Related papers: Zeros of harmonic polynomials, critical lemniscate…

The efficacy of numerical methods like integral estimates via Gaussian quadrature formulas depends on the localization of the zeros of the associated family of orthogonal polynomials. In this regard, following the renewed interest in…

We prove a sharp inequality for the modulus of the logarithmic derivative of a polynomial in the lemniscate components containing no critical points.

First a formula for the number of zeros of the orthogonal polynomial in the intervals is presented. Then a criteria about the appearance of a zero in a gap is given. Finally a necessary and sufficient condition is derived such that the…

In this article we study the limiting empirical measure of zeros of higher derivatives for sequences of random polynomials. We show that these measures agree with the limiting empirical measure of zeros of corresponding random polynomials.…

D. Khavinson and G. Swiatek proved that harmonic polynomials p(z)+q(z), where p is holomorphic, q is antiholomorphic, and deg p = n > 1 = deg q, can have at most 3n-2 complex zeros. We show that this bound is sharp for all n by proving a…

We examine the maximal number of zeros a polynomial of degree at most n with constrained coefficients may have at 1. Our results are essentially sharp and extend earlier results of this variety. An interesting connection to certain…

The complex or non-hermitian orthogonal polynomials with analytic weights are ubiquitous in several areas such as approximation theory, random matrix models, theoretical physics and in numerical analysis, to mention a few. Due to the…

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…

We study the number of real zeros of trigonometric polynomials in a period and the number of zeros of self-reciprocal algebraic polynomials on the unit circle under the assumption that their coefficients are in a fixed finite set of real…

Polynomials whose zeros are symmetric either to the real line or to the unit circle are very important in mathematics and physics. We can classify them into three main classes: the self-conjugate polynomials, whose zeros are symmetric to…

This is primarily a survey of the developments in the theory of harmonic maps of finite uniton number (or unitons) which have taken place since the introduction of extended solutions by Uhlenbeck. Such maps include all harmonic maps from…

There is considered the problem of describing up to linear conformal equivalence those harmonic cubic homogeneous polynomials for which the squared-norm of the Hessian is a nonzero multiple of the quadratic form defining the Euclidean…

We review some recent results on random polynomials and their generalizations in complex and symplectic geometry. The main theme is the universality of statistics of zeros and critical points of (generalized) polynomials of degree $N$ on…

We examine the maximal domain of radial harmonic functions on harmonic spaces in the context of positive, zero, and negative curvature.

We consider the sensitivity of real zeros of structured polynomial systems to perturbations of their coefficients. In particular, we provide explicit estimates for condition numbers of structured random real polynomial systems, and extend…

In this work, previous results concerning the infinitely many zeros of single stochastic oscillators driven by random forces are extended to the general class of coupled stochastic oscillators. We focus on three main subjects: 1) the…

It is known that polynomials over quaternions may have spherical zeros and isolated left and right zeros. These zeros along with appropriately defined multiplicities form the zero structure of a polynomial. In this paper, we equivalently…

The authors present a unified method for calculating the zeros of the classical orthogonal polynomials based upon the electrostatic interpretation and its connection to the energy minimization problem. Examples are given with error…

Locating the zeros of quaternionic polynomials is a fundamental problem with significant implications across scientific and engineering disciplines, yet the noncommutative nature of quaternion multiplication makes it fundamentally more…

In this paper we focus on two new families of polynomials which are connected with exponential polynomials and geometric polynomials. We discuss their generalizations and show that these new families of polynomials and their generalizations…