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2021 Zeros of complex random polynomials spanned by Bergman polynomials
Marianela Landi, Kayla Johnson, Garrett Moseley, Aaron Yeager
Involve 14(2): 271-281 (2021). DOI: 10.2140/involve.2021.14.271

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.

Citation

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Marianela Landi. Kayla Johnson. Garrett Moseley. Aaron Yeager. "Zeros of complex random polynomials spanned by Bergman polynomials." Involve 14 (2) 271 - 281, 2021. https://doi.org/10.2140/involve.2021.14.271

Information

Received: 7 July 2020; Revised: 11 January 2021; Accepted: 12 January 2021; Published: 2021
First available in Project Euclid: 25 June 2021

Digital Object Identifier: 10.2140/involve.2021.14.271

Subjects:
Primary: 30C15, 30C40, 30E15
Secondary: 60B99

Rights: Copyright © 2021 Mathematical Sciences Publishers

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