Value distribution of meromorphic functions or holomorphic curves and its applications

(A)对于不同的复数λ_1、…，λ_&lt；n+1&gt；(n≧2)，我们研究了指数曲线Fe=[eλ^1^2，…]的亏量关系，eλ^&lt；n+1&gt；^2]，得到了如下定理。这是从一般理论中得不到的。设X是一般位置的c^&lt；n+1&gt；-{0}的子集，X^+是X的子集，X的每个元素的亏量都是正的。进一步设d_i是X^+中AEI的个数，D是λ_1，…的凸包，λ_&lt；n+1；(B)设f是从C到n维复射影空间的一条超越全纯曲线，X是N维次一般位置(N&gt；)上的一个子集，X是f_e)≦n+1-Σ^^&lt；n+1&gt；__&lt；I=1&gt；α_I(1-d_I)(Σ_i&gt；0)中的一条超越全纯曲线，如果D是n+L-∈，则0&lt；d_i&lt；1和∫(a，α_i&gt；N≧2)。进一步，我们假设存在{a_1，…满足(Q≦∞).Σ^^q_&lt；i=1&gt；∫(a_i，f)=2N-n+1)。如果N&gt；n=2m(m为正整数)，则{a_1，…中有[(2n-n+L)/(n+L)]+1个元素，a_q}满足δ(a_j，f)=1.c作为亚纯函数的值分布的应用，我们得到如下结果：(I)我们将Wiman-Valron理论应用于一些q-差分方程解的存在性和增长性，得到了与微分方程解的存在性和增长性相同的结果.(Ii)我们考虑了具有亚纯系数的Riccati微分方程亚纯解，得到了方程w‘+w^2+aγ(Z)=0的解都是复平面上的单值亚纯的.这里，γ(Z)是魏尔斯特拉斯γ函数，其中g_3≠为0，a=(1-m^2)/4，m(&gt；2)是满足m≠6n的整数。

(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.