## Abstract

Let $F:\mathbb{R}\times \mathbb{R}\to \mathbb{R}$ 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 $f(x)\in \mathbb{R}[x]$, $x\in \mathbb{R}$, we will find a polynomial solution $y(x)\in \mathbb{R}[x]$ to satisfy the following equation: ($\mathrm{*}$): $F(x,y(x))=af(x)$, where $a\in \mathbb{R}$ 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 $c\in \mathbb{R}$, provided the equation $(\mathrm{*})$ has infinitely many quasi-coincidence (point) solutions.

## Citation

Yi-Chou Chen. Hang-Chin Lai. "New Quasi-Coincidence Point Polynomial Problems." J. Appl. Math. 2013 (SI21) 1 - 8, 2013. https://doi.org/10.1155/2013/959464