Related papers: A-Tint: A polymake extension for algorithmic tropi…
In this text we develop the formalism of products and powers of linear codes under componentwise multiplication. As an expanded version of the author's talk at AGCT-14, focus is put mostly on basic properties and descriptive statements that…
This short article, which appeared in the MSRI Fall 2004 newletter, is intended for a general mathematical audience. It describes some mathematical themes from the Winter 2004 program at MSRI in topological aspects of real algebraic…
We study the intersection of tropical psi-classes on tropical heavy/light Hassett spaces, generalising a result of Kerber--Markwig for tropical moduli spaces of rational stable curves with distinct marked points. Our computation reveals…
We review results of papers written on the topic of polynomial amoebas with an emphasis on computational aspects of the topic. The polynomial amoebas have a lot of applications in various domains of science. Computation of the amoeba for a…
The notion of convexity in tropical geometry is closely related to notions of convexity in the theory of affine buildings. We explore this relationship from a combinatorial and computational perspective. Our results include a convex hull…
The main purpose of this paper is to report on the state of the art of computing integer hulls and their facets as well as counting lattice points in convex polytopes. Using the polymake system we explore various algorithms and…
We propose to take a look at a new approach to the study of integral polyhedra. The main idea is to give an integral representation, or matrix model representation, for the key combinatorial characteristics of integral polytopes. Based on…
In this review article, we report on some recent advances on the computational aspects of cohomology intersection numbers of GKZ systems developed in \cite{GM}, \cite{MH}, \cite{MT} and \cite{MT2}. We also discuss the relation between…
We study rational double Hurwitz cycles, i.e. loci of marked rational stable curves admitting a map to the projective line with assigned ramification profiles over two fixed branch points. Generalizing the phenomenon observed for double…
In these notes we survey the tropical intersection theory on R^n by deriving the properties for tropical cycles from the corresponding properties in Chow cohomology. For this we review the stable intersection product introduced by Mikhalkin…
Line intersection with convex and un-convex polygons or polyhedron algorithms are well known as line clipping algorithms and very often used in computer graphics. Rendering of geometrical problems often leads to ray tracing techniques, when…
We study the relationship between line bundles on tropical compactifications of a very affine variety $Y$ and toric b-divisors on the associated tropical variety ${\rm Trop}(Y)$. By focusing on numerical equivalence classes, we construct a…
After endowing the space of diagrams of probability spaces with an entropy distance, we study its large-scale geometry by identifying the asymptotic cone as a closed convex cone in a Banach space. We call this cone the tropical cone, and…
Cycle polytopes of matroids have been introduced in combinatorial optimization as a generalization of important classes of polyhedral objects like cut polytopes and Eulerian subgraph polytopes associated to graphs. Here we start an…
We study the tropical version of the contraction morphism $\mathcal{T}$ between moduli spaces of stable and pseudostable curves. By promoting $\mathcal{T}$ to a logarithmic morphism, we obtain a piecewise linear function between the…
We study the combinatorial properties of a tropical hyperplane arrangement. We define tropical oriented matroids, and prove that they share many of the properties of ordinary oriented matroids. We show that a tropical oriented matroid…
The problem deals with an exact calculation of the intersection area of a circle arbitrary placed on a grid of square shaped elements with gaps between them (finite fill factor). Usually an approximation is used for the calculation of the…
We present the Julia interface Polymake.jl to polymake, a software for research in polyhedral geometry. We describe the technical design and how the integration into Julia makes it possible to combine polymake with state-of-the-art…
Twisted period integrals are ubiquitous in theoretical physics and mathematics, where they inhabit a finite-dimensional vector space governed by an inner product known as the intersection number. In this work, we uncover the associated…
We investigate, using purely combinatorial methods, structural and algorithmic properties of linear equivalence classes of divisors on tropical curves. In particular, an elementary proof of the Riemann-Roch theorem for tropical curves,…