Related papers: The critical points of a polynomial

In this paper, we obtain new results on the critical points of a polynomial. We discuss the Sendov conjecture for polynomials of degree nine.

Closely following recent ideas of J. Borcea, we discuss various modifications and relaxations of Sendov's conjecture about the location of critical points of a polynomial with complex coefficients. The resulting open problems are formulated…

Sendov's conjecture, which was first introduced in the last 50s, asserts that if all the zeros of a polynomial $p$ lie in the closed unit disk then for each zero there must be a critical point of $p$ within unit distance. This paper…

The Sendovs conjecture asserts that if all the zeros of a polynomial p(z) lie in the closed unit disk, then there must be a critical point of p(z) within unit distance of each zero. The conjecture has been proved to be true for many special…

In this paper, we prove the Sendov conjecture for polynomials of degree nine. We use a new idea to obtain new upper bound for the $\sigma-$sum to zeros of the polynomial.

In this paper we first prove that a simple root of a polynomial satisfies the Sendov's conjecture. As the multiple roots trivially satisfy the Sendov's conjecture we conclude that the Sendov's conjecture holds true.

Sendov conjecture tells that if $P$ denotes a complex polynomial having all his zeros in the closed unit disk and $a$ denote a zero of $P$, the closed disk of center $a$ and radius 1 contains a zero of the derivative $P'$. The main result…

The Sendov conjecture asserts that if all the zeros of a polynomial p lie in the closed unit disk then there must be a zero of p ' within unit distance of each zero. In this paper we give a partial result when p has simple zeros.

Sendov's conjecture states that if all the zeroes of a complex polynomial $P(z)$ of degree at least two lie in the unit disk, then within a unit distance of each zero lies a critical point of $P(z)$. In a paper that appeared in 2014,…

We consider the set S(n,0) of monic complex polynomials of degree $n\ge 2$ having all their zeros in the closed unit disk and vanishing at 0. For $p\in S(n,0)$ we let $|p|_{0}$ denote the distance from the origin to the zero set of $p'$. We…

In this paper we introduce and develop the concept of expansivity of a tuple whose entries are elements from the polynomial ring $\mathbb{C}[x]$. As an inverse problem, we examine how to recover a tuple from the expanded tuple at any given…

We give an upper bound in O(d ^((n+1)/2)) for the number of critical points of a normal random polynomial with degree d and at most n variables. Using the large deviation principle for the spectral value of large random matrices we obtain…

A proof of Sendov's conjecture is given.

A critical value of a function is the value of this function at one of its critical points. Each critical point of a differentiable multivariate function is described by the equations which consist in equating to zero all of its partial…

In this article, we study critical points (zeros of derivative) of random polynomials. Take two deterministic sequences $\{a_n\}_{n\geq1}$ and $\{b_n\}_{n\geq1}$ of complex numbers whose limiting empirical measures are same. By choosing…

We announce numerous new results in the theory of orthogonal polynomials on the unit circle.

In this paper, we prove an inequality regarding the differential polynomial. This improves some recent results.

In this paper we study a particular class of polynomials. We study the distribution of their zeros, including the zeros of their derivatives as well as the interaction between this two. We prove a weak variant of the sendov conjecture in…

We obtain new partial results supporting the spectral set conjecture in dimension 1.

In the geometry of polynomials, Schoenberg's conjecture, now a theorem, is a quadratic inequality between the zeros and critical points of a polynomial whose zeros have their centroid at the origin. We call its generalizations to other…