theory presented as follows, such that in these models, $x^\dagger$ is always the multiplicative inverse of $x$.
Question. What ...
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6
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1
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Hodge Bundles on Tropical Spaces
I am not sure that this question even makes sense, which I suppose is part of the questions itself.
In any case, I attended a talk recently wherin there was some discussion about a "tropical ...
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1
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Tropical polynomial Positivstellensatz
In real algebraic geometry, Stengle's Positivstellensatz can be used to characterize polynomials that are positive on a semialgebraic set.
Say that a tropical semialgebraic set is a subset of $\...
- 1,590
2
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1
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212
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group structure on (subsets of) tropicalizations of Abelian varieties
In this paper Vigeland shows how one can define a group law on subset of a tropical elliptic curve, so that this group is homeomorphic to $S^1$. It is not clear to me what is the relationship between ...
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2
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483
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Can Hausdorff dimension make sets into a Tropical Semiring?
If $X$ is a metric space, we construct Hausdorff $d$ measure from the outer measure
\begin{equation}
H^d(U) = \lim_{\delta \to 0}\inf\left\{\sum_{i=1}^\infty \left(\text{diam}(E_i)\right)^d : \...
- 1,124
1
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1
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Boundary of a tropical variety.
For a variety X (over some proper fields), if Trop(X) is a tropicalization of X, then
we know that Trop(X) is a polyhedral complex. If we consider the interior of the support of that polyhedral ...
user42154
7
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1
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820
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Has this generalization of a determinant (assigning multiplicities to the rows) been studied?
I'm working on some questions in tropical geometry, and my problem led me to create the following generalization of a determinant:
Let $A$ be an $m \times n$ matrix with $m \le n$, and positive ...
- 1,509
2
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1
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387
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Gradient of Ronkin function
I have a complex curve $P(z,w)=z+w-1=0$. I get the amoeba map
$$(z,w)\rightarrow (\log|z|,\log|w|)$$ of this curve. It's look like this http://en.wikipedia.org/wiki/Amoeba_(mathematics) (the first ...
0
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1
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Volume in tropical geometry as compared to volume in convex geometry
In tropical geometry, is there a notion of volume. Maybe one with some of the properties as found in classical convex geometry? If so, is there a good reference that elaborates on this question.
I ...
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What is Tropicalization, and how is it applied
My question is:
What is Tropicalization, how is it done, and what are some basic applications of it?
motivation
I am interested especially in how questions about enumerative algebraic geometry ...
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5
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0
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710
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$L_1$ and $L_\infty$ Voronoi diagrams and tropical geometry: Connection?
I just realized that there is a visual similarity between Voronoi diagrams in
the $L_1$ and $L_\infty$ metrics (two images below)
Left: O'Rourke, "Computing Relative Neighborhood graph ...
- 151k
8
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0
answers
204
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Chow ring of extended tropicalizations
In Allermann-Rau '09, the authors define the Chow groups of an arbitrary abstract tropical cycle. In particular, one may take the tropical cycle to be the tropicalization of a subvariety of a torus. ...
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0
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327
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Asymptotics vs Puiseux series
Define asymptotic as a class of sequences {$ x_i$},$_{i\in\mathbb N}$ modulo equivalence {$x_i$}={$y_i$} if $\lim_{i\to\infty} (x_i/y_i)=c\in\mathbb R,c\ne 0$.
More, we define $X= \{x_i\} \lt Y= \{ ...
- 5,063
1
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0
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102
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Notion of transversality over the field of Puiseux series.
To a given a Laurent polynomial $f$ over the field of Puiseux seris with parameter $t$, $f \in \mathbb{C} \lbrace\lbrace t \rbrace\rbrace[z_1^{\pm1},...,z_n^{\pm 1}]$, one can associate the ...
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424
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Tropicalization of the Grassmannian
Let $Trop(Gr(m,n))$ denote the tropicalization of the grassmannian $Gr(n,m)$. Let $\phi^m : \mathbb R^{n \choose 2} \rightarrow \mathbb R^{n \choose m}$ such that $X_{i,j} \rightarrow X_{i_1,...,i_m}$...
- 1,277
4
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1
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Family of hypersurfaces in (C^*)^2 corresponding to tropical family
Edit: I realize the mathematics below is lacking a precise phrasing. I hope that the intuitiion behind the question is clear enough that a reader will understand the question and provide guidance. The ...
2
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1
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594
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Picture of a 3 dimensional amoeba.
On Wikipedia there some pictures of two dimensional amoebas (Thanks to Oleg Alexandrov for the pictures and the Matlab code he gives to build them). I was wondering if somewhere there are pictures ...
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852
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Which manifolds decompose into pants?
In this nice paper Mikhalkin uses certain (more geometrical than algebraic) aspects of tropical geometry to prove that every complex projective hypersurface in $\mathbb C \mathbb P ^n$ decomposes as a ...
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3
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1
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How to recover toric invariants tropically?
My excuses in advance in case my question is too vague (which is mainly due to the fact that I'm not really familiar with tropical/toric geometry, but at least I still believe that the content of the ...
