Vol. 14, No. 2, 2021

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Zeros of complex random polynomials spanned by Bergman polynomials

Marianela Landi, Kayla Johnson, Garrett Moseley and Aaron Yeager

Vol. 14 (2021), No. 2, 271–281
Abstract

We study the expected number of zeros of

Pn(z) = k = 0nηkpk(z),

where {ηk} are complex-valued independent and identically distributed standard Gaussian random variables, and {pk(z)} are polynomials orthogonal on the unit disk. When pk(z) = (k + 1)πzk , k {0,1,,n}, we give an explicit formula for the expected number of zeros of Pn(z) in a disk of radius r (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 2n3. Generalizing our basis functions {pk(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 (0,1) centered at the origin, and show that asymptotically the expected number of zeros in the unit disk is 2n3.

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