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.
69
questions
2
votes
0answers
59 views
A structure sheaf for real analytification of semialgebraic sets in the context of signed tropicalization
Let $X=Spec(A)$ be an affine scheme, where $A$ be a commutative algebra over a non-archimedean valued field $K$. Assume that $K$ is a real closed field with the unique ordering $<$, which should be ...
6
votes
0answers
88 views
Tropical abelian variety as a limit
A tropical abelian variety is given by a quotient of a real vector space $V \cong \mathbb{R}^g$ with a fixed integral structure $\Gamma_2$, by a lattice $\Gamma_1$, equipped with some aditional ...
15
votes
1answer
310 views
Categorical description of log as approximate rig homomorphism?
Summary
The base-$\beta$ logarithm gives an isomorphism of topological spaces
$$
\log_\beta\colon\mathbb{R}_{\geq0}\xrightarrow{\cong}[-\infty,\infty).
$$
This continuous map preserves some algebraic ...
1
vote
0answers
40 views
Essential rays in fan structure
Let $|\Sigma|$ be the underlying set of some fan $\Sigma$ in $\mathbb{R}^n$. It is well known that in general there is no coarsest fan structure on $|\Sigma|$. However, there may be some special rays ...
1
vote
0answers
78 views
Local toric varieties and tropicalization
Let $K$ be a valued field, and consider the ring $R=K((x_1,\dots,x_m))$ of formal Laurent series. This is "the germ of the torus at $0$". Is there a theory of "local toric varieties" where $R$ ...
1
vote
0answers
31 views
Generators for Ideals in ring of multivariate Laurent Polynomials
Consider the following problem:
Find an ideal $I \subset \mathbb{Q}[x^{\pm}_1,x^{\pm}_2,x^{\pm}_3]$ such that $I_{aff} \subset \mathbb{Q}[x_1, x_2, x_3] = I \cap k[x_1, x_2, x_3]$ requires more ...
2
votes
0answers
196 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
56 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
89 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
120 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
63 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
52 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
327 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
963 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
184 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
1answer
210 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
504 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
101 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 ...
4
votes
1answer
120 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 ...
13
votes
0answers
438 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
192 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
162 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
126 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
378 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
70 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
809 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 ...
2
votes
0answers
171 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
266 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
565 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
250 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
224 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
306 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
204 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
148 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
379 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
176 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
616 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
252 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
196 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
4k 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
479 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 ...
8
votes
0answers
185 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
296 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
400 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
234 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
537 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
790 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
421 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?
