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| Main Authors: | , , |
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| Format: | Preprint |
| Published: |
2023
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| Subjects: | |
| Online Access: | https://arxiv.org/abs/2312.14519 |
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| _version_ | 1866929303563796480 |
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| author | Henriksen, Christian Petersen, Carsten Lunde Uhre, Eva |
| author_facet | Henriksen, Christian Petersen, Carsten Lunde Uhre, Eva |
| contents | 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 $μ$ we provide conditions on $C$ and $μ$ assuring the sequence of $m$th derivatives $(q_k^{(m)})_k$ also has asymptotic root distribution $μ$ for any $m\geq 1$. This complements recent results of Totik. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2312_14519 |
| institution | arXiv |
| publishDate | 2023 |
| record_format | arxiv |
| spellingShingle | Zero distributions of derivatives of polynomial families centering on a set Henriksen, Christian Petersen, Carsten Lunde Uhre, Eva Complex Variables Dynamical Systems 30C15 (Primary) 37F10, 42C05 (Secondary) 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 $μ$ we provide conditions on $C$ and $μ$ assuring the sequence of $m$th derivatives $(q_k^{(m)})_k$ also has asymptotic root distribution $μ$ for any $m\geq 1$. This complements recent results of Totik. |
| title | Zero distributions of derivatives of polynomial families centering on a set |
| topic | Complex Variables Dynamical Systems 30C15 (Primary) 37F10, 42C05 (Secondary) |
| url | https://arxiv.org/abs/2312.14519 |