代数体函数与其k阶导数的唯一性

Uniqueness of an Algebraic Function and Its k-th Derivatives

利用现有的亚纯函数与其一阶导数和k阶导数的唯一性结论,结合代数体函数与其一阶导数的唯一性相关结论,将Frank和Weissenborn研究的亚纯函数与其k阶导数存在的唯一性定理推广到代数体函数,研究代数体函数与其k阶导数存在的唯一性问题,得到结果:v(v?2)值代数体函数与其k阶导函数至少CM分担2 v个小函数且IM分担∞,则二者相等。由此,可得推论:对v(v?2)值代数体函数与其k阶导函数CM分担2 v个小函数且IM分担∞,则二者相等。对v值代数体函数与其一阶导函数而言,当v?3时,分担值的个数可以减为2v-1个,即得到:v(v?3)值代数体函数与其一阶导函数至少CM分担包括0在内的2v-1个有限复数且IM分担∞,则二者相等。

By using the conclusion of uniqueness of meromorphic function and its first and k-th derivative, and combining the conclusion of the uniqueness of algebroid function and its first derivative,the uniqueness theorem of meromorphic function and its k-th derivative studied by Frank and Weissenborn is extended to algebroid function. The results show that if a v-valued algebroid function(v≥2) and its k-th derivative function at least CM share 2 v small functions and IM share ∞, then the two are equal. Herefrom, it is inferenced that if a v-valued algebroid function(v≥2) and its k-th derivative function CM share 2 v small functions and IM share ∞, then the two are equal. The number of shared values can be reduced to 2 v-1, when the v-value is equal to or greater than 3(v≥3), that is, if an algebroid body function and its derivative functions CM share 2 v-1 finite complex numbers including 0 and IM share ∞, then the two are equal.