Let \(\{\varphi _k\}_{k=0}^\infty \) be a sequence of orthonormal polynomials on the unit circle with respect to a probability measure \( \mu \). We study the variance of the number of zeros of random linear combinations of the form$$\begin{aligned} P_n(z)=\sum _{k=0}^{n}\eta _k\varphi _k(z), \end{aligned}$$where \(\{\eta _k\}_{k=0}^n \) are complex-valued random variables. Under the assumption that the distribution for each \(\eta _k\) satisfies certain uniform bounds for the fractional and logarithmic moments, for the cases when \(\{\varphi _k\}\) are regular in the sense of Ullman–Stahl–Totik or are such that the measure of orthogonality \(\mu \) satisfies \(d\mu (\theta )=w(\theta )d\theta \) where \(w(\theta )=v(\theta )\prod _{j=1}^J|\theta - \theta _j|^{\alpha _j}\), with \(v(\theta )\ge c>0\), \(\theta ,\theta _j\in [0,2\pi )\), and \(\alpha _j>0\), we give a quantitative estimate on the variance of the number of zeros of \(P_n\) in sectors that intersect the unit circle. When \(\{\varphi _k\}\) are real-valued on the real-line from the Nevai class and \(\{\eta _k\}\) are i.i.d. complex-valued standard Gaussian, we obtain 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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OPUC跨越的复数随机多项式的零个数的方差

令\（\ {\ varphi _k \} _ {k = 0} ^ \ infty \）是关于概率度量\（\ mu \）的单位圆上的正交多项式序列。我们研究形式为$$ \ begin {aligned} P_n（z）= \ sum _ {k = 0} ^ {n} \ eta _k \ varphi _k（z）的随机线性组合的零个数的方差， \ end {aligned} $$其中\（\ {\ eta _k \} _ {k = 0} ^ n \）是复数值随机变量。假设对于\（\ {\ varphi _k \} \）在Ullman–Stahl意义上是正则的情况下，每个\（\ eta _k \）的分布都满足分数和对数矩的某些统一界线–Totik或正交度\（\ mu \）满足\（d \ mu（\ theta）= w（\ theta）d \ theta \）其中\（w（\ theta）= v（\ theta）\ prod _ {j = 1} ^ J | \ theta-\ theta _j | ^ {\ alpha _j} \），其中\（v（\ theta）\ ge c> 0 \），\（\ theta，\ theta _j \ in [0,2 \ pi）\）和\ （\ alpha _j> 0 \），我们给出了与单位圆相交的扇区中\（P_n \）的零个数的方差的定量估计。当\（\ {\ varphi _k \} \）在Nevai类的实线上是实值，并且\（\ {\ eta _k \} \）是iid复数值标准高斯时，我们得到一个公式\（P_n \）的零个数方差的极限值 在不包含单位圆的环中。