Related papers: Tropical Scaling of Polynomial Matrices
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…
We consider multicriteria problems of evaluating absolute ratings (scores, priorities, weights) of given alternatives for making decisions, which are compared in pairs under several criteria. Given matrices of pairwise comparisons of…
As a new concept tropical halfspaces are introduced to the (linear algebraic) geometry of the tropical semiring (R,min,+). This yields exterior descriptions of the tropical polytopes that were recently studied by Develin and Sturmfels in a…
A novel way to use SMT (Satisfiability Modulo Theories) solvers to compute the tropical prevariety (resp. equilibrium) of a polynomial system is presented. The new method is benchmarked against a naive approach that uses purely polyhedral…
In this paper we further develop the theory of matrices over the extended tropical semiring. Introducing a notion of tropical linear dependence allows for a natural definition of matrix rank in a sense that coincides with the notions of…
This work is divided into three parts. The first part concerns polynomials in one variable with all real roots. We consider linear transformations that preserve real rootedness, as well as matrices that preserve interlacing. The second part…
This paper provides an accurate method to obtain the bidiagonal factorization of many generalized Pascal matrices, which in turn can be used to compute with high relative accuracy the eigenvalues, singular values and inverses of these…
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…
In many applications it is important to understand the sensitivity of eigenvalues of a matrix polynomial to perturbations of the polynomial. The sensitivity commonly is described by condition numbers or pseudospectra. However, the…
We calculate the autocorrelation function for the characteristic polynomial of a random matrix in the microscopic scaling regime. While results fitting this description have be proved before, we will cover all values of inverse temperature…
Given a curve defined over an algebraically closed field which is complete with respect to a nontrivial valuation, we study its tropical Jacobian. This is done by first tropicalizing the curve, and then computing the Jacobian of the…
Tropical varieties capture combinatorial information about how coordinates of points in a classical variety approach zero or infinity. We present algorithms for computing the rays of a complex and real tropical curve defined by polynomials…
We prove identities on compound matrices in extended tropical semirings. Such identities include analogues to properties of conjugate matrices, powers of matrices and~$\adj(A)\det(A)^{ -1}$, all of which have implications on the eigenvalues…
Inspired by numerical homotopy methods we propose a combinatorial homotopy algorithm for finding all isolated solutions to a tropical polynomial systems of n tropical polynomials in n variables. In particular, a tropicalisation of the…
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…
We give a new method to calculate the universal cohomology classes of coincident root loci. We show a polynomial behavior of them and apply this result to prove that generalized Pl\"ucker formulas are polynomials in the degree, just as the…
We describe a new incomplete but terminating method for real root finding for large multivariate polynomials. We take an abstract view of the polynomial as the set of exponent vectors associated with sign information on the coefficients.…
The purpose of this paper is to show how the problem of finding the zeros of unilateral n-order quaternionic polynomials can be solved by determining the eigen-vectors of the corresponding companion matrix. This approach, probably…
It is well known that for two univariate polynomials over complex number field the number of their common roots is equal to the order of their resultant. In this paper, we show that this fundamental relationship still holds for the tropical…
Tropical geometry has recently found several applications in the analysis of neural networks with piecewise linear activation functions. This paper presents a new look at the problem of tropical polynomial division and its application to…