Finitely generated function fields and complexity in potential theory in the plane

We prove that the Bergman kernel function associated to a finitely connected domain in the plane is given as a rational combination of only three basic functions of one complex variable: an Alhfors map, its derivative, and one other function whose existence is deduced by means of the field of meromorphic functions on the double of the domain. Because many other functions of conformal mapping and potential theory can be expressed in terms of the Bergman kernel, our results shed light on the complexity of these objects. We also prove that the Bergman kernel is an algebraic function of a single Ahlfors map and its derivative. It follows that many objects of potential theory associated to a multiply connected domain are algebraic if and only if the domain is a finite branched cover of the unit disc via an algebraic holomorphic mapping.