Cubic post-critically finite polynomials defined over $\mathbb{Q}$

TL;DR

This paper presents an algorithm to classify all post-critically finite cubic polynomials over the rationals, implementing a finite search that identifies fifteen such polynomials.

Contribution

The authors develop a normal form classification and an algorithmic approach to find all rational PCF cubic polynomials, expanding understanding of their structure.

Findings

Identified fifteen rational PCF cubic polynomials.

Developed a finite search method based on coefficient bounds.

Established normal forms respecting field of definition.

Abstract

We describe and implement an algorithm to find all post-critically finite (PCF) cubic polynomials defined over $\mathbb{Q}$, up to conjugacy over $\text{PGL}_2(\bar{\mathbb{Q}})$. We describe normal forms that classify equivalence classes of cubic polynomials while respecting the field of definition. Applying known bounds on the coefficients of post-critically bounded polynomials to these normal forms simultaneously at all places of $\mathbb{Q}$, we create a finite search space of cubic polynomials over $\mathbb{Q}$ that may be PCF. Using a computer search of these possibly PCF cubic polynomials, we find fifteen which are in fact PCF.