Infinitely Many Quasi-Coincidence Point Solutions of Multivariate Polynomial Problems

Abstract

Let $F:{ℝ}^n\times ℝ\to ℝ$ 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 $f(\overline{x})\in ℝ[\overline{x}]$, $\overline{x}\in {ℝ}^n$, we will find a polynomial solution $y(\overline{x})\in ℝ[\overline{x}]$ to satisfy the following equation ($\mathrm{\star }$): $F(\overline{x},y(\overline{x}))=af(\overline{x})$ where $a\in ℝ$ 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 ${\mathrm{deg}}_{y}F(\overline{x},y)\left((2^{\mathrm{deg}f(\overline{x})}+s+3)·2^{\mathrm{deg}f(\overline{x})}+1\right)$ 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.