Abstract :
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 polynomial g(x) Q [ x ], with its constant term equal to 0, such that F(x, y) = ∑j = 0ncj(y − g(x))jfor some rational numbers cj, hence, F(x, g(x) + a) Q for all a Q, or there are at most t distinct polynomials g1(x), , gt(x), t ≤ n, such that F(x, gi(x)) Q for 1 ≤ i ≤ t. Suppose that F(x, y) is a polynomial of two variables. The polynomial g(x) for the first case, or g1(x), ,gt(x) for the second case, are approximate solutions of F(x, y), respectively. There is also a polynomial time algorithm to find all of these approximate solutions. We then use Kronecker’s substitution to solve the case of F(x, y).
