Related papers: Expected number of real zeros for random orthogona…

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…

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 show that the zeros of consecutive orthogonal polynomials $p_n$ and $p_{n-1}$ are linearly connected by a doubly stochastic matrix for which the entries are explicitly computed in terms of Christoffel numbers. We give similar results for…

Let ${z_n}$ be a sequence in the unit disk ${z\in\mathbb{C}:|z|<1}$. It is known that there exists a unique positive Borel measure in the unit circle ${z\in\mathbb{C}:|z|=1}$ such that the orthogonal polynomials ${\Phi_n}$ satisfy…

We consider random polynomials of the form $H_n(z)=\sum_{j=0}^n\xi_jq_j(z)$ where the $\{\xi_j\}$ are i.i.d non-degenerate complex random variables, and the $\{q_j(z)\}$ are orthonormal polynomials with respect to a compactly supported…

We are concerned with zeros of random power series with coefficients being a stationary, centered, complex Gaussian process. We show that the expected number of zeros in every smooth domain in the disk of convergence is less than that of…

We study the simultaneous zeros of a random family of $d$ polynomials in $d$ variables over the $p$-adic numbers. For a family of natural models, we obtain an explicit constant for the expected number of zeros that lie in the $d$-fold…

For a regular compact set $K$ in $C^m$ and a measure $\mu$ on $K$ satisfying the Bernstein-Markov inequality, we consider the ensemble $P_N$ of polynomials of degree $N$, endowed with the Gaussian probability measure induced by $L^2(\mu)$.…

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.…

Koiran's real $\tau$-conjecture claims that the number of real zeros of a structured polynomial given as a sum of $m$ products of $k$ real sparse polynomials, each with at most $t$ monomials, is bounded by a polynomial in $m,k,t$. This…

Let $P_{n}(x)= \sum_{i=0}^n \xi_i x^i$ be a Kac random polynomial where the coefficients $\xi_i$ are iid copies of a given random variable $\xi$. Our main result is an optimal quantitative bound concerning real roots repulsion. This leads…

Consider random polynomials of the form $G_n = \sum_{i=0}^n \xi_i p_i$, where the $\xi_i$ are i.i.d.\ non-degenerate complex random variables, and $\{p_i\}$ is a sequence of orthonormal polynomials with respect to a regular measure $\tau$…

We utilize Cauchy's argument principle in combination with the Jacobian of a holomorphic function in several complex variables and the first moment of a ratio of two correlated complex normal random variables to prove explicit formulas for…

In this note we study asymptotic isotopy of random real algebraic plane curves. More precisely, we obtain a Kac-Rice type formula that gives the expected number of two-sided components (i.e.\ ovals) of a random real algebraic plane curve…

In this contribution, the behavior of zeros of orthogonal polynomials associated with canonical linear spectral transformations of measures supported in the real line is analyzed. An electrostatic interpretation of them is given.

We consider random polynomials of the form $G_n(z):= \sum_{|\alpha|\leq n} \xi^{(n)}_{\alpha}p_{n,\alpha}(z)$ where $\{\xi^{(n)}_{\alpha}\}_{|\alpha|\leq n}$ are i.i.d. (complex) random variables and $\{p_{n,\alpha}\}_{|\alpha|\leq n}$ form…

Consider a polynomial of large degree n whose coefficients are independent, identically distributed, nondegenerate random variables having zero mean and finite moments of all orders. We show that such a polynomial has exactly k real zeros…

We propose a way to find the asymptotic distribution of zeros of orthogonal polynomials p_n(x) satisfying a difference equation of the form B(x)p_n(x+\delta)-C(x,n)p_n(x)+D(x)p_n(x-\delta)=0. We calculate the asymptotic distribution of…

We give the asymptotic behavior of the zeros of orthogonal polynomials, after appropriate scaling, for which the orthogonality measure is supported on the $q$-lattice $\{q^k, k=0,1,2,3,\ldots\}$, where $0 < q < 1$. The asymptotic…

The expected number of real projective roots of orthogonally invariant random homogeneous real polynomial systems is known to be equal to the square root of the B\'ezout number. A similar result is known for random multi-homogeneous…