Visualization of Polynomial Maps from the Complex Plane to Itself

We show a way to cut up the complex plane into regions that are mapped 1-1 onto the complex plane by a polynomial. This is done for any finite number of ramification points with any multiplicity. For any polynomial map with only one or two ramification points we can do this explicitly (with minor adjustments). Most of the figures are drawn by approximating solutions to polynomial equations using Newton’s method. However, some of the special cases are computed exactly. At ramification points the plane is cut up by equally spaced arcs and the mapping there acts as if it is a hinge which opens to map to the full plane. In order to show the full extent of possibilities, our last example is a degree 12 polynomial with 5 ramification points of varying degrees.