A Non-NP-Complete Algorithm for a Quasi-Fixed Polynomial Problem

and Applied Analysis 3 To complete the following algorithm, we need the following elementary property. Lemma 7. Let F(x, y) ∈ R[x, y] and p(x) be an irreducible polynomial in R[x]. Consider the module equation F (x, y) = 0 (mod p (x)) . (20) The number of all solutions y = y(x) (modp(x)) is thus at most deg y F. Assumption. Throughout this algorithm, for any F(x, y) ∈ R[x, y], one can solve all solutions y = g(x) (modp(x)) of the equation F (x, g (x)) = 0 mod p (x) . (21) The procedure of our main algorithm is described below. Procedures of Main Algorithm (Algorithm 5) Step 0. If cont y F | p m (x) and cont y F = 1, then let F 1 (x, y) = F(x, y) and move to (Step 1, (1.2.2)). Step 1. For convenience, we let F (x, y) = (cont y F)F 1 (x, y) , (22) where F 1 (x, y) ∈ R[x, y] is a primitive polynomial. Step 1.1. If cont y F ∤ p m (x), we would have the problem F (x, y) = cp m (x) , (23) to deduce that cont y F | c ⋅ p m (x). But cont y F ∤ p m (x), then c = 0. Consequently, (23) becomes F (x, y) = 0. (24) We can then solve all solutions y(x) for F(x, y) = 0 to get a solution set Y 0 = {y (x) : F (x, y (x)) = 0} . (25) Step 1.2. If cont y F | p m (x) and cont y F ̸ = 1, then cont y F = p l (x) with l 1 ≤ m. In this case, (23) becomes F 1 (x, y) = cp m−l 1 (x) . (26) Step 1.2.1. In case l 1 = m, then (26) becomes F 1 (x, y) = c which can be solved by Algorithm 3 to obtain all solutions for the equation F 1 (x, y) = c to get y = y(x) and obtain a set W 0 = {y (x) : F 1 (x, y (x)) = c} . (27) Step 1.2.2. If l 1 < m, thenm − l 1 > 0 and we can divide both sides of (26) by p(x); consequently, F 1 (x, y) = 0 (mod p (x)) . (28) According to Lemma 7, the solution number does not exceed deg y F, thus we may assume that y = a 0 (x) is a solution of (28) with deg a 0 (x) < degp(x); please note that the choice of a 0 (x) may be larger than 1 and we may define a solution set T 0 (F)which collects such a 0 (x) by setting as the following form: a 0 (x) ∈ T 0 (F) = {a 0