On Zeroes of Random Polynomials and an Application to Unwinding

Let μ be a probability measure in C with a continuous and compactly supported density function, let z1, zn be independent random variables, zi ∼ μ, and consider the random polynomial pn(z) = ∏ k=1n(z-zk).We determine the asymptotic distribution of left z C: pn(z) = pn(0). In particular, if mu is radial around the origin, then those solutions are also distributed according to mu as n. 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.