Related papers: Roots of polynomials over division rings

The degree polynomial of a multigraph $G$ is given by $\sum _{v \in V(G)} x^{\mbox{deg}(v)}$. We investigate here properties of the roots of such polynomials. In addition to examining the roots for some families of graphs with few and many…

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

In this article, the ring of polynomials is studied in a systematic way through the theory of monoid rings. As a consequence, this study provides natural and canonical approaches in order to find easy and rigorous proofs and methods for…

We study the roots of a random polynomial over the field of p-adic numbers. For a random monic polynomial with coefficients in $\mathbb{Z}_p$, we obtain an asymptotic formula for the factorial moments of the number of roots of this…

In this article we shall study some basic properties of posynomial rings with particular emphasis on rings ${\rm Pos}(K,\mathbb{Q})[\bar x]$, and ${\rm Pos}(K,\mathbb{Z})[\bar x]$. The latter ring is the well known ring of Laurent…

We clarify the question whether for a smooth curve of polynomials one can choose the roots smoothly and related questions. Applications to perturbation theory of operators are given.

In this paper, we give some counting results on integer polynomials of fixed degree and bounded height whose distinct non-zero roots are multiplicatively dependent. These include sharp lower bounds, upper bounds and asymptotic formulas for…

The roots of any polynomial of degree m with complex integer coefficients can be computed by manipulation of sequences made from distinct symbols and counting the different symbols in the sequences. This method requires only primitive…

We prove some separation results for the roots of the generalized Fibonacci polynomials and their absolute values

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…

The ring of dual numbers over a ring $R$ is $R[\alpha] = R[x]/(x^2)$, where $\alpha$ denotes $x+(x^2)$. For any finite commutative ring $R$, we characterize null polynomials and permutation polynomials on $R[\alpha]$ in terms of the…

This is a straightforward introduction to the properties of polynomials in many variables that do not vanish in the open upper half plane. Such polynomials generalize many of the well-known properties of polynomials with all real roots.

In this note, we provide a wide range of upper bounds for the moduli of the zeros of a complex polynomial. The obtained bounds complete a series of previous papers on the location of zeros of polynomials.

We consider polynomials with integer coefficients and discuss their factorization properties in Z[[x]], the ring of formal power series over Z. We treat polynomials of arbitrary degree and give sufficient conditions for their reducibility…

In this paper, as an extension of the integer case, we define polynomial functions over the residue class rings of Dedekind domains, and then we give canonical representations and counting formulas for such polynomial functions. In…

We present new classes of permutation polynomials over finite fields.

We construct a family of root-finding algorithms which exploit the branched covering structure of a polynomial of degree $d$ with a path-lifting algorithm for finding individual roots. In particular, the family includes an algorithm that…

The paper studies constructions of irreducible polynomials over finite fields using polynomial composition method.

We give two determinantal representations for a bivariate polynomial. They may be used to compute the zeros of a system of two of these polynomials via the eigenvalues of a two-parameter eigenvalue problem. The first determinantal…

In this article we use a method of finding the index of a complex-valued function by determined number of arithmetic operations to describe an algorithm of localization of roots of square-free polynomials. We give an estimation of the…