The Density of Primes in Orbits of zd + c

Given a polynomial f (z)= z(d) + c is an element of K[z] over a global field K and a(0) is an element of K, we study the density of prime ideals of K dividing at least one element of the orbit of a(0) under f. 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 giving criteria to ensure 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(xi(d)).