## Abstract

Let $F:{\mathbb{R}}^{n}\times \mathbb{R}\to \mathbb{R}$ be a real-valued polynomial function of the form $F(\overline{x},y)={a}_{s}(\overline{x}){y}^{s}+{a}_{s-1}(\overline{x}){y}^{s-1}+\cdots +{a}_{0}(\overline{x})$ where the degree $s$ of $y$ in $F(\overline{x},y)$ is greater than $1$. For arbitrary polynomial function $f(\overline{x})\in \mathbb{R}[\overline{x}]$, $\overline{x}\in {\mathbb{R}}^{n}$, we will find a polynomial solution $y(\overline{x})\in \mathbb{R}[\overline{x}]$ to satisfy the following equation ($\mathrm{\star}$): $F(\overline{x},y(\overline{x}))=af(\overline{x})$ where $a\in \mathbb{R}$ is a constant depending on the solution $y(\overline{x})$, namely a quasi-coincidence (point) solution of ($\mathrm{\star}$), and $a$ is called a quasi-coincidence value of ($\mathrm{\star}$). In this paper, we prove that $(i)$ the number of all solutions in ($\mathrm{\star}$) does not exceed ${\mathrm{deg}}_{y}F(\overline{x},y)\left(({2}^{\mathrm{deg}f(\overline{x})}+s+3)\xb7{2}^{\mathrm{deg}f(\overline{x})}+1\right)$ provided those solutions are of finitely many exist, $(ii)$ if all solutions are of infinitely many exist, then any solution is represented as the form $y(\overline{x})=-{a}_{s-1}(\overline{x})/s{a}_{s}(\overline{x})+\lambda p(\overline{x})$ where $\lambda $ is arbitrary and $p(\overline{x})=(f(\overline{x})/{a}_{s}(\overline{x}){)}^{1/s}$ is also a factor of $f(\overline{x})$, provided the equation ($\mathrm{\star}$) has infinitely many quasi-coincidence (point) solutions.

## Citation

Yi-Chou Chen. "Infinitely Many Quasi-Coincidence Point Solutions of Multivariate Polynomial Problems." Abstr. Appl. Anal. 2013 (SI09) 1 - 9, 2013. https://doi.org/10.1155/2013/307913

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