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.
86
questions
2
votes
0
answers
80
views
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 ...
- 305
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votes
0
answers
61
views
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 $F\circ G$?
...
- 121
0
votes
0
answers
64
views
Tropical limit and Jacobians
Consider a collection of $n$ rational functions $f_i,\ i=1,\dots,n$ in $n$ variables $t_i,\ i=1,\dots,n$.
Let $J^f$ be the rational function defined implicitly by
$$\bigwedge_{i=1}^n \frac{dt_i}{t_i}J^...
- 1,305
1
vote
0
answers
39
views
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 ...
- 1,305
2
votes
0
answers
89
views
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 ...
- 5,961
4
votes
0
answers
133
views
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 ...
- 211
7
votes
4
answers
728
views
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 ...
- 211
6
votes
0
answers
185
views
Motives in tropical geometry
Is there a notion of motives in tropical geometry? Similar like the notion introduced by Grothendieck in algebraic geometry.
- 153
1
vote
0
answers
23
views
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
7
votes
0
answers
103
views
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\...
- 2,524
2
votes
0
answers
173
views
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
4
votes
0
answers
165
views
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}(\...
- 9,387
6
votes
1
answer
319
views
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 ...
- 1,733
2
votes
0
answers
90
views
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 ...
- 2,661
2
votes
0
answers
90
views
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
12
votes
1
answer
471
views
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 ...
- 1,733
10
votes
0
answers
341
views
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 $g: Z \to Z$ ...
- 3,499
2
votes
0
answers
106
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 ...
- 21
6
votes
0
answers
181
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 ...
- 61
16
votes
1
answer
362
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 ...
- 8,077
1
vote
0
answers
52
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 ...
- 2,718
2
votes
0
answers
184
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 $R$ ...
- 2,718
1
vote
0
answers
76
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 ...
- 11
2
votes
0
answers
530
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?
- 475
1
vote
0
answers
71
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, ...
- 11
4
votes
0
answers
127
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? ...
- 41
1
vote
0
answers
207
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-...
- 2,661
2
votes
0
answers
78
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?
- 21
1
vote
0
answers
58
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, ...
- 18.8k
4
votes
1
answer
395
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 ...
- 2,661
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 ...
- 32.1k
3
votes
0
answers
274
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 ...
- 2,116
2
votes
1
answer
255
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 ...
- 2,116
2
votes
1
answer
960
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 ...
- 2,537
3
votes
0
answers
113
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 ...
- 2,116
4
votes
1
answer
207
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 ...
- 3,532
13
votes
0
answers
481
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 ...
- 4,661
3
votes
0
answers
203
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 ...
- 15k
4
votes
1
answer
173
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 \...
- 15k
4
votes
0
answers
141
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 \...
- 2,116
6
votes
1
answer
782
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 ...
- 2,116
2
votes
0
answers
82
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
914
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 ...
- 51.4k
2
votes
0
answers
179
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
287
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 ...
- 6,212
11
votes
1
answer
633
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 ...
- 3,225
2
votes
1
answer
266
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 ...
- 3,613
6
votes
1
answer
374
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
244
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 $\...
- 1,532