Zeros of complex random polynomials spanned by Bergman polynomials

where ${η_{k}}$ are complex-valued independent and identically distributed standard Gaussian random variables, and ${p_{k} (z)}$ are polynomials orthogonal on the unit disk. When $p_{k} (z) = \sqrt{(k + 1) ∕ π} z^{k}$ , $k \in {0, 1, \dots, n}$ , we give an explicit formula for the expected number of zeros of $P_{n} (z)$ in a disk of radius $r \in (0, 1)$ centered at the origin. From our formula we establish the limiting value of the expected number of zeros, the expected number of zeros in a radially expanding disk, and show that the expected number of zeros in the unit disk is $2 n ∕ 3$ . Generalizing our basis functions ${p_{k} (z)}$ to be regular in the sense of Ullman–Stahl–Totik and that the measure of orthogonality associated to polynomials is absolutely continuous with respect to planar Lebesgue measure, we give the limiting value of the expected number of zeros in a disk of radius $r \in (0, 1)$ centered at the origin, and show that asymptotically the expected number of zeros in the unit disk is $2 n ∕ 3$ .

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Keywords

random polynomials, Bergman polynomials, Ullman–Stahl–Totik regular

Mathematical Subject Classification

Primary: 30C15, 30E15, 30C40

Secondary: 60B99

Milestones

Received: 7 July 2020

Revised: 11 January 2021

Accepted: 12 January 2021

Published: 6 April 2021

Communicated by John C. Wierman

Authors

Marianela Landi
Department of Mathematics and Data Science College of Coastal Georgia Brunswick, GA United States

Kayla Johnson
Department of Mathematics and Data Science College of Coastal Georgia Brunswick, GA United States

Garrett Moseley
Department of Mathematics and Data Science College of Coastal Georgia Brunswick, GA United States

Aaron Yeager
Department of Mathematics and Data Science College of Coastal Georgia Brunswick, GA United States

Vol. 14, No. 2, 2021

Marianela Landi, Kayla Johnson, Garrett Moseley and Aaron Yeager

Abstract

PDF Access Denied

Keywords

Mathematical Subject Classification

Milestones

Authors