Tropical Arithmetic and Matrix Algebra

This article introduces a new structure of commutative semiring, generalizing the tropical semiring, and having an arithmetic that modifies the standard tropical operations, i.e., summation and maximum. Although our framework is combinatorial, notions of regularity and invertibility arise naturally for matrices over this semiring; we show that a tropical matrix is invertible if and only if it is regular.     

Tropical Cryptography

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.     

Tropical cryptography II: Extensions by homomorphisms

We use extensions of tropical algebras as platforms for very efficient public key exchange protocols.     

Identities of the Semigroup of Upper Triangular Tropical Matrices

A new family of identities satisfied by the semigroups U _n (𝕋) of n × n upper triangular tropical matrices is constructed and an elementary proof is given.     

Tropical differential equations

Tropical differential equations are introduced and an algorithm is designed which tests solvability of a system of tropical linear differential equations within the complexity polynomial in the size of the system and in the absolute values of its coefficients. Moreover, we show that there exists a minimal solution, and the algorithm constructs it (in case of solvability). This extends a similar complexity bound established for tropical linear systems. In case of tropical linear differential systems in one variable a polynomial complexity algorithm for testing its solvability is designed.We prove also that the problem of solvability of a system of tropical non-linear differential equations in one variable is NP-hard, and this problem for arbitrary number of variables belongs to NP. Similar to tropical algebraic equations, a tropical differential equation expresses the (necessary) condition on the dominant term in the issue of solvability of a differential equation in power series.     

Singular Tropical Hypersurfaces

We study the notion of singular tropical hypersurfaces of any dimension. We characterize the singular points in terms of tropical Euler derivatives and we give an algorithm to compute all singular points. We also describe non-transversal intersection points of planar tropical curves.     

Tropical Surface Singularities

In this paper, we study tropicalizations of singular surfaces in toric threefolds. We completely classify singular tropical surfaces of maximal-dimensional geometric type, show that they can generically have only finitely many singular points, and describe all possible locations of singular points. More precisely, we show that singular points must be either vertices, or generalized midpoints and barycenters of certain faces of singular tropical surfaces, and, in some case, there may be additional metric restrictions to faces of singular tropical surfaces.     

Tropical hyperelliptic curves

We study the locus of tropical hyperelliptic curves inside the moduli space of tropical curves of genus g. We define a harmonic morphism of metric graphs and prove that a metric graph is hyperelliptic if and only if it admits a harmonic morphism of degree 2 to a metric tree. This generalizes the work of Baker and Norine on combinatorial graphs to the metric case. We then prove that the locus of 2-edge-connected genus g tropical hyperelliptic curves is a (2g?1)-dimensional stacky polyhedral fan whose maximal cells are in bijection with trees on g?1 vertices with maximum valence 3. Finally, we show that the Berkovich skeleton of a classical hyperelliptic plane curve satisfying a certain tropical smoothness condition is a standard ladder of genus g.     

Tropical matrix groups

We study the structure of groups of finitary tropical matrices under multiplication. We show that the maximal groups of \(n \times n\) tropical matrices are precisely the groups of the form \(G \times \mathbb {R}\) where G is a group admitting a 2-closed permutation representation on n points. Each such maximal group is also naturally isomorphic to the full linear automorphism group of a related tropical polytope. Our results have numerous corollaries, including the fact that every automorphism of a projective (as a module) tropical polytope of full rank extends to an automorphism of the containing space.     

Tropical convexity via cellular resolutions

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. Tropical cyclic polytopes are also presented.     

Submathematics and Tropical Mathematics

Mathematical Notes - The notions of “tropical mathematics” and “subtropical mathematics” are studied. The main principles of tropical analysis and examples of their...     

Tropical hyperplane arrangements and oriented matroids

We study the combinatorial properties of a tropical hyperplane arrangement. We define tropical oriented matroids, and prove that they share many of the properties of ordinary oriented matroids. We show that a tropical oriented matroid determines a subdivision of a product of two simplices, and conjecture that this correspondence is a bijection.     

Linear Pencils of Tropical Plane Curves

Linear pencils of tropical plane curves are parameterized by tropical lines (i.e. trees) in the space of coefficients. We study pencils of tropical curves with n-element support that pass through n?2 general points in the plane. Richter-Gebert et al. proved that such trees are compatible with their support set, and they conjectured that every compatible tree can be realized by a point configuration. In this article, we prove this conjecture. Our approach is based on a characterization of the fixed loci of tropical linear pencils.     

Tropical Hurwitz numbers

Hurwitz numbers count genus g, degree d covers of ℙ¹ with fixed branch locus. This equals the degree of a natural branch map defined on the Hurwitz space. In tropical geometry, algebraic curves are replaced by certain piece-wise linear objects called tropical curves. This paper develops a tropical counterpart of the branch map and shows that its degree recovers classical Hurwitz numbers. Further, the combinatorial techniques developed are applied to recover results of Goulden et al. (in Adv. Math. 198:43–92, 2005) and Shadrin et al. (in Adv. Math. 217(1):79–96, 2008) on the piecewise polynomial structure of double Hurwitz numbers in genus 0.     

Tropical quadrics through three points

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 objects inside the tropical Grassmannian of planes in projective 5-space, and we explore a small tropical Hilbert scheme.     

Tropical curves with a singularity in a fixed point

In this paper, we study tropicalisations of families of plane 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 tropical varieties. We show that a singularity tropicalises either to a vertex of higher valence or of higher multiplicity, or to an edge of higher weight. We then classify maximal dimensional types of singular tropical curves. For those, the singularity is either a crossing of two edges, or a 3-valent vertex of multiplicity 3, or a point on an edge of weight 2 whose distances to the neighbouring vertices satisfy a certain metric condition. We also study generic singular tropical curves enhanced with refined tropical limits and construct canonical simple parameterisations for them, explaining the above metric condition.     

Tropical Bisectors and Voronoi Diagrams

Foundations of Computational Mathematics - In this paper we initiate the study of tropical Voronoi diagrams. We start out with investigating bisectors of finitely many points with respect to...