论文标题
具有独立根的随机多项式衍生物越来越多的零
Zeros of a growing number of derivatives of random polynomials with independent roots
论文作者
论文摘要
令$ x_1，x_2，\ ldots $是独立的，并且在$ \ mathbb {c} $中相同分布的随机变量从概率度量$μ$中选择，并定义随机多项式$$ p_n（z）=（z-x_1）\ ldots（z-x_n）\，。 $$我们表明，对于任何序列$ k = k（n）$满足$ k \ leq \ log n /（5 \ log \ log n）$，$ p_n $的$ k $ th衍生物的零是根据同一度量$ $ $ $ $ $ $ $ $ $ $ $ $ $ $ $ $ $ $ $。这扩展了Kabluchko的工作，该工作证明了$ K = 1 $ case，以及证明固定$ K $案的Byun，Lee和Reddy。
Let $X_1,X_2,\ldots$ be independent and identically distributed random variables in $\mathbb{C}$ chosen from a probability measure $μ$ and define the random polynomial $$ P_n(z)=(z-X_1)\ldots(z-X_n)\,. $$ We show that for any sequence $k = k(n)$ satisfying $k \leq \log n / (5 \log\log n)$, the zeros of the $k$th derivative of $P_n$ are asymptotically distributed according to the same measure $μ$. This extends work of Kabluchko, which proved the $k = 1$ case, as well as Byun, Lee and Reddy who proved the fixed $k$ case.
