Related papers: Computing tropical points and tropical links
A novel way to use SMT (Satisfiability Modulo Theories) solvers to compute the tropical prevariety (resp. equilibrium) of a polynomial system is presented. The new method is benchmarked against a naive approach that uses purely polyhedral…
We study tropical line arrangements associated to a three-regular graph $G$ that we refer to as \emph{tropical graph curves}. Roughly speaking, the tropical graph curve associated to $G$, whose genus is $g$, is an arrangement of $2g-2$…
In this paper, we investigate the computational complexity of the knapsack problem and subset sum problem for the following tropical algebraic structures. We consider the semigroup of square matrices of size $k \times k$ with non-negative…
In this article, we look into the tree gonality of genus $3$ metric graphs $\Gamma$ which is defined as the minimum of degrees of all tropical morphisms from any tropical modification of $\Gamma$ to any metric tree. It is denoted by…
We introduce quivers of valuated matroids and study their tropical parameter spaces. We define quiver Dressians, which parametrize containment of tropical linear spaces after tropical matrix multiplication, and show that tropicalizations of…
For certain tropical quartic curves the existing techniques could not predict the lifting behavior of their bitangents over the real numbers. We close this gap by using patchworking techniques. Further, this paper provides an analysis of…
Let $K$ be a field equipped with a valuation. Tropical varieties over $K$ can be defined with a theory of Gr\"obner bases taking into account the valuation of $K$. Because of the use of the valuation, this theory is promising for stable…
We provide some new local obstructions to approximating tropical curves in smooth tropical surfaces. These obstructions are based on the relation between tropical and complex intersection theories which is also established here. We give two…
In this paper we bring together tropical linear algebra and convex 3-dimensional bodies. We show how certain convex 3-dimensional bodies having 20 vertices and 12 facets can be encoded in a $4\times 4$ integer zero-diagonal matrix $A$. A…
We define transversal tropical triangles (affine and projective) and characterize them via six inequalities to be satisfied by the coordinates of the vertices. We prove that the vertices of a transversal tropical triangle are tropically…
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 study pairs of mutually orthogonal normal matrices with respect to tropical multiplication. Minimal orthogonal pairs are characterized. The diameter and girth of three graphs arising from the orthogonality equivalence relation are…
This is a guide on how to create 3d printable models of tropical surfaces, curves, and combinations thereof. It uses Polymake to construct bounded models of the tropical objects, and OpenSCAD to thicken and export them to any common 3D…
Neural networks are important tools in machine learning. Representing piecewise linear activation functions with tropical arithmetic enables the application of tropical geometry. Algorithms are presented to compute regions where the neural…
Polytropes are both ordinary and tropical polytopes. We show that tropical types of polytropes in $\mathbb{TP}^{n-1}$ are in bijection with cones of a certain Gr\"{o}bner fan $\mathcal{GF}_n$ in $\mathbb{R}^{n^2 - n}$ restricted to a small…
This paper supplements [17], showing that categorically the layered theory is the same as the theory of ordered monoids (e.g. the max-plus algebra) used in tropical mathematics. A layered theory is developed in the context of categories,…
We introduce delta-forms on tropical toric varieties generalizing the construction of Mihatsch for $R^n$. These delta-forms will be used to define the star-product with Green functions of piecewise smooth type on a tropical toric variety.…
We consider discrete best approximation problems in the setting of tropical algebra, which is concerned with the theory and application of algebraic systems with idempotent operations. Given a set of input--output pairs of an unknown…
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 describe recent work connecting combinatorics and tropical/non-Archimedean geometry to Diophantine geometry, particularly the uniformity conjectures for rational points on curves and for torsion packets of curves. The method of…