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Questions tagged [tropical-geometry]

For questions related to tropical geometry.

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Tropical algebra and tropical determinate

We know that a determinant of the matrix is the sum of permutation of the elements Multiplied by "(-1)" number of times equal to changes of matrix rows. Why the signal is not important in defining the ...
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Construct Polytopes from Arbitrary Edge Sets

I was wondering if given an arbitrary set of edges in 3-space, if there is a way to determine if those edges can be used to construct a single polytope (allowing for the scaling of edge lengths). In $\...
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56 views

Are tropical polynomials differentiable?

I know that for a function $f$ to be differentiable at $a$, the following equality must hold. $$f'(a)=\lim_{x\to a}{\frac{f(x)-f(a)}{x-a}}$$ I also know that the left hand limit differs from the ...
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28 views

Tropical vector bundles?

Is there a nice notion of tropical vector bundles? A cursory google turns up https://arxiv.org/abs/0911.2909 but it is unclear to me how these objects are like usual vector bundles.
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24 views

Tropical Lie algebra

In this article https://arxiv.org/pdf/1705.01075.pdf are we mean that Lie semialgebras over semirings with a negation map is tropical version of Lie algebra?. And what we do when we define lifting? ...
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18 views

Tropical algebra and tropical semiring

I want to know what is the difference between Tropical algebra and min-plus algebra and the difference between Tropical semiring and semiring . I need Reference explains these differences and ...
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20 views

Tropical algebra

When we want to study tropical version to any algebraic structure we need to apply tropicalization. what is the difference between lifting (ring and map such that some properties hold) and ...
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9 views

Tropical algebraic structure

What is the difference between tropical lie semialgebra and lie semialgebra with anegation map? and How can I build another algebraic structure in tropical algebra?
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62 views

Definition of Tropical Hypersurface

Given the tropical semiring $(\mathbb{T},\oplus,\otimes)$, a tropical hypersurface associated to a tropical polynomial is the set of points where it is non-differentiable. I'm wondering how ...
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91 views

Combinatorial or polyhedral description for tropicalization of the positive subset of a real linear subspace

I had two questions: one regarding a definition of tropicalized linear subspaces, and the second about how to find similar characterizations for the logarithmic limit set of the positive subset of a ...
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1answer
121 views

Very affine varieties form basis of Zarisky Topology

I am currently working through a paper (related to tropical geometry, but this is not important in the following context) which utilizes the concept of very affine varieties in the following way (I'll ...
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71 views

Can one induce the Dual Subdivision of Tropical Signomials Similarly to Tropical Polynomials?

Preliminaries. Consider the semiring $\mathbb{T} = \{\mathbb{R} \cup \{-\infty\}, \oplus, \odot \}$, where the $a \oplus b = \max\{a,b\}$ and that $a \odot b = a + b$. The addition and scalar ...
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71 views

Maximal polyhedra of a polyhedral complex

According to this paper by Diane Maclagan (https://arxiv.org/abs/1207.1925, p.10), a polyhedral complex is pure if the dimension of every maximal polyhedron is the same, which is shown in the figure ...
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74 views

Tropical geometry

I studied algebraic geometry and looked for the development of some concepts and found tropical geometry is an important field. I need a simple reference to explain tropical geometry and its ...
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28 views

Tropical geometry over fields other than the real numbers

Tropical geometry studies the semiring $(\mathbb R, \min,+)$. I'm wondering if anything is lost by studying semirings over some field other than $\mathbb R$? It's obvious that finite fields and $\...
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1answer
34 views

Add conditionals to plot in sage

I am trying to implement an algorithm which plots a tropical curve given a tropical polynomial $P(x,y)$. So for instance the graph of $P(x,y)=x+y+0$ should be a union of three half rays, starting at ...
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1answer
92 views

definition of tropicalization map

I'm fairly new to the whole topic of tropicalization and I'm having trouble to even understand the following introductory definition: [General setting: Let $K$ be a field which is complete with ...
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1answer
181 views

Min-Plus Algebra (Operation Question)

I was reading about distance matrices, and they discuss a concept known as min-plus algebra. I am unsure if the operator + means addition in this context, and also unsure what operation goes in the ...
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50 views

Algorithm to find the convex polyhedron

In the context of my research, I've encountered the problem described below. I wonder if this is a standard problem in convex optimization or linear programming and would appreciate any comments. I ...
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1answer
63 views

Can median be expressed using linear combinations and max?

