  In this paper we study the asymptotic behavior of the maximum magnitude of a
complex random polynomial with i.i.d. uniformly distributed random roots on the
unit circle. More specifically, let $\{n_k\}_{k=1}^{\infty}$ be an infinite
sequence of positive integers and let $\{z_{k}\}_{k=1}^{\infty}$ be a sequence
of i.i.d. uniform distributed random variables on the unit circle. The above
pair of sequences determine a sequence of random polynomials $P_{N}(z) =
\prod_{k=1}^{N}{(z-z_k)^{n_k}}$ with random roots on the unit circle and their
corresponding multiplicities. In this work, we show that subject to a certain
regularity condition on the sequence $\{n_k\}_{k=1}^{\infty}$, the log maximum
magnitude of these polynomials scales as $s_{N}I^{*}$ where
$s_{N}^{2}=\sum_{k=1}^{N}{n_{k}^{2}}$ and $I^{*}$ is a strictly positive random
variable.
