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
3
votes
1
answer
231
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 ...
- 3,793
3
votes
1
answer
256
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 $\...
- 1,590
3
votes
1
answer
515
views
When is the quotient of a tropical curve also a tropical curve?
A plane tropical curve $\Gamma$ is the corner locus of a tropical polynomial in two variables. That is, it is the set of points at which the tropical polynomial, which is a piecewise-linear concave ...
- 166
3
votes
0
answers
315
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 ...
- 2,246
3
votes
0
answers
127
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 ...
- 2,246
3
votes
0
answers
209
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 ...
- 15.4k
3
votes
0
answers
327
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= \{ ...
- 5,063
3
votes
0
answers
424
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 $X_{i,j} \rightarrow X_{i_1,...,i_m}$...
- 1,277
2
votes
1
answer
212
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 ...
- 4,070
2
votes
1
answer
594
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 ...
- 183
2
votes
1
answer
296
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 ...
- 2,246
2
votes
1
answer
387
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 ...
2
votes
1
answer
276
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
2
votes
1
answer
172
views
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 ...
- 91
2
votes
0
answers
104
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 ...
- 315
2
votes
0
answers
87
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 $F\circ G$?
...
- 306
2
votes
0
answers
97
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 ...
- 6,211
2
votes
0
answers
249
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
2
votes
0
answers
103
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 ...
- 2,789
2
votes
0
answers
92
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
2
votes
0
answers
125
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
2
votes
0
answers
205
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$ ...
- 2,788
2
votes
0
answers
875
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
2
votes
0
answers
80
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
2
votes
0
answers
94
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 $...
- 357
2
votes
0
answers
190
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
2
votes
0
answers
164
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 ...
- 1,509
2
votes
0
answers
185
views
Simple topological question on taking complements inside a simplex
We would like to know if the following claim is true:
(If you don't know the definition of a tropical hyperplane, then please consider the case when d=3)
Let $P_1,\cdots,P_d$ be full dimensional (...
- 113
1
vote
1
answer
1k
views
How to Tropicalize a Polynomial in Two Variables?
Trying to draw the Amoeba
With Mathematica, it's possible to graph $e^{-k x} + e^{-k y} = 1,e^{-k x} - e^{-k y} = 1$ and $e^{-k x} + e^{-k y} = -1$ to get the amoeba of 1 + x + y when k = 1. Then by ...
- 22.8k
1
vote
1
answer
199
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
1
vote
0
answers
162
views
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$...
- 91
1
vote
0
answers
43
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 ...
- 1,435
1
vote
0
answers
29
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
1
vote
0
answers
55
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 ...
- 2,788
1
vote
0
answers
94
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
1
vote
0
answers
87
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
1
vote
0
answers
256
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-...
- 2,789
1
vote
0
answers
66
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, ...
- 19.7k
1
vote
0
answers
102
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 ...
- 41
1
vote
0
answers
138
views
Bases of Ideals With no Monomials
Let $K$ be an algebraically closed field and $K[\underline{x}]$ its ring of polynomials in $n$ variables $x_1,\cdots, x_n$. Let $J\leq K[\underline{x}]$ be an ideal such that there are no monomials in ...
- 345
0
votes
1
answer
241
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 ...
- 329
0
votes
0
answers
82
views
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 ...
- 85
0
votes
0
answers
27
views
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-...
- 364
0
votes
0
answers
279
views
Tropical varieties correspondence to varieties over a non-archimedean valuation field.
I am a mathematical physicist and I am studying certain discrete dynamical systems defined in terms of piecewise linear mappings, which may be expressed in terms of expressions over the max-plus semi-...
- 106