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

258 Appendix by Khovanskii
all curves forms a group, which is called the monodromy group of the
holonomic system.
Kolchin generalized the Picard–Vessiot theory to the case of holonomic
systems of differential equations. From the Kolchin theory we obtain two
corollaries concerning the solvability by quadratures of the holonomic
systems of differential equations. As in the one-dimensional case, a holonomic
system is said to be regular if approaching the singular set and
infinity its solutions grow at most as some power.
THEOREM 1. A regular holonomic system of linear differential equations
is soluble by quadratures and by generalized quadrature if its monodromy
group is, respectively, soluble and almost soluble.
Kolchin’s theory proves at the same time two results.
1) If the monodromy group of a regular holonomic system of linear
differential equations is soluble (almost soluble) then this system is
solvable by quadratures (by generalized quadratures).
2) If the monodromy group of a regular holonomic system of linear
differential equations is not soluble (is not almost soluble) then this
system is not solvable by quadratures (by generalized quadratures).
Our theorem makes the result (2) stronger.
THEOREM 2. If the monodromy group of a holonomic system of equations
of linear differential equations is not soluble (is not almost soluble),
then every germ of almost all solutions of this system cannot be expressed
in terms of the germs of single-valued having an analytic set
of singular points by means of compositions, meromorphic operations, integrations
and differentiations (by means of compositions, meromorphic
operations, integrations, differentiations and solutions of algebraic equations).
A.16.2 Holonomic systems of equations of linear
differential equations with small coefficients
Consider a system of linear differential equations completely integrable of
the following form