Abstract: Let $\{\varphi_k\}_{k=0}^\infty $ be a sequence of orthonormal polynomials on the unit circle (OPUC) with respect to a probability measure $ \mu $. We study the variance of the number of zeros of random linear combinations of the form
$$
P_n(z)=\sum_{k=0}^{n}\eta_k\varphi_k(z),
$$
where $\{\eta_k\}_{k=0}^n $ are complex-valued random variables. Under the assumption that the measure of orthogonality $\mu$ is absolutely continuous with respect to
arclength measure and its Radon-Nikodym derivative is non-vanishing on the unit circle, and the distribution for each $\eta_k$ satisfies certain uniform bounds for the fractional and logarithmic moments, we show that the variance of the number of zeros of $P_n$ in annuli that contain the unit circle is at most of the order $n\sqrt{n\log n}$ as $n\rightarrow \infty$. When the measure of orthogonality $\mu$ is symmetric with respect to conjugation and in the Nevai class, and $\{\eta_k\}_{k=0}^n$ are i.i.d.~complex-valued standard Gaussian, we prove a formula for the limiting value of variance of the number of zeros of $P_n$ in annuli that do not contain the unit circle.
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