相关论文: Completions, Reversals, and Duality for Tropical V…

We give an overview of recently implemented polymake features for computations in tropical geometry. The main focus is on explicit examples rather than technical explanations. Our computations employ tropical hypersurfaces, moduli of…

We present a new algorithmic framework which utilizes tropical geometry and homotopy continuation for solving systems of polynomial equations where some of the polynomials are generic elements in linear subspaces of the polynomial ring.…

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 its coefficients.…

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 give a complete description of Green's D relation for the multiplicative semigroup of all n-by-n tropical matrices. Our main tool is a new variant on the duality between the row and column space of a tropical matrix (studied by Cohen,…

We present an algorithm for computing zero-dimensional tropical varieties based on triangular decomposition and Newton polygon methods. From it, we derive algorithms for computing points on and links of higher-dimensional tropical…

Finding a common factor of two multivariate polynomials with approximate coefficients is a problem in symbolic-numeric computing. Taking a tropical view on this problem leads to efficient preprocessing techniques, applying polyhedral…

We develop the algebraic polynomial theory for "supertropical algebra," as initiated earlier over the real numbers by the first author. The main innovation there was the introduction of "ghost elements," which also play the key role in our…

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…

An algorithm is designed which decomposes a tropical univariate rational function into a composition of tropical binomials and trinomials. When a function is monotone, the composition consists just of binomials. Similar algorithms are…

Ducros, Hrushovski, and Loeser gave maps from families of archimedean diffrential forms to non-archiemedean (or tropical) ones, which are compatible with integrals on algebraic varieties. In this paper, we introduce slight modifications of…

Tropical implicitization means computing the tropicalization of a unirational variety from its parametrization. In the case of a hypersurface, this amounts to finding the Newton polytope of the implicit equation, without computing its…

We establish a partial extension of the Poincar\'e duality theorem of Jell-Rau-Shaw to tropical hypersurfaces arising from non-primitive triangulations. We introduce a notion of level of primitivity for triangulations of lattice polytopes…

We develop a collection of numerical algorithms which connect ideas from polyhedral geometry and algebraic geometry. The first algorithm we develop functions as a numerical oracle for the Newton polytope of a hypersurface and is based on…

We discuss the tropical analogues of several basic questions of convex duality. In particular, the polar of a tropical polyhedral cone represents the set of linear inequalities that its elements satisfy. We characterize the extreme rays of…

We study the tropicalization of the variety of symmetric rank two matrices. Analogously to the result of Markwig and Yu for general tropical rank two matrices, we show that it has a simplicial complex structure as the space of symmetric…

We study tropical degree bounds, stable tropical intersections, and tropical B\'ezout-type estimates through the geometry of Newton polytopes, mixed subdivisions, and lattice indices. We establish an upper bound for the tropical degree of a…

This paper presents a unified mathematical framework for inference in graphical models, building on the observation that graphical models are algebraic varieties. From this geometric viewpoint, observations generated from a model are…

We consider the question of when points in tropical affine space uniquely determine a tropical hypersurface. We introduce a notion of multiplicity of points so that this question may be meaningful even if some of the points coincide. We…

In this work, we examine the process of Tropical Polynomial Division, a geometric method which seeks to emulate the division of regular polynomials, when applied to those of the max-plus semiring. This is done via the approximation of the…