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

Appendix by A. Khovanskii:
Solvability of equations by
explicit formulae
(Liouville’s theory,
differential Galois theory,
and topological obstructions)
This Appendix is dedicated to the study of the solvability of differential
equations by explicit formulae. This is a quite old problem: the first idea
for solving it dates back to Abel. Today one knows three approaches to
solving this problem. The first belongs to Liouville; the second approach
considers the problem from the point of view of Galois theory: it is related
to the names of Picard, Vessiot, Kolchin, and others; the third approach,
topological, was first introduced in the case of functions of one variable
in my thesis. I am infinitely grateful to my research director V.I. Arnold
who aroused my interest in this subject.
I had always believed that the topological approach cannot be completely
applied to the case of many variables. Only recently I discovered
that this is not true and that in the multi-dimensional case one can obtain
absolutely analogous results [25]–[27].
This Appendix contains the subject of my lectures to the Mathematical
Society of Moscow and to the students of the École Normale Supérieure
at the Independent University of Moscow (October 1994).
The section, concerning the functions of many variables, was added
for this Appendix in autumn 2002.
I would like to thank T.V. Belokrinitska for her help during the editing
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