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| Main Authors: | , , |
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| Format: | Preprint |
| Published: |
2025
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| Subjects: | |
| Online Access: | https://arxiv.org/abs/2501.05203 |
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| _version_ | 1866915095991287808 |
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| author | Henriksen, Christian Petersen, Carsten Lunde Uhre, Eva |
| author_facet | Henriksen, Christian Petersen, Carsten Lunde Uhre, Eva |
| contents | Given a family $(q_k)_k$ of polynomials, we call an open set $U$ root-sparse if the number of zeros of $q_k$ is locally uniformly bounded on $U$. We study the interplay between the individual zeros of the polynomials $q_k$ and those of the $m$th derivatives $q_k^{(m)}$, in a root-sparse open set $U$, as $k\to\infty$. More precisely, if the root distributions $μ_k$ of $q_k$ converge weak* to some compactly supported measure $μ$, whose potential is nowhere locally constant on a root-sparse open set $U$, then we link the roots of the $m$th derivative $q_k^{m}$, for an arbitrary $m>0$, to the roots of $q_k$ and the critical points of the potential $p_μ$ on compact subsets of $U$.
We apply this result in a polynomial dynamics setting to obtain convergence results for the roots of the $m$th derivative of iterates of a polynomial outside the filled-in Julia set. We also apply our result in the setting of extremal polynomials. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2501_05203 |
| institution | arXiv |
| publishDate | 2025 |
| record_format | arxiv |
| spellingShingle | Roots of polynomial sequences in root-sparse regions Henriksen, Christian Petersen, Carsten Lunde Uhre, Eva Complex Variables 42C05, 37F10, 31A15 Given a family $(q_k)_k$ of polynomials, we call an open set $U$ root-sparse if the number of zeros of $q_k$ is locally uniformly bounded on $U$. We study the interplay between the individual zeros of the polynomials $q_k$ and those of the $m$th derivatives $q_k^{(m)}$, in a root-sparse open set $U$, as $k\to\infty$. More precisely, if the root distributions $μ_k$ of $q_k$ converge weak* to some compactly supported measure $μ$, whose potential is nowhere locally constant on a root-sparse open set $U$, then we link the roots of the $m$th derivative $q_k^{m}$, for an arbitrary $m>0$, to the roots of $q_k$ and the critical points of the potential $p_μ$ on compact subsets of $U$. We apply this result in a polynomial dynamics setting to obtain convergence results for the roots of the $m$th derivative of iterates of a polynomial outside the filled-in Julia set. We also apply our result in the setting of extremal polynomials. |
| title | Roots of polynomial sequences in root-sparse regions |
| topic | Complex Variables 42C05, 37F10, 31A15 |
| url | https://arxiv.org/abs/2501.05203 |