## Infinitely many quasi-coincidence point solutions of multivariate polynomial problems.(English)Zbl 1381.39020

Summary: Let $$F\:\mathbb R^n\times\mathbb R\rightarrow\mathbb R$$ be a real-valued polynomial function of the form $$F(\overline x,y)=a_s(\overline x)y^s+a_{s-1}(\overline x)y^{s-1}+\dots+a_0(\overline x)$$ 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 $$(*)$$: $$F(\overline x,y(\overline x))=af(\overline x)$$ where $$a\in\mathbb R$$ is a constant depending on the solution $$y(\overline x)$$, namely a quasi-coincidence (point) solution of $$(*)$$, and $$a$$ is called a quasi-coincidence value of $$(*)$$. In this paper, we prove that (i) the number of all solutions in $$(*)$$ does not exceed $$\deg_yF(\overline x,y)((2^{\deg f(\overline x)} +s+3)\cdot2^{\deg f(\overline x)}+1)$$ 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 $$y(\overline x)=-a_{s-1}(\overline x)/sa_s(\overline x)+ \lambda p(\overline x)$$ 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 $$(*)$$ has infinitely many quasi-coincidence (point) solutions.

### MSC:

 39B22 Functional equations for real functions 47H10 Fixed-point theorems
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### References:

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