New Quasi-Coincidence Point Polynomial Problems

Abstract

Let $F:ℝ\times ℝ\to ℝ$ be a real-valued polynomial function of the form $F(x,y)=a_s(x)y^s+a_{s-\mathrm{1}}(x)y^{s-\mathrm{1}}+\cdots +a_{\mathrm{0}}(x)$, where the degree $s$ of $y$ in $F(x,y)$ is greater than or equal to $\mathrm{1}$. For arbitrary polynomial function $f(x)\in ℝ[x]$, $x\in ℝ$, we will find a polynomial solution $y(x)\in ℝ[x]$ to satisfy the following equation: ($\mathrm{*}$): $F(x,y(x))=af(x)$, where $a\in ℝ$ is a constant depending on the solution $y(x)$, namely, a quasi-coincidence (point) solution of ($\mathrm{*}$), and $a$ is called a quasi-coincidence value. In this paper, we prove that (i) the leading coefficient $a_s(x)$ must be a factor of $f(x)$, and (ii) each solution of ($\mathrm{*}$) is of the form $y(x)=-a_{s-\mathrm{1}}(x)/s{a}_s(x)+\lambda p(x)$, where $\lambda$ is arbitrary and $p(x)=c(f(x)/a_s(x){)}^{\mathrm{1}/s}$ is also a factor of $f(x)$, for some constant $c\in ℝ$, provided the equation $(\mathrm{*})$ has infinitely many quasi-coincidence (point) solutions.