Some Applications of Dynamical Belyi Polynomials

TL;DR

This paper characterizes when certain polynomials have persistent bad reduction at primes and addresses questions about conservative polynomials using dynamical Belyi polynomials.

Contribution

It provides necessary and sufficient conditions for post-critically finite polynomials to have persistent bad reduction and answers questions about conservative polynomials.

Findings

Criteria for persistent bad reduction at primes.

Negative answer to Silverman's questions on conservative polynomials.

Use of dynamical Belyi polynomials as key examples.

Abstract

We give necessary and sufficient conditions for post-critically finite polynomials to have persistent bad reduction at a given prime. We also answer in the negative a pair of questions posed by Silverman about conservative polynomials. Our proofs rely on conservative dynamical Belyi polynomials as exemplars of PCF (resp. conservative) maps.