Related papers: Tropical Cramer Determinants Revisited
In this paper we present two intrinsic algebraic definitions of tropical variety motivated by the classical Zariski correspondence, one utilizing the algebraic structure of the coordinate semiring of an affine supertropical algebraic set,…
Tropical geometry is a piecewise linear "shadow" of algebraic geometry. It allows for the computation of several cohomological invariants of an algebraic variety. In particular, its application to enumerative algebraic geometry led to…
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…
We develop the basic theory of projective modules and splitting in the more general setting of systems. Systems provide a common language for most tropical algebraic approaches including supertropical algebra, hyperrings (specifically…
We consider arrangements of tropical hyperplanes where the apices of the hyperplanes are taken to infinity in certain directions. Such an arrangement defines a decomposition of Euclidean space where a cell is determined by its `type' data,…
Tropical and idempotent analysis with their relations to the Hamilton-Jacobi and matrix Bellman equations are discussed. Some dequantization procedures are important in tropical and idempotent mathematics. In particular, the…
Let K be a field equipped with a valuation. Tropical varieties over K can be defined with a theory of Gr{\"o}bner bases taking into account the valuation of K. Because of the use of the valuation, the theory of tropical Gr{\"o}bner bases…
We show factorization of polynomials in one variable over the tropical semiring is in general NP-complete, either if all coefficients are finite, or if all are either 0 or infinity (Boolean case). We give algorithms for the factorization…
In this paper, we survey and study definitions and properties of tropical polynomials, tropical rational functions and in general, tropical meromorphic functions, emphasizing practical techniques that can really carry out computations. For…
This article is a sequel of [4], where we introduced quadratic forms on a module~ $V$ over a supertropical semiring $R$ and analysed the set of bilinear companions of a quadratic form $q: V \to R$ in case that the module $V$ is free, with…
The aim of this article is to solve the system $XA=Y$ where $A=(a_{ij})\in M_{m\times n}(S)$, $Y\in S^{m}$ and $X$ is an unknown vector of size $n$, being $S$ an additively idempotent semiring. If the system has solutions then we completely…
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…
We find geometric and arithmetic conditions in order to characterize the irreducibility of the determinant of the generic Vandermonde matrix over the algebraic closure of any field k. We also characterize those determinants whose…
We introduce a sheaf-theoretic approach to tropical homology, especially for tropical homology with potentially non-compact supports. Our setup is suited to study the functorial properties of tropical homology, and we show that it behaves…
Let $K$ be a field equipped with a valuation. Tropical varieties over $K$ can be defined with a theory of Gr\"obner bases taking into account the valuation of $K$. Because of the use of the valuation, this theory is promising for stable…
H. J. S. Smith proved Fermat's two-square theorem using the notion of palindromic continuants. In this paper we extend Smith's approach to proper binary quadratic form representations in some commutative Euclidean rings, including rings of…
We state the fundamental theorem of projective geometry for semimodules over semirings, which is facilitated by recent work in the study of bases in semimodules defined over semirings. In the process we explore in detail the linear algebra…
Tropical linear algebra has been recently put forward by Grigoriev and Shpilrain as a promising platform for implementation of protocols of Diffie-Hellman and Stickel type. Based on the CSR expansion of tropical matrix powers, we suggest a…
In this paper we propose some continuation theorems for the periodic problem \begin{equation*} \begin{cases} \, x_{i}' = g_{i}(t,x_{i+1}), &i=1,\ldots,n-1, \\ \, x_{n}' = h(t,x_{1},\ldots,x_{n}), \\ \, x_{i}(0)=x_{i}(T), &i=1,\ldots,n,…
We prove the existence of a common eigenvector for commutative, nilpotent and quasinilpotent semigroups of matrices with complex or real nonnegative entries both in the conventional and tropical linear algebra.