Construction of Polynomials with prescribed divisibility conditions on the critical orbit

TL;DR

This paper studies the distribution of primitive prime divisors in the critical orbits of polynomials of the form $x^d + c$, showing positive density results and constructing examples with arbitrarily many primitive divisors, with implications for Galois groups.

Contribution

It provides explicit density calculations for primitive prime divisors in polynomial orbits and constructs polynomials with unbounded primitive divisors, advancing understanding of their arithmetic properties.

Findings

Positive density of primes as primitive divisors for fixed degree

Existence of polynomials with arbitrarily many primitive divisors

New results on post-critically finite polynomials over local fields

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.