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

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