For questions related to tropical geometry.

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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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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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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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Generalizing all singular cubic curves in the projective tropical plane with genus zero

How can I qualitatively classify all these curves under the condition that they must all be fully supported with no double lines touching infinity?
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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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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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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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Significance of eigen vectors in Max-Plus algebra

In Max-Plus algebra we have notion of eigen values and eigen vectors. Given a matrix $A \in \mathbb{R}_{max}^{n\times n}$ , $\mu$ is an eigen value of $A$ if it satisfies $A \otimes v = \mu \otimes v$ ...
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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
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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 ...
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1answer
89 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} ...
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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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1answer
124 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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238 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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191 views

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 ...