AI 中文总结
研究形如\(P(z)=p(z^m)\)多项式的重复求导，在\(m\)和\(n\)很大时，简化相关结果证明，扩展到\(z^a(d/dz)^b\)的重复应用，并计算\(m\)固定\(n\)趋于无穷时的极限根分布。
AI 中文摘要
近期Galligo、Najnudel和Vu（2025年）以及Najnudel和Vu（2026年）的工作研究了形如\(P(z)=p(z^m)\)的多项式的重复求导，其中\(p\)是具有实非负根的\(n\)次确定性多项式，在\(m\)和\(n\)很大的情况下。若\(m\gg \log(n)\)且\(P\)的根分布收敛到一个紧支撑的径向概率测度\(\mu_0\)，这些工作表明对于\(0\le t<1\)，\(P\)的\(\lfloor nmt\rfloor\)阶导数的根分布收敛到一个紧支撑概率测度\(\mu_t\)，由其径向分位数函数的显式公式给出。我们给出了该结果的大幅简化证明，并将结果从重复求导扩展到微分算子\(z^a(d/dz)^b\)的重复应用。我们还计算了\(m\)固定且\(n\)趋于无穷时的极限根分布。
英文摘要
Recent works of Galligo, Najnudel, and Vu (2025) and Najnudel and Vu (2026) study repeated differentiation for polynomials of the form $P(z)=p(z^m)$, where $p$ is a deterministic polynomial of degree $n$ with real, non-negative roots, in the regime where $m$ and $n$ are large. If $m\gg \log(n)$ and the root distribution of $P$ converges to a compactly supported, radial probability measure $μ_0$, these works show that for $0\le t<1$, the root distribution of the $\lfloor nmt\rfloor$-th derivative of $P$ converges to a compactly supported probability measure $μ_t$ given by an explicit formula for its radial quantile function. We give a substantially simplified proof of this result and also extend the result from repeated differentiation to repeated applications of the differential operator $z^a(d/dz)^b$. We also compute the limiting root distribution in the case when $m$ is fixed and $n$ tends to infinity.
