Topological obstructions to the representability of
functions by quadratures

A.G.Khovanskii

Attempts to solve explicitly differential equations usually fail.  The
first rigorous proofs that some differential equations are not solvable by
quadrature were obtained in the 1830's by Liouville.  Liouville was
undoubtedly inspired by the results of Lagrange, Abel and Galois on the
nonsolvability of algebraic equations by radicals.  Unlike in Galois
theory, automorphism groups do not play a central role in Liouville's
method, even though Liouville uses ``infinitely small automorphisms''.
His results have the following character: Liouville shows that ``simple''
equations cannot have solutions written by complicated formulas.
``Simple'' equations either have solutions of a sufficiently simple kind,
or cannot be solved by quadrature.

Another approach to the problem of solvability of linear differential
equations by quadrature was developed by Picard. Picard generalized Galois
theory to the case of linear differential equations. Vessiot finished in
1910 the work started by Picard, and proved that a linear differential
equation is solvable by quadrature if and only if its Galois group has a
solvable normal subgroup of finite index.  This theorem of Picard-Vessiot
is analogous to the Galois theorem on the solvability of algebraic
equations by radicals.

It is interesting that the Picard-Vessiot approach is close to the
Liouville approach.  Namely, the Galois group of a linear differential
equation has a solvable normal subgroup of finite index if and only if the
equation has solutions of a very specific simple kind.  One can therefore
state the Picard-Vessiot theorem without mentioning Galois groups.  In
that form, for second-order equations, it was discovered and proved by
Liouville, and for $n$-th order equations, by Mordukhai-Boltovskii.
Mordukhai-Boltovskii obtained this result by Liouville's method in 1910,
independently of and simultaneously with Vessiot.

In this paper we describe a third approach to the problem of representing
functions by quadratures.  We consider functions that are representable by
quadratures as multi-valued functions of one complex variable.  It turns
out that there are topological restrictions on the way the Riemann surface
of a function representable by quadratures covers the complex plane.  If
the function does not satisfy these restrictions, then it is not
representable by quadratures.