Related papers: Tropical geometry of PERT
We introduce a scheme-theoretic enrichment of the principal objects of tropical geometry. Using a category of semiring schemes, we construct tropical hypersurfaces as schemes over idempotent semirings such as $\mathbb{T} = (\mathbb{R}\cup…
This paper surveys {\it tropical modifications}, which have already become a folklore in tropical geometry. Tropical modifications are used in tropical intersection theory, tropical Hodge theory, and in the study of singularities. They…
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…
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…
We investigate the system of polynomial equations, known as $QQ$-systems, which are closely related to the so-called Bethe ansatz equations of the XXZ spin chain, using the methods of tropical geometry.
We introduce the notion of tropical area of a tropical curve defined in an open subset of $\mathbb R^n$. We prove that the number of vertices of a tropical curve is bounded by the area of the curve. The approach is totally elementary yet…
We prove that if X, X' are closed subschemes of a torus T over a non-Archimedean field K, of complementary codimension and with finite intersection, then the stable tropical intersection along a (possibly positive-dimensional, possibly…
The tropical convex hull of a finite set of points in tropical projective space has a natural structure of a cellular free resolution. Therefore, methods from computational commutative algebra can be used to compute tropical convex hulls.…
Phylogenetic trees are the fundamental mathematical representation of evolutionary processes in biology. They are also objects of interest in pure mathematics, such as algebraic geometry and combinatorics, due to their discrete geometry.…
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.…
In this paper we study algorithmic aspects of tropical intersection theory. We analyse how divisors and intersection products on tropical cycles can actually be computed using polyhedral geometry. The main focus of this paper is the study…
Generalizing supertropical algebras, we present a "layered" structure, "sorted" by a semiring which permits varying ghost layers, and indicate how it is more amenable than the "standard" supertropical construction in factorizations of…
We study representations of tropical linear spaces as intersections of tropical hyperplanes of circuits. For several classes of matroids, we describe minimal tropical bases. We also show that every realizable tropical linear space has a…
We employ tropical algebras as platforms for several cryptographic schemes that would be vulnerable to linear algebra attacks were they based on "usual" algebras as platforms.
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…
The main aim of this paper is to show the interconnections between {\L}ukasiewicz logic and algebraic geometry using algebraic, geometric and logical instruments. We continue our investigation into a new algebraic geometry based on…
We show that points in the intersection of the tropicalizations of subvarieties of a torus lift to algebraic intersection points with expected multiplicities, provided that the tropicalizations intersect in the expected dimension. We also…
This is a survey on an analogue of tropical convexity developed over the max-min semiring, starting with the descriptions of max-min segments, semispaces, hyperplanes and an account of separation and non-separation results based on…
We use extensions of tropical algebras as platforms for very efficient public key exchange protocols.
The trophic levels of nodes in directed networks can reveal their functional properties. Moreover, the trophic coherence of a network, defined in terms of trophic levels, is related to properties such as cycle structure, stability and…