Related papers: The Fundamental Theorem of Tropical Differential A…

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…

We construct a general framework for tropical differential equations based on idempotent semirings and an idempotent version of differential algebra. Over a differential ring equipped with a non-archimedean norm enhanced with additional…

In this paper we give an elementary proof of the Fundamental Theorem of Algebra for polynomials over the rational tropical semi-ring. We prove that, tropically, the rational numbers are algebraically closed. We provide a simple algorithm…

Let $K$ be a real closed field with a nontrivial non-archimedean absolute value. We study a refined version of the tropicalization map, which we call real tropicalization map, that takes into account the signs on $K$. We study images of…

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,…

Kapranov's theorem is a foundational result in tropical geometry. It states that the set of tropicalisations of points on a hypersurface coincides precisely with the tropical variety of the tropicalisation of the defining polynomial. The…

In this paper we continue the program to develop the algebraic foundations of tropical (algebraic) geometry. We give strong characterizations of prime congruences containing a given congruence on a toric semiring. We give four applications…

We introduce and study a special class of ideals, called tropical ideals, in the semiring of tropical polynomials, with the goal of developing a useful and solid algebraic foundation for tropical geometry. The class of tropical ideals…

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…

Let $G$ be a connected reductive algebraic group. We develop a Gr\"obner theory for multiplicity-free $G$-algebras, as well as a tropical geometry for subschemes in a spherical homogeneous space $G/H$. We define the notion of a spherical…

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…

In earlier papers it was shown that the generic tropical variety of an ideal can contain information on algebraic invariants as for example the depth in a direct way. The existence of generic tropical varieties has so far been proved in the…

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…

Tropical geometry is used to develop a new approach to the theory of discriminants and resultants in the sense of Gel'fand, Kapranov and Zelevinsky. The tropical A-discriminant, which is the tropicalization of the dual variety of the…

We continue, in this second article, the study of the the algebraic tools which play a role in tropical algebra. We especially examine here the polynomial algebras over idempotent semi-fields. this work is motivated by the development of…

Tropical algebraic geometry offers new tools for elimination theory and implicitization. We determine the tropicalization of the image of a subvariety of an algebraic torus under any homomorphism from that torus to another torus.

In this paper, the tropical differential Gr\"obner basis is studied, which is a natural generalization of the tropical Gr\"obner basis to the recently introduced tropical differential algebra. Like the differential Gr\"obner basis, the…

We prove a fundamental theorem for tropical partial differential equations, analogous to the fundamental theorem of tropical geometry in this context. We extend results from Aroca et al., Falkensteiner et al. and from Fink and Toghani for…

A correspondence exists between affine tropical varieties and algebraic objects, following the classical Zariski correspondence between irreducible affine varieties and the prime spectrum of the coordinate algebra in affine algebraic…

Tropical ideals, introduced in arXiv:1609.03838, define subschemes of tropical toric varieties. We prove that the top-dimensional parts of their varieties are balanced polyhedral complexes of the same dimension as the ideal. This means that…