Finitely generated function fields and complexity in potential theory in the plane
On an \(n\)-connected domain \(\Omega\) such that no boundary component is a point, the author shows that the Bergman kernel \(K(z,w)\) associated to \(\Omega\) is a rational combination of only three basic functions of one complex variable. That is, \(K(z,w)=R(f_a(z)\), \(f_a'(z)\), \(K(z,b)\), \(\overline{f_a(w)}\), \(\overline{f_a'(w)}\), \(\overline{K(w,b)})\) where \(R\) is a rational function on \(\mathbb{C}^6\), \(f_a\) is an Ahlfors map (which is a branched \(n\)-to-one covering map of \(\Omega\) onto the unit disc with \(f_a(a)=0\) and \(f_a'(a)>0\) and maximal), and \(a\) and \(b\) are fixed points in \(\Omega\).
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