Preperiodic portraits for unicritical polynomials over a rational function field

TL;DR

This paper investigates the existence of special primes in a rational function field where a given point becomes periodic after a fixed number of steps under a family of unicritical polynomials, extending previous constant point results.

Contribution

It provides a complete answer to the existence of such primes for unicritical polynomials over rational function fields, including explicit counterexamples.

Findings

Most points have such primes where they enter an N-cycle after M steps.

Explicit characterization of counterexamples where such primes do not exist.

Extension of previous results from constant points to general points.

Abstract

Let $K$ be an algebraically closed field of characteristic zero, and let $\mathcal{K} := K(t)$ be the rational function field over $K$. For each $d \ge 2$, we consider the unicritical polynomial $f_d(z) := z^d + t \in \mathcal{K}[z]$, and we ask the following question: If we fix $\alpha \in \mathcal{K}$ and integers $M \ge 0$, $N \ge 1$, and $d \ge 2$, does there exist a place $\mathfrak{p} \in \operatorname{Spec} K[t]$ such that, modulo $\mathfrak{p}$, the point $\alpha$ enters into an $N$-cycle after precisely $M$ steps under iteration by $f_d$? We answer this question completely, concluding that the answer is generally affirmative and explicitly giving all counterexamples. This extends previous work by the author in the case that $\alpha$ is a constant point.