相关论文: Roots and polynomials as homeomorphic spaces

This paper deals with the use of numerical methods based on random root sampling techniques to solve some theoretical problems arising in the analysis of polynomials. These methods are proved to be practical and give solutions where…

In this paper, we study properties of polynomials over division rings. Moreover, we present formulas for finding roots of some polynomials

In an isomorphic copy of the ring of symmetric polynomials we study some families of polynomials which are indexed by rational weight vectors. These families include well known symmetric polynomials, such as the elementary, homogeneous, and…

Some cubic polynomials over the integers have three distinct real roots with continued fractions that all have the same common tail. We characterize the polynomials for which this happens, and then investigate the situation for other…

In this short note, we give a new sufficient condition for a linear map from a product of copies of a field to endomorphisms of a finite dimensional vector space over the same field to be an algebra homomorphism. We expect that this result…

We prove that projective spaces of Lorentzian and real stable polynomials are homeomorphic to closed Euclidean balls. This solves a conjecture of June Huh and the author. The proof utilizes and refines a connection between the symmetric…

Let P_{k, n}^l be the space consisting of monic complex polynomials f(z) of degree k and such that the number of n-fold roots of f(z) is at most l. In this paper, we determine the integral homology groups of P_{k, n}^l.

We give a concise proof of a classification of lens spaces up to orientation-preserving homeomorphisms. The chief ingredient in our proof is a study of the Alexander polynomial of ` symmetric' links in $S^3$.

Hyperbolic polynomials are monic real-rooted polynomials. By Bronshtein's theorem, the increasingly ordered roots of a hyperbolic polynomial of degree $d$ with $C^{d-1,1}$ coefficients are locally Lipschitz and the solution map…

We give some general criteria of being a homeomorphism for continuous mappings of topological manifolds, as well as criteria of being a diffeomorphism for smooth mappings of smooth manifolds. As an illustration, we apply these criteria to…

We give deterministic polynomial-time algorithms that, given an order, compute the primitive idempotents and determine a set of generators for the group of roots of unity in the order. Also, we show that the discrete logarithm problem in…

For univariate polynomials over arbitrary field the degree gives an upper bound on the number of roots (factor theorem) and as a related result for any finite point-set one can construct a polynomial of degree equal to the cardinality…

A "Littlewood polynomial" is a polynomial whose coefficients are all 1 or -1. The set of all complex roots of all Littlewood polynomials exhibits many complicated, beautiful and fascinating patterns. Some fractal regions of this set closely…

We evaluate the number of monic polynomials (of arbitrary degree $N$) the zeros of which equal their coefficients when these are allowed to take arbitrary complex values. In the following, we call polynomials with this property {\em…

Recently, it was found that there is a remarkable intuitive similarity between studies in theoretical computer science dealing with large data sets on the one hand, and categorical methods of topology and geometry in pure mathematics, on…

We develop a unified framework for the study of properties involving diagonalizations of dense families in topological spaces. We provide complete classification of these properties. Our classification draws upon a large number of methods…

Revisiting canonical integration of the classical solid near a uniform rotation, canonical action angle coordinates, hyperbolic and elliptic, are constructed in terms of various power series with coefficients which are polynomials in a…

We consider stable manifolds of a holomorphic diffeomorphism of a complex manifold. Using a conjugation of the dynamics to a (non-stationary) polynomial normal form, we show that typical stable manifolds are biholomorphic to complex…

Towards building tropical analogues of adic spaces, we study certain spaces of prime congruences as a topological semiring replacement for the space of continuous valuations on a topological ring. This requires building the theory of…

In the present study, we propose necessary and sufficient assumptions on the coefficients in order to only get distinct real roots of polynomials.