Theory of algebraic functions on the Riemann Sphere

The Riemann sphere (S) is defined as the complex plane together with the point at infinity. Algebraic functions are defined as subsets of S × S such that a bivariate polynomial on S is zero. It is shown that the set of algebraic functions is closed under addition, multiplication, composition, inversion, union, and differentiation. Singular points are defined as points where the function is not locally 1 to 1. A general method is given for calculating the singular point parameters i.e. a topological winding number ratio, a strength coefficient, and location in S × S, and it is argued that the topology of an algebraic function de- pends only on the winding number ratios of all its singular points. After showing how most of these singular point parameters can be calculated under the closure operations and that a function without singular points is linear, it follows that the set of all quadruples of singular point pa- rameters uniquely determine an algebraic function.