- 2,029
26
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1
answer
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Important open questions in the field of Tropical geometry
What are some of the important unanswered questions in the field of tropical geometry?
2
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0
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164
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Algorithms for "Ideals" in polynomial algebras over the max-plus semi-ring
I'm a beginner in tropical geometry, and I'm running into the following question:
In the usual polynomial ring over a field, one has algorithms (i.e. using a Groebner basis) for determining whether ...
- 1,509
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1
answer
614
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Hypersurfaces in Toric Varieties, Help understand a proof from Mikhalkin's paper
Hello,
in G. Mikhalkin's Papaer "DECOMPOSITION INTO PAIRS-OF-PANTS FOR
COMPLEX ALGEBRAIC HYPERSURFACES":
http://arxiv.org/pdf/math/0205011.pdf
There is a lemma about the relation between intersection ...
- 1,893
21
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2
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6k
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Learning Tropical geometry
I'm interested in learning tropical geometry. But my background in algebraic geometry is limited. I know basic facts about varieties in affine and projective space, but nothing about sheaves, schemes ...
- 1,277
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5
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Why tropical geometry?
Tropical geometry can be described as "algebraic geometry" over the semifield $\mathbb{T}$ of tropical numbers. As a set, $\mathbb{T}=\mathbb{R}\cup \{ -\infty\}$; this is endowed with addition being ...
9
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1
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615
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Properties from Tropical Geometry that do not imply their algebraic counterpart.
One of the motivations to study tropical geometry is that there are some hard Algebraic Questions that can be answered by proving them in the Tropical World. For example one can show that tropical ...
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46
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Mikhalkin's tropical schemes versus Durov's tropical schemes
In Mikhalkin's unfinished draft book on tropical geometry, (available here) (page 26) he defines a notion of tropical schemes. It seems to me that this definition is not just a wholesale adaptation of ...
- 6,229
1
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0
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138
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Bases of Ideals With no Monomials
Let $K$ be an algebraically closed field and $K[\underline{x}]$ its ring of polynomials in $n$ variables $x_1,\cdots, x_n$. Let $J\leq K[\underline{x}]$ be an ideal such that there are no monomials in ...
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Intersection of curves on projective toric surface and some enumerative questions
Reading on the tropical approach to enumerative geometry I have come across the claim:
given a projective toric surface from a polygon P, we can consider a tautological bundle of algebraic / ...
- 1,893
4
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0
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577
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Tropical Properties From Algebraic Geometry
What properties of tropical geometry (Starting from a valued Field) can be proven to be true using their analogue in algebraic geometry? For example, using the valuation on the Puiseux series $\mathbb{...
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1
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Tropical homological algebra
Has anyone out there thought about homological algebra over the tropical semifield $\mathbb{T}$? For example, I'm interested in the Hochschild homology and cyclic homology of tropical algebras, if ...
- 6,229
3
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1
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515
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When is the quotient of a tropical curve also a tropical curve?
A plane tropical curve $\Gamma$ is the corner locus of a tropical polynomial in two variables. That is, it is the set of points at which the tropical polynomial, which is a piecewise-linear concave ...
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4
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4k
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What can we learn from the tropicalization of an algebraic variety?
I often hear people speaking of the many connections between algebraic varieties and tropical geometry and how geometric information about a variety can be read off from the associated tropical ...
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2
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1k
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Weight filtration and Hodge theory for tropical varieties
Many concepts is algebraic geometry have tropical analogues.
Question: Is there an analogue of the weight filtration or Hodge filtration for tropical varieties?
A tropical curve ends up being ...
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1
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How to Tropicalize a Polynomial in Two Variables?
Trying to draw the Amoeba
With Mathematica, it's possible to graph $e^{-k x} + e^{-k y} = 1,e^{-k x} - e^{-k y} = 1$ and $e^{-k x} + e^{-k y} = -1$ to get the amoeba of 1 + x + y when k = 1. Then by ...
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279
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Tropical varieties correspondence to varieties over a non-archimedean valuation field.
I am a mathematical physicist and I am studying certain discrete dynamical systems defined in terms of piecewise linear mappings, which may be expressed in terms of expressions over the max-plus semi-...
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Simple topological question on taking complements inside a simplex
We would like to know if the following claim is true:
(If you don't know the definition of a tropical hyperplane, then please consider the case when d=3)
Let $P_1,\cdots,P_d$ be full dimensional (...
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2
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647
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Is anything known about the enumeration of degree d, genus g curves in CP^2 where g >1 ?
I wanted to know if there is something analogous to Kontsevich's recursion formula for
enumeration of genus zero curves in $\mathbb{C}\mathbb{P}^2$, for higher genus curves.
There is a
similar ...
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5
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4k
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How should one approach tropical mathematics?
Let me preface this by saying that my background is pretty meagre (i.e. solid undergrad). However, a few months ago I came across Litvinov - The Maslov dequantization, idempotent and tropical ...
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4
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How is tropicalization like taking the classical limit?
There is a folk — I can't call it a theorem — "fact" that the mathematical relationship between Complex and Tropical geometry is analogous to the physical relationship between Quantum and ...
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