Abel's theorem in problems and solutions - School of Mathematics

232 Appendix by Khovanskii
most countable. An analytic multi-valued function is called an
if each one of its regular germs is an
We proved the following theorem.
THEOREM ON THE CLOSURE OF THE CLASS OF (see
[6],[8],[10]). The class of all is closed with respect to the
following operations:
1) differentiation, i.e., if then
2) integration, i.e., if then
3) composition, i.e., if then
4) meromorphic operation, i.e., if
is a meromorphic function of variables and
then
5) solution of algebraic equations, i.e., if and
then
6) solution of linear differential equations, i.e., if
and then
COROLLARY. If the multi-valued function can be obtained from
single-valued by the operations of integration, differentiation,
meromorphic operations, compositions, solutions of algebraic and linear
differential equations, then the function has at most a countable set of
singular points. In particular, a function having a non countable set of
singular points is not representable by generalized quadratures.
A.6 Monodromy group
The monodromy group of an with a set A of singular points
is the group of all permutations of the sheets of the Riemann surface of
which are visited when one moves around the points of set A.
4 More precisely, the meromorphic operation defined by the meromorphic function
puts into correspondence with the functions a new function
The arithmetic operations and the exponential are examples of meromorphic
operations, corresponding to the functions
and