Abstract
We study the expected number of zeros of
where are complex-valued independent and identically distributed standard Gaussian random variables, and are polynomials orthogonal on the unit disk. When , , we give an explicit formula for the expected number of zeros of in a disk of radius 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 . Generalizing our basis functions 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 centered at the origin, and show that asymptotically the expected number of zeros in the unit disk is .
Citation
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
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