A finiteness property of postcritically finite unicritical polynomials

TL;DR

This paper proves finiteness of certain algebraic parameters for postcritically finite unicritical polynomials over number fields, under specific conditions, contributing to the understanding of their distribution in moduli space.

Contribution

It establishes a finiteness result for $S$-integral PCF parameters in the moduli space of unicritical polynomials, assuming a technical hypothesis on $eta$.

Findings

Finiteness of $S$-integral PCF parameters under certain conditions

Finiteness holds for a broad class of unicritical polynomials

Conjecture extends the result without the technical hypothesis

Abstract

Let $k$ be a number field with algebraic closure $\bar{k}$, and let $S$ be a finite set of places of $k$ containing all the archimedean ones. Fix $d\geq 2$ and $\alpha \in \bar{k}$ such that the map $z\mapsto z^d+\alpha$ is not postcritically finite. Assuming a technical hypothesis on $\alpha$, we prove that there are only finitely many parameters $c\in\bar{k}$ for which $z\mapsto z^d+c$ is postcritically finite and for which $c$ is $S$-integral relative to $(\alpha)$. That is, in the moduli space of unicritical polynomials of degree d, there are only finitely many PCF $\bar{k}$-rational points that are $((\alpha),S)$-integral. We conjecture that the same statement is true without the technical hypothesis.