相关论文: A tropical morphism related to the hyperplane arra…
In this article, we present a massively parallel framework for computing tropicalizations of algebraic varieties which can make use of finite symmetries. We compute the tropical Grassmannian TGr$_0(3,8)$, and show that it refines the…
We study tropicalisations of quasi-automorphisms of cluster algebras and show that their induced action on the g-vectors can be realized by tropicalising their action on the homogeneous $\hat{y}$ (or $\mathcal{X}$) variables of a chosen…
An $(n,m)$-graph is a graph with $n$ types of arcs and $m$ types of edges. A homomorphism of an $(n,m)$-graph $G$ to another $(n,m)$-graph $H$ is a vertex mapping that preserves the adjacencies along with their types and directions. The…
A biconvex polytope is a classical and tropical convex hull of finitely many points. Given a biconvex polytope, for each vertex of it we construct a directed bigraph and a gammoid so that the collection of base polytopes of those gammoids…
We consider the question of when points in tropical affine space uniquely determine a tropical hypersurface. We introduce a notion of multiplicity of points so that this question may be meaningful even if some of the points coincide. We…
Scaffolds are the one-dimensional skeleta of high-dimensional flag simplicial complexes of nonpositive curvature. They generalize the phylogenetic trees of Trop G(2,n) to arbitrary $k$, drawing together SL(k)-web bases, affine buildings,…
Tropical geometry has recently found several applications in the analysis of neural networks with piecewise linear activation functions. This paper presents a new look at the problem of tropical polynomial division and its application to…
For an integer $k\ge 2$, let $G$ be a graph with $m$ edges and without cycles of length $2k$. The pivotal Alon-Krivelevich-Sudakov Theorem on Max-Cuts states that $G$ has a bipartite subgraph with at least $m/2+\Omega(m^{(2k+1)/(2k+2)})$…
We prove that the topological complexity of (a motion planning algorithm on) the complement of generic complex essential hyperplane arrangement of $n$ hyperplanes in an $r$-dimensional linear space is min$\{n+1,2r\}$.
We prove that, if $m$ is sufficiently large, every graph on $m+1$ vertices that has a universal vertex and minimum degree at least $\lfloor \frac{2m}{3} \rfloor$ contains each tree $T$ with $m$ edges as a subgraph. Our result confirms, for…
We present an algorithm for computing zero-dimensional tropical varieties based on triangular decomposition and Newton polygon methods. From it, we derive algorithms for computing points on and links of higher-dimensional tropical…
In a previous paper, we announced a formula to compute Gromov-Witten and Welschinger invariants of some toric varieties, in terms of combinatorial objects called floor diagrams. We give here detailed proofs in the tropical geometry…
Using the notion of a root datum of a reductive group $G$ we propose a tropical analogue of a principal $G$-bundle on a metric graph. We focus on the case $G=\mathrm{GL}_n$, i.e. the case of vector bundles. Here we give a characterization…
A bipartite graph on 2n vertices is bipancyclic if it contains cycles of all even lengths from 4 to 2n. In this paper we prove that the random bipartite graph $G(n,n,p)$ with $p(n)\gg n^{-2/3}$ asymptotically almost surely has the following…
Let $K$ be a real closed field with a nontrivial non-archimedean absolute value. We study a refined version of the tropicalization map, which we call real tropicalization map, that takes into account the signs on $K$. We study images of…
This paper considers a hyperplane arrangement constructed with a subset of a set of all simple paths in a graph. A connection of the constructed arrangement to the maximum matching problem is established. Moreover, the problem of finding…
A graph $G$ is {\it weakly semiregular} if there are two numbers $a,b$, such that the degree of every vertex is $a$ or $b$. The {\it weakly semiregular number} of a graph $G$, denoted by $wr(G)$, is the minimum number of subsets into which…
We continue, in this second article, the study of the the algebraic tools which play a role in tropical algebra. We especially examine here the polynomial algebras over idempotent semi-fields. this work is motivated by the development of…
Given a group $G$ of automorphisms of a matroid $M$, we describe the representations of $G$ on the homology of the independence complex of the dual matroid $M^*$. These representations are related with the homology of the lattice of flats…
The graph of a real symplectic linear transformation is an R-Lagrangian subspace of a complex symplectic vector space. The restriction of the complex symplectic form is thus purely imaginary and may be expressed in terms of the generating…