Critical Points Of Polynomial Research Articles

On Variance and Average Moduli of Zeros and Critical Points of Polynomials

This paper investigates various aspects of the distribution of roots and critical points of a complex polynomial, including their variance and the relationships between their moduli using an inequality due to de Bruijn. Making use of two other inequalities-again due to de Bruijn-we derive two probabilistic results concerning upper bounds for the average moduli of the imaginary parts of zeros and those of critical points, assuming uniform distribution of the zeros over a unit disc and employing the Markov inequality. The paper also provides an explicit formula for the variance of the roots of a complex polynomial for the case when all the zeros are real. In addition, for polynomials with uniform distribution of roots over the unit disc, the expected variance of the zeros is computed. Furthermore, a bound on the variance of the critical points in terms of the variance of the zeros of a general polynomial is derived, whereby it is established that the variance of the critical points of a polynomial cannot exceed the variance of its roots. Finally, we conjecture a relation between the real parts of the zeros and the critical points of a polynomial.

Distribution of Zeros and Critical Points of a Polynomial, and Sendov’s Conjecture

According to the Gauss–Lucas theorem, the critical points of a complex polynomial p(z):=sum_{j=0}^{n}a_{j}z^{j} where a_{j}inmathbb{C} always lie in the convex hull of its zeros. In this paper, we prove certain relations between the distribution of zeros of a polynomial and its critical points. Using these relations, we prove the well-known Sendov’s conjecture for certain special cases.

Critical Points of Polynomials With Roots on the Unit Circle

Abstract Let ${\mathcal {L}}(u)$ be a polynomial with all its roots on the unit circle. We establish an “angle duality” theorem relating the roots and critical points of ${\mathcal {L}}(u)$. As an application, we prove that if there is a large gap between roots of ${\mathcal {L}}(u)$, then most critical points of ${\mathcal {L}}(u)$ lie close to the unit circle. We also use our result to strengthen Sendov’s well-known conjecture for these special polynomials. Finally, we prove a sharp estimate for the location of the critical point near a small gap between roots of ${\mathcal {L}}(u)$.

Skew-circulant matrix and critical points of polynomials

Skew-circulant matrix and critical points of polynomials

Roots and critical points of polynomials over Cayley–Dickson algebras

The roots of polynomials over Cayley–Dickson algebras over an arbitrary field and of arbitrary dimension are studied. It is shown that the spherical roots of a polynomial f(x) are also roots of its companion polynomial . We generalize the classical theorems for complex and real polynomials by Gauss–Lucas and Jensen to locally-complex Cayley–Dickson algebras: it is proved that the spherical roots of belong to the convex hull of the roots of , and we also show that all roots of are contained in the snail of f(x), as defined by Ghiloni and Perotti.

Remarks on circulant matrices and critical points of polynomials

In Kushel and Tyaglov (2016) [10], by making use of a connection between critical points and circulant matrices, the authors reproved, in an elegant manner, Schoenberg's conjecture, de Bruin and Sharma's conjecture. Some further questions and open problems are proposed in their paper. In this article, we provide answers to them.

Critical points of polynomials

This paper is devoted to the problem of where the critical points of a polynomial are relative to their zeros. Classical and new developments are surveyed along with illustrative examples. The paper finishes with a short proof of the sector theorem of Sendov and Sendov.

The location of critical points of polynomials

Given a set of points in the complex plane, an incomplete polynomial is defined as one which has these points as zeros except one of them. Recently, the classical result known as Gauss–Lucas theorem on the location of zeros of polynomials and their derivatives was extended to the linear combinations of incomplete polynomials. In this paper, a simple proof of this result is given, and some results concerning the critical points of polynomials due to Jensen and others have extended the linear combinations of incomplete polynomials.

Geometry of a Family of Complex Polynomials

Let Pa be the family of complex-valued polynomials of the form p(z)=(z-a)(z-r)(z-s) with a in [0,1] and r and s on the unit circle. The Gauss-Lucas Theorem implies that the critical points of a polynomial in Pa lie in the unit disk. This paper characterizes the location and structure of these critical points. We show that the unit disk contains an open circular disk in which critical points of polynomials in Pa do not occur. Furthermore, almost every c inside the unit disk and outside of the desert region is the critical point of a unique polynomial in Pa.

Distances between zeroes and critical points for random polynomials with i.i.d. zeroes

Consider a random polynomial $Q_n$ of degree $n+1$ whose zeroes are i.i.d. random variables $\xi _0,\xi _1,\ldots ,\xi _n$ in the complex plane. We study the pairing between the zeroes of $Q_n$ and its critical points, i.e. the zeroes of its derivative $Q_n'$. In the asymptotic regime when $n\to \infty $, with high probability there is a critical point of $Q_n$ which is very close to $\xi _0$. We localize the position of this critical point by proving that the difference between $\xi _0$ and the critical point has approximately complex Gaussian distribution with mean $1/(nf(\xi _0))$ and variance of order $\log n \cdot n^{-3}$. Here, $f(z)= \mathbb E [\frac 1 {z-\xi _k}]$ is the Cauchy–Stieltjes transform of the $\xi _k$’s. We also state some conjectures on critical points of polynomials with dependent zeroes, for example the Weyl polynomials and characteristic polynomials of random matrices.

On the Zeros and Critical Points of Polynomials with Nonnegative Coefficients: A Nonconvex Analogue of the Gauss–Lucas Theorem

In this work, we present a nonconvex analogue of the classical Gauss–Lucas theorem stating that the critical points of a polynomial with complex coefficients are in the convex hull of its zeros. We show that if the polynomial p(z) of degree n has nonnegative coefficients and zeros in the sector $$\{z \in \mathcal C: |\arg (z)| \ge \varphi \}$$ , for some $$\varphi \in [0,\pi ]$$ , then the critical points of p(z) are also in that sector. Clearly, when $$\varphi \in [\pi /2,\pi ]$$ , our result follows from the classical Gauss–Lucas theorem. But when $$\varphi \in [0,\pi /2)$$ , we obtain a nonconvex analogue.

Dynamic Geometry of Cubic Polynomial

In the paper we consider the dynamic behavior of the critical points of some cubic polynomials, with the motion of one of the roots of the polynomial along a given trajectory. Some dynamic property of polynomials is investigated. The statements about traces of critical points of some polynomials are proved. The equations of curves, on which critical points move, are obtained.

Joseph L. Walsh: Selected papers, edited by Theodore J. Rivlin and Edward B. Saff. Pp. 682. £86. 2000. ISBN 0 387 98782 7 (Springer Verlag).

Joseph L. Walsh: Selected papers, edited by Theodore J. Rivlin and Edward B. Saff. Pp. 682. £86. 2000. ISBN 0 387 98782 7 (Springer Verlag).

Parameter-Generated Loci of Critical Points of Polynomials

Parameter-Generated Loci of Critical Points of Polynomials