# Zeros of repeated derivatives of random polynomials

## TL;DR

This paper investigates how the zeros of derivatives of random polynomials, particularly Kac and elliptic types, behave asymptotically depending on the ratio of derivatives taken to the polynomial degree, revealing clustering and rescaling phenomena.

## Contribution

It analyzes the asymptotic zero distribution of derivatives of random polynomials with varying derivative order, extending known results to cases where the derivative order grows with degree.

## Key findings

- Zeros cluster near the unit circle when the derivative order ratio tends to zero.
- Clustering breaks down when the ratio is positive, indicating a phase transition.
- Rescaling phenomena occur when the derivative order ratio approaches one.

## 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 $K^{(l)}_n(z)$ 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 global 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.