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 \(K=\mathbb{Q}(\zeta_d)\).


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