Abstract: Consider polynomials over a number field with exactly two finite ramification points. We use single-cycle normalized Belyi maps to define a normal form for such polynomials. Furthermore, if the ramification points have distinct ramification indices then normalizing the polynomials preserves the field of definition. Using this normal form, we can then begin to describe the conjugacy classes of post-critically finite bicritical polynomials over the rational numbers. We will build up the background necessary to understand the concepts and techniques used.
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