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24
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0answers
1k 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 ...
19
votes
0answers
649 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 ...
11
votes
0answers
326 views

Is there a Galois theory for $\mathbb R_{\geq 0}$?

The broadest version of my question is the following: Where can I find algebrogeometric abstract nonsense that handles "rings" and "fields" like $\mathbb R_{\geq 0}$ in which there is no ...
6
votes
0answers
146 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. ...
4
votes
0answers
212 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 ...
4
votes
0answers
494 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 ...
3
votes
0answers
327 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 ...
2
votes
0answers
91 views

holomorphic curves in almost toric fibration and their relation to tropical curves

My goal is to get better understanding how the projection of holomorphic curves converge to tropical disks. We are given an almost toric fibration $X\rightarrow B$ with special Lagrangian fibers with ...
2
votes
0answers
130 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
votes
0answers
201 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= \{ ...
2
votes
0answers
120 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
0answers
173 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 ...
1
vote
0answers
83 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 ...
1
vote
0answers
128 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 ...
0
votes
0answers
237 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 ...