Infinitely Many Quasi-Coincidence Point Solutions of Multivariate Polynomial Problems

Yi-Chou Chen
2013 Abstract and Applied Analysis
Let 𝐹 : R 𝑛 × R → R be a real-valued polynomial function of the form 𝐹(𝑥, 𝑦) = 𝑎 𝑠 (𝑥)𝑦 𝑠 + 𝑎 𝑠−1 (𝑥)𝑦 𝑠−1 + ⋅ ⋅ ⋅ + 𝑎 0 (𝑥) where the degree 𝑠 of 𝑦 in 𝐹(𝑥, 𝑦) is greater than 1. For arbitrary polynomial function 𝑓(𝑥) ∈ R[𝑥], 𝑥 ∈ R 𝑛 , 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
more » ... In this paper, we prove that (i) the number of all solutions in (⋆) does not exceed deg 𝑦 𝐹(𝑥, 𝑦)((2 deg 𝑓(𝑥) + 𝑠 + 3) ⋅ 2 deg 𝑓(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) = −𝑎 𝑠−1 (𝑥)/𝑠𝑎 𝑠 (𝑥) + 𝜆𝑝(𝑥) where 𝜆 is arbitrary and 𝑝(𝑥) = (𝑓(𝑥)/𝑎 𝑠 (𝑥)) 1/𝑠 is also a factor of 𝑓(𝑥), provided the equation (⋆) has infinitely many quasi-coincidence (point) solutions.