The zeros of random polynomials cluster uniformly near the unit circle

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 variables distributed uniformly in a disk of radius r < 1 and \alpha_{n-1} another random variable independent of the previous ones and distributed uniformly on the unit circle. The previous recurrence relation gives a sequence of random p...

In this paper, we study how the roots of the so-called Kac polynomial $W_n(z) = \sum_{k=0}^{n-1} \xi_k z^k$ are concentrating to the unit circle when its coefficients of $W_n$ are independent and identically distributed non-degenerate real random variables. It is well-known that the roots of a Kac polynomial are concentrating around the unit circle as $n\to\infty$ if and only if $\mathbb{E}[\log( 1+ |\xi_0|)]<\infty$. Under the condition of $\mathbb{E}[\xi^2_0]<\infty$, we sh...

For a system of Laurent polynomials f_1,..., f_n \in C[x_1^{\pm1},..., x_n^{\pm1}] whose coefficients are not too big with respect to its directional resultants, we show that the solutions in the algebraic n-th dimensional complex torus of the system of equations f_1=\dots=f_n=0, are approximately equidistributed near the unit polycircle. This generalizes to the multivariate case a classical result due to Erdos and Turan on the distribution of the arguments of the roots of a ...

This article is divided in two parts. In the first part we review some recent results concerning the expected number of real roots of random system of polynomial equations. In the second part we deal with a different problem, namely, the distribution of the roots of certain complex random polynomials. We discuss a recent result in this direction, which shows that the associated points in the sphere (via the stereographic projection) are surprisingly well-suited with respect t...

In this paper, we study the number of real roots of random trigonometric polynomials with iid coefficients. When the coefficients have zero mean, unit variance and some finite high moments, we show that the variance of the number of real roots is asymptotically linear in terms of the expectation; furthermore, the multiplicative constant in this linear relationship depends only on the kurtosis of the common distribution of the polynomial's coefficients. This result is in sharp...

In this paper we find the asymptotic main term of the variance of the number of roots of Kostlan-Shub-Smale random polynomials and prove a central limit theorem for the number of roots as the degree goes to infinity.

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. Various models of random polynomials are considered by introducing randomness through multiplying a factor with a random zero or removing a zero at random for a given sequence of deterministic polynomials. We also obtain similar results for...

Let p_N be a random degree N polynomial in one complex variable whose zeros are chosen independently from a fixed probability measure mu on the Riemann sphere S^2. This article proves that if we condition p_N to have a zero at some fixed point xi in , then, with high probability, there will be a critical point w_xi a distance 1/N away from xi. This 1/N distance is much smaller than the one over root N typical spacing between nearest neighbors for N i.i.d. points on S^2. Moreo...

We consider random polynomials $p_n(x)=\xi_0+\xi_1+\dots+\xi_n x^n$ whose coefficients are independent and identically distributed with zero mean, unit variance, and bounded $(2+\epsilon)^{th}$ moment (for some $\epsilon>0$), also known as the Kac polynomials. Let $N_n$ denote the number of real roots of $p_n$. In this paper, motivated by a question from Igor Pritsker, we prove that almost surely the following convergence holds: \begin{eqnarray*} \lim_{n\to\infty} \frac{N_n([...

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 connected analytic plane domain $\Omega \subset C$ relative to an analytic weight $\rho(z) |dz|$. In the simplest case where $\Omega$ is the unit disk and $\rho=1$, so that $P_j(z) = z^j$, it is known that the average distribution of zeros is...