Abstract: Post-critically finite polynomials are of great interest to arithmetic dynamicists. Very little is known about PCF polynomials with more than one critical point. Using normalized single-cycle Belyi maps we can describe a two parameter normal form for bicritical polynomials. Using this normal form and results of Ingram we can determine height bounds for the parameters for PCF bicritical polynomials. This allows us to completely describe the set of post-critically finite cubic polynomials over $\mathbb{Q}$ and partially describe the set of PCF bicritical polynomials of arbitrary degree $d>3$.
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