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arXiv · 2312.14519

Zero distributions of derivatives of polynomial families centering on a set

Christian Henriksen·Carsten Lunde Petersen·Eva Uhre
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Abstract

Suppose $C \subset \mathbb{C}$ is compact. Let $q_k$ be a sequence of polynomials of degree $n_k \to \infty$, such that the locus of roots of all the polynomials is bounded, and the number of roots of $q_k$ in any closed set $L$ not meeting $C$ is uniformly bounded. Supposing that $(q_k)_k$ has an asymptotic root distribution $\mu$ we provide conditions on $C$ and $\mu$ assuring the sequence of $m$th derivatives $(q_k^{(m)})_k$ also has asymptotic root distribution $\mu$ for any $m\geq 1$. This complements recent results of Totik.

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Christian Henriksen, Carsten Lunde Petersen, Eva Uhre. 2023-12-22. Zero distributions of derivatives of polynomial families centering on a set. https://arxiv.org/abs/2312.14519

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