Zero distribution of random polynomials

Abstract

We study global distribution of zeros for a wide range of ensembles of random polynomials. Two main directions are related to almost sure limits of the zero counting measures, and to quantitative results on the expected number of zeros in various sets. In the simplest case of Kac polynomials, given by the linear combinations of monomials with i.i.d. random coefficients, it is well known that their zeros are asymptotically uniformly distributed near the unit circumference under mild assumptions on the coefficients. We give estimates of the expected discrepancy between the zero counting measure and the normalized arclength on the unit circle. Similar results are established for polynomials with random coefficients spanned by different bases, e.g., by orthogonal polynomials. We show almost sure convergence of the zero counting measures to the corresponding equilibrium measures for associated sets in the plane, and quantify this convergence. Random coefficients may be dependent and need not have identical distributions in our results.

Equidistribution of zeros of random polynomials

“…If the random coefficients satisfy mild assumptions such as in Theorem 2.1, then the zero counting measures of random polynomials P n (z) = n k=0 A k B k (z) converge almost surely to µ E for very general sets E and associated bases {B k } ∞ k=0 . We direct the reader to the recent papers [8], [3,4] and [22,23], and to references found therein. However, the necessity part of Theorem 2.1 seems to be open in such general setting.…”

“…Sufficient conditions for almost sure equidistribution of zeros of random orthogonal polynomials were considered by Shiffman and Zelditch [26] and [27], Bloom [6] and [7], Bloom and Shiffman [9], Bloom and Levenberg [8], Bayraktar [3] and [4], and others. Pritsker [22] and [23] considered zero distribution for random polynomials spanned by general bases.…”

An asymptotic expansion for the expected number of real zeros of real random polynomials spanned by OPUC

“…Many recent (and already not so recent) results on random polynomials are concerned with the behavior of counting measures of zeros of random polynomials spanned by various deterministic bases with random coefficients that are not necessarily Gaussian nor i.i.d. [5,6,7,8,9,10,11,12,13,14,15,16,17,18,19]. In the case of Kac polynomials these normalized counting measures almost surely converge to the arclength distribution on the unit circle (log n real zeros are clearly negligible when normalized by 1/n).…”

Asymptotic Zero Distribution of Random Polynomials Spanned by General Bases

“…One of our main goals is to remove unnecessary restrictions, and prove results on zeros of polynomials whose coefficients need not have identical distributions and may be dependent. We continue the line of research from the papers [37], [38] and [36].…”