Related papers: Obstructions to lifting tropical curves in surface…
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…
Duality of curves is one of the important aspects of the ``classical'' algebraic geometry. In this paper, using this foundation, the duality of tropical polynomials is constructed to introduce the duality of Non-Archimedean curves. Using…
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…
A realisation of a graph in the plane as a bar-joint framework is rigid if there are finitely many other realisations, up to isometries, with the same edge lengths. Each of these finitely-many realisations can be seen as a solution to a…
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…
We develop a number of general techniques for comparing analytifications and tropicalizations of algebraic varieties. Our basic results include a projection formula for tropical multiplicities and a generalization of the Sturmfels-Tevelev…
Tropical algebraic geometry is an active new field of mathematics that establishes and studies some very general principles to translate algebro-geometric problems into purely combinatorial ones. This expository paper gives an introduction…
In this paper, we study tropicalisations of singular surfaces in toric threefolds. We completely classify singular tropical surfaces of maximal-dimensional type, show that they can generically have only finitely many singular points, and…
We analyse the dynamics of the pullback of the map $z \longmapsto z^m$ on the complex tori and toric varieties. We will observe that tropical objects naturally appear in the limit, and review several theorems in tropical geometry.
Tropical curves in $\mathbb{R}^2$ correspond to metric planar graphs but not all planar graphs arise in this way. We describe several new classes of graphs which cannot occur. For instance, this yields a full combinatorial characterization…
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…
Tropical mathematics redefines the rules of arithmetic by replacing addition with taking a maximum, and by replacing multiplication with addition. After briefly discussing a tropical version of linear algebra, we study polynomials build…
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 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 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…
This is a follow-up paper of arXiv:1805.00115, where rational curves in surfaces that satisfy general positioned point and cross-ratio conditions were enumerated. A suitable correspondence theorem provided in arXiv:1509.07453 allowed us to…
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…
In this paper, we study tropicalisations of families of curves with a singularity in a fixed point. The tropicalisation of such a family is a linear tropical variety. We describe its maximal dimensional cones using results about linear…
The note introduces a novel concept of non-Abelian patchworking arising as real locus of non-Abelian complex-phase tropical hypersurfaces, the theory of which is now developed enough to allow the proposed spin-off. Although, non-Abelian…
The paper establishes a formula for enumeration of curves of arbitrary genus in toric surfaces. It turns out that such curves can be counted by means of certain lattice paths in the Newton polygon. The formula was announced earlier in…