New quasi-coincidence point polynomial problems

Let f: r × r → r be a real-valued polynomial function of the form f (x,y) = as (x) ys + as-1 (x) y s-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) ε r [ x ],x ε r,we will find a polynomial solution y (x) ε r [ x ] to satisfy the following equation: (*): f (x,y (x)) = a f (x),where a ε 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 a s (x) must be a factor of f (x),and (ii) each solution of (*) is of the form y (x) = - as-1 (x) / s as (x) + p (x),where is arbitrary and p (x) = c (f (x) / as (x))1/s is also a factor of f (x),for some constant c ε r,provided the equation (*) has infinitely many quasi-coincidence (point) solutions. © 2013 yi-chou chen and hang-chin lai.