Topological obstructions to the representability of functions by quadratures. (Q1972682)

The author considers the question of when the germ of an analytic function is representable by generalized quadratures, that is, when it can be represented by applying (to constant functions) arithmetic operations, differentiation, exponentiation, integration, and solving algebraic equations. His first result is that the functions representable by generalized quadratures can have no more than a countable number of singular points, that is, points to which there is no analytic continuation. Furthermore, the class of functions having at most a countable number of singular points is closed with respect to composition, integration and solutions of algebraic equations. Given a germ \(f_a\) of an analytic function at a point \(a\), one can consider the set of all of its possible analytic continuations and associate a Riemann surface to this set. Analytic continuation around closed curves at \(a\) in the Riemann sphere \(S^2\) (minus the singular points \(A\) of \(f_a\)) permutes the possible determinations of \(f_a\) at \(a\). This gives a representation of the fundamental group of \(S^2_ A\) into the permutation group on this set, called the monodromy group of \(f_a\). The author shows that if the monodromy group of \(f_a\) is discrete and \(f_a\) is representable by quadratures, then the monodromy group is solvable. He also develops criteria when the monodromy group is not discrete and for various restricted notions of quadrature. The final part of the paper is devoted to mappings of the half plane to polygons bounded by arcs of circles. The author gives a classification of the polygons for which the corresponding function is representable by quadratures. The author's approach uses neither the techniques of Liouville (or their algebraic counterparts developed by Ostrowski, Rosenlicht and others) nor the Galois theory developed by Kolchin. His topological approach allows him to give nonrepresentability results when one even allows a composition with arbitrary meromorphic functions. On the other hand, his approach cannot be used to show that a single-valued meromorphic function is nonrepresentable.