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范楚君,刘晓俊.涉及导曲线与分担超平面的正规定则[J].上海理工大学学报,2022,44(5):490-496.
涉及导曲线与分担超平面的正规定则
Normal criteria relating to derived curves and shared hyperplanes
投稿时间:2021-07-07  
DOI:10.13255/j.cnki.jusst.20210707001
中文关键词:  正规族  全纯曲线  t次一般位置  导曲线  分担超平面
英文关键词:normal family  holomorphic curves  t-subgeneral position  derived curves  shared hyperplanes
基金项目:国家自然科学基金资助项目(11871216)
作者单位E-mail
范楚君 上海理工大学 理学院 上海 200093  
刘晓俊 上海理工大学 理学院 上海 200093 xiaojunliu2007@hotmail.com 
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中文摘要:
      基于值分布和正规族理论以及高等代数相关知识,研究了全纯曲线族及其导曲线分担处于$ t $次一般位置的超平面的正规定则。设$ \mathcal{F} $是一族从区域$ D \subset \mathbb{C} $到${\mathbb{P}}^{N}(\mathbb{C})$的全纯曲线,${H_\ell } = \rhbr \left\{ {{\bm{x}} \in {\mathbb{P}^N}(\mathbb{C}):} \right.\left. {\left\langle {{\bm{x}},{{\bm{\alpha}} _\ell }} \right\rangle = {\text{0}}} \right\}$是$ {\mathbb{P}^N}(\mathbb{C}) $中处于$ t $次一般位置的超平面,${{\bm{\alpha}} _\ell } = {\left( {{a_{\ell 0}},{a_{\ell 1}}, \cdots ,{a_{\ell N}}} \right)^{\text{T}}},{\text{ }}\ell = 1,2, \cdots ,3t + 1$,$ {H_0} = \left\{ {{x_0} = {\text{0}}} \right\} $,$t\geqslant N$。假定对任意的$ f \in \mathcal{F} $满足条件:若$ f(z) \in {H_\ell } $,则$ \nabla f(z) \in {H_\ell } $,$ \ell = 1,2, \cdots ,3t + 1 $;若$f(z) \in \displaystyle \bigcup\limits_{\ell = 1}^{3t + 1} {{H_\ell }}$,则$\dfrac{\left|\langle f(z),{H}_{0}\rangle \right|}{\Vert f(z)\Vert \cdot \Vert {H}_{0}\Vert }\geqslant\delta$,其中,$ \delta \in \left(0,1\right) $且为常数。那么,$ \mathcal{F} $在$ D $上正规。对于$ N = 3 $,$ t = 3,4,5 $的特殊情形,本文有效降低了所分担超平面的个数。
英文摘要:
      Based on the theories of value distribution, normal family and the knowledge of advanced algebra, the normal criteria for the holomorphic curves and their derived curves sharing hyperplanes located in t-subgeneral position was studied. Let $ \mathcal{F} $ be a family of holomorphic curves of a domain $ D \subset \mathbb{C} $ to $ {\mathbb{P}^N}(\mathbb{C}) $, ${H_\ell } =\{ {{\bm{x}} \in {\mathbb{P}^N}(\mathbb{C}):} \left. {\left\langle {{\bm{x}},{{\bm{\alpha}} _\ell }} \right\rangle = 0} \right\}$ be hyperplanes in $ {\mathbb{P}^N}(\mathbb{C}) $ located in t-subgeneral position, where ${{\bm{\alpha}} _\ell } = {\left( {{a_{\ell 0}},{a_{\ell 1}}, \cdots ,{a_{\ell N}}} \right)^{\text{T}}}$, $ \ell = 1,2, \cdots ,3t + 1 $, $ {H_0} = \left\{ {{x_0} = 0} \right\} $, $t\geqslant N$. Assume the following conditions hold for every $ f \in \mathcal{F} $: If $ f(z) \in {H_\ell } $, then $ \nabla f(z) \in {H_\ell } $, $ \ell = 1,2, \cdots ,3t + 1 $; If $f(z) \in\displaystyle \bigcup\limits_{\ell = 1}^{3t + 1} {{H_\ell }}$, then $\dfrac{\left|\langle f(z),{H}_{0}\rangle \right|}{\Vert f(z)\Vert \cdot \Vert {H}_{0}\Vert }\geqslant \delta$, where $ \delta \in \left(0,1\right) $ is a constant. Then $ \mathcal{F} $ is normal on $ D $. For the special case of $ N = 3 $ and $ t = 3,4,5 $, the number of shared hyperplanes can be effectively reduced through this research.
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