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.
77
questions
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141
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Enlightening examples of tropical skeletons of Berkovich spaces
Let $K$ be a complete non-archimedean field and let $X$ be a $K$-analytic space in the sense of Berkovich of pure dimension $d$. Let $\varphi \colon X \to \mathbf{G}_m^r$ be a moment map to an ...
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32
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Conditions on $g$ which ensure the function $\phi(t):=\int_0^{2\pi} g(\cos\theta)g(\cos(\theta-\arccos(t)))\,\mathrm{d}\theta$ is $C^k$ on $(-1,1)$
Given an almost-everywhere continuous function $g:[-1,1] \to \mathbb R$, define another function $\phi_g:[-1,1] \to \mathbb R$ by
$$
\phi_g(t) := \int_0^{2\pi} g(\cos\theta)g(\cos(\theta-\arccos(t)))\,...
- 5,728
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votes
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answers
151
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Does the tropical semiring admit a universal property?
Each of the rings $\mathbf{Q}$, $\mathbf{Z}_p$, $\mathbf{Q}_p$, $\mathbf{R}$ admits a universal construction: the rationals are the field of fractions of the integers: $\mathbf{Q}:=\mathrm{Frac}(\...
- 614
6
votes
1
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285
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Does solving polynomial equations commute with tropicalization? (particularly for the field of Puiseux series)
The field of Puiseux series over an algebraically closed field of characteristic zero is also an algebraically closed field, and furthermore it has a valuation so that our Puiseux series can be ...
- 1,025
2
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0
answers
86
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Amoeba for a K3 surface in $\mathbb {CP}^3$
Let $X=X_\Delta$ be the toric variety associated to a reflexive polyhedron $\Delta$.
Consider a Calabi-Yau hypersurface $Y\subset X$, and the image of $Y$ under the moment map $\mu:X\to \Delta$ has ...
- 2,611
2
votes
0
answers
89
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Rational sections of tropical conics
Let us consider the family of Fermat conics in $(\mathbb{C}^*)^2\subset\mathbb{C}^2$ given by
$$\pi\colon V(ax^2+by^2-1)\subset(\mathbb{C}^*)^2_{a,b}\times(\mathbb{C}^*)^2_{x,y}\to(\mathbb{C}^*)^2_{a,...
- 5,613
12
votes
1
answer
406
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Is there a straightforward generalization of min(x,y) to positive-semidefinite Hermitian matrices?
This is an open-ended question I have. Is there a function of two positive-semidefinite hermitian operators $\min(A,B)$ returning another positive-semidefinite Hermitian operator such that:
If A and ...
- 1,025
10
votes
0
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328
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Does the tropicalization of a curve remember the curve's automorphism group?
For a tropical curve $Z$, let us call $Z_0$ this curve with its 1-valent points removed.
(Def [5] of Joyner-Ksir-Grant Melles) Let the automorphism group of a tropical curve $Z$ be a map $g: Z \to Z$ ...
- 3,379
2
votes
0
answers
98
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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 ...
- 21
6
votes
0
answers
161
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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 ...
- 61
16
votes
1
answer
350
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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 ...
- 7,857
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vote
0
answers
50
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 ...
- 2,665
2
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157
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 $R$ ...
- 2,665
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vote
0
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61
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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 ...
- 11
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394
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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?
- 475
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0
answers
64
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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, ...
- 11
4
votes
0
answers
115
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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? ...
- 41
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0
answers
168
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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-...
- 2,611
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0
answers
74
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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?
- 21
1
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0
answers
57
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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, ...
- 17.3k
4
votes
1
answer
381
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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 ...
- 2,611
24
votes
3
answers
1k
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 ...
- 30.8k
3
votes
0
answers
257
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 ...
- 2,016
2
votes
1
answer
247
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 ...
- 2,016
1
vote
1
answer
838
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 ...
- 2,477
3
votes
0
answers
111
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 ...
- 2,016
4
votes
1
answer
200
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 ...
- 3,492
13
votes
0
answers
474
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 ...
- 4,601
3
votes
0
answers
201
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 ...
- 14.7k
4
votes
1
answer
168
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 \...
- 14.7k
4
votes
0
answers
138
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 \...
- 2,016
6
votes
1
answer
702
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 ...
- 2,016
2
votes
0
answers
77
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 $...
- 337
15
votes
1
answer
887
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 ...
- 49.4k
2
votes
0
answers
178
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 ...
- 21
10
votes
1
answer
280
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 ...
- 6,092
11
votes
1
answer
607
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 ...
- 3,195
2
votes
1
answer
263
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 ...
- 51
3
votes
1
answer
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 ...
- 3,513
6
votes
1
answer
350
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 ...
- 255
3
votes
1
answer
224
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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 $\...
- 1,484
2
votes
1
answer
166
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 ...
- 3,982
5
votes
2
answers
418
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 : \...
- 889
1
vote
1
answer
185
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 ...
user42154
7
votes
1
answer
723
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 ...
- 1,471
2
votes
1
answer
316
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
1
answer
217
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 ...
- 319
20
votes
2
answers
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 ...
- 23.5k
5
votes
0
answers
597
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 ...
- 144k
8
votes
0
answers
199
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. ...
- 166