伯格曼多项式生成的复随机多项式的零点
Zeros of Complex Random Polynomials Spanned by Bergman Polynomials
摘要：
研究了$$P_n(z)=\sum_{k=0}^nη_kp_k(z),$$的零点期望个数，其中$\{η_k\}$是复值独立同分布的标准高斯随机变量，$\{p_k(z)\}$是单位圆盘上正交的多项式.当$p_k(z)=\sqrt{(k+1)/π} z^k$，$k\in \{0,1,\dots, n\}$，我们给出了一个明确的公式$P_n(z)$的零的期望数在一个半径为$r\in (0,1)$的圆盘中心的原点。从我们的公式中，我们建立了期望的零的数量的极限值，期望的零的数量在一个径向膨胀的磁盘，并表明，期望的零的数量在单位磁盘是$2n/3$。将基函数$\{p_k(z)\}$推广为Ullman-Stahl-Totik意义下的正则基函数，并证明多项式的正交性测度关于平面勒贝格测度是绝对连续的，给出了以原点为中心的半径为$r\in (0,1)$的圆盘中$P_n(z)$的期望零点数的极限值，并证明了单位圆盘中零点的期望个数渐近为$2n/3$。
英文摘要：
We study the expected number of zeros of $$P_n(z)=\sum_{k=0}^nη_kp_k(z),$$ where $\{η_k\}$ are complex-valued i.i.d standard Gaussian random variables, and $\{p_k(z)\}$ are polynomials orthogonal on the unit disk. When $p_k(z)=\sqrt{(k+1)/π} z^k$, $k\in \{0,1,\dots, n\}$, we give an explicit formula for the expected number of zeros of $P_n(z)$ in a disk of radius $r\in (0,1)$ centered at the origin. From our formula we establish the limiting value of the expected number of zeros, the expected number of zeros in a radially expanding disk, and show that the expected number of zeros in the unit disk is $2n/3$. Generalizing our basis functions $\{p_k(z)\}$ to be regular in the sense of Ullman--Stahl--Totik, and that the measure of orthogonality associated to polynomials is absolutely continuous with respect to planar Lebesgue measure, we give the limiting value of the expected number of zeros of $P_n(z)$ in a disk of radius $r\in (0,1)$ centered at the origin, and show that asymptotically the expected number of zeros in the unit disk is $2n/3$.
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