# Questions tagged [tropical-geometry]

For questions about tropical geometry, piece-wise linear functions with integer slopes, tropical degenerations and applications of tropical geometry, max-plus algebra, and tropical semifields.

63 questions
0answers
111 views

### How to draw tropical curves?

In paper arXiv:1311.2360v3, there are a lot of tropical curves. I want to know how to draw them by using some softwares/algorithms?
0answers
49 views

### Tropical Arithmetic and Numeral Systems - Number systems [closed]

Is there some paper about Numeral Systems / Number Systems, using tools of Tropical Geometry to represent numbers? Maybe through Continuous fractions, triangular numbers, arithmetic functions, ...
0answers
68 views

### Tropical lie algebra

In this article https://arxiv.org/pdf/1705.01075.pdf are we mean that Lie semialgebras over semirings with a negation map is tropical version of Lie algebra?. And what we do when we define lifting? ...
0answers
79 views

### Explicit description of rigid analytification of torus

It is known that in non-archimedean world there is also a GAGA-functor from the category of $K$-schemes of locally finite type to the category of rigid $K$-spaces. Here $K$ is a field with a non-...
0answers
62 views

### Tropical algebraic structures

What is the difference between tropical lie semialgebra and lie semialgebra with anegation map? and How can I build another algebraic structure in tropical algebra?
0answers
46 views

### Conjugating the tropical Lyness 5-cycle into a rotation of the plane

In his response to my question Conjugating the Lyness 5-cycle into a rotation of the plane, Francois Brunault provided an explicit conjugacy between the Lyness order-5 map and a 72-degree rotation, ...
1answer
286 views

### Tropical charts (coordinates) and differential forms in non-archimedean geometry

Chambert-Loir and Ducros have introduced real differential forms and currents on Berkovich spaces.(See Gubler's survey for example). In that survey, a tropical chart $V$ is defined on an ...
3answers
819 views

### Is the Ford-Fulkerson algorithm a tropical rational function?

The Ford-Fulkerson algorithm Let me recall the standard scenario of flow optimization (for integer flows at least): Let $\mathbb{N} = \left\{0,1,2,\ldots\right\}$. Consider a digraph $D$ with vertex ...
0answers
132 views

### Factorization of tropical polynomials

I am referring to the definition of a tropical polynomial given on page 8 (top of section 3) here in this review, https://arxiv.org/pdf/math/0306366.pdf. I understand that here a tropical polynomial ...
0answers
122 views

### Can we have “tropical polynomials” with arbitrary real powers?

I am referring to the definition of a tropical polynomial given on page 8 (top of section 3) here in this review, https://arxiv.org/pdf/math/0306366.pdf. I understand that here the notion of a ...
1answer
333 views

### Is there any “fundamental” distinction between min-plus, max-plus, min-product, and max-product algebras?

In the paper Faster Algorithms for Max-Product Message Passing by McAuley and Caetano (see e.g. here or here), several statements are made which seem mathematically questionable to me. For ...
0answers
88 views

### Max-Plus algebra and hyperplane arrangements

Given an expression in the Max-Plus algebra is it possible to recognize if it represents a continuous piecewise linear (CPWL) function whose polyhedral complex is a hyperplane arrangement? Or ...
1answer
94 views

### Poles of an integral of a meromorphic function with toric poles

Suppose I have a meromorphic function in several variables $f(x_1,\ldots,x_k,y_1,\ldots,y_m)$ and I want to integrate along the torus $T^m$ given by $|y_1|=\cdots=|y_m|=1$. It is not true in general ...
0answers
413 views

### Strange formula in arithmetic dynamic

Added: another function like that is $S_p f(z) = f(z)+\frac{f(\sqrt{zp})^2}{f(p)}$ in a field of characteristic two. We discovered the following operator which acts on the space of polynomials (or ...
0answers
186 views

### Can MacLane's notion of universality inform $\mathbb{F}_1$?

MacLane (1939) calls a field $F$ universal if every other field $F'$ of the same cardinality and characteristic as $F$ is a subfield of $F$. He then exhibits an example, viz. a field of generalized ...
1answer
157 views

1answer
287 views

1answer
756 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 ...
0answers
165 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 ...
1answer
248 views

### What can we say about tropical maps $\mathbb{P}^1 \to A$ for an Abelian variety $A$?

It's well known that maps $\mathbb{P}^1_\mathbb{C} \to A$ are constant for any Abelian variety $A$ (in fact, for any complex torus). Is there any similar statement in the tropical case? Naively, the ...
1answer
531 views

### Is there a tropical geometric proof for counting genus g curves in any n dimensional projective space?

Consider the following question: Let $X$ be a compact complex manifold and $\beta \in H_2(X, \mathbb{Z})$ a fixed homology class. Let $\mu_1, \mu_2, \ldots, \mu_k$ denote certain generic ...
1answer
245 views

### Are schemes which agree on open set and its complement equal? - w/ applications to initial ideals/tropical basis

I appreciate the comments so far and am modifying based on something closer to the problem I'm interested in. I started out with something far too general. This is probably easy, but I have been ...
1answer
221 views

### Looking for interesting, natural models of this algebraic theory in which $x^\dagger$ is not always the multiplicative inverse of $x$

It is easy to think up interesting, natural models of the algebraic theory presented as follows, such that in these models, $x^\dagger$ is always the multiplicative inverse of $x$. Question. What ...
1answer
283 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 ...
1answer
193 views

0answers
91 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 ...
0answers
390 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 $X_{i,j} \rightarrow X_{i_1,...,i_m}$...
1answer
223 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 ...
1answer
514 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 ...
0answers
766 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 ...
1answer
402 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 ...
1answer
2k views

### Important open questions in the field of Tropical geometry

What are some of the important unanswered questions in the field of tropical geometry?
0answers
143 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 ...
1answer
494 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 ...
2answers
4k 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 ...
5answers
13k 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 ...
1answer
486 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 ...
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 ...