Related papers: Computing tropical points and tropical links
We develop a novel framework to construct and analyze finite valued, multidimensional mechanisms using tropical convex geometry. We geometrically characterize incentive compatibility using cells in the tropical convex hull of the type set.…
We study the geometry of metrics and convexity structures on the space of phylogenetic trees, which is here realized as the tropical linear space of all \ ultrametrics. The ${\rm CAT}(0)$-metric of Billera-Holmes-Vogtman arises from the…
The software TrIm offers implementations of tropical implicitization and tropical elimination, as developed by Tevelev and the authors. Given a polynomial map with generic coefficients, TrIm computes the tropical variety of the image. When…
We show that super Gromov-Witten invariants can be defined and computed by methods of tropical geometry. When the target is a point, the super invariants are descendant invariants on the moduli space of curves, which can be computed…
We give an introduction to Tropical Geometry and prove some results in Tropical Intersection Theory. The first part of this paper is an introduction to tropical geometry aimed at researchers in Algebraic Geometry from the point of view of…
This paper proposes a tropical geometry-based edge detection framework that reformulates convolution and gradient computations using min-plus and max-plus algebra. The tropical formulation emphasizes dominant intensity variations,…
We present an algorithm to compute all $n$ nondominated points of a multicriteria discrete optimization problem with $d$ objectives using at most $\mathcal{O}(n^{\lfloor d/2 \rfloor})$ scalarizations. The method is similar to algorithms by…
We study parameterized versions of classical algorithms for computing shortest-path trees. This is most easily expressed in terms of tropical geometry. Applications include shortest paths in traffic networks with variable link travel times.
This thesis delves into the geometry of abstract tropical curves, exploring their complete linear system and associated tropical submodules. We establish a lower bound on the dimension of tropical submodules in terms of the Baker-Norine…
We study singularities in tropical hypersurfaces defined by a valuation over a field of positive characteristic. We provide a method to compute the set of singular points of a tropical hypersurface in positive characteristic and the p-adic…
We present tools and definitions to study abstract tropical manifolds in dimension 2, which we call simply tropical surfaces. This includes explicit descriptions of intersection numbers of 1-cycles, normal bundles to some curves and…
The eigenvalues of a matrix polynomial can be determined classically by solving a generalized eigenproblem for a linearized matrix pencil, for instance by writing the matrix polynomial in companion form. We introduce a general scaling…
We tropicalize the rational map that takes triples of points in the projective plane to the plane of quadrics passing through these points. The image of its tropicalization is contained in the tropicalization of its image. We identify these…
In this paper, we consider weighted counts of tropical plane curves of particular combinatorial type through a certain number of generic points. We give a criterion, derived from tropical intersection theory on the secondary fan, for a…
An unconstrained optimization problem is formulated in terms of tropical mathematics to minimize a functional that is defined on a vector set by a matrix and calculated through multiplicative conjugate transposition. For some particular…
This paper illustrates a computational approach to Culler-Morgan-Shalen theory using ideal triangulations, spun-normal surfaces and tropical geometry. Certain affine algebraic sets associated to the Whitehead link complement as well as…
A Laman graph $G$ is a minimally rigid graph in dimension two, and its realization number is its number of distinct embeddings with fixed generic edge lengths. While conjectured to grow exponentially in the number of vertices of $G$, the…
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…
We propose new algorithms for computing triangular decompositions of polynomial systems incrementally. With respect to previous works, our improvements are based on a {\em weakened} notion of a polynomial GCD modulo a regular chain, which…
We give a framework for constructing generically optimal homotopies for parametrized polynomial systems from tropical data. Here, generically optimal means that the number of paths tracked is equal to the generic number of solutions. We…