For questions related to tropical geometry.

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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!
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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 ...
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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: ...
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Tropicalization of a line in the projective plane 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....
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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 ...
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Linear and Gondran-Minoux independence in (Max,+) Algebra

I'm reading Peter Butkovic's monograph Max-Linear Systems: Theory and Algorithms. In Chapter 6, linear independence and Gondran-Minoux independence are introduced. A set of vectors $\{a_1, a_2, \dots ...
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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 ...
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32 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}^{...
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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?
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Weighted and Generic vector

I am writing a project about tropicalization of toric varieties and in my work I come up with 2 expressions I do not know the meaning of. A weighted vector This is an example of where it is used. ...
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38 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 ...
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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 ...
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Minimal equations defining a linear subspace (or: how I forgot linear algebra).

I have a question that may be trivial, but I just can't find the answer in the internet (nor in my head). Given a linear vector subspace of dimension $d$ and given its Plücker coordinates I can ...
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The definition of face - in regard to polyhedral fans

Question: How is face defined rigorously for bullet point 2 below. Definition: A convex polyhedral fan, $F$, of polyhedral cones, all living in the same vector space, requires two things: If $\...
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1answer
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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 ...
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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?
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64 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 $(...
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Some questions about tropical geometry: graphs of tropical curves.

I am reading the file about tropical geometry. I have some questions about the file. The questions are in the following. On page 33 of the file, why the tropical version of $$ 0.001 + 1000 x + 100 ...
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“No cone of this fan contains a nonzero linear space”

On page 393 of this paper, Speyer and Sturmfels produce a quotient of a particular geometric object (which happens to be a polyhedral fan), and claim that, as in the question title, "no cone in this ...
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266 views

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 $>...
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1answer
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Subjects or recent progress in Tropical geometry or similar suitable for undergraduate investigation

I'm worried this might not be fitting for this forum, but it's basically a literature and reference request. I'm looking to do a project in algebra where we are supposed to research some topic (...
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Hodge Bundles on Tropical Spaces

I am not sure that this question even makes sense, which I suppose is part of the questions itself. In any case, I attended a talk recently wherin there was some discussion about a "tropical ...
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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 ...
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1answer
90 views

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 ...
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1answer
138 views

Is the residue field of an algebraically closed field with respect to a non-trivial valuation infinite?

Let $K$ be an algebraically closed field with a non-trivial valuation. Let $\Bbb k$ be the residue field, i.e. $\Bbb k = R/\mathfrak m$ where $R$ is the valuation ring $R:=\{a\in K\mid \text{val} (a)\...
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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 ...
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Motivation for the study of amoebas.

What was the primary motivation for the study of the amoebas?
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167 views

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?
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What is the link of the origin of a polyhedral fan?

I am trying to understand the article http://arxiv.org/abs/0804.3651 by David Helm and Eric Katz. There the link of the origin of a polyhedral fan is mentioned. I googled and only found definitions ...
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Transition functions of toric projective bundle (Proposition in [Cox, Toric Varieties])

My reference: David Cox's "Toric Varieties" My question is the proof of Proposition 7.3.3. Proposition 7.3.3. The cones {$\sigma_i$ | $\sigma \in \Sigma$, $i = 0,\dots,r$} and their faces form ...
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262 views

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 ...
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The structure theorem of Tropical geometry

The Structure Theorem of Tropical geometry states that "Let $X$ be an irreducible $d$-dimensional subvariety of $\mathbb T^n$ . Then $\operatorname{trop}(X)$ is the support of a balanced weighted $Γ_{...