Related papers: Weighted digraphs and tropical cones

We establish a connection between linear codes and hyperplane arrangements using the Thomas decomposition of polynomial systems and the resulting counting polynomial. This yields both a generalization and a refinement of the weight…

We study the combinatorics of tropical hyperplane arrangements, and their relationship to (classical) hyperplane face monoids. We show that the refinement operation on the faces of a tropical hyperplane arrangement, introduced by Ardila and…

We define tropical analogues of the notions of linear space and Plucker coordinate and study their combinatorics. We introduce tropical analogues of intersection and dualization and define a tropical linear space built by repeated…

An algorithm is designed which decomposes a tropical univariate rational function into a composition of tropical binomials and trinomials. When a function is monotone, the composition consists just of binomials. Similar algorithms are…

In a metric space, the Fermat-Weber points of a sample are statistics to measure the central tendency of the sample and it is well-known that the Fermat-Weber point of a sample is not necessarily unique in the metric space. We investigate…

We construct a functorial decomposition of de Rham cohomology sheaves, called weight decomposition, for smooth analytic spaces over non-Archimedean fields embeddable into $\mathbf{C}_p$, which generalizes a construction of Berkovich and…

The family of complex projective surfaces in projective three space of degree $d$ having precisely $\delta$ nodes as their only singularities has codimension $\delta$ in the linear system of surfaces of degree $d$ for sufficiently large $d$…

It is known that any tropical polytope is the image under the valuation map of ordinary polytopes over the Puiseux series field. The latter polytopes are called lifts of the tropical polytope. We prove that any pure tropical polytope is the…

In this paper, we give an explicit description of tropical cohomology of smooth algebraic varieties over trivially valued fields. We also construct ``monodromy weight'' spectral sequences for tropical cohomology of geometric strictly…

We study some basic algorithmic problems concerning the intersection of tropical hypersurfaces in general dimension: deciding whether this intersection is nonempty, whether it is a tropical variety, and whether it is connected, as well as…

This paper is devoted to the bounding and computation of the dimension of deformation spaces of tropical curves and hypersurfaces. This characteristic is interesting in light of the fact that it often coincides with the dimension of…

We show that, once translated to the dual setting of convex triangulations of lattice polytopes, results and methods from previous tropical works by Arnal-Renaudineau-Shaw, Renaudineau-Shaw, Renaudineau-Rau-Shaw, and Jell-Rau-Shaw extend to…

We enumerate rational curves in toric surfaces passing through points and satisfying cross-ratio constraints using tropical and combinatorial methods. Our starting point is arXiv:1509.07453, where a tropical-algebraic correspondence theorem…

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 present an algorithm for computing zero-dimensional tropical varieties based on triangular decomposition and Newton polygon methods. From it, we derive algorithms for computing points on and links of higher-dimensional tropical…

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…

We revisit the problem of hard particles on planar random tetravalent graphs in view of recent combinatorial techniques relating planar diagrams to decorated trees. We show how to recover the two-matrix model solution to this problem in…

The map which takes a square matrix to its tropical eigenvalue-eigenvector pair is piecewise linear. We determine the cones of linearity of this map. They are simplicial but they do not form a fan. Motivated by statistical ranking, we also…

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…

We establish a new notion of tropical convexity for signed tropical numbers. We provide several equivalent descriptions involving balance relations and intersections of open halfspaces as well as the image of a union of polytopes over…