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2019 Zeros of repeated derivatives of random polynomials
Renjie Feng, Dong Yao
Anal. PDE 12(6): 1489-1512 (2019). DOI: 10.2140/apde.2019.12.1489

Abstract

It has been shown that zeros of Kac polynomials Kn(z) of degree n cluster asymptotically near the unit circle as n under some assumptions. This property remains unchanged for the l-th derivative of the Kac polynomials Kn(l)(z) for any fixed order l. So it’s natural to study the situation when the number of the derivatives we take depends on n, i.e., l=Nn. We will show that the limiting behavior of zeros of Kn(Nn)(z) depends on the limit of the ratio Nnn. In particular, we prove that when the limit of the ratio is strictly positive, the property of the uniform clustering around the unit circle fails; when the ratio is close to 1, the zeros have some rescaling phenomenon. Then we study such problem for random polynomials with more general coefficients. But things, especially the rescaling phenomenon, become very complicated for the general case when Nnn1, where we compute the case of the random elliptic polynomials to illustrate this.

Citation

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Renjie Feng. Dong Yao. "Zeros of repeated derivatives of random polynomials." Anal. PDE 12 (6) 1489 - 1512, 2019. https://doi.org/10.2140/apde.2019.12.1489

Information

Received: 21 October 2017; Revised: 22 August 2018; Accepted: 18 October 2018; Published: 2019
First available in Project Euclid: 12 March 2019

zbMATH: 07061132
MathSciNet: MR3921311
Digital Object Identifier: 10.2140/apde.2019.12.1489

Subjects:
Primary: 60E05

Rights: Copyright © 2019 Mathematical Sciences Publishers

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Vol.12 • No. 6 • 2019
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