Related papers: Detecting tropical defects of polynomial equations
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 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.…
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…
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…
A polynomial complexity algorithm is designed which tests whether a point belongs to a given tropical linear variety.
We study some basic algorithmic problems concerning the intersection of tropical hypersurfaces in general dimension: deciding whether this intersection is nonempty, whether it is a tropical variety, and whether it is connected, as well as…
This paper introduces the foundations of the polynomial algebra and basic structures for algebraic geometry over the extended tropical semiring. Our development, which includes the tropical version for the fundamental theorem of algebra,…
We present algorithms for computing zero-dimensional tropical varieties as implemented in OscarZerodimensionalTropicalization.jl. The algorithms include a mathematical workaround for a common practical issue arising when working with…
Tropical counting tools are useful for many enumerative questions. We count tropical multinodal surfaces using floor plans, looking at the case when two nodes are tropically close together, i.e., unseparated. We generalize tropical floor…
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…
Tropical ideals are a class of ideals in the tropical polynomial semiring that combinatorially abstracts the possible collections of supports of all polynomials in an ideal over a field. We study zero-dimensional tropical ideals I with…
The purpose of this paper is fourfold. The first is to develop the theory of tropical differential algebraic geometry from scratch; the second is to present the tropical fundamental theorem for differential algebraic geometry, and show how…
We introduce new discrete best approximation problems, formulated and solved in the framework of tropical algebra, which deals with semirings and semifields with idempotent addition. Given a set of samples, each consisting of the input and…
We give an explicit upper bound for the degree of a tropical basis of a homogeneous polynomial ideal. As an application f-vectors of tropical varieties are discussed. Various examples illustrate differences between Gr\"obner and tropical…
We consider discrete best approximation problems in the setting of tropical algebra, which is concerned with the theory and application of algebraic systems with idempotent operations. Given a set of input--output pairs of an unknown…
We explore the concept of real tropical basis of an ideal in the field of real Puiseux series. We show explicit tropical bases of zero-dimensional real radical ideals, linear ideals and hypersurfaces coming from combinatorial patchworking.…
Tropical algebra is an emerging field with a number of applications in various areas of mathematics. In many of these applications appeal to tropical polynomials allows to study properties of mathematical objects such as algebraic varieties…
Already for bivariate tropical polynomials, factorization is an NP-Complete problem. In this paper, we give an efficient algorithm for factorization and rational factorization of a rich class of tropical polynomials in $n$ variables.…
We present two effective tools for computing the positive tropicalization of algebraic varieties. First, we outline conditions under which the initial ideal can be used to compute the positive tropicalization, offering a real analogue to…
We introduce tropical Newton-Puiseux polynomials admitting rational exponents. A resolution of a tropical hypersurface is defined by means of a tropical Newton-Puiseux polynomial. A polynomial complexity algorithm for resolubility of a…