Infinitely Many Quasi-Coincidence Point Solutions of Multivariate Polynomial Problems

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)(2degf(x¯)+s+3)·2degf(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(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.