# 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). ...
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 ...
• 384
1 vote
40 views

### 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 ...
• 81
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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, ...
30 views

### 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 ...
37 views

### 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 ...
1 vote
48 views

### 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 ...
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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1 vote
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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 ...
70 views

### 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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1 vote
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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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1 vote
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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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### 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 ...
44 views

### 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
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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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1 vote
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### 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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1 vote
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