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arXiv · 2010.15941

A finiteness property of postcritically finite unicritical polynomials

Robert L. Benedetto·Su-Ion Ih
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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.

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Robert L. Benedetto, Su-Ion Ih. 2020-10-29. A finiteness property of postcritically finite unicritical polynomials. https://arxiv.org/abs/2010.15941

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