Primitive prime divisors in the critical orbits of one-parameter families of rational polynomials

For a polynomial f (x) epsilon Q[x] and rational numbers c, u, we put f(c)(x) := f (x)+ c, and consider the Zsigmondy set Z(f(c), u) associated to the sequence {f(n) (c) (u) - u}(n >= 1), see Definition 1.1, where f(c)(n) is the n-st iteration of f(c). In this paper, we prove that if u is a rational critical point of f, then there exists an M-f > 0 such that M-f >= max(c is an element of Q) {#Z(f(c), u)}.