Approximate solutions of polynomial equations

摘要

In this paper, we introduce "approximate solutions" to solve the following problem: given a polynomial F(x, y)over Q, where x represents an n-tuple of variables, can we find all the polynomials G(x)such that F(x, G(x))is identically equal to a constant c in Q? We have the following: let F(x, y)be a polynomial over Q and the degree of y in F(x, y)be n. Either there is a unique polynomials g(x)∈Q[x], with its constant term equal to 0, such that F(x, y)=∑~n_j=0c_j(x)~j for some rational numbers c)j, hence, F(x, g(x)+a)∈Q for all a∈Q, or there are at most t distinct polynomials g1(x),..., g_t(x_, t≤n, such that F(x, g_i(x))∈Q for 1≤i≤t.