(3,n)型多项式函数方程xf^2(x)+xg^2(x)=h^2(x)的解

Solutions of Polynomial Functional Equation xf^2(x)+xg^2(x)=h^2(x) with Type(3,n)

讨论在复数域上,当f(x)与g(x)的次数都等于3,并且g(x)的次数不超过3时,多项式函数方程xf2(x)+xg2(x)=h2(x)的解的情况,得到部分结果。主要结果为:如果h(x)的次数等于1,那么这个函数方程无解;如果h(x)的次数等于2,那么这个函数方程一共有8组解;如果h(x)的次数等于3,那么h(x)的1次项系数等于零时,这个函数方程一共有24组解;当h(x)的2次项系数等于零时,但1次项系数不等于零时,这个函数方程一共有36组解。

In this paper, we discuss the situation of solutions of the polynomial functional equation x f （x）＋xg^2 （x）=h^2（x） not more than 3, and obtain partial results. The main results are fellows： if the degreeof h （x） isequal to 1, there is not any solution for the functional equation above;if the degree of h （x） is queal to 2,there are altogether 8 solutions for the same functional equation;if the degree of h （x） is queal to 3,there are altogether 24 solutions for it when the coeffcient of term of the first degree of h（x） is queal to zero,and there are altogether 36 solutions for it when the coeffcient of term ofzhe second degree of h（x） is equal to zero,but the coeffcient of term of the first degree ofh（x） is not to zero.