Classification of Rational Functions with Newton Map Möbius Conjugate to a Polynomial

Newton’s iterative root finding method and the dynamics of the Newton maps of rational functions have been studied in recent years, resulting in the classification of rational functions whose Newton maps are Möbius conjugate to polynomials of degree 1, 2, and 3. In this dissertation, we describe a new approach for obtaining the same results, then classify the sets of rational functions whose Newton maps are Möbius conjugate to polynomials of degree 4, polynomials of degree 5, and polynomials of any finite degree.