New Quasi‐Coincidence Point Polynomial Problems

Yi-Chou Chen and Hang-Chin Lai

Journal of Applied Mathematics, 2013, vol. 2013, issue 1

Abstract: Let F : ℝ × 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 or equal to 1. For arbitrary polynomial function f(x) ∈ ℝ[x], x ∈ ℝ, 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. In this paper, we prove that (i) the leading coefficient as(x) must be a factor of f(x), and (ii) each solution of (*) is of the form y(x) = −as−1(x)/sas(x) + λp(x), where λ is arbitrary and p(x) = c(f(x)/as(x)) 1/s is also a factor of f(x), for some constant c ∈ ℝ, provided the equation (*) has infinitely many quasi‐coincidence (point) solutions.

Date: 2013
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https://doi.org/10.1155/2013/959464

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