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

256 Appendix by Khovanskii
COROLLARY. If a germ of a function can be obtained from the germs
of single-valued having an analytic set of singular points by
means of integrations, of differentiations, meromorphic operations, compositions,
solutions of algebraic equations, and solutions of holonomic systems
of linear differential equations, then this germ of is an
In particular, a germ which is not an cannot be represented by
generalized quadratures.
A.15 Topological obstructions for the
representability by quadratures
of functions of several variables
This section is dedicated to the topological obstructions for the representability
by quadratures and by generalized quadratures of functions
of several complex variables. These obstructions are analogous to those
holding for functions of one variable considered in §§A.7–A.9.
THEOREM 1. The class of all in having a soluble monodromy
group, is closed with respect to the operations of integration and of
differentiation. Moreover, this class is closed with respect to the composition
with the of variables having soluble monodromy
groups.
RESULT ON QUADRATURES. The monodromy group of any germ of a
function representable by quadratures is soluble. Moreover, every germ
of a function, representable by the germs of single-valued having
an analytic set of singular points is also soluble by means of integrations,
of differentiations, and compositions.
COROLLARY. If the monodromy group of the algebraic equation
in which the are rational functions of variables is not soluble, then
any germ of its solutions not only is not representable by radicals, but
cannot be represented in terms of the germs of single-valued
having an analytic set of singular points by means of integrations, of differentiations,
and compositions.
This corollary represents the strongest version of the Abel theorem.