Quantitative bounds on integrality for post-critically finite maps

TL;DR

This paper provides quantitative bounds on the number of algebraic parameters that are both post-critically finite and S-integral relative to a fixed point, extending understanding of their distribution in polynomial families.

Contribution

It establishes explicit quantitative bounds on the count of S-integral post-critically finite parameters in the generalized Mandelbrot set, under certain conditions.

Findings

Bounds depend on the degree of the number field and the set of places.

Results apply to unicritical polynomial families with non-PCF parameters.

Provides a quantitative refinement of previous distribution results.

Abstract

Let $K$ be a number field with algebraic closure $\overline{K}$ and let $S$ be a finite set of places of $K$ that contain all the archimedean places. For an integer $d \ge 2$, consider the unicritical polynomial family $f_{d,c}(z) = z^d + c$. Recently, Benedetto and Ih studied the distribution of post-critically finite parameters $c$ that are $S$-integral relative to a fixed point $\alpha \in \overline{K}$ such that $f_{d, \alpha}$ is not post-critically finite. In this paper, we study the quantitative aspects of their result. In particular, under some additional assumptions we establish quantitative bounds on the number of $S$-integral post-critically finite parameters in the generalized Mandelbrot set $\mathcal{M}_{d, v}$ relative to a non post-critically finite parameter $\alpha$ as $\alpha$ varies over number fields of bounded degree.