The median of two numbers is their mean, $(a+b)/2$, the median of three numbers is their sum less their maximum and minimum, i.e. $$Median(a,b,c) = a + b + c - \max(a,\max(b,c)) - \min(a,\min(b,c)) $...
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343 views

Definition/Example Polyhedral Geometry: Lineality space, pure, simplicial

I'm having some trouble understanding definitions concerning Polyhedral Geometry (more specific, tropical geometry, but that doesn't make a difference right now.) A cone is called simplicial iff all ...
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1answer
447 views

Four-point condition for tree distances: is there a detropicalization proof?

Theorem 1. Let $G$ be a tree. Let $x$, $y$, $z$ and $w$ be four vertices of $G$. For any two vertices $s$ and $t$ of $G$, let $d \left(s, t\right)$ denote the minimum length of a path from $s$ to $...
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1answer
255 views

Kapranov's Theorem for tropical hypersurfaces: Understanding closure

I am working with the book Introduction to Tropical Geometry by Sturmfels and Maclagan, and am trying to understand Kapranov's Theorem (Theorem 3.1.3 in the book), which states (I only mention the ...
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258 views

definition of specialization vs degeneration : power series vs polynomial

While reading the literature on toric degeneration problems, I notice that sometimes the degeneration problem is described as following: Whether there exist a flat family $\pi:\mathcal{F} \to \text{...
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1answer
28 views

From exponential to squareless monomial

Currently, I am going through Geometry of the Restricted Boltzmann Machine by Cueto et al. In section 2, the authors defines $\psi (v, h)$ as follows. $$ \psi (v, h) = e^{(h^\top W v + b^\top v + c^\...
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69 views

What is 'identifying restricted Boltzmann machine'?

Currently, I am going through Geometry of the Restricted Boltzmann Machine by Cueto et al. In this paper (in the abstract, multiple places in Section 1, etc.), the authors use the concept of the ...
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1answer
98 views

Explain the Zero Tension Condition of a Tropical Curve $\mathcal{H}(x)$

Right now I am studying tropical mathematics and I just arrived at a statement that I'm having trouble understanding. Here's how its stated: For a tropic polynomial in two variables. $p$, the ...
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1answer
48 views

A question about the solutions of $x+y-1$

Let us try and tropicalize $f(x,y)=x+y-1$. Ley $X=V(f)\subset G_m^2$. SO $X$ is $\Bbb{P}^1$ minus $3$ points. How is this? I understand that $(G_m)^2$ is a torus in two variables. I suppose $V(f)$ ...
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2answers
60 views

Is there some notation and name for this norm?

Given a field $K$ with an absolute value (you may imagine $\mathbb{R},\mathbb{C}$ or a $p$-adic field), I wonder if there is some notation and name (like $\|\|_\infty$ for the infinite [or supremum, ...
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228 views

Newton Polytopes of Tropical Polynomials

recently I've been reading about tropical polynomials and stumbled upon Andreas Gathmann's lecture notes. In section 1.4, it is stated that a tropical polynomial in the form of $$ g(x, y) = \max_i \...
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45 views

Tropical top self-intersection numbers of boundary divisors in toroidal embeddings

Let $X_0 \subset X$ a toroidal embedding without self intersections and denote by $\overline{\Sigma}$ its corresponding (weakly embedded) extended conical simplicial complex. Let $D$ be a divisor on $...
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215 views

Software Plotting Tropical Curves

I'm an undergrad student and currently working on my paper focusing on Tropical Math. Can anyone suggest for any software that could plot tropical curves easily? Help please!
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The standard tropical lines - Pointwise valuation

Let $K$ be an algebraically closed field with valuation, and its value group $\Gamma_\text{val}^n$ is dense in $\mathbb{R}$. Consider the polynomial $ f(x,y)=x+y+1. $ It can easy by shown that the ...
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28 views

