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

Abstract EN

Let F:ℝ×ℝ→ℝ be a real-valued polynomial function of the form F(x,y)=∑i=0sfi(x)yi, with degree of y in F(x,y)=s≥1,x∈ℝ.

An irreducible real-valued polynomial function p(x) and a nonnegative integer m are given to find a polynomial function y(x)∈ℝ[x] satisfying the following expression: F(x,y(x))=cpm(x) for some constant c∈ℝ.

The constant c is dependent on the solution y(x), namely, a quasi-fixed (polynomial) solution of the polynomial-like equation (*).

In this paper, we will provide a non-NP-complete algorithm to solve all quasi-fixed solutions if the equation (*) has only a finite number of quasi-fixed solutions.