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2
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0answers
76 views

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 ...
2
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0answers
49 views

Tropical polynomial Positivstellensatz

In real algebraic geometry, Stengle's Positivstellensatz can be used to characterize polynomials that are positive on a semialgebraic set. Say that a tropical semialgebraic set is a subset of ...
2
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1answer
70 views

group structure on (subsets of) tropicalizations of Abelian varieties

In this paper Vigeland shows how one can define a group law on subset of a tropical elliptic curve, so that this group is homeomorphic to $S^1$. It is not clear to me what is the relationship between ...
5
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2answers
237 views

Can Hausdorff dimension make sets into a Tropical Semiring?

If $X$ is a metric space, we construct Hausdorff $d$ measure from the outer measure \begin{equation} H^d(U) = \lim_{\delta \to 0}\inf\left\{\sum_{i=1}^\infty \left(\text{diam}(E_i)\right)^d : ...
33
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5answers
4k views

Why tropical geometry?

Tropical geometry can be described as "algebraic geometry" over the semifield $\mathbb{T}$ of tropical numbers. As a set, $\mathbb{T}=\mathbb{R}\cup \{ -\infty\}$; this is endowed with addition being ...
1
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1answer
117 views

Boundary of a tropical variety.

For a variety X (over some proper fields), if Trop(X) is a tropicalization of X, then we know that Trop(X) is a polyhedral complex. If we consider the interior of the support of that polyhedral ...
2
votes
1answer
77 views

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 ...
6
votes
1answer
251 views

Has this generalization of a determinant (assigning multiplicities to the rows) been studied?

I'm working on some questions in tropical geometry, and my problem led me to create the following generalization of a determinant: Let $A$ be an $m \times n$ matrix with $m \le n$, and positive ...
1
vote
1answer
132 views

Volume in tropical geometry as compared to volume in convex geometry

In tropical geometry, is there a notion of volume. Maybe one with some of the properties as found in classical convex geometry? If so, is there a good reference that elaborates on this question. I ...
14
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2answers
1k views

What is Tropicalization, and how is it applied

My question is: What is Tropicalization, how is it done, and what are some basic applications of it? motivation I am interested especially in how questions about enumerative algebraic geometry ...
4
votes
0answers
163 views

$L_1$ and $L_\infty$ Voronoi diagrams and tropical geometry: Connection?

I just realized that there is a visual similarity between Voronoi diagrams in the $L_1$ and $L_\infty$ metrics (two images below)     Left: O'Rourke, "Computing Relative Neighborhood graph ...
6
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0answers
123 views

Chow ring of extended tropicalizations

In Allermann-Rau '09, the authors define the Chow groups of an arbitrary abstract tropical cycle. In particular, one may take the tropical cycle to be the tropicalization of a subvariety of a torus. ...
3
votes
1answer
413 views

When is the quotient of a tropical curve also a tropical curve?

A plane tropical curve $\Gamma$ is the corner locus of a tropical polynomial in two variables. That is, it is the set of points at which the tropical polynomial, which is a piecewise-linear concave ...
1
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0answers
172 views

Asymptotics vs Puiseux series

Define asymptotic as a class of sequences {$ x_i$},$_{i\in\mathbb N}$ modulo equivalence {$x_i$}={$y_i$} if $\lim_{i\to\infty} (x_i/y_i)=c\in\mathbb R,c\ne 0$. More, we define $X= \{x_i\} \lt Y= \{ ...
1
vote
0answers
71 views

Notion of transversality over the field of Puiseux series.

To a given a Laurent polynomial $f$ over the field of Puiseux seris with parameter $t$, $f \in \mathbb{C} \lbrace\lbrace t \rbrace\rbrace[z_1^{\pm1},...,z_n^{\pm 1}]$, one can associate the ...
3
votes
0answers
301 views

Tropicalization of the Grassmannian

Let $Trop(Gr(m,n))$ denote the tropicalization of the grassmannian $Gr(n,m)$. Let $\phi^m : \mathbb R^{n \choose 2} \rightarrow \mathbb R^{n \choose m}$ such that $X_{i,j} \rightarrow ...
4
votes
1answer
180 views

Family of hypersurfaces in (C^*)^2 corresponding to tropical family

Edit: I realize the mathematics below is lacking a precise phrasing. I hope that the intuitiion behind the question is clear enough that a reader will understand the question and provide guidance. The ...
2
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1answer
356 views

Picture of a 3 dimensional amoeba.

On Wikipedia there some pictures of two dimensional amoebas (Thanks to Oleg Alexandrov for the pictures and the Matlab code he gives to build them). I was wondering if somewhere there are pictures ...
17
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0answers
554 views

Which manifolds decompose into pants?

In this nice paper Mikhalkin uses certain (more geometrical than algebraic) aspects of tropical geometry to prove that every complex projective hypersurface in $\mathbb C \mathbb P ^n$ decomposes as a ...
3
votes
1answer
255 views

How to recover toric invariants tropically?

