相关论文: Tropical Fermat-Weber points
We show that the tropical projective Grassmannian of planes is homeomorphic to a closed subset of the analytic Grassmannian in Berkovich's sense by constructing a continuous section to the tropicalization map. Our main tool is an explicit…
We explain how to tropicalize scalar quantum field theory and show that tropicalized massive scalar quantum field theory is exactly solvable. This exact solution manifests as a non-linear recursion equation fulfilled by the expansion…
A key issue in tropical geometry is the lifting of intersection points to a non-Archimedean field. Here, we ask: Where can classical intersection points of planar curves tropicalize to? An answer should have two parts: first, identifying…
In this paper we study general tropical linear spaces locally: For any basis B of the matroid underlying a tropical linear space L, we define the local tropical linear space L_B to be the subcomplex of L consisting of all vectors v that…
Weighted Hurwitz numbers arise as coefficients in the power sum expansion of deformed hypergeometric $\tau$--functions. They specialise to essentially all known cases of Hurwitz numbers, including classical, monotone, strictly monotone and…
We consider arrangements of tropical hyperplanes where the apices of the hyperplanes are taken to infinity in certain directions. Such an arrangement defines a decomposition of Euclidean space where a cell is determined by its `type' data,…
In this paper we use the connections between tropical algebraic geometry and rigid analytic geometry in order to prove two main results. We use tropical methods to prove a theorem about the Newton polygon for convergent power series in…
In this paper we generalize the $j$-invariant criterion for the semistable reduction type of an elliptic curve to superelliptic curves $X$ given by $y^{n}=f(x)$. We first define a set of tropical invariants for $f(x)$ using symmetrized…
We investigate the complexity of counting the number of integer points in tropical polytopes, and the complexity of calculating their volume. We study the tropical analogue of the outer parallel body and establish bounds for its volume. We…
In this paper, we introduce the Fermat-Steiner-Frechet problem for a given $\frac{N(N+1)}{2}-$tuple of positive real numbers determining the edge lengths of an $N-$simplex in $\mathbb{R}^{N},$ in order to study its solution called the…
We compare the phylogenetic tensors for various trees and networks for two, three and four taxa. If the probability spaces between one tree or network and another are not identical then there will be phylogenetic tensors that could have…
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…
We develop a geometric framework that unifies several different combinatorial fixed-point theorems related to Tucker's lemma and Sperner's lemma, showing them to be different geometric manifestations of the same topological phenomena. In…
Let $P_1,P_2,P_3$ be three given points in $\mathbf{R}^2$, and $P$ be an arbitrary point in $\mathbf{R}^2$. The classical Fermat's problem to Torricelli asks for the location of the point $P$ such that $|PP_1|+|PP_2|+|PP_3|$ is a minimum.…
Metric graphs are important models for capturing the structure of complex data across various domains. While much effort has been devoted to extracting geometric and topological features from graph data, computational aspects of metric…
We study integral plane curves meeting at a single unibranch point and show that such curves must satisfy two equivalent conditions. A numeric condition: the local invariants of the curves at the contact point must be arithmetically…
The geometric median as well as the Frechet mean of points in an Hadamard space are important in both theory and applications. Surprisingly, no algorithms for their computation are hitherto known. To address this issue, we use a split…
The tropical semiring is an algebraic system with addition ``$\max$'' and multiplication ``$+$''. As well as in conventional algebra, linear programming in the tropical semiring has been developed. In this study, we introduce a new type of…
Asymptotic properties of matrices are, in general, difficult to analyze with classical mathematical techniques. In very specific cases, there is a well-known connection between the asymptotic behavior of a matrix's leading eigenvector and…
The uniform probability measure on a convex polytope induces piecewise polynomial densities on its projections. For a fixed combinatorial type of simplicial polytopes, the moments of these measures are rational functions in the vertex…