Two New Normals Forms for Polynomial Endomorphisms of the Projective Line with Applications to Postcritically Finite Maps

We explore two normal forms for polynomial endomorphisms of the projective line. The first is a normal form for degree 3 polynomials in terms of the multipliers of the fixed points. This normal form allows for an enumeration of all $K$-rational conjugacy classes in the moduli space of degree 3 polynomials. The second normal form is for polynomials of arbitrary degree with $n$ critical points. As an application, we give an algebraic proof of Thurston transversality in the periodic case for bicritical polynomials of any degree.Comment: minor updates and clarification of results and example