A finiteness property of postcritically finite unicritical polynomials

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 $d\ge 2$ 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.

14 pages