关于杨重骏的一个猜测

On a Conjecture of Yang C C

杨重骏在研究多项式的惟一性时曾提出如下猜测:对于两个次数相同的非常数多项式,若它们在两个不同点值分布相同则这两个多项式相等或者它们的和退化为一个常数(等于这两个数的和).这个猜测不仅是亚纯函数惟一性中的问题,莫宗坚在研究多项式群结构时,也提出过这个问题.本文用代数的方法得到了n次多项式在不同两点的值分布至少有n+1个.根据这些值的重数关系及这两个多项式的部分系数关系得到猜测成立的一些充分条件,同时得到了当n小于或等于3时猜测成立.

When researching the uniquenss of polynomials, Yang C C. had had a conjecture： Given two polynomials with equal degrees,if they are of the same value distribution in two different places,then they are either equal or the sum of the two polynomials degenerate to a constant. This is the problem arisen in the uniqueness question of meromorphic function and it had been also considered by Moh T T when he studied a certain group construction of polynomial. We prove that the value distribution of the nth polynomials at different points is at least n＋ 1. We also have got some sufficient conditions for the holding of the conjecture according to the value distributions of the two polynomials at the two different places. Meanwhile, we prove that the conjecture is true when n is smaller or equal to 3.