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We consider the family of polynomials $f_{d,c}(x)=x^d+c$ over the rationalfield $\Q$. Fixing integers $d, n\ge 2$, we show that the density of primesthat 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 thelatter density when $d=2$. Furthermore, fixing $d,n\ge 2$, we show that for agiven 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 predeterminedpower. This shows that there is no uniform upper bound on the number ofprimitive prime divisors in the critical orbit of $\f(x)$ that does not dependon $c$. The developed results provide a method to construct polynomials of theform $\f(x)$ 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, wegive explicit new results on post-critically finite polynomials $\f(x)$ overlocal fields.

Construction of Polynomials with prescribed divisibility conditions on the critical orbit