  Given a polynomial f(z) = z^d + c over a global field K and a_0 in K, we
study the density of prime ideals of K dividing at least one element of the
orbit of a_0 under f. The density of such sets for linear polynomials has
attracted much study, and the second author has examined several families of
quadratic polynomials, but little is known in the higher-degree case. We show
that for many choices of d and c this density is zero for all a_0, assuming K
contains a primitive dth root of unity. The proof relies on several new
results, including some ensuring the number of irreducible factors of the nth
iterate of f remains bounded as n grows, and others on the ramification above
certain primes in iterated extensions. Together these allow for nearly complete
information when K is a global function field or when K=Q(zeta_d).
