Related papers: A Topological Representation Theorem for Tropical …
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…
Subject to hypotheses based on the matroid structure theory of Geelen, Gerards, and Whittle, we completely characterize the highly connected members of the class of golden-mean matroids and several other closely related classes of…
This paper is a combinatorial and computational study of the moduli space of tropical curves of genus g, the moduli space of principally polarized tropical abelian varieties, and the tropical Torelli map. These objects were introduced…
We study the geometry of tropical extensions of hyperfields, including the ordinary, signed and complex tropical hyperfields. We introduce the framework of 'enriched valuations' as hyperfield homomorphisms to tropical extensions, and show…
Let A be an n by d matrix having full rank n. An orthogonal dual A^{\perp} of A is a (d-n) by d matrix of rank (d-n) such that every row of A^{\perp} is orthogonal (under the usual dot product) to every row of A. We define the orthogonal…
We develop a theory of representations of (discrete) polymatroids over tracts in terms of Pl\"ucker coordinates and suitable Pl\"ucker relations. As special cases, we recover polymatroids themselves as polymatroid representations over the…
We find restrictions on the topology of tropical varieties that arise from a certain natural class of varieties. We develop a theory of tropical degenerations that is a nonconstant coefficient analogue of Tevelev's theory of tropical…
In this note we generalize the convolution formula for the Tutte polynomial of Kook-Reiner-Stanton and Etienne-Las Vergnas to a more general setting that includes both arithmetic matroids and delta-matroids. As corollaries, we obtain new…
We characterize the shifted simple graphs and the $3$-uniform shifted hypergraphs whose inverse image under exterior shifting is the set of bases of a matroid: those are exactly the hypergraphs whose hyperedges form an initial lex-segment.…
We study Jacobian varieties for tropical curves. These are real tori equipped with integral affine structure and symmetric bilinear form. We define tropical counterpart of the theta function and establish tropical versions of the…
To each prime ideal in a polynomial ring over a field we associate an algebraic matroid and show that it is preserved under tropicalization. This gives a necessary condition for a tropical variety to be set-theoretically realizable from a…
We extend the fundamentals for tropical convexity beyond the tropically positive orthant expanding the theory developed by Loho and V\'egh (ITCS 2020). We study two notions of convexity for signed tropical numbers called 'TO-convexity'…
We introduce a sheaf-theoretic approach to tropical homology, especially for tropical homology with potentially non-compact supports. Our setup is suited to study the functorial properties of tropical homology, and we show that it behaves…
Morphisms of matroids are combinatorial abstractions of linear maps and graph homomorphisms. We introduce the notion of basis for morphisms of matroids, and show that its generating function is strongly log-concave. As a consequence, we…
L\'opez de Medrano, Rinc\'on and Shaw defined the Chern classes on tropical manifolds as an extension of their theory of the Chern-Schwartz-MacPherson cycles on matroids. This makes it possible to define the Riemann-Roch number of tropical…
A matroid is a combinatorial structure that captures and generalizes the algebraic concept of linear independence under a broader and more abstract framework. Matroids are closely related with many other topics in discrete mathematics, such…
In this paper we further develop the theory of geometric tropicalization due to Hacking, Keel and Tevelev and we describe tropical methods for implicitization of surfaces. More precisely, we enrich this theory with a combinatorial formula…
Oriented matroids (often called order types) are combinatorial structures that generalize point configurations, vector configurations, hyperplane arrangements, polyhedra, linear programs, and directed graphs. Oriented matroids have played a…
Using Tutte's combinatorial definition of a map we define a $\Delta$-matroid purely combinatorially and show that it is identical to Bouchet's topological definition.
The algebraic foundation of tropical polynomial algebra provides the framework for the geometric construction of the supplement and the reversal of tropical varieties, thereby inducing a duality of reduced tropical varieties; for classes of…