A remark on post-critically finite compositions of polynomials

TL;DR

This paper investigates the boundedness of heights in specific classes of polynomials related to post-critical finiteness, extending previous results and providing new bounds for certain polynomial transformations.

Contribution

It proves that polynomials with a composition involving a power map that are post-critically finite also have bounded height, complementing earlier results on general post-critically finite polynomials.

Findings

Set of such polynomials has bounded height

Established a lower bound on the critical height of g(z^d)

Extended understanding of polynomial dynamics and height bounds

Abstract

The second author proved that the set of post-critically finite polynomials of given degree is a set of bounded height, up to change of variables. Motivated by an observation about unicritical polynomials, we complement this by proving that the set of monic polynomials g(z) of given degree with the property that there exists a d > 1 such that g(z^d) is post-critically finite, is also a set of bounded height. Moreover, we establish a lower bound on the critical height of g(z^d).