Questions tagged [tropical-geometry]

For questions related to tropical geometry.

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Why operations in tropical geometry defined as it is?

I'm a deep learning researcher, and these days studying algebraic geometry for my research and for personal interest. I'm noob to this field, and I found a research area called tropical geometry (TG). ...
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2 votes
1 answer
131 views

Reference for Hausdorff metric

Consider $\mathbb{R}^n$ with the Hausdorff metric, $$d(A,B) = \max(\sup_{a\in A}\inf_{b \in B} ||a-b||,\sup_{b\in B} \inf_{a\in A}||a-b||).$$ I'm looking for a reference containing a statement like ...
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Tropical Variety of $f = y^2 - x^3 + 3x^2 -2x$

I am currently working on tropical varieties and came across the lecture series by the Fields Institute (https://youtu.be/jMnJ3E0axjc). Here, the polynomial $f = y^2 - x^3 + 3x^2 -2x$ is considered. I ...
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How to plot tropical varieties?

I'm reading "Introduction to Tropical Geometry" by Maclagan and Sturmfels and wanted to do some plotting myself of different tropical varieties in both 2D and 3D. On my computer, I have ...
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What is the definition of $( C^*)^n$

Let $C^* = C\setminus \{ 0 \}$ where $C$ is a field, what does this notation $( C^*)^n$ mean? I kept encountering this notation many times where people don’t even bother giving its definition. For ...
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Why is $\max\{0,x\}$ not a function at $x=0$? [closed]

Please someone tell me why $\max\{0,x\}$ is not a function at $x=0$. I always learned that to fail the vertical line test the function's graph should have different values for the same input. However, ...
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Tropical hypersurface definition (book picture included).

I don't entirely get how this is picture is the locus of points where the function is not linear. I don't get what function is supposed to not be linear. I can take any help that I can get right now ...
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What are the most important applications of Kapranov's theorem in Tropical Geometry?

One of the most important and interesting theorems in tropical geometry is Kapranov's theorem which is extended to the fundamental theorem of tropical geometry( as far as I understood, but I am not ...
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Tropicalization $min(2x,2y,0)$ is a piecewise linear function.

Please instruct me on how to interpret $min(2x,2y,0)$ as a piecewise linear function. I have been working up to it for a while because I really wanted insight towards computing blowups in algebraic ...
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34 views

Tropical semiring and p-adic numbers correspondence

There are some striking new developments in tropical algebra and geometry. Recently I recalled the relation with valued fields from tropical semirings. However, is there any direct relationship ...
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How do I interpret the tropicalized curves (one of degree $6$ and one of degree $8$)?

For me, the topic of tropicalization is new and I am trying to understand what new insights and perspectives tropicalization could provide me on the following two curves (degree 6 and 8). So it is a ...
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3 votes
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What are the basic next steps to tropicalize a curve of degree $6$ (ideally using any CAS)?

First of all, I must admit that the topic of Tropicalization is new to me. I have a rough outline and I'm by no means asking for a ready-made solution here, but rather some pointers on how to get an ...
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4 votes
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125 views

Does the tropical semiring admit a universal property?

Each of the rings $\mathbf{Q}$, $\mathbf{Z}_p$, $\mathbf{Q}_p$, $\mathbf{R}$ admits a universal construction: the rationals are the field of fractions of the integers: $\mathbf{Q}:=\mathrm{Frac}(\...
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The dimension of the normal cone of a face in a polytope

Let $P$ is a polytope in $\mathbb{R}^n$ if $F$ is one of its faces of dimension $d$ then the dimension of its normal cone $\mathcal{N}(F)$ is $n-d$.\ This seems to be intuitively obvious but I can't ...
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Min-plus tensor broadcast addition?

Let $A \in \mathbb{R}^{(2n+1) \times (2n+1) \times n \times n}, B \in \mathbb{R}^{(2n+1)\times n}$. Define $C = A\bigoplus B, C \in \mathbb{R}^{(2n+1) \times (2n+1) \times n \times n}$. And $C_{ijkl} =...
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Express a combinatorics problem using polynomial coefficients.

Suppose, i have a combinatoric problem: given a set of $n$ tuples $Z =\{(a_1, b_1), \cdots, (a_n, b_n)\}, a_i \in \mathbb{R}^+, b_i \in \mathbb{R}^+$. My goal is to find a $S \in \mathcal{P}(Z)$, ...
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initial degenerations

I am trying to understand the article "Realization spaces for tropical fans" https://arxiv.org/pdf/0909.4582.pdf by Eric Katz and Sam Payne. Let $X\subset T$ be a subvariety of the torus. ...
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2 votes
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How many tropical polynomials give rise to the same variety? (reference request)

An n-variable polynomial $p(x)=\bigoplus_{i=1}^n \beta_i \textbf{x}^{\alpha_i}$ (here $\textbf{x}$ is the tuple $(x_1,x_2,...,x_n)\in \mathbb{R}^n$) in the tropical (max-plus) semiring has an ...
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3 votes
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Is the basic definition of the tropical semiring related to the elementary behaviour of degrees of polynomials?

Background: I'm a teacher preparing a precalculus course for a group of mature students with wildly differing backgrounds, so I constantly have an eye out for more sophisticated topics to point the ...
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1 vote
1 answer
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De-tropicalization of multiplication

Does there exist a function $f(x)$ so that $$ \lim_{t \to \infty} \frac{\log \,f(e^{tx})}{t} = x^2 ~~~ ? $$ Motivation: The "tropicalization" of a function $f(x_1,\dots,x_n)$ is the function ...
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4 votes
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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^{...
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6 votes
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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 ...
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3 votes
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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 ...
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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^...
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6 votes
1 answer
350 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_\...
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1 vote
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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/...
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2 votes
1 answer
84 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 ...
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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?
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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}...
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1 answer
137 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 $R-$...
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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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2 votes
1 answer
267 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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3 votes
0 answers
47 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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  • 121
0 votes
1 answer
327 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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4 votes
2 answers
158 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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1 answer
356 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 ...
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1 vote
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132 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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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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2 votes
1 answer
222 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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1 vote
1 answer
160 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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2 votes
1 answer
208 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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2 votes
1 answer
599 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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1 vote
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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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2 votes
2 answers
239 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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1 vote
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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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3 votes
1 answer
1k 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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4 votes
1 answer
439 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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2 votes
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367 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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1 answer
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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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0 votes
1 answer
80 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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