差分多项式的一个新的Clunie型定理
A new Clunie type theorem for difference polynomials
摘要：
证明了若w（z）是方程H（z，w）P（z，w）= Q（z，w）的有限级亚纯解，其中P（z，w）= P（z，w（z），w（z+c_1），.，w（z+c_n））是具有亚纯系数的齐次差分多项式，H（z，w）= H（z，w（z））和Q（z，w）= Q（z，w（z））是在w（z）中具有亚纯系数的多项式，它们没有公共因子使得max{deg_w（H），deg_w（Q）- deg_w（P）} > min{deg_w（P），ord_0（Q）} - ord_0（P），其中ord_0（P）表示P（z，x_0，x_1，...，x_n），则Nevanlinna计数函数N（r，w）满足N（r，w）> S（r，w）。这意味着w（z）具有相对大的极点数量。对于更小的一类方程，得到了更强的断言N（r，w）= T（r，w）+S（r，w），这意味着w（z）的极点密度基本上与w（z）的增长所允许的一样高。作为应用，根据解的值分布模式给出了一个简单的充要条件，该条件可作为排除一类包含若干已知Painleve型差分方程的差分方程存在特殊有限阶Riccati解的工具.
英文摘要：
It is shown that if w(z) is a finite-order meromorphic solution of the equation H(z,w) P(z,w) = Q(z,w), where P(z,w) = P(z,w(z),w(z+c_1),...,w(z+c_n)) is a homogeneous difference polynomial with meromorphic coefficients, and H(z,w) = H(z,w(z)) and Q(z,w) = Q(z,w(z)) are polynomials in w(z) with meromorphic coefficients having no common factors such that max{deg_w(H), deg_w(Q) - deg_w(P)} > min{deg_w(P), ord_0(Q)} - ord_0(P), where ord_0(P) denotes the order of zero of P(z,x_0,x_1,...,x_n) at x_0=0 with respect to the variable x_0, then the Nevanlinna counting function N(r,w) satisfies N(r,w) > S(r,w). This implies that w(z) has a relatively large number of poles. For a smaller class of equations a stronger assertion N(r,w) = T(r,w)+S(r,w) is obtained, which means that the pole density of w(z) is essentially as high as the growth of w(z) allows. As an application, a simple necessary and sufficient condition is given in terms of the value distribution pattern of the solution, which can be used as a tool in ruling out the possible existence of special finite-order Riccati solutions within a large class of difference equations containing several known difference equations considered to be of Painleve type.
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