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

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F是一族从区域D■C到P^(N)(C)的全纯曲线,H_(l)={x∈P^(N)(C):<x,a_(l)>=0}是P^(N)(C)中处于t次一般位置的超平面,α_(l)=(a_(l0),a_(l1),···,a^(lN))^(T),l=1,2,···,3t+1,H_(0)={x_(0)=0},t≥N。假定对任意的f∈F满足条件:若f(z)∈H,则■f(z)∈H_(l),l=1,2,···,3t+1;若f(z)∈∪^(3t+1)_(l=1),则H_(l),其中|<f(z),H_(0)>|/||f(z)·H_(0)||≥δ,δ∈(0,1)且为常数。那么,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。

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 F be a family of holomorphic curves of a domain D■C to P^(N)(C),H_(l)={x∈P^(N)(C):<x,a_(l)>=0}be hyperplanes in P^(N)(C)located in t-subgeneral position,whereα_(l)=(a_(l0),a_(l1),···,a^(lN))^(T),l=1,2,···,3t+1,H_(0)={x_(0)=0},t≥N.Assume the following conditions hold for every f∈F:If f(z)∈Hl,then■f(z)∈H_(l),l=1,2,···,3t+1;If f(z)∈∪^(3t+1)_(l=1),then H_(0)>|/||f(z)·H_(0)||≥δ,whereδ∈(0,1)is a constant.Then 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.