We utilize Cauchy's argument principle in combination with the Jacobian of a holomorphic function in several complex variables and the first moment of a ratio of two correlated complex normal random variables to prove explicit formulas for the density and the mean distribution of complex zeros of random polynomials spanned by orthogonal polynomials on the unit circle and on the unit disk. We then inquire into the consequences of their asymptotical evaluations.
翻译：我们利用Cauchy的论据原则,结合数种复杂变量的全貌函数和两个相关复杂正常随机变量之比的第一个时刻,来证明单位圆和单位磁盘上通过正方形多数值圆形圆形圆形圆形圆形圆形圆形圆形圆形圆形圆形圆形圆形圆形圆形圆形圆形圆形圆形圆形圆形圆形圆形圆形圆形圆形圆形圆形圆形圆形圆形圆形圆形圆形圆形圆形圆形圆形圆形圆形圆形圆形圆形圆形圆形圆形圆形圆形圆形圆形圆形圆形圆形圆形圆形圆形圆形圆形圆形圆形圆形圆形圆形圆形圆形圆形圆形圆形圆形圆形圆形圆形圆形圆形圆形圆形圆形圆形圆形圆形圆形圆形圆形圆形圆形圆形圆形圆形圆形圆形圆形圆形圆形圆形圆形圆形圆形圆形圆形圆形圆形圆形圆形圆形圆形圆形圆形圆形圆形圆形圆形圆形圆形圆形圆形圆形圆形圆形圆形圆形圆形圆形圆形圆形圆形圆形圆形圆形圆形圆形圆形圆形圆形圆形圆形圆形圆形圆形圆形圆形圆形圆形圆形圆形圆形圆形圆形圆形圆形圆形圆形圆形圆形圆形圆形圆形圆形圆形圆形圆形圆形圆形圆形圆形圆形圆形圆形圆形圆形圆形圆形圆形圆形圆形圆形圆形圆形圆形圆形圆形圆形圆形圆形圆形圆形圆形圆形圆形圆形圆形圆形圆形圆形圆形圆形圆形圆形圆形圆形圆形圆形圆形圆形圆形圆形圆形圆形圆形圆形圆形圆形圆形圆形圆形圆形圆形圆形圆形圆形圆形圆形圆形圆形圆形圆形圆形圆形圆形圆形圆形圆形圆形
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