Belyi Maps and Bicritical Polynomials

Abstract

Let $K$ be a number field. We will show that any bicritical polynomial $f(z) \in K[z]$ is conjugate to a polynomial of the form $a\mathcal{B}_{d,k}(z) +c \in \bar{K}[z]$ where $\mathcal{B}_{d,k}(z)$ is a normalized single-cycle Belyi map with combinatorial type $(d; d-k, k+1, d)$. We use results of Ingram \cite{ingram2010} to determine height bounds on pairs $(a,c)$ such that $a\mathcal{B}_{d,k}(z) +c$ is post-critically finite. Using these height bounds, we completely describe the set of post-critically finite cubic polynomials of the form $a\mathcal{B}_{d,k}(z)+c \in \mathbb{Q}[z]$, up to conjugacy over $\mathbb{Q}$. We give partial results for post-critically finite polynomials over $\mathbb{Q}$ of arbitrary degree $d>3$.