On Zeroes of Random Polynomials and Applications to Unwinding

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 $n\to \infty$. 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.