Related papers: Tropical linear maps on the plane
We prove for a tropical rational map that if for any point the convex hull of Jacobian matrices at smooth points in a neighborhood of the point does not contain singular matrices then the map is an isomorphism. We also show that a tropical…
Let $p',q'\in R^n$. Write $p'\sim q'$ if $p'-q'$ is a multiple of $(1,\ldots,1)$. Two different points $p$ and $q$ in $R^n/\sim$ uniquely determine a tropical line $L(p,q)$, passing through them, and stable under small perturbations. This…
This is a survey article written for the Jahresberichte der DMV. Tropical geometry can be viewed as an efficient combinatorial tool to study degenerations in algebraic geometry. Abstract tropical curves are essentially metric graphs, and…
Complex algebraic varieties become easy piecewise-linear objects after passing to the so-called tropical limit. Geometry of these limiting objects is known as tropical geometry. In this short survey we take a look at motivation and…
We state the fundamental theorem of projective geometry for semimodules over semirings, which is facilitated by recent work in the study of bases in semimodules defined over semirings. In the process we explore in detail the linear algebra…
Tropical Geometry and Mathematical Morphology share the same max-plus and min-plus semiring arithmetic and matrix algebra. In this chapter we summarize some of their main ideas and common (geometric and algebraic) structure, generalize and…
This paper is about the combinatorics of finite point configurations in the tropical projective space or, dually, of arrangements of finitely many tropical hyperplanes. Moreover, arrangements of finitely many tropical halfspaces can be…
The tropical Stiefel map associates to a tropical matrix A its tropical Pluecker vector of maximal minors, and thus a tropical linear space L(A). We call the L(A)s obtained in this way Stiefel tropical linear spaces. We prove that they are…
In tropical geometry, one studies algebraic curves using combinatorial techniques via the tropicalization procedure. The tropicalization depends on a map to an algebraic torus and the combinatorial methods are most useful when the…
We give a definition of tropical orbit spaces and their morphisms. We show that, under certain conditions, the weighted number of preimages of a point in the target of such a morphism does not depend on the choice of this point. We equip…
We study cubic rational maps that take lines to plane curves. A complete description of such cubic rational maps concludes the classification of all planarizations, i.e., maps taking lines to plane curves.
The discriminant of a polynomial map is central to problems from affine geometry and singularity theory. Standard methods for characterizing it rely on elimination techniques that can often be ineffective. This paper concerns polynomial…
The first secant variety of a projective monomial curve is a threefold with an action by a one-dimensional torus. Its tropicalization is a three-dimensional fan with a one-dimensional lineality space, so the tropical threefold is…
We study a notion of tropical linear series on metric graphs that combines two essential properties of tropicalizations of linear series on algebraic curves: the Baker-Norine rank and the independence rank. Our main results relate the local…
We show that every tropical totally positive matrix can be uniquely represented as the transfer matrix of a canonical totally connected weighted planar network. We deduce a uniqueness theorem for the factorization of a tropical totally…
In this paper we study tropicalization of Grassmannian and linear varieties. In particular, we study the tropical linear spaces cor- responding to the phylogenetic trees. We prove that corresponding to each subtree of the phylogenetic tree…
We consider multidimensional optimization problems, which are formulated and solved in terms of tropical mathematics. The problems are to minimize (maximize) a linear or nonlinear function defined on vectors over an idempotent semifield,…
We study algebraic and combinatorial aspects of (classical) projections of $m$-dimensional tropical varieties onto $(m+1)$-dimensional planes. Building upon the work of Sturmfels, Tevelev, and Yu on tropical elimination as well as the work…
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…
We present a tropical geometric description of a piecewise linear map whose invariant curve is a concave polygon. In contrast to convex polygons, a concave one is not directly related to tropical geometry; nevertheless the description is…