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.
94 questions
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How can complex abelian varieties degenerate to tropical abelian varieties
There is a similar interesting question here
which has not been answered. I therefore ask this question in the hope to get an answer. I wonder how a family of complex abelian varieties can exactly ...
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27
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Projection onto polytopes as tropical polynomial
Let $C$ be a convex polytope in $\mathbb{R}^n$ with $m$ extremal points. Let $p\in \{1,2\}$.
Can the $\ell^p$-projection $\Pi_C:\mathbb{R}^n\to C$
$$
\Pi_C(x) \in \operatorname{argmin}_{z\in C}\, \|x-...
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Actual zeros of tropical Laurent polynomial
I consider the tropical semi-ring $(\mathbb{R},\oplus,\odot)=(\mathbb{R},\max,+)$. I know that the tropiclisation of any (Laurent) polynomial $p\in\mathbb{R}[x_1^{\pm1},...,x_n^{\pm1}]$ given a ...
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Integral points on "complex exponential surface" in $\mathbb{C}^3$
I encountered the following object in $\mathbb{C}^3$ defined for $m\in\mathbb{N}$ by
$$A_m:=\lbrace (z_1,z_2,z_3)\in\mathbb{C}^3|(2^{2z_3}m-1)2^{2z_1+z_2+1}+3^{z_2-1}(2^{2z_1}-2^2-3^{z_3+1}m)=0\rbrace$...
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The proof of the fundamental theorem of tropical algebraic geometry in Maclagan-Sturmfels
I am trying to understand the proof of the fundamental theorem of tropical algebraic geometry from Maclagan-Sturmfels (Introduction to Tropical Geometry, Section 3.2 of the 2015 edition). Is the ...
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Do amoebas obtain extra tentacles as we take the tropical limit?
Original Question
In this question, we'll restrict ourselves to plane curves.
Define the $t$-amoeba of a polynomial $p(z,w) = \sum_{i,j \in \mathbb{N}} a_{ij} z^i w^j$ to be the set $\mathcal{A}_t(p) =...
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Is there a direct translation between Tropical and Algebraic geometry?
I am thinking of doing research in Tropical Geometry for my senior thesis, and I was wondering to what degree we can translate problems in Algebraic Geometry to Tropical Geometry.
We know that there ...
- 177
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Motivational distinctions between max and min conventions in tropical geometry
I am aware that algebraically, there is no real distinction between the tropical semirings
$A = (\mathbb{R} \cup \{ \infty \}, \text{min}, \infty, +, 0)$
$B = (\mathbb{R} \cup \{ - \infty \}, \text{...
- 177
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Economic equilibrium and tropical geometry
There is a famous saying in economics: When everyone pursues his or her own interests, there is an invisible hand that brings the market to equilibrium. However, this is not always the case. Here is ...
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Origin of tropical mathematics
On Wikipedia, it is claimed without a source that Imre Simon founded tropical mathematics.
The first work of his I was able to find on the subject is Limited subsets of a free monoid which uses the ...
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Kouchnirenko's theorem for non-generic polynomials
In Polyèdres de Newton et nombres de Milnor (Theorem 1.18), Kouchnirenko proved that given Laurent polynomials $f_1, \dotsc, f_k$ in $k$ variables, the number of isolated solutions is less than or ...
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Motives in tropical geometry
Is there a notion of motives in tropical geometry? Similar like the notion introduced by Grothendieck in algebraic geometry.
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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 ...
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History of tropical mathematics
This is a follow-up to this question about the origin of tropical mathematics.
Are there any articles, websites or books which deal with the history of tropical mathematics?
I have been trying to find ...
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Does tropicalization commute with composition?
Say we have two polynomials
$$
f = \sum d_n x^n, \quad g = \sum_n b_n x^n
$$
that tropicalize to
$$
F = \max(d_n + nx), \quad G = \max(b_n + nx).
$$
Can we say $f\circ g$ tropicalizes to $F\circ G$?
...
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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 ...
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What is the closed cone generated by constant and coordinate functions and closed under taking $f\mapsto\max(f,0)$?
Let $C$ be the smallest closed convex cone of functions from $\mathbb{R}^n$ to $\mathbb{R}$ that contains all constant functions, all coordinate functions, and such that $\max(f,0)\in C$ whenever $f\...
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How to define the limit of an infinite sequence of Newton polytopes rigorously?
Let $A_1 \subset A_2 \subset \cdots$ and each $A_i$ is a finite set of polynomials over variables $x_1, \ldots, x_n$. For each $i$, let $N_i$ be the Newton polytope of $A_i$. Since $A_{\infty}$ has ...
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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 ...
- 1,947
6
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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 ...
- 1,947
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Parametrize regions of positivity of a polynomial
I realize that this problem is extremely generic, so I am pessimistic that there may be concrete solutions, but let me try...
Consider a multi-variate polynomial $P(x)$, is it possible to find ...
- 1,435
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249
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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 ...
user478030
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5
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How should one approach tropical mathematics?
Let me preface this by saying that my background is pretty meagre (i.e. solid undergrad). However, a few months ago I came across Litvinov - The Maslov dequantization, idempotent and tropical ...
4
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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}(\...
- 11.8k
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Proving equivalence of tropical polynomials
I am new to the world of tropical mathematics. I am wondering if there is an algorithm to prove the equivalence of two tropical polynomials (in the plus-min semiring let's say), say over multivaribles?...
- 11
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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 ...
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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 ...
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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 $g: Z \to Z$ ...
- 3,539
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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,...
user39380
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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 ...
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What can we learn from the tropicalization of an algebraic variety?
I often hear people speaking of the many connections between algebraic varieties and tropical geometry and how geometric information about a variety can be read off from the associated tropical ...
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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 $R$ ...
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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 ...
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How is tropicalization like taking the classical limit?
There is a folk — I can't call it a theorem — "fact" that the mathematical relationship between Complex and Tropical geometry is analogous to the physical relationship between Quantum and ...
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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 ...
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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 ...
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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?
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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 ...
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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 ...
- 6,229
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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 ...
- 2,246
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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 ...
- 54.6k
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1
answer
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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 ...
- 2,246
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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-...
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6k
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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 ...
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55
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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 ...
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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 ...
- 3,245
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0
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94
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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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163
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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? ...
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2
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1k
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Weight filtration and Hodge theory for tropical varieties
Many concepts is algebraic geometry have tropical analogues.
Question: Is there an analogue of the weight filtration or Hodge filtration for tropical varieties?
A tropical curve ends up being ...
- 6,229
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0
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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