相关论文: Tropical Linear Series and Tropical Independence

In classical geometry, a linear space is a space that is closed under linear combinations. In tropical geometry, it has long been a consensus that tropical varieties defined by valuated matroids are the tropical analogue of linear spaces.…

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 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 give an algorithm, with a singly exponential complexity, deciding whether a tropical linear prevariety is a tropical linear variety. The algorithm relies on a criterion to be a tropical linear variety in terms of a duality between the…

We define the tropical moduli space of covers of a tropical line in the plane as weighted abstract polyhedral complex, and the tropical branch map recording the images of the simple ramifications. Our main result is the invariance of the…

This paper supplements [17], showing that categorically the layered theory is the same as the theory of ordered monoids (e.g. the max-plus algebra) used in tropical mathematics. A layered theory is developed in the context of categories,…

We prove a specialization inequality relating the dimension of the complete linear series on a variety to the tropical complex of a regular semistable degeneration. Our result extends Baker's specialization inequality to arbitrary…

We present a simple and elementary procedure to sketch the tropical conic given by a degree--two homogeneous tropical polynomial. These conics are trees of a very particular kind. Given such a tree, we explain how to compute a defining…

Tropical geometry is a relatively recent field in mathematics created as a simplified model for certain problems in algebraic geometry. We introduce the definition of abstract and planar tropical curves as well as their properties,…

In this survey, we discuss linear series on tropical curves and their relation to classical algebraic geometry, describe the main techniques of the subject, and survey some of the recent major developments in the field, with an emphasis on…

In this paper we develop a combinatorial abstraction of tropical linear programming. This generalizes the search for a feasible point of a system of min-plus-inequalities. It is based on the polyhedral properties of triangulations of the…

We show that tropicalization of linear series on curves gives rise to two-parameter families of tilings by polymatroids, with one parameter arising from the theory of divisors on tropical curves and the other from the reduction of linear…

We prove that the existence of a divisor of degree $3$ and Baker-Norine rank at least $1$ on a $3$-edge connected tropical curve is equivalent to the existence of a non-degenerate harmonic morphism of degree $3$ from a tropical modification…

In this article we study the tropicalization of the Hilbert scheme and its suitability as a parameter space for tropical varieties. We prove that the points of the tropicalization of the Hilbert scheme have a tropical variety naturally…

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…

Tropical Differential Algebraic Geometry considers difficult or even intractable problems in Differential Equations and tries to extract information on their solutions from a restricted structure of the input. The Fundamental Theorem of…

A polynomial complexity algorithm is designed which tests whether a point belongs to a given tropical linear variety.

This paper is a follow-up of a previous work in which we show that, for a $3$-edge connected tropical curve $\Gamma$, the existence of a divisor of degree $3$ and Baker-Norine rank at least $1$ in $\Gamma$ is equivalent to the existence of…

We define transversal tropical triangles (affine and projective) and characterize them via six inequalities to be satisfied by the coordinates of the vertices. We prove that the vertices of a transversal tropical triangle are tropically…

We construct a space classifying divisor classes of a fixed degree on all tropical curves of a fixed combinatorial type and show that the function taking a divisor class to its rank is upper semicontinuous. We extend the definition of the…