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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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 ...
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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?
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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 ...
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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 ...
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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 ...
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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 : \...
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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, ...
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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 ...
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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 ...
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Important open questions in the field of Tropical geometry
What are some of the important unanswered questions in the field of tropical geometry?
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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 ...
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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 ...
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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 \...
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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 \...
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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 ...
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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 ...
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Is anything known about the enumeration of degree d, genus g curves in CP^2 where g >1 ?
I wanted to know if there is something analogous to Kontsevich's recursion formula for
enumeration of genus zero curves in $\mathbb{C}\mathbb{P}^2$, for higher genus curves.
There is a
similar ...
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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= \{ ...
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Tropical homological algebra
Has anyone out there thought about homological algebra over the tropical semifield $\mathbb{T}$? For example, I'm interested in the Hochschild homology and cyclic homology of tropical algebras, if ...
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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 ...
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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 $\...
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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 ...
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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 $...
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Picture of a 3 dimensional amoeba.
On Wikipedia there some pictures of two dimensional amoebas (Thanks to Oleg Alexandrov for the pictures and the Matlab code he gives to build them). I was wondering if somewhere there are pictures ...
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$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 ...
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Properties from Tropical Geometry that do not imply their algebraic counterpart.
One of the motivations to study tropical geometry is that there are some hard Algebraic Questions that can be answered by proving them in the Tropical World. For example one can show that tropical ...
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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 ...
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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 ...
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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 ...
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Hypersurfaces in Toric Varieties, Help understand a proof from Mikhalkin's paper
Hello,
in G. Mikhalkin's Papaer "DECOMPOSITION INTO PAIRS-OF-PANTS FOR
COMPLEX ALGEBRAIC HYPERSURFACES":
http://arxiv.org/pdf/math/0205011.pdf
There is a lemma about the relation between intersection ...
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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
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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 ...
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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 $X_{i,j} \rightarrow X_{i_1,...,i_m}$...
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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 ...
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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 ...
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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 ...
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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. ...
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Intersection of curves on projective toric surface and some enumerative questions
Reading on the tropical approach to enumerative geometry I have come across the claim:
given a projective toric surface from a polygon P, we can consider a tautological bundle of algebraic / ...
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Tropical Properties From Algebraic Geometry
What properties of tropical geometry (Starting from a valued Field) can be proven to be true using their analogue in algebraic geometry? For example, using the valuation on the Puiseux series $\mathbb{...
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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 ...
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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 ...
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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-...
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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 ...
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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 (...
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