On Zeroes of Random Polynomials and an Application to Unwinding

Abstract

Abstract Let $\mu $ be a probability measure in $\mathbb{C}$ with a continuous and compactly supported density function, let $z_1, \dots , z_n$ be independent random variables, $z_i \sim \mu $, and consider the random polynomial $ p_n(z) = \prod _{k=1}^{n}{(z - z_k)}.$ We determine the asymptotic distribution of $\left \{z \in \mathbb{C}: p_n(z) = p_n(0)\right \}$. In particular, if $\mu $ is radial around the origin, then those solutions are also distributed according to $\mu $ as $n \rightarrow \infty $. Generally, the distribution of the solutions will reproduce parts of $\mu $ and condense another part on curves. We use these insights to study the behavior of the Blaschke unwinding series on random data.