Related papers: A Tropical Toolkit
A very brief introduction to tropical and idempotent mathematics is presented. Applications to classical mechanics and geometry are especially examined.
We apply ideas from intersection theory on toric varieties to tropical intersection theory. We introduce mixed Minkowski weights on toric varieties which interpolate between equivariant and ordinary Chow cohomology classes on complete toric…
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…
Tropical mathematics is used to establish a correspondence between certain microscopic and macroscopic objects in statistical models. Tropical algebra gives a common framework for macrosystems (subsets) and their elementary constituents…
A key issue in tropical geometry is the lifting of intersection points to a non-Archimedean field. Here, we ask: Where can classical intersection points of planar curves tropicalize to? An answer should have two parts: first, identifying…
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…
We establish first parts of a tropical intersection theory. Namely, we define cycles, Cartier divisors and intersection products between these two (without passing to rational equivalence) and discuss push-forward and pull-back. We do this…
We introduce the notion of resultant of two planar curves in the tropical geometry framework. We prove that the tropicalization of the algebraic resultant can be used to compute the stable intersection of two tropical plane curves. It is…
We present the Macaulay2 package TropicalToric.m2 for toric intersection theory computations using tropical geometry.
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 study a construction of algebraic curves from combinatorial data. In the study of algebraic curves through degeneration, graphs usually appear as the dual intersection graph of the central fiber. Properties of such graphs…
Numerical equivalence of algebraic cycles is defined abstractly by intersection numbers. Classically, for smooth complex proper toric varieties, the quotients by numerical equivalence with rational coefficients can be described…
The notions of convexity and convex polytopes are introduced in the setting of tropical geometry. Combinatorial types of tropical polytopes are shown to be in bijection with regular triangulations of products of two simplices. Applications…
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…
In recent years a series of remarkable advances in tropical geometry and in non-archimedean geometry have brought new insights to the moduli theory of algebraic curves and their Jacobians. The goal of this survey, an expanded version of my…
In this paper we use the connections between tropical algebraic geometry and rigid analytic geometry in order to prove two main results. We use tropical methods to prove a theorem about the Newton polygon for convergent power series in…
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…
Given a tropical divisor $D$ in the intersection of two tropical plane curves, we study when it can be realized as the tropicalization of the intersection of two algebraic curves, and give a sufficient condition. We show that under a…
We study algebraic and combinatorial aspects of (classical) projections of $m$-dimensional tropical varieties onto $(m+1)$-dimensional planes. Building upon the work of Sturmfels, Tevelev, and Yu on tropical elimination as well as the work…
In this paper, we study the deformation theory of degenerate algebraic curves on singular varieties which appear as the degenerate limit of families of varieties. For this purpose, we systematically develop a new method to calculate the…