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

Abstract

Let $F:ℝ\times ℝ\to ℝ$ be a real-valued polynomial function of the form $F(x,y)={\sum }_{i=\mathrm{0}}^s\mathrm{‍}{f}_i(x)y^i$, with degree of $\mathrm{}y\mathrm{}\mathrm{}$ in $F(x,y)=s\ge \mathrm{}\mathrm{}\mathrm{1},\mathrm{}\mathrm{}x\in ℝ\mathbb{}.$ An irreducible real-valued polynomial function $p(x)$ and a nonnegative integer $m$ are given to find a polynomial function $y(x)\in ℝ[x]$ satisfying the following expression: $F(x,y(x))=c{p}^m(x)$ for some constant $c\in ℝ$. The constant $c$ is dependent on the solution $y(x)$, namely, a quasi-fixed (polynomial) solution of the polynomial-like equation $(\mathrm{*})$. In this paper, we will provide a non-NP-complete algorithm to solve all quasi-fixed solutions if the equation $(\mathrm{*})$ has only a finite number of quasi-fixed solutions.