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Questions tagged [tropical-geometry]

For questions related to tropical geometry.

Tropical geometry is essentially the piecewise-linear version of algebraic geometry, where algebraic varieties are replaced by polyhedral complexes. According to Grigory Mikhalkin, "tropical geometry describes worst possible degenerations of the complex structure on an $n$-fold $X$." It is a relatively new and growing field of mathematics that has strong applications in enumerative algebraic geometry.

98 questions
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Motivation for the study of amoebas.

What was the primary motivation for the study of the amoebas?
Gilles Bonnet
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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 rephrasing equations with "max" or "min"…
Brendan Mallery
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Tropical Machinery

Recently I heard of a recent field in mathematics called tropical geometry. Having read the wiki page on it it seems like it is combinatorial algebraic geometry. My question is what are the benefits of applying tropical geometry to problems in…
Eugene
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The structure theorem of Tropical geometry

The Structure Theorem of Tropical geometry states: Let $X$ be an irreducible $d$-dimensional subvariety of $\mathbb T^n$ . Then $\operatorname{trop}(X)$ is the support of a balanced weighted $Γ_{\rm val}$ -rational polyhedral complex pure of…
Mohan
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The tropical integers

Let \begin{align} \oplus_\mathbb{N} &= + \\ 0_\mathbb{N} &= 0 \\ \odot_\mathbb{N} &= \cdot \\ 1_\mathbb{N} &= 1 \end{align} Then $(\mathbb{N}, \oplus_\mathbb{N}, 0_\mathbb{N}, \odot_\mathbb{N}, 1_\mathbb{N})$ is the ordinary rig of…
user76284
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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 $t$. Then, the two largest ones among the three…
darij grinberg
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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 $>1$. Is there some other relation connecting these…
TheWanderer
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Understanding divisors in tropical geometry

I've been studying tropical geometry recently, and I found the divisor idea, but I don't really get it, and also, I didn't find good examples. If you can explain me this idea, and give me some intuitive examples, I'd be so grateful. Thanks in…
Gyadso
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Defining Derivative on Tropical Semiring

I have tried to find references that are related to calculus on tropical semiring, but I was not able to find appropriate references. So, I used to Thompson's approach to define deriviative on tropical semiring as follows: $\;$ Let $ \forall x \in…
30412
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Are tropical polynomials "tropically" differentiable?

In this question, one asks whether tropical polynomials are differentiable, and the answer is "yes" in the classical sense. However, I'm wondering whether tropical polynomials are "tropically differentiable". This means that is there any tropical…
Ryoungwoo Jang
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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 empirical predictions in applied sciences. Recently, for…
Laurent Duval
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Why are linear functions the natural analogue of exponential functions in a tropical semiring?

I was reading a blog post on the Fourier transform and the Legendre transform as being the same thing over different semirings, in which the author says It's not obvious how to interpret the exponential function in (R',min,+) but it turns out…
grautur
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Semirings which cannot be extended to semifields

Definitions By a commutative $\textit{semiring}$ (with 1 and without 0), I mean a triple $(S,+,\cdot)$ where $(S,\cdot)$ is a commutative monoid, $(S,+)$ is a commutative semigroup, and $\cdot$ distributes over $+$. If $(S,\cdot)$ is in fact a…
Antoine de Saint Germain
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Tropical geometry for high school

Recently, I've found out about tropical geometry and its application in algebraic geometry. As I read about it, I thought that this would be a subject that can be taught in school as it is very simple to understand what is the tropical semiring is…
bml64
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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: $\mathbf{Q}:=\mathrm{Frac}(\mathbf{Z})$; the $p$-adic integers $\mathbf{Z}_p$ and…
Emily
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