Related papers: Distribution of roots of random real generalized p…

This paper extends earlier work on the distribution in the complex plane of the roots of random polynomials. In this paper, the random polynomials are generalized to random finite sums of given "basis" functions. The basis functions are…

Let $G_n(z)=\xi_0+\xi_1z+...+\xi_n z^n$ be a random polynomial with i.i.d. coefficients (real or complex). We show that the arguments of the roots of $G_n(z)$ are uniformly distributed in $[0,2\pi]$ asymptotically as $n\to\infty$. We also…

We identify the scaling region of a width O(n^{-1}) in the vicinity of the accumulation points $t=\pm 1$ of the real roots of a random Kac-like polynomial of large degree n. We argue that the density of the real roots in this region tends…

In this article, we investigate the statistical distribution and asymptotic behavior of the family of monic integer polynomials of degree $n$ having at least one root in a fixed number field $K$. Although the framework of thin sets implies…

This paper seeks to further explore the distribution of the real roots of random polynomials with non-centered coefficients. We focus on polynomials where the typical values of the coefficients have power growth and count the average number…

We introduce a family of real random polynomials of degree n whose coefficients a_k are symmetric independent Gaussian variables with variance <a_k^2> = e^{-k^\alpha}, indexed by a real \alpha \geq 0. We compute exactly the mean number of…

For each $\alpha \in (0, 1)$, we construct a bounded monotone deterministic sequence $(c_k)_{k \geq 0}$ of real numbers so that the number of real roots of the random polynomial $f_n(z) = \sum_{k=0}^n c_k \varepsilon_k z^k$ is $n^{\alpha +…

We study the asymptotic distribution of roots of Lommel polynomials as polynomials of the order with a variable and purely imaginary argument. The roots are complex and accumulate on certain curves in the complex plane. We prove existence…

We consider random orthonormal polynomials $$ F_{n}(x)=\sum_{i=0}^{n}\xi_{i}p_{i}(x), $$ where $\xi_{0}$, \dots, $\xi_{n}$ are independent random variables with zero mean, unit variance and uniformly bounded $(2+\ep)$ moments, and…

Many statistics of roots of random polynomials have been studied in the literature, but not much is known on the concentration aspect. In this note we present a systematic study of this question, aiming towards nearly optimal bounds to some…

Suppose $C \subset \mathbb{C}$ is compact. Let $q_k$ be a sequence of polynomials of degree $n_k \to \infty$, such that the locus of roots of all the polynomials is bounded, and the number of roots of $q_k$ in any closed set $L$ not meeting…

It has been shown by Ibragimov and Zaporozhets [In Prokhorov and Contemporary Probability Theory (2013) Springer] that the complex roots of a random polynomial $G_n(z)=\sum_{k=0}^n\xi_kz^k$ with i.i.d. coefficients $\xi_0,\ldots,\xi_n$…

Let $\{\eta_{j}\}_{j = 0}^{N}$ be a sequence of independent, identically distributed random complex Gaussian variables, and let $\{f_{j} (z)\}_{j = 0}^{N}$ be a sequence of given analytic functions that are real-valued on the real number…

The study of random polynomials has a long and rich history. This paper studies random algebraic polynomials $P_n(x) = a_0 + a_1 x + \ldots + a_{n-1} x^{n-1}$ where the coefficients $(a_k)$ are correlated random variables taken as the…

Consider a random polynomial $G_n(z)=\xi_nz^n+...+\xi_1z+\xi_0$ with i.i.d. complex-valued coefficients. Suppose that the distribution of $\log(1+\log(1+|\xi_0|))$ has a slowly varying tail. Then the distribution of the complex roots of…

Let \( \{\varphi_i\}_{i=0}^\infty \) be a sequence of orthonormal polynomials on the unit circle with respect to a probability measure \( \mu \). We study zero distribution of random linear combinations of the form \[…

We study the zero distribution of non-orthogonal polynomials attached to $g(n)=s(n)=n^2$: \begin{equation*} Q_n^g(x)= x \sum_{k=1}^n g(k) \, Q_{n-k}^g(x), \quad Q_0^g(x):=1. \end{equation*} It is known that the case $g=id$ involves…

We consider random orthonormal polynomials $$ P_{n}(x)=\sum_{i=0}^{n}\xi_{i}p_{i}(x), $$ where $\xi_{0}$, . . . , $\xi_{n}$ are independent random variables with zero mean, unit variance and uniformly bounded $(2+\ep_0)$-moments, and…

We study the asymptotic distribution of zeros for the random polynomials $P_n(z) = \sum_{k=0}^n A_k B_k(z)$, where $\{A_k\}_{k=0}^{\infty}$ are non-trivial i.i.d. complex random variables. Polynomials $\{B_k\}_{k=0}^{\infty}$ are…

Let $\{f_j\}_{j=0}^n$ be a sequence of orthonormal polynomials where the orthogonality relation is satisfied on either the real line or on the unit circle. We study zero distribution of random linear combinations of the form…