Zeros of repeated derivatives of random polynomials

Abstract

It has been shown that zeros of Kac polynomials $K_n\left(z\right)$ 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 $K_n^{\left(l\right)}\left(z\right)$ 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 $K_n^{\left(N_n\right)}\left(z\right)$ 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.