Related papers: Random Polynomials in Several Complex Variables

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 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…

There is a natural pluripotential-theoretic extremal function V_{K,Q} associated to a closed subset K of C^m and a real-valued, continuous function Q on K. We define random polynomials H_n whose coefficients with respect to a related…

Let $ p_n(x) $ be a random polynomial of degree $n$ and $\{Z^{(n)}_j\}_{j=1}^n$ and $\{X^{n, k}_j\}_{j=1}^{n-k}, k<n$, be the zeros of $p_n$ and $p_n^{(k)}$, the $k$th derivative of $p_n$, respectively. We show that if the linear statistics…

In this work, we study asymptotic zero distribution of random multi-variable polynomials which are random linear combinations $\sum_{j}a_jP_j(z)$ with i.i.d coefficients relative to a basis of orthonormal polynomials $\{P_j\}_j$ induced by…

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…

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…

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…

We show that the average characteristic polynomial P_n(z) = E [\det(zI-M)] of the random Hermitian matrix ensemble Z_n^{-1} \exp(-Tr(V(M)-AM))dM is characterized by multiple orthogonality conditions that depend on the eigenvalues of the…

A sum of a large-dimensional random matrix polynomial and a fixed low-rank matrix polynomial is considered. The main assumption is that the resolvent of the random polynomial converges to some deterministic limit. A formula for the limit of…

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 $\tau$…

Let a sequence $(P_n)$ of polynomials in one complex variable satisfy a recurre ce relation with length growing slowlier than linearly. It is shown that $(P_n) $ is an orthonormal basis in $L^2_{\mu}$ for some measure $\mu$ on $\C$, if and…

This paper puts forward a new generalized polynomial dimensional decomposition (PDD), referred to as GPDD, comprising hierarchically ordered measure-consistent multivariate orthogonal polynomials in dependent random variables. Unlike the…

We calculate the expectation value of an arbitrary product of characteristic polynomials of complex random matrices and their hermitian conjugates. Using the technique of orthogonal polynomials in the complex plane our result can be written…

Polynomials known as Multiple Orthogonal Polynomials in a single variable are polynomials that satisfy orthogonality conditions concerning multiple measures and play a significant role in several applications such as Hermite-Pad\'e…

A complex harmonic polynomial is the sum of a complex polynomial and a conjugated complex polynomial, of degrees $n$ and $m$ respectively. Li and Wei (2009) presented a formula for the expected number of zeros of a random harmonic…

For a polynomial $P_n$ of degree $n$, Bernstein's inequality states that $\|P_n'\| \le n \|P_n\|$ for all $L^p$ norms on the unit circle, $0<p\le\infty,$ with equality for $P_n(z)= c z^n.$ We study this inequality for random polynomials,…

We consider polynomials of the form $\operatorname{h}_m(y_1^{[\varkappa_1]},\ldots,y_n^{[\varkappa_n]})$, where $\operatorname{h}_m$ is the complete homogeneous polynomial of degree $m$ and $y_j^{[\varkappa_j]}$ denotes $y_j$ repeated…

The paper investigates the problem of performing correlation analysis when the number of observations is very large. In such a case, it is often necessary to combine the random observations to achieve dimensionality reduction of the…

The paper deals with different properties of polynomials in random elements: bounds for characteristics functionals of polynomials, stochastic generalization of the Vinogradov mean value theorem, characterization problem, bounds for…