Topological obstructions to the representability of functions by quadratures.
type: Item
instance of: scholarly article
title: Topological obstructions to the representability of functions by quadratures.
zbMATH Open document ID: 1071.34521
DOI: 10.1007/BF02254657
published in: Journal of Dynamical and Control Systems
publication date: 2000-04-13T00:00:00Z
review text: 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…
Mathematics Subject Classification ID: 34M35
Mathematics Subject Classification ID: 12H05
Mathematics Subject Classification ID: 30F99
zbMATH DE Number: 1431780
author: A. G. Khovanskii
reviewed by: Michael F. Singer
MaRDI profile type: Publication
cites work: Q5787786
cites work: Q3255437
cites work: Formal solutions of differential equations
cites work: Q3290722
cites work: Q5797273
description: scientific article; zbMATH DE number 1431780