Misiurewicz polynomials for rational maps with nontrivial automorphisms II

TL;DR

This paper extends the understanding of Misiurewicz polynomials for certain rational maps, proving their irreducibility under broader conditions and showing they always have a large irreducible factor.

Contribution

It generalizes previous results by providing new sufficient conditions for irreducibility and demonstrates the existence of large irreducible factors in Misiurewicz polynomials.

Findings

Misiurewicz polynomials are irreducible under extended conditions.

They always possess an irreducible factor of large degree.

The results apply to rational maps with cyclic automorphism groups.

Abstract

This paper continues discussions in the author's previous paper about the Misiurewicz polynomials defined for a family of degree $d \ge 2$ rational maps with an automorphism group containing the cyclic group of order $d$. In particular, we extend the sufficient conditions that the Misiurewicz polynomials are irreducible over $\mathbb{Q}$. We also prove that the Misiurewicz polynomials always have an irreducible factor of large degree.