The Variety X(K) as a subset of the analytification of X

I am working with Sam Payne's artical Analytification is the limit of all tropicalizations, and because of my limited understanding of analytification it gives me some difficulties. The article: ...
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126 views

Tropicalization of a line in the projective plane P^2

Lets assume that our field $K$ is the Puiseux series. I have been working with tropicalization from the book "Introduction to tropical geometry" link : https://homepages.warwick.ac.uk/staff/D....
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26 views

Condition for global cascade

Assume a unidirectional, unweighted network generated according to a degree distribution. Each node is given a value between 0 and 1 called threshold $\phi$. We topple some nodes, the neighbours will ...
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84 views

Some Basic Properties of Tropical Amoebas

I am working through Sturmfels' new book on Tropical Geometry with another student, and we are stuck at a pretty important concept, namely the basic properties of amoebas. Let me reproduce a bit of ...
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1answer
290 views

Initial form of a polynomial

I am reading some tropical geometry and came up with the concept of the initial form of a polynomial. The definition says that the initial form of f with respecto to a weight vector $w \in \mathbb{R}^{...
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49 views

Non-convex subdivisions of newton polygon of a tropical plane curve

This is probably an elementary question, but how come the Newton polygon of a tropical plane curve can't have non-convex subdivisions? Or can it?
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60 views

Convex Hull given set of planes.

If I have some finite amount of planes, for example \begin{equation} z_1=2x, \\ z_2=2y, \\ z_3=3+x+y, \\ z_4= 2+x, \\ z_5=2+y \\ z_6 =3 \end{equation} And I wish to find the convex hull in order to ...
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231 views

Zeros of a Tropical polynomial

Consider the Tropical semiring $(\mathbb{R}\cup\{-\infty\},\max,+)$. We define $x$ as a zero of a tropical polynomial $f(x)$ if $f$ attains its maximum twice at point $x$ in its linear parts. Why is ...
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1answer
90 views

Zeroes of prime polynomials in the algebraic torus (A Hilbert's Nullstellensatz for Laurent polynomials?)

Let $Q\in\mathbb C[z_1,\dots,z_D]$ be a prime polynomial and let $Z(Q)$ be the algebraic hypersurface of its zeroes. Assume that $P\in\mathbb C[z_1,\dots,z_D]$ is a polynomial which has zeroes at ...
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97 views

Amoeba of a line in the plane: An example

Let $z+w+1=0$ a line in $\mathbb{C}^2$ and let $x=log|z| \ge 0$ and $y=log|w|$. I have to show that $$ log(e^x-1) \le y \le 1+e^x $$ But I can't do it! Can you help me, please?
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134 views

Maximal weight of path in directed graph (using max-plus algebra)

I am working with matrices over the max-plus algebra $(\mathbb{R}_\max,\oplus,\otimes)$. For $A \in \mathbb{R}_\max^{n\times n}$, the graph $\mathcal{G}(A)$ has vertex set $\{1,\dots,n\}$ and edges $(...
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614 views

Some questions about tropical geometry: graphs of tropical curves.

I am reading the file about tropical geometry. I have some questions about the file. The questions are in the following. On page 33 of the file, why the tropical version of $$ 0.001 + 1000 x + 100 x^...
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“No cone of this fan contains a nonzero linear space”

On page 393 of this paper, Speyer and Sturmfels produce a quotient of a particular geometric object (which happens to be a polyhedral fan), and claim that, as in the question title, "no cone in this ...
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727 views

Riemann-Hurwitz formula generalization in higher dimension

In "Basic algebraic geometry 2", Shafarevich finds a relation between the Euler characteristic and the genus of the curve. At page 139 he says that there's no analogue for varieties of dimension $>...
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144 views

Subjects or recent progress in Tropical geometry or similar suitable for undergraduate investigation

I'm worried this might not be fitting for this forum, but it's basically a literature and reference request. I'm looking to do a project in algebra where we are supposed to research some topic (...
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143 views

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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147 views

How Max plus algebra is different from conventional algebra?

Here I have some basic questions about max-plus algebra. How this is useful? Why we need to define a different algebra? What aspects are highlighted in this which were untouched in conventional ...