Related papers: Real zeros of algebraic polynomials with nonidenti…

We investigate the asymptotics of the expected number of real roots of random trigonometric polynomials $$ X_n(t)=u+\frac{1}{\sqrt{n}}\sum_{k=1}^n (A_k\cos(kt)+B_k\sin(kt)), \quad t\in [0,2\pi],\quad u\in\mathbb{R} $$ whose coefficients…

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 consider random polynomials whose coefficients are independent and identically distributed on the integers. We prove that if the coefficient distribution has bounded support and its probability to take any particular value is at most…

We use a method based on the division algorithm to determine all the values of the real parameters $b$ and $c$ for which the hypergeometric polynomials $_2F_1(-n, b; c; z)$ have $n$ real, simple zeros. Furthermore, we use the…

We consider random systems of equations over the reals, with $m$ equations and $m$ unknowns $P_i(t)+X_i(t)=0$, $t\in\mathbb{R}^m$, $i=1,...,m$, where the $P_i$'s are non-random polynomials having degrees $d_i$'s (the "signal") and the…

Let $f(q)=a_rq^r+\cdots+a_sq^s$, with $a_r\neq 0$ and $a_s\neq 0$, be a real polynomial. It is a palindromic polynomial of darga $n$ if $r+s=n$ and $a_{r+i}=a_{s-i}$ for all $i$. Polynomials of darga $n$ form a linear subspace…

We obtain exact analytical expressions for correlations between real zeros of the Kac random polynomial. We show that the zeros in the interval $(-1,1)$ are asymptotically independent of the zeros outside of this interval, and that the…

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…

Consider a random polynomial $Q_n$ of degree $n+1$ whose zeroes are i.i.d. random variables $\xi_0,\xi_1,\ldots,\xi_n$ in the complex plane. We study the pairing between the zeroes of $Q_n$ and its critical points, i.e. the zeroes of its…

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…

We prove that the expectation of the logarithm of the condition number of each of the zeros of the characteristic polynomial of a complex standard Gaussian matrix is ${\Omega}(n)$. This may provide an explanation for the common wisdom in…

The interplay among the time-evolution of the coefficients and the zeros of a generic time-dependent (monic) polynomial provides a convenient tool to identify certain classes of solvable dynamical systems. Recently this tool has been…

A probabilistic approach to the study of the number of zeros of complex harmonic polynomials was initiated by W. Li and A. Wei (2009), who derived a Kac-Rice type formula for the expected number of zeros of random harmonic polynomials with…

In this paper we study admissible polynomials. We establish an estimate for the number of admissible polynomials of degree $n$ with coeffients $a_i$ satisfying $0\leq a_i\leq H$ for a fixed $H$, for $i=0,1,2, \ldots, n-1$. In particular,…

This paper investigates the expected number of complex roots of nonlinear equations. Those equations are assumed to be analytic, and to belong to certain inner product spaces. Those spaces are then endowed with the Gaussian probability…

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…

A partition polynomial is a refinement of the partition number p(n) whose coefficients count some special partition statistic. Just as partition numbers have useful asymptotics so do partition polynomials. In fact, their asymptotics…

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…

We study the number of real zeros of finite combinations of $K+1$ consecutive normalized Laguerre polynomials of the form $$ q_n(x)=\sum_{j=0}^K\gamma_j\tilde L^\alpha_{n-j}(x),\quad n\ge K, $$ where $\gamma_j$, $j=0,\cdots ,K$, are real…

Let $f(x)=x^n+a_{n-1}x^{n-1}+\dots+a_0$ be an irreducible polynomial with integer coefficients. For a prime $p$ for which $f(x)$ is fully splitting modulo $ p$, we consider $n$ roots $r_i$ of $f(x)\equiv 0\bmod p$ with $0 \le r_1\le\dots\le…