The zeros of random polynomials cluster uniformly near the unit circle

We compute the precise leading asymptotics of the variance of the number of real roots for random polynomials whose coefficients have polynomial asymptotics. This class of random polynomials contains as special cases the Kac polynomials, hyperbolic polynomials, and any linear combinations of their derivatives. Prior to this paper, such asymptotics was only established in the 1970s for the Kac polynomials, with the seminal contribution of Maslova. The main ingredients of the p...

We prove, informally put, that it is not a coincidence that $\cos{(n \theta)} + 1 \geq 0$ and that the roots of $z^n + 1 =0$ are uniformly distributed in angle -- a version of the statement holds for all trigonometric polynomials with `few' real roots. The Erd\H{o}s-Tur\'an theorem states that if $p(z) =\sum_{k=0}^{n}{a_k z^k}$ is suitably normalized and not too large for $|z|=1$, then its roots are clustered around $|z| = 1$ and equidistribute in angle at scale $\sim n^{-1/2...

In this note, we prove a central limit theorem for smooth linear statistics of zeros of random polynomials which are linear combinations of orthogonal polynomials with iid standard complex Gaussian coefficients. Along the way, we obtain Bergman kernel asymptotics for weighted $L^2$-space of polynomials endowed with varying measures of the form $e^{-2n\varphi_n(z)}dz$ under suitable assumptions on the weight functions $\varphi_n$.

Let $X_N$ be a random trigonometric polynomial of degree $N$ with iid coefficients and let $Z_N(I)$ denote the (random) number of its zeros lying in the compact interval $I\subset\mathbb{R}$. Recently, a number of important advances were made in the understanding of the asymptotic behaviour of $Z_N(I)$ as $N\to\infty$, in the case of standard Gaussian coefficients. The main theorem of the present paper is a universality result, that states that the limit of $Z_N(I)$ does not ...

In this paper, we establish some local universality results concerning the correlation functions of the zeroes of random polynomials with independent coefficients. More precisely, consider two random polynomials $f =\sum_{i=1}^n c_i \xi_i z^i$ and $\tilde f =\sum_{i=1}^n c_i \tilde \xi_i z^i$, where the $\xi_i$ and $\tilde \xi_i$ are iid random variables that match moments to second order, the coefficients $c_i$ are deterministic, and the degree parameter $n$ is large. Our re...

Let $X_1,X_2,\ldots$ be independent and identically distributed random variables in $\mathbb{C}$ chosen from a probability measure $\mu$ and define the random polynomial $$ P_n(z)=(z-X_1)\ldots(z-X_n)\,. $$ We show that for any sequence $k = k(n)$ satisfying $k \leq \log n / (5 \log\log n)$, the zeros of the $k$th derivative of $P_n$ are asymptotically distributed according to the same measure $\mu$. This extends work of Kabluchko, which proved the $k = 1$ case, as well as By...

We study the asymptotic distribution of the number $Z_{N}$ of zeros of random trigonometric polynomials of degree $N$ as $N\to\infty$. It is known that as $N$ grows to infinity, the expected number of the zeros is asymptotic to $\frac{2}{\sqrt{3}}\cdot N$. The asymptotic form of the variance was predicted by Bogomolny, Bohigas and Leboeuf to be $cN$ for some $c>0$. We prove that $\frac{Z_{N}-\E Z_{N}}{\sqrt{cN}}$ converges to the standard Gaussian. In addition, we find that t...

Let $\mu$ be a probability measure in $\mathbb{C}$ with a continuous and compactly supported density function, let $z_1, \dots, z_n$ be independent random variables, $z_i \sim \mu$, and consider the random polynomial $$ p_n(z) = \prod_{k=1}^{n}{(z - z_k)}.$$ We determine the asymptotic distribution of $\left\{z \in \mathbb{C}: p_n(z) = p_n(0)\right\}$. In particular, if $\mu$ is radial around the origin, then those solutions are also distributed according to $\mu$ as $n \righ...

It has been shown that zeros of Kac polynomials $K_n(z)$ of degree $n$ cluster asymptotically near the unit circle as $n\to\infty$ under some assumptions. This property remains unchanged for the $l$-th derivative of the Kac polynomials $K^{(l)}_n(z)$ for any fixed order $l$. So it's natural to study the situation when the number of the derivatives we take depends on $n$, i.e., $l=N_n$. We will show that the limiting global behavior of zeros of $K_n^{(N_n)}(z)$ depends on the ...

In the study of the cyclicity of a function $f$ in reproducing kernel Hilbert spaces an important role is played by sequences of polynomials $\{p_n\}_{n\in \mathbb{N}}$ called \emph{optimal polynomial approximants} (o.p.a.). For many such spaces and when the functions $f$ generating those o.p.a. are polynomials without zeros inside the disk but with some zeros on its boundary, we find that the weakly asympotic distribution of the zeros of $1-p_nf$ is the uniform measure on th...