Related papers: Tropical planar networks

Exploded layered tropical (ELT) algebra is an extension of tropical algebra with a structure of layers. These layers allow us to use classical algebraic results in order to easily prove analogous tropical results. Specifically we study the…

In this note we characterize tropical bases as sets of circuits that by orthogonality determine the set of cocircuits of a simple matroid. Furthermore, we show that any circuit, which itself is closed, must be contained in any tropical…

We show how for every integer n one can explicitly construct n distinct plane quartics and one hyperelliptic curve over the complex numbers all of whose Jacobians are isomorphic to one another as abelian varieties without polarization. When…

Let $\chi(A)$ denote the characteristic polynomial of a matrix $A$ over a field; a standard result of linear algebra states that $\chi(A^{-1})$ is the reciprocal polynomial of $\chi(A)$. More formally, the condition $\chi^n(X)…

A tropical matrix is a matrix defined over the max-plus semiring. For such matrices, there exist several non-coinciding notions of rank: the row rank, the column rank, the Schein/Barvinok rank, the Kapranov rank, or the tropical rank, among…

For each positive integer n the HOMFLY polynomial of links specializes to a one-variable polynomial that can be recovered from the representation theory of quantum sl(n). For each such n we build a doubly-graded homology theory of links…

Simple finite dimensional Kantor triple systems over the complex numbers are classified in terms of Satake diagrams. We prove that every simple and linearly compact Kantor triple system has finite dimension and give an explicit presentation…

We classify the optimal mock Jacobi forms of weight one with rational coefficients. The space they span is thirty-four-dimensional, and admits a distinguished basis parameterized by genus zero groups of isometries of the hyperbolic plane.…

We show that the weights on a tropical variety can be recovered from the tropical scheme structure proposed by the Giansiracusas in arXiv:1308.0042, so there is a well-defined Hilbert-Chow morphism from a tropical scheme to the underlying…

Given a smooth complex toric variety we will compare real Lagerberg forms and currents on its tropicalization with invariant complex forms and currents on the toric variety. Our main result is a correspondence theorem which identifies the…

Let $\mathfrak{g}$ be a symmetrizable Kac-Moody Lie algebra with Cartan subalgebra $\mathfrak{h}$. We prove a unique factorization property for tensor products of parabolic Verma modules. More generally, we prove unique factorization for…

Let $X$ be a closed algebraic subset of $\mathbb{A}^{n}(K)$ where $K$ is an algebraically closed field complete with respect to a nontrivial non-Archimedean valuation. We show that there is a surjective continuous map from the Berkovich…

We prove the existence of a common eigenvector for commutative, nilpotent and quasinilpotent semigroups of matrices with complex or real nonnegative entries both in the conventional and tropical linear algebra.

Let $\mathbf{v}_1,\ldots,\mathbf{v}_m$ be points in a metric space with distance $d$, and let $w_1,\ldots,w_m$ be positive real weights. The weighted Fermat-Weber points are those points $\mathbf{x}$ which minimize $\sum w_i d(\mathbf{v}_i,…

We introduce the notion of tropical curves of hyperelliptic type. These are tropical curves whose Jacobian is isomorphic to that of a hyperelliptic tropical curve, as polarized tropical abelian varieties. We show that this property depends…

The study of open quantum systems relies on the notion of unital completely positive semigroups on $C^*$-algebras representing physical systems. The natural generalisation would be to consider the unital completely positive semigroups on…

In this paper, we survey and study definitions and properties of tropical polynomials, tropical rational functions and in general, tropical meromorphic functions, emphasizing practical techniques that can really carry out computations. For…

An alternative proof of the duality of generalized Lie bialgebroid is given and proved a canonical Jacobi structure can be defined on the base of it. We also introduce the notion of morphism between generalized Lie bialgebroids and proved…

We prove that the ideal generated by the maximal minors of the higher-order Jacobian matrix of a weighted homogeneous polynomial is also weighted homogeneous. As an application, we give a partial answer to a conjecture concerning the…

In this paper we consider Jacobi forms of half-integral index for any positive definite lattice L (classical Jacobi forms from the book of Eichler and Zagier correspond to the lattice A_1=<2>). We give a lot of examples of Jacobi forms of…