Approximate Solutions of Polynomial Equations

✍ Scribed by Shih Ping Tung

• Publisher: Elsevier Science
• Year: 2002
• Tongue: English
• Weight: 268 KB
• Volume: 33
• Category: Article
• ISSN: 0747-7171
• DOI: 10.1006/jsco.2001.0505

✦ Synopsis

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) = n j=0 c j (y -g(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 g 1 (x), . . . , gt(x), t ≤ n, such that F (x, g i (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 g 1 (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).