My excuses in advance in case my question is too vague (which is mainly due to the fact that I'm not really familiar with tropical/toric geometry, but at least I still believe that the content of the ...
25
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1answer
1k views

Important open questions in the field of Tropical geometry

What are some of the important unanswered questions in the field of tropical geometry?
10
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2answers
612 views

Weight filtration and Hodge theory for tropical varieties

Many concepts is algebraic geometry have tropical analogues. Question: Is there an analogue of the weight filtration or Hodge filtration for tropical varieties? A tropical curve ends up being ...
2
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0answers
106 views

Algorithms for “Ideals” in polynomial algebras over the max-plus semi-ring

I'm a beginner in tropical geometry, and I'm running into the following question: In the usual polynomial ring over a field, one has algorithms (i.e. using a Groebner basis) for determining whether ...
2
votes
1answer
356 views

Hypersurfaces in Toric Varieties, Help understand a proof from Mikhalkin's paper

Hello, in G. Mikhalkin's Papaer "DECOMPOSITION INTO PAIRS-OF-PANTS FOR COMPLEX ALGEBRAIC HYPERSURFACES": http://arxiv.org/pdf/math/0205011.pdf There is a lemma about the relation between intersection ...
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2answers
2k views

Learning Tropical geometry

I'm interested in learning tropical geometry. But my background in algebraic geometry is limited. I know basic facts about varieties in affine and projective space, but nothing about sheaves, schemes ...
30
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4answers
2k views

What can we learn from the tropicalization of an algebraic variety?

I often hear people speaking of the many connections between algebraic varieties and tropical geometry and how geometric information about a variety can be read off from the associated tropical ...
7
votes
1answer
400 views

Properties from Tropical Geometry that do not imply their algebraic counterpart.

One of the motivations to study tropical geometry is that there are some hard Algebraic Questions that can be answered by proving them in the Tropical World. For example one can show that tropical ...
22
votes
0answers
906 views

Mikhalkin's tropical schemes versus Durov's tropical schemes

In Mikhalkin's unfinished draft book on tropical geometry, (available here) (page 26) he defines a notion of tropical schemes. It seems to me that this definition is not just a wholesale adaptation of ...
4
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0answers
477 views

Tropical Properties From Algebraic Geometry

What properties of tropical geometry (Starting from a valued Field) can be proven to be true using their analogue in algebraic geometry? For example, using the valuation on the Puiseux series ...
1
vote
0answers
120 views

Bases of Ideals With no Monomials

Let $K$ be an algebraically closed field and $K[\underline{x}]$ its ring of polynomials in $n$ variables $x_1,\cdots, x_n$. Let $J\leq K[\underline{x}]$ be an ideal such that there are no monomials in ...
4
votes
1answer
425 views

Intersection of curves on projective toric surface and some enumerative questions

Reading on the tropical approach to enumerative geometry I have come across the claim: given a projective toric surface from a polygon P, we can consider a tautological bundle of algebraic / ...
14
votes
1answer
1k views

Tropical homological algebra

Has anyone out there thought about homological algebra over the tropical semifield $\mathbb{T}$? For example, I'm interested in the Hochschild homology and cyclic homology of tropical algebras, if ...
2
votes
1answer
742 views

How to Tropicalize a Polynomial in Two Variables?

Trying to draw the Amoeba With Mathematica, it's possible to graph $e^{-k x} + e^{-k y} = 1,e^{-k x} - e^{-k y} = 1$ and $e^{-k x} + e^{-k y} = -1$ to get the amoeba of 1 + x + y when k = 1. Then by ...
0
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0answers
216 views

Tropical varieties correspondence to varieties over a non-archimedean valuation field.

I am a mathematical physicist and I am studying certain discrete dynamical systems defined in terms of piecewise linear mappings, which may be expressed in terms of expressions over the max-plus ...
2
votes
0answers
169 views

Simple topological question on taking complements inside a simplex

We would like to know if the following claim is true: (If you don't know the definition of a tropical hyperplane, then please consider the case when d=3) Let $P_1,\cdots,P_d$ be full dimensional ...
5
votes
2answers
495 views

Is anything known about the enumeration of degree d, genus g curves in CP^2 where g >1 ?

I wanted to know if there is something analogous to Kontsevich's recursion formula for enumeration of genus zero curves in $\mathbb{C}\mathbb{P}^2$, for higher genus curves. There is a similar ...
30
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4answers
2k views

How should one approach tropical mathematics?

Let me preface this by saying that my background is pretty meagre (i.e. solid undergrad). However, a few months ago I came across this paper which presented an idea that struck me as really ...
16
votes
4answers
866 views

How is tropicalization like taking the classical limit?

There is a folk — I can't call it a theorem — "fact" that the mathematical relationship between Complex and Tropical geometry is analogous to the physical relationship between Quantum and ...