Infinitely Many Quasi‐Coincidence Point Solutions of Multivariate Polynomial Problems

Yi-Chou Chen

Abstract and Applied Analysis, 2013, vol. 2013, issue 1

Abstract: Let F : ℝn × ℝ → ℝ be a real‐valued polynomial function of the form F(x¯,y)=as(x¯)ys+as-1(x¯)ys-1+⋯+a0(x¯) where the degree s of y in F(x¯,y) is greater than 1. For arbitrary polynomial function f(x¯)∈ℝ[x¯], x¯∈ℝn, we will find a polynomial solution y(x¯)∈ℝ[x¯] to satisfy the following equation (⋆): F(x¯,y(x¯))=af(x¯) where a ∈ ℝ is a constant depending on the solution y(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 degyF(x¯,y)(2deg f(x¯)+s+31)·2deg f(x¯)+ 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(x¯)=-as-1(x¯)/sas(x¯)+λp(x¯) where λ is arbitrary and p(x¯)=(f(x¯)/as(x¯)) 1/s is also a factor of f(x¯), provided the equation (⋆) has infinitely many quasi‐coincidence (point) solutions.

Date: 2013
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https://doi.org/10.1155/2013/307913

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