Cubic post-critically finite polynomials defined over ℚ

2020/01/28 by Jacqueline Anderson, Anderson, Jacqueline, Michelle Manes +3 · 2 citations
Computer Science · Mathematics · #37P05 #Algebraic Geometry and Number Theory #Coding theory and cryptography #Dynamical Systems (math.DS) #FOS: Mathematics #Number Theory (math.NT) #Polynomial and algebraic computation

paper · pdf · doi:10.48550/arxiv.2001.10471

openalex publication_date 2020/01/28 · openalex created_date 2025/10/10 · openalex updated_date 2026/07/28

Abstract

We describe and implement an algorithm to find all post-critically finite (PCF) cubic polynomials defined over ℚ, up to conjugacy over PGL₂(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 ℚ, we create a finite search space of cubic polynomials over ℚ that may be PCF. Using a computer search of these possibly PCF cubic polynomials, we find fifteen which are in fact PCF.

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