Primitive prime divisors for unicritical polynomials

We prove the finiteness of the Zsigmondy set associated to critical orbits of polynomials. In the case of unicritical polynomials over the rational numbers, we find a uniform bound on the size of the Zsigmondy set. We prove further that there exists an effectively computable bound on the largest element of the Zsigmondy set, and that, under mild additional hypotheses, that bound is small.