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

Solvability of Equations 253
the question posed. I discovered the class of relatively recently:
up to that time I believed the answer were negative.
In the case of functions of a single variable it was useful to introduce
the class of the Let us start with a direct generalization of
the class of to the multi-dimensional case.
A subspace in a connected analytic manifold
M is said to be thin if there exists a countable set of open subsets
and a countable set of analytic subspaces in these open subsets
such that An analytic multi-valued function on the manifold
M is called an if the set of its singular points is thin. Let us
make this definition more precise.
Two regular germs and given at the points and of the manifold
M, are said to be equivalent if the germ is obtained by a regular
continuation of the germ along some curve. Every germ equivalent
to the germ is also called a regular germ of the analytic multi-valued
function generated by the germ
A point is said to be singular for the germ if there exists
a curve such that the germ cannot
be regularly continued along this curve, but for every this
germ can be continued along the shortened curve It is easy
to see that the sets of singular points for equivalent germs coincide.
A regular germ is said to be an if the set of its singular points
is thin. An analytic multi-valued function is called an if every
one of its regular germs is an
REMARK. For functions of one complex variable we have already given
two definitions of The first one is the above definition, the
second one is given by the theorem in §A.5. These definitions obviously
coincide.
For of several variables the notions of monodromy group
and of monodromy pair are automatically translated.
Let us clarify way the multi-dimensional case is more complicated than
the one-dimensional case.
Imagine the following situation. Let be a multi-valued analytic
function of two variables with a set A of branch points, where is
an analytic curve on the complex plane. It can happen that at one of the
points there exists an analytic germ of the multi-valued analytic
function (by the definition of the set A of branch points, at the point
not every germ of the function exists; yet some of them can exist).
Now let and be two analytic functions of the complex variable