On Zeroes of Random Polynomials and an Application to Unwinding
On Zeroes of Random Polynomials and an Application to Unwinding
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DOI：
10.1093/imrn/rnz096
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发表时间：
2018-07
影响因子：
1
通讯作者：
S. Steinerberger;Hau‐Tieng Wu
中科院分区：
文献类型：
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作者：
S. Steinerberger;Hau‐Tieng Wu
设$\mu $是$\mathbb{C}$中具有连续紧支撑密度函数的概率测度，$z_1，\dots，z_n$是独立的随机变量，$z_i \sim \mu $，考虑随机多项式$ p_n（z）= \prod _{k=1}^{n}{（z-z_k）}。我们确定了$\left \{z \in \mathbb{C}的渐近分布：p_n（z）= p_n（0）\right \}$。特别地，如果$\mu $是围绕原点的径向，那么这些解也根据$\mu $分布为$n \rightarrow \infty $。一般来说，溶液的分布将重现$\mu $的一部分，并将另一部分浓缩在曲线上。我们使用这些见解来研究随机数据上的Blaschke展开序列的行为。
Let $\mu $ be a probability measure in $\mathbb{C}$ with a continuous and compactly supported density function, let $z_1, \dots , z_n$ be independent random variables, $z_i \sim \mu $, and consider the random polynomial $ p_n(z) = \prod _{k=1}^{n}{(z - z_k)}.$ We determine the asymptotic distribution of $\left \{z \in \mathbb{C}: p_n(z) = p_n(0)\right \}$. In particular, if $\mu $ is radial around the origin, then those solutions are also distributed according to $\mu $ as $n \rightarrow \infty $. Generally, the distribution of the solutions will reproduce parts of $\mu $ and condense another part on curves. We use these insights to study the behavior of the Blaschke unwinding series on random data.