Repeated differentiation of deterministic polynomials with asymptotically radial root distributions

Abstract

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 $\hat{I}{¼}_0$, 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 $\hat{I}{¼}_t$ 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.