Approximate solutions of polynomial equations

• Authors: Shih Ping Tung
• Venue: Journal of Symbolic Computation
• Year: 2002

In this paper, we introduce "approximate solutions' to solve thefollowing problem: given a polynomial F(x, y) over Q, where xrepresents an n -tuple of variables, can we find all thepolynomials G(x) such that F(x, G(x)) is identically equal to aconstant c in Q ? We have the following: let F(x, y) be apolynomial over Q and the degree of y in F(x, y) be n. Either thereis a unique polynomial g(x)∈Q [x], with its constant termequal to 0, such that F(x, y)Σ=j=0ncj(y-g(x))j for somerational numbers cj, hence, F(x, g(x) +a) ∈Q forall a∈Q, or there are at most t distinct polynomialsg1(x),..., gt(x), t≤n, such that F(x,gi(x)) ∈Q for 1 ≤i≤t. Suppose that F(x,y) isa polynomial of two variables. The polynomial g(x) for the firstcase, or g1(x),..., gt(x) for the secondcase, are approximate solutions of F(x, y), respectively. There isalso a polynomial time algorithm to find all of these approximatesolutions. We then use Kronecker's substitution to solve the caseof F(x, y). Copyright 2002 Academic Press