#### Vol. 4, No. 1, 2020

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

### Jacqueline Anderson, Michelle Manes and Bella Tobin

Vol. 4 (2020), No. 1, 23–38
##### Abstract

We find all post-critically finite (PCF) cubic polynomials defined over $ℚ$, up to conjugacy over ${PGL}_{2}\left(\stackrel{̄}{ℚ}\right)$. 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 $ℚ$, 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.

##### Keywords
arithmetic dynamics, post-critically finite, cubic polynomials
Primary: 37P05