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

Solvability of Equations 257
THEOREM 2. The class of all in having an almost soluble
monodromy pair is closed with respect to the operations of integration,
differentiation, and solution of algebraic equations. Moreover, this class
is closed with respect to the composition with the of variables
having an almost soluble monodromy pair.
RESULT ON GENERALIZED QUADRATURES. The monodromy pair of a
germ of a function representable by generalized quadratures, is almost
soluble. Moreover, the monodromy pair of every germ of a function
representable in terms of the germs of single-valued having an
analytic set of singular points by means of integrations, differentiations,
compositions, and solutions of algebraic equations is also almost soluble.
A.16 Topological obstruction for the
solvability of the holonomic systems
of linear differential equations
A.16.1 The monodromy group of a holonomic
system of linear differential equations
Consider a holonomic system of N differential equations
where is the unknown function, and the coefficients are rational
functions of the complex variables
One knows that for any holonomic system there exists a singular algebraic
surface in the space that have the following properties. Every
solution of the system can be analytically continued along an arbitrary
curve avoiding the hypersurface Let V be the finite-dimensional space
of the solutions of a holonomic system near a point which lies outside
the hypersurface Consider an arbitrary curve in the space
with the initial point not crossing the hypersurface The solutions
of the system can be analytically continued along the curve remaining
solutions of the system. Consequently to every curve of this type
there corresponds a linear transformation of the space of solutions V
in itself. The totality of the linear transformations corresponding to