  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.
