Abstract

We find all post-critically finite (PCF) cubic polynomials defined over $\mathbb{Q}$, up to conjugacy over ${PGL}_2\left(\bar{\mathbb{Q}}\right)$. 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.

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Mathematical Subject Classification 2010

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