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.
2
votes
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?
1
vote
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, ...
4
votes
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? ...
1
vote
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-...
1
vote
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?
1
vote
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, ...
4
votes
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 ...
24
votes
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 ...
3
votes
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 ...
2
votes
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 ...
1
vote
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 ...
3
votes
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 ...
3
votes
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 ...
12
votes
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 ...
3
votes
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 ...
4
votes
1answer
157 views
Does a nontrivial notion of integral under logarithmic deformations of $\mathbb{R}_+$ exist?
Background
The upper and lower Maslov dequantizations are respectively the limits $h \downarrow 0$ and $h \uparrow 0$ of deformations of the semifield $(\mathbb{R}_+,+,\cdot)$ defined for $0 \ne h \...
4
votes
0answers
104 views
Two variants of the Littlewood-Offord theorem
I found two different looking things being called the Littlewood-Offord theorem,
If $\vec{a} \in \mathbb{R}^k \setminus 0$ and $t \in \mathbb{R}$ then there are $O(\frac{2^k}{\sqrt{k}})$ points $x \...
5
votes
1answer
287 views
Is there any structure theorem for piecewise linear functions?
I was wondering if such statements are known like "any piecewise linear function from $\mathbb{R}^d \rightarrow \mathbb{R}$ can be written as $\sum_{i=1}^k \alpha_i (\text{ some $2$ piece linear ...
2
votes
0answers
64 views
Tropical self intersection number of boundary divisor on toroidal embedding
Let $X_0 \subset X$ a toroidal embedding without self intersections and denote by $\overline{\Sigma}$ its corresponding (weakly embedded) extended conical simplicial complex. Let $D$ be a divisor on $...
14
votes
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 ...
2
votes
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 ...
10
votes
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 ...
9
votes
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 ...
2
votes
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 ...
3
votes
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 ...
6
votes
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 ...
3
votes
1answer
193 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
votes
1answer
132 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 ...
4
votes
2answers
362 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 : \...
1
vote
1answer
172 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 ...
7
votes
1answer
548 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 ...
2
votes
1answer
238 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 ...
0
votes
1answer
185 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 ...
19
votes
2answers
3k 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 ...
5
votes
0answers
423 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 ...
7
votes
0answers
182 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. ...
2
votes
0answers
282 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
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 ...
3
votes
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 $X_{i,j} \rightarrow X_{i_1,...,i_m}$...
4
votes
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 ...
2
votes
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 ...
19
votes
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 ...
3
votes
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 ...
26
votes
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?
2
votes
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 ...
3
votes
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 ...
19
votes
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 ...
53
votes
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 ...
7
votes
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 ...
34
votes
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 ...
