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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.

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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 relevant…
johnnycrab
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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 tropicalization (which can be viewed as a morphism from the…
Mehrema
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Gradient of Ronkin function

I have a complex curve $P(z,w)=z+w-1=0$. I get the amoeba map $$(z,w)\rightarrow (\log|z|,\log|w|)$$ of this curve. It's look like this http://en.wikipedia.org/wiki/Amoeba_(mathematics) (the first picture). The Ronkin function is defined…
Vasili_Petrov
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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 versus max-plus

Is the max-plus algebra an example of tropical mathematics or is it an independent structure? How can we turn a max-plus algebra into tropical geometry or viceversa-if possible?
riemannium
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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!
Crunchy
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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 installed Polymake and SAGE (which should contain Gfan). I have…
ThLD
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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 geometry... according to nlab though it is also…
Ohio skateboarder . 7
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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?
flower
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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 $w \in \mathbb{R}^{n+1}$ is \begin{equation} in_w(f) =…
user284639
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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 algebra?
Sourabh
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Operations in Max-Plus Algebra

I am a undergraduate student. I am doing a project on tropical geometry and Max-Plus Algebra. So I started reading about max plus algebra. First thing I came across is that here we don't use conventional algebra. Instead of that we define a new…
Sourabh
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Tropical Geometry Mikhalkin

In the paper below it describes Proposition 3.2 as “ Let p ∈ C ∩C′ which is a vertex of neither C nor C′. Then the number of intersection points of Ct and C′t whose image under Logt converges to p is exactly equal to m( p).”. I don’t understand how…
Isabelle
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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 thanks .
Ohio skateboarder . 7
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Why is $\max\{0,x\}$ not a function at $x=0$?

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, this function apparently has those different…
Ohio skateboarder . 7
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