低次多项式函数方程xf2(x)+xg2(x)=h2(x)解的讨论

Solution of Polynomial Functional Equation xf^2(x)+xg^2(x)=h^2(x) with Lower Degrees

讨论复数域上多项式函数方程xf2(x)+xg2(x)=h2(x),得到这个函数方程的一些基本性质,以及当f(x),g(x),h(x)的次数都不超过2时,该函数方程的所有解。其解的情况如下:在复数域上,如果上述三个多项式的次数都不超过2,那么该函数方程有解当且仅当下列3个条件之一成立:(1)h(x)是零多项式;(2)f(x),g(x),h(x)都是1次多项式;(3)f(x),g(x),h(x)都是2次多项式。更进一步地,满足条件(1)的解只有1组;满足条件(2)的解一共有4组;满足条件(3)的解一共有16组。

In this paper,we discuss the polynomial functional equation xf 2（x） ＋ xg2（x） = h2（x） over complex number eld,and obtain some fundamental properties of this functional equation as well as the whole solutions of it when the degrees of f（x）;g（x） and h（x） are not more than 2,whose situation of solutions is as follows： if the degrees of the three polynomials mentioned above are not more than 2,there are solutions of this functional equation over complex number eld if and only if one of the following conditions holds：（1） h（x） is the zero polynomial;（2） f（x）;g（x） and h（x） are the polynomials with degree 1;（3） f（x） g（x） and h（x） are the polynomials with degree 2.Furthermore,there is only one solution satisfying the condition（1）,and there are altogether 4 and 16 solutions satisfying the conditions（2） and（3） respectively.