Zeros of repeated derivatives of random polynomials

Feng, Renjie; Yao, Dong

doi:10.2140/apde.2019.12.1489

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Zeros of repeated derivatives of random polynomials

Renjie Feng and Dong Yao

Vol. 12 (2019), No. 6, 1489–1512

DOI: 10.2140/apde.2019.12.1489

Abstract

It has been shown that zeros of Kac polynomials $K_{n} (z)$ of degree $n$ cluster asymptotically near the unit circle as $n \to \infty$ under some assumptions. This property remains unchanged for the $l$ -th derivative of the Kac polynomials $K_{n}^{(l)} (z)$ 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., $l = N_{n}$ . We will show that the limiting behavior of zeros of $K_{n}^{(N_{n})} (z)$ depends on the limit of the ratio $N_{n} ∕ n$ . 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 $N_{n} ∕ n \to 1$ , where we compute the case of the random elliptic polynomials to illustrate this.

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Keywords

derivatives of random polynomials, empirical measure

Mathematical Subject Classification 2010

Primary: 60E05

Milestones

Received: 21 October 2017

Revised: 22 August 2018

Accepted: 18 October 2018

Published: 7 February 2019

Authors

Renjie Feng
Beijing International Center for Mathematical Research Peking University Beijing China

Dong Yao
Duke University Durham, NC United States

Vol. 12, No. 6, 2019