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arXiv · 2312.14655

Value Distributions of Derivatives of $K$-regular Polynomial Families

Christian Henriksen·Carsten Lunde Petersen·Eva Uhre
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Abstract

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 $m = 0, 1, \ldots$

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Christian Henriksen, Carsten Lunde Petersen, Eva Uhre. 2024-11-13. Value Distributions of Derivatives of $K$-regular Polynomial Families. https://doi.org/10.1017/etds.2026.10278

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