论文标题
由伯格曼多项式跨越的复杂随机多项式的零
Zeros of Complex Random Polynomials Spanned by Bergman Polynomials
论文作者
论文摘要
我们研究$$ p_n（z）= \ sum_ {k = 0}^nη_kp_k（z）的预期零数，其中$ \ {η_k\} $是复杂的I.I.D标准Gaussian随机变量，以及$ \ {p_k（z）$ polynomals或polynomials。当$ p_k（z）= \ sqrt {（k+1）/π} z^k $，$ k \ in \ {0,1，\ dots，n \} $，我们给出了预期的$ p_n（z）$的预期零数的明确公式。从我们的公式中，我们建立了预期零数的限制值，径向扩展磁盘中的预期零数，并表明单位磁盘中的预期零为$ 2N/3 $。从ullman-stahl--totik的意义上概括我们的基础功能$ \ {p_k（z）\} $，并且与多项式相关的正交性的度量绝对是在平面lebesgue量度方面是绝对连续的来源，并表明单位磁盘中预期的零数为$ 2N/3 $。
We study the expected number of zeros of $$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 $2n/3$.
