Construction of Polynomials with prescribed divisibility conditions on the critical orbit

Abstract

We consider the family of polynomials $f_{d,c}(x)=x^d+c$ over the rational field $\Q$. Fixing integers $d, n\ge 2$, we show that the density of primes that can appear as primitive prime divisors of $f_{d,c}^n(0)$ for some $c\in\Q$ is positive. In fact, under certain assumptions, we explicitly calculate the latter density when $d=2$. Furthermore, fixing $d,n\ge 2$, we show that for a given integer $N>0$, there is $c\in \Q$ such that $\f^n(0)$ has at least $N$ primitive prime divisors each of which is appearing up to any predetermined power. This shows that there is no uniform upper bound on the number of primitive prime divisors in the critical orbit of $\f(x)$ that does not depend on $c$. The developed results provide a method to construct polynomials of the form $\f(x)$ for which the splitting field of the $m$-th iteration, $m\ge1$, has Galois group of maximal possible order. During the course of this work, we give explicit new results on post-critically finite polynomials $\f(x)$ over local fields.