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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/2312.14655 |
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| _version_ | 1866908851299680256 |
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| author | Henriksen, Christian Petersen, Carsten Lunde Uhre, Eva |
| author_facet | Henriksen, Christian Petersen, Carsten Lunde Uhre, Eva |
| contents | Let $Ω\in \mathbb{C}$ be a domain such that $K:= \mathbb{C} \setminus Ω$ is compact and non-polar. Let $g_Ω$ be the Green's function with a logarithmic pole at infinity, and let $ω= ω_K$ be the equilibrium distribution on $K$. Let $(q_k)_{k>0}$ be a sequence of polynomials with $n_k$, the degree of $q_k$ satisfying $n_k \to \infty$, and let $(q_k^m)_k$ denote the sequence of $m$-th derivatives. We provide conditions, which ensure that the preimages $(q_k^m)^{-1}(\{a\})$ uniformly equidistribute on $\partial Ω$, as $k \to \infty$, for every $a \in \mathbb{C}$ and every $m = 0, 1, \ldots$ |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2312_14655 |
| institution | arXiv |
| publishDate | 2023 |
| record_format | arxiv |
| spellingShingle | Value Distributions of Derivatives of $K$-regular Polynomial Families Henriksen, Christian Petersen, Carsten Lunde Uhre, Eva Complex Variables 42C05 (Primary) 37F10, 31A15 (Secondary) Let $Ω\in \mathbb{C}$ be a domain such that $K:= \mathbb{C} \setminus Ω$ is compact and non-polar. Let $g_Ω$ be the Green's function with a logarithmic pole at infinity, and let $ω= ω_K$ be the equilibrium distribution on $K$. Let $(q_k)_{k>0}$ be a sequence of polynomials with $n_k$, the degree of $q_k$ satisfying $n_k \to \infty$, and let $(q_k^m)_k$ denote the sequence of $m$-th derivatives. We provide conditions, which ensure that the preimages $(q_k^m)^{-1}(\{a\})$ uniformly equidistribute on $\partial Ω$, as $k \to \infty$, for every $a \in \mathbb{C}$ and every $m = 0, 1, \ldots$ |
| title | Value Distributions of Derivatives of $K$-regular Polynomial Families |
| topic | Complex Variables 42C05 (Primary) 37F10, 31A15 (Secondary) |
| url | https://arxiv.org/abs/2312.14655 |