Mathematical Research Letters

Volume 30 (2023)

Number 2

A finiteness property of postcritically finite unicritical polynomials

Pages: 295 – 317



Robert L. Benedetto (Department of Mathematics and Statistics, Amherst College, Amherst, Massachusetts, U.S.A.)

Su-Ion Ih (Department of Mathematics, University of Colorado, Boulder, Co., U.S.A.; and Korea Institute for Advanced Study, Seoul, South Korea)


Let $k$ be a number field with algebraic closure $\overline{k}$, and let $S$ be a finite set of places of $k$ containing all the archimedean ones. Fix $d \geq 2$ and $\alpha \in \overline{k}$ 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 $c \in \overline{k}$ 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 $\overline{k}$-rational points that are $((\alpha), S)$-integral. We conjecture that the same statement is true without the technical hypothesis.

In memory of Lucien Szpiro

Received 30 October 2020

Received revised 13 May 2023

Accepted 23 June 2023

Published 13 September 2023