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Value distribution of meromorphic functions or holomorphic curves and its applications

亚纯函数或全纯曲线的值分布及其应用

基本信息

批准号：
11640164
负责人：
TODA Nobushige
金额：
$ 2.18万
依托单位：
Nagoya Institute of Technology
依托单位国家：
日本
项目类别：
Grant-in-Aid for Scientific Research (C)
财政年份：
1999
资助国家：
日本
起止时间：
1999 至 2001
项目状态：
已结题

来源：
https://kaken.nii.ac.jp/grant/KAKENHI-PROJECT-11640164/
关键词：
meromorphic function holomorphic curve value distribution algebroid function Nevanlinna theory complex differential equation entire solution complex projective space 微分方程式

项目摘要

(A) For different complex numbers λ_1, … ,λ_<n+1>(n≧2), we investigated the defect relation for the exponential curve fe=[eλ^1^2, …,eλ^<n+1>^2] and we obtained the following theorem. This cannot be obtained from the general theory. Let X be a subset of c^<n+1>-{0} in general position and X^+ be the subset of X the defect of each element of which is positive. Further let d_I is the number of aei in X^+ and D be the convex hull of λ_1, … ,λ_<n+1>.Theorem. If D is n+l-gon, then 0<d_I<1 andΣ__<a∈X^+>∫(a,f_e)≦n+1-Σ^^<n+1>__<I=1>α_I(1-d_I) (α_I>0).(B) Let f be a transcendental holomorphic curve from C into the n-dimensional complex projective space, X be a subset of C^<n+1>-{0} in N-subgeneral position(N>n≧2). Further, we suppose that there exists {a_1, … , a_q} in X satisfying (q≦∞).Σ^^q__<I=1>∫(a_I,f)=2N-n+1.Then, we have the following theorem.Theorem. If N>n=2m (m is a positive integer), then there are [(2N-n+l)/(n+l)]+1 elements in {a_1, … , a_q} satisfying δ(a_j,f)=1.c As applications of value distribution of meromorphic functions, we obtained the followings:(I)We applied the Wiman-Valiron theory to some q-difference equations to obtain the same results on the existence of solutions and the growth of solutions as in the case of differential equations.(ii)We considered meromorphic solutions of the Riccati differential equation with meromorphic coefficients to obtain that the solutions of the equation w' +w^2 +aγ(z)=0 are all one-valued meromorphic in the complex plane. Here, γ(z) is the Weierstrass γ-function with g_3≠0, a=(1-m^2)/4 and m(>2) is integer satisfying m≠6n.

(A)对于不同的复数λ_1，…，λ_<n+1>(n≧2)，研究了指数曲线fe=[λ^1^2，…，λ ^<n+1>^2]的缺陷关系，得到了以下定理。这是不能从一般理论得到的。设X为一般位置c^<n+1>-{0}的子集，X^+为X的子集，其中每个元素的缺陷为正。进一步设d_I为X^+中aei的个数，D为λ_1，…，λ_<n+1>的凸包。如果n + l-gon D,然后0 < d_I < 1Σ__ <∈X ^ + >∫(,f_e)≦n + 1 -Σ^ ^ < n + 1 > __ < I = 1 >α_I (1-d_I)(α_I > 0)。(B)设f是一条从C到n维复射影空间的超越全纯曲线，X是C^<n+1>-{0}在n次一般位置（n >n≧2）的子集。此外,我们假设存在{a_1,…,a_q}在X满足(q≦∞)。Σ^ ^ q__ < I = 1 >∫(ai, f) = 2 n n + 1。然后，我们有下面的定理，定理。如果N> N =2m （m为正整数），则{a_1，…，a_q}中满足δ(a_j,f)=1.c的元素有[(2N-n+l)/(N +l)]+1个。作为亚纯函数值分布的应用，我们得到了以下结果：(I)将Wiman-Valiron理论应用于一些q-差分方程，得到了与微分方程相同的解的存在性和解的生长性结果。（ii）考虑具有亚纯系数的Riccati微分方程的亚纯解，得到方程w′+w^2 +aγ(z)=0的解在复平面上都是一值亚纯。其中，γ(z)是Weierstrass γ-函数，其中g_3≠0，a=（1-m²）/4,m（bbb2）是满足m≠6n的整数。