Related papers: A note on the polynomial-like iterative equations …

We give two elementary proofs, at a level understandable by students with only pre-calculus knowledge of Algebra, of the well known fact that an irreducible irrational n-th root of a positive rational number cannot be solution of a…

In this paper, we prove a number of results providing either necessary or sufficient conditions guaranteeing that the number of real roots of real polynomials of a given degree is either less or greater than a given number. We also provide…

We present a technique for reducing the order of polynomial-like iterative equations; in particular, we answer a question asked by Wenmeng Zhang and Weinian Zhang. Our method involves the asymptotic behaviour of the sequence of consecutive…

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

The analysis of solutions to algebraic equations is further simplified. A couple of functions and their analytic continuation or root findings are required.

We show that the roots of any smooth curve of polynomials with only real roots can be parametrized twice differentiably (but not better).

A novel method with two variations is proposed with which the number of positive and negative zeros of a polynomial with real coefficients and degree $n$ can be restricted with significantly better determinacy than that provided by the…

An expression for the coefficients of a linear iterative equation in terms of the parameters of the source equation is given both for equations in standard form and for equations in reduced normal form. The operator generating an iterative…

We study real univariate polynomials with non-zero coefficients and with all roots real, out of which exactly two positive. The sequence of coefficients of such a polynomial begins with $m$ positive coefficients followed by $n$ negative…

Some easily verifiable sufficient conditions for the nonexistence of iterative roots for multifunctions on arbitrary nonempty sets are presented. Typically if the graph of the multifunction has a distinguished point with a relatively large…

We consider real univariate polynomials with all roots real. Such a polynomial with $c$ sign changes and $p$ sign preservations in the sequence of its coefficients has $c$ positive and $p$ negative roots counted with multiplicity. Suppose…

Let $c_1(x),c_2(x),f_1(x),f_2(x)$ be polynomials with rational coefficients. With obvious exceptions, there can be at most finitely many roots of unity among the zeros of the polynomials $c_1(x)f_1(x)^n+c_2(x)f_2(x)^n$ with $n=1,2\ldots$.…

We will show that the roots of a polynomial equation in one variable of degree n are related to the solutions of a symmetric quadratic form in n-1 variables with constant positive integer coefficients. The classic polynomial notation will…

We devise a simple but remarkably accurate iterative routine for calculating the roots of a polynomial of any degree. We demonstrate that our results have significant improvement in accuracy over those obtained by methods used in popular…

Univariate polynomial root-finding is a classical subject, still important for modern computing. Frequently one seeks just the real roots of a polynomial with real coefficients. They can be approximated at a low computational cost if the…

A fundamental problem in the theory of linearized and projective polynomials over finite fields is to characterize the number of roots in the coefficient field directly from the coefficients. We prove results of this type, of a recursive…

In this paper, we obtain several new factorization results for certain classes of polynomials having integer coefficients. In doing so, we use the information about prime factorization of the value taken up by such polynomials and their…

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 classical Descartes' rule of signs limits the number of positive roots of a real polynomial in one variable by the number of sign changes in the sequence of its coefficients. One can ask the question which pairs of nonnegative integers…