## The density of primes in orbits of $$z^d+c$$.(English)Zbl 1395.11128

Summary: Given a polynomial $$f(z)=z^d+c\in K[z]$$ 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 $$f$$. We show that for many choices of $$d$$ and $$c$$ this density is zero for all $$a_0$$, assuming $$K$$ contains a primitive $$d$$th root of unity. The proof relies on several new results, including some giving criteria to ensure the number of irreducible factors of the $$n$$th 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 $$K=\mathbb{Q}(\zeta_d)$$.

### MSC:

 11R45 Density theorems 11R09 Polynomials (irreducibility, etc.) 11R58 Arithmetic theory of algebraic function fields
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