Related papers: Correlations between zeros of a random polynomial

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 study the root distribution of some univariate polynomials satisfying a recurrence of order two with linear and quadratic polynomial coefficients. We show that the set of non-isolated limits of zeros of the polynomials is the closure of…

By random complex zeroes we mean the zero set of a random entire function whose Taylor coefficients are independent complex-valued Gaussian variables, and the variance of the k-th coefficient is 1/k!. This zero set is distribution invariant…

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.…

We obtain some results on the asymptotic behaviour of Geometric polynomials in both the complex plane minus $[-1,0]$ and the interval $(-1,0)$. We also find the distance of consecutive zeros of these polynomials in the bulk of the interval…

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…

We investigate the number of real zeros of a univariate $k$-sparse polynomial $f$ over the reals, when the coefficients of $f$ come from independent standard normal distributions. Recently B\"urgisser, Erg\"ur and Tonelli-Cueto showed that…

We study concentration inequalities for the number of real roots of the classical Kac polynomials $$f_{n} (x) = \sum_{i=0}^n \xi_i x^i$$ where $\xi_i$ are independent random variables with mean 0, variance 1, and uniformly bounded…

This paper shows that one needs to be careful when making statements on potential links between correlation and coskewness. Specifically, we first show that, on the one hand, it is possible to observe any possible values of coskewness among…

We give results on zeros of a polynomial of $\zeta(s),\zeta'(s),\ldots,\zeta^{(k)}(s)$. First, we give a zero free region and prove that there exist zeros corresponding to the trivial zeros of the Riemann zeta function. Next, we estimate…

The expected number of real zeros of an algebraic polynomial $a_0+a_1x+a_2x^2+a_3x^3+....+a_{n-1}x^{n-1}$ depends on the types of random coefficients, with large $n.$ In this article, we show that when the random coefficients…

Hayes equivalence is defined on monic polynomials over a finite field $\fq$ in terms of the prescribed leading coefficients and the residue classes modulo a given monic polynomial $Q$. We study the distribution of the number of zeros in a…

We consider the zero sets $Z_N$ of systems of $m$ random polynomials of degree $N$ in $m$ complex variables, and we give asymptotic formulas for the random variables given by summing a smooth test function over $Z_N$. Our asymptotic…

Li and Wei (2009) studied the density of zeros of Gaussian harmonic polynomials with independent Gaussian coefficients. They derived a formula for the expected number of zeros of random harmonic polynomials as well as asymptotics for the…

We investigate the probability that a random polynomial with independent, mean-zero and finite variance coefficients has no real zeros. Specifically, we consider a random polynomial of degree $2n$ with coefficients given by an i.i.d.…

We study asymptotic clustering of zeros of random polynomials, and show that the expected discrepancy of roots of a polynomial of degree $n$, with not necessarily independent coefficients, decays like $\sqrt{\log n/n}$. Our proofs rely on…

For $0<\alpha<1$, we study the zeros of the sequence of polynomials $\left\{ P_{m}(z)\right\} _{m=0}^{\infty}$ generated by the reciprocal of $(1-t)^{\alpha}(1-2zt+t^{2})$, expanded as a power series in $t$. Equivalently, this sequence is…

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…

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…

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…