多复变差分Clunie型定理

Difference Clunie type theorem in several complex variables

令P(z,w)是一个系数为亚纯函数的齐次偏差分多项式,H(z,w)和Q(z,w)是系数为亚纯函数的关于w(z)的多项式且没有公因子,本文主要研究了m上满足方程H(z,w)P(z,w)=Q(z,w)的亚纯解w(z)的性质。首先,若lim supr→∞logT(r,w)/r=0且max{deg_(w)(H),deg_(w)(Q)-deg_(w)(P)}>min{deg_(w)(P),ord_(0)(Q)}-ord_(0)(P),我们得到N(r,w)≠ο(T((r,w)))(r■E_(1)且densE_(1)=0)。另外,若lim supr→∞logT(r,w)/r=0且2κ(P)≤max{deg_(w)(Q),deg_(w)(H)+deg_(w)(P)}-min{deg_(w)(P),ord_(0)(Q)},我们得到m(r,w)=o(T(r,w))+O(T(r))(r■E且densE=0),其中deg_(w)(P)为P(z,w)在w(z)和w(z+c_(i))(i=1,…,m)的总次数,ord_(0)(P)为P(z,x_(0),x_(1),…,x_(m))在x_(0)=0处关于变量x_(0)的零点的阶,T(r)是P(z,w),Q(z,w)和H(z,w)的系数的特征函数的最大值。将Korhonen的结果[13]推广到高维的情形。

It mainly study the property of meromorphic functions w(z)(on C^(m))satisfying the equation H(z,w)P(z,w)=Q(z,w),where P(z,w)is a homogeneous partial difference polynomial with meromorphic coefficients,and H(z,w)and Q(z,w)are polynomials in w(z)with meromorphic coefficients having no common factors.First,if lim supr→∞logT(r,w)/r=0 and max{deg_(w)(H),deg_(w)(Q)-deg_(w)(P)}>min{deg_(w)(P),ord_(0)(Q)}-ord_(0)(P),we get that N(r,w)≠o(T((r,w)))(r■E_(1) and densE_(1)=0).Second,if lim supr→∞log T(r,w)r=0 and 2κ(P)≤max{deg_(w)(Q),deg_(w)(H)+degw(P)}-min{deg_(w)(P),ord_(0)(Q)},we obtain thatm(r,w)=o(T(r,w))+O(T(r))(r■E and densE=0),where degw(P)denotes the total degree of P(z,w)in w(z)and w(z+c_(i))(i=1,…,m),or d0(P)denotes the order of a zero of P(z,x_(0),x_(1),…,x_(m))as a function of x0at x0=0.Those are extension of results of Korhonen.