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Belyi Maps and Bicritical Polynomials

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Title:Belyi Maps and Bicritical Polynomials
Authors:Tobin, Isabella Olympia
Contributors:Manes, Michelle (advisor)
Mathematics (department)
Arithmetic Dynamics
Number Theory
Date Issued:2019
Publisher:University of Hawai'i at Manoa
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 $d>3$.
Pages/Duration:65 pages
Appears in Collections: Ph.D. - Mathematics

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