Questions tagged [tropical-geometry]
For questions related to tropical geometry.
65
questions
3
votes
1answer
60 views
Do exponents in tropical polynomials have to be integers?
Recently I've been learning about tropical geometry, and every time I see a definition of a tropical polynomial in e.g. $k$ variables $x_1,x_2,...,x_k$ such as $\bigoplus_{i=1}^n a_i x_1^{b_{i1}}x_2^{...
5
votes
0answers
94 views
Nontrivial applications of tropical mathematics to optimization (soft question)
I have been looking into tropical algebra/geometry for a research problem I'm working on in optimization. Tropical math gets referenced a lot in the literature, but it seems to me that its mostly just ...
2
votes
0answers
54 views
Is Tropicalization a Functor?
Sorry if this is a basic or incoherent question, I can't seem to find any literature (written at my level of understanding of category theory) that addresses the "tropicalization" (unsure if ...
2
votes
0answers
37 views
Linear regions of tropical rational functions via Newton polytopes
Let $f: \mathbb{R}^d \rightarrow \mathbb{R}$ be a tropical polynomial. Below we use the following notation: $a \oplus b = \text{max}(a,b)$, and if $\alpha_i=(\alpha_{i1}, \cdots, \alpha_{id})$ then $x^...
6
votes
1answer
309 views
The tropical integers
Let
\begin{align}
\oplus_\mathbb{N} &= + \\
0_\mathbb{N} &= 0 \\
\odot_\mathbb{N} &= \cdot \\
1_\mathbb{N} &= 1
\end{align}
Then $(\mathbb{N}, \oplus_\mathbb{N}, 0_\...
1
vote
0answers
21 views
Order of a point in the amoeba complement is integer valued
I'm currently trying to understand the paper "Laurent Determinants and Arrangements of
Hyperplane Amoebas" by Forsberg, Passare and Tsikh, which can be found here https://core.ac.uk/download/...
2
votes
1answer
35 views
Q about torus automorphism example
I am working through the book "introduction to tropical algebra" and page 70 and 71 example 2.2.10 states
"An automorphism of the torus is an invertible map specified by
Laurent monomials. Thus the ...
1
vote
0answers
35 views
Real number has valuation?
https://en.m.wikipedia.org/wiki/Valuation_(algebra)
I know rational number has valuation, p adic valuation but i think that`s not work in real number you know
anyone know?
1
vote
0answers
34 views
Two definitions of embedded tropicalization
Melody Chan gave two definitions of embedded tropicalization in her Lectures on Tropical Curves and Their Moduli Spaces. Let $K$ be a nonarchimedean valued field and $X$ be a subvariety of $(\mathbb{G}...
1
vote
1answer
91 views
Is the category of semimodules over a semiring abelian?
We consider a semimodule over commutative semiring. It's a well-known fact that the category of $R-$modules (over a ring) is an abelian category. Does this result generalize to the category of $R-$...
1
vote
0answers
22 views
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 $\...
1
vote
1answer
167 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 ...
2
votes
0answers
33 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 ...
0
votes
1answer
208 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 ...
4
votes
2answers
128 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 ...
1
vote
1answer
235 views
Very affine varieties form basis of Zariski 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 ...
1
vote
0answers
108 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 ...
0
votes
1answer
98 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 ...
2
votes
1answer
157 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 ...
1
vote
1answer
110 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 ...
2
votes
1answer
153 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 ...
2
votes
1answer
364 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 ...
1
vote
0answers
61 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 ...
1
vote
1answer
161 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.
$$Median(a,b,c) = a + b + c - \max(a,\max(b,c)) - \min(a,\min(b,c)) $...
1
vote
0answers
771 views
Definition/Example Polyhedral Geometry: Lineality space, pure, simplicial
I'm having some trouble understanding definitions concerning Polyhedral Geometry (more specific, tropical geometry, but that doesn't make a difference right now.)
A cone is called simplicial iff all ...
3
votes
1answer
844 views
Four-point condition for tree distances: is there a detropicalization proof?
Theorem 1. Let $G$ be a tree.
Let $x$, $y$, $z$ and $w$ be four vertices of $G$.
For any two vertices $s$ and $t$ of $G$, let $d \left(s, t\right)$ denote the minimum length of a path from $s$ to $...
4
votes
1answer
356 views
Kapranov's Theorem for tropical hypersurfaces: Understanding closure
I am working with the book Introduction to Tropical Geometry by Sturmfels and Maclagan, and am trying to understand Kapranov's Theorem (Theorem 3.1.3 in the book), which states (I only mention the ...
2
votes
0answers
352 views
definition of specialization vs degeneration : power series vs polynomial
While reading the literature on toric degeneration problems, I notice that sometimes the degeneration problem is described as following:
Whether there exist a flat family $\pi:\mathcal{F} \to \text{...
0
votes
1answer
31 views
From exponential to squareless monomial
Currently, I am going through Geometry of the Restricted Boltzmann Machine by Cueto et al.
In section 2, the authors defines $\psi (v, h)$ as follows.
$$
\psi (v, h) = e^{(h^\top W v + b^\top v + c^\...
0
votes
1answer
76 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 ...
0
votes
1answer
125 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 ...
0
votes
1answer
53 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 $V(f)$ ...
1
vote
2answers
71 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, ...
2
votes
0answers
256 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 \...
2
votes
0answers
53 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 $...
2
votes
0answers
341 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!
1
vote
0answers
45 views
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 ...
1
vote
0answers
36 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: ...
4
votes
0answers
157 views
Tropicalization of a line in the projective plane $\mathbb 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....
0
votes
0answers
29 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 ...
5
votes
1answer
880 views
Tropical geometry: practical applications?
In 1960, E. Wigner published a paper entitled "The Unreasonable Effectiveness of Mathematics in the Natural Sciences". Theoretical mathematical structures pave the way to further advances and ...
1
vote
0answers
96 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 ...
1
vote
1answer
502 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}^{...
1
vote
0answers
52 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?
1
vote
0answers
67 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 ...
4
votes
1answer
324 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 ...
2
votes
1answer
105 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 ...
3
votes
1answer
113 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
$$
But I can't do it! Can you help me, please?
0
votes
1answer
157 views
Maximal weight of path in directed graph (using max-plus algebra)
I am working with matrices over the max-plus algebra $(\mathbb{R}_\max,\oplus,\otimes)$. For $A \in \mathbb{R}_\max^{n\times n}$, the graph $\mathcal{G}(A)$ has vertex set $\{1,\dots,n\}$ and edges $(...
3
votes
1answer
854 views
Some questions about tropical geometry: graphs of tropical curves.
I am reading Kontsevich's slides about tropical geometry. I have some questions about the slides. The questions are the following:
On page 33 of the file, why the tropical version of
$$
0.001 + 1000 ...
