Higher-Order, Polar and Sz.-Nagy’s Generalized Derivatives of Random Polynomials with Independent and Identically Distributed Zeros on the Unit Circle

Abstract

For random polynomials with independent and identically distributed (i.i.d.) zeros following any common probability distribution μ with support contained in the unit circle, the empirical measures of the zeros of their first and higher-order derivatives will be proved to converge weakly to μ almost surely (a.s.). This, in particular, completes a recent work of Subramanian on the first-order derivative case where μ was assumed to be non-uniform. The same almost sure weak convergence will also be shown for polar and Sz.-Nagy’s generalized derivatives, assuming some mild conditions.