Related papers: Limits of Zeros of Orthogonal Polynomials on the C…

We estimate the number of zeros of a polynomial in $\mathbb{C}[z]$ within any small circular disc centered on the unit circle, which improves and comprehensively extends a result established by Borwein, Erd{\'e}lyi, and Littmann~\cite{BE1}…

Let ${z_n}$ be a sequence in the unit disk ${z\in\mathbb{C}:|z|<1}$. It is known that there exists a unique positive Borel measure in the unit circle ${z\in\mathbb{C}:|z|=1}$ such that the orthogonal polynomials ${\Phi_n}$ satisfy…

We consider polynomials on the unit circle defined by the recurrence relation \Phi_{k+1}(z) = z \Phi_{k} (z) - \bar{\alpha}_{k} \Phi_k^{*}(z) for k \geq 0 and \Phi_0=1. For each n we take \alpha_0, \alpha_1, ...,\alpha_{n-2} i.i.d. random…

We consider ensembles of random polynomials of the form $p(z)=\sum_{j = 1}^N a_j P_j$ where $\{a_j\}$ are independent complex normal random variables and where $\{P_j\}$ are the orthonormal polynomials on the boundary of a bounded simply…

We derive a useful result about the zeros of the $k$-polar polynomials on the unit circle; in particular we obtain a ring shaped region containing all the zeros of these polynomials. Some examples are presented.

We study global distribution of zeros for a wide range of ensembles of random polynomials. Two main directions are related to almost sure limits of the zero counting measures, and to quantitative results on the expected number of zeros in…

This article presents some interesting and novel results concerning the average modulus of random polynomials on the unit circle and the unit disc, with coefficients distributed as standard normal variates. The paper also introduces new…

We prove a lower bound on the spacing of zeros of paraorthogonal polynomials on the unit circle, based on continuity of the underlying measure as measured by Hausdorff dimensions. We complement this with the analog of the result from…

Orthogonal polynomials on the unit circle are completely determined by their reflection coefficients through the Szeg\H{o} recurrences. We assume that the reflection coefficients converge to some complex number a with 0 < |a| < 1. The…

We study the expected number of zeros of $$P_n(z)=\sum_{k=0}^n\eta_kp_k(z),$$ where $\{\eta_k\}$ are complex-valued i.i.d standard Gaussian random variables, and $\{p_k(z)\}$ are polynomials orthogonal on the unit disk. When…

In this work, we are dealing with some properties relating the zeros of a polynomial and its Mahler measure. We provide estimates on the number of real zeros of a polynomial, lower bounds on the distance between the zeros of a polynomial…

In this paper we obtain new results about the orthogonality measure of orthogonal polynomials on the unit circle, through the study of unitary truncations of the corresponding unitary multiplication operator, and the use of the…

We investigate two distinct formulations of Laurent multiple orthogonal polynomials on the unit circle, introduced in arXiv:2410.12094 and arXiv:2601.04783 respectively. For the first formulation, we prove that all zeros lie strictly within…

We establish a discrepancy theorem for signed measures, with a given positive part, which are supported on an arbitrary convex curve. As a main application, we obtain a result concerning the distribution of zeros of polynomials orthogonal…

Let $\mu$ be a probability measure with an infinite compact support on $\mathbb{R}$. Let us further assume that $(F_n)_{n=1}^\infty$ is a sequence of orthogonal polynomials for $\mu$ where $(f_n)_{n=1}^\infty$ is a sequence of nonlinear…

Schur studied limits of the arithmetic means $s_n$ of zeros for polynomials of degree $n$ with integer coefficients and simple zeros in the closed unit disk. If the leading coefficients are bounded, Schur proved that $\limsup_{n\to\infty}…

In this paper we deduce a universal result about the asymptotic distribution of roots of random polynomials, which can be seen as a complement to an old and famous result of Erdos and Turan. More precisely, given a sequence of random…

Given a probability measure $\mu$ with infinite support on the unit circle $\partial\mathbb{D}=\{z:|z|=1\}$, we consider a sequence of paraorthogonal polynomials $\h_n(z,\lambda)$ vanishing at $z=\lambda$ where $\lambda \in \T$ is fixed. We…

We consider a problem of bounding the maximal possible multiplicity of a zero at of some expansions $\sum a_i F_i(x)$, at a certain point $c,$ depending on the chosen family $\{F_i \}$. The most important example is a polynomial with $c=1.$…

We introduce a new family of orthogonal polynomials on the disk that has emerged in the context of wave propagation in layered media. Unlike known examples, the polynomials are orthogonal with respect to a measure all of whose even moments…