Expected discrepancy for zeros of random algebraic polynomials

Abstract

Abstract. 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 √ log n/n. Our proofs rely on discrepancy results for deterministic polynomials, and order statistics of a random variable. We also consider the expected number of zeros lying in certain subsets of the plane, such as circles centered on the unit circumference, and polygons inscribed in the unit circumference.

Zero distribution of random polynomials

“…, are not necessarily independent nor identically distributed. It is convenient to first discuss the simplest case of the unit circle, which originated in [26]. A standard way to study the deviation of τ n from µ T is to consider the discrepancy of these measures in the annular sectors of the form…”

“…The proofs of Theorem 3.1 and Corollary 3.2 are sketched in Section 4 for convenience of the reader. Papers [26] and [27] explain how one can obtain quantitative results about the expected number of zeros of random polynomials in various sets, see Propositions 2.3-2.5 of [27]. The basic observation here is that the number of zeros of P n in a set S ⊂ C denoted by N n (S) is equal to nτ n (S), and the estimates for E[N n (S)] readily follow from Theorem 3.1 and Corollary 3.2.…”

“…Section 2 deals with almost sure convergence of the zero counting measures for polynomials with random coefficients that satisfy only weak log-integrability assumptions. Section 3 develops the discrepancy results of [26] and [27], and establishes expected rates of convergence of the zero counting measures to the equilibrium measures. Again, the random coefficients in Section 3 are neither independent nor identically distributed, and their distributions only satisfy certain uniform bounds for the fractional and logarithmic moments.…”

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].…”

Zeros of polynomials with random coefficients

“…The arguments of [15] now give quantitative results about the expected number of zeros of random polynomials in various sets. We first consider sets separated from T. Proposition 2.3.…”