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

Solvability of Equations 249
up to now there had been no proof that there exist algebraic functions of
several variables which are not representable by algebraic functions of a
single variable.
We know, however, the following result:
THEOREM ([4], [5]). An entire function of two variables defined
by the equation
cannot be expressed in terms of entire functions of a single variable by
means of compositions, additions, and subtractions.
The reason is the following. To every singular point of an algebraic
function one can associate a local monodromy group, i.e., the group of
the permutations of the sheets of the Riemann surface which is obtained
by going around the singularities of the function along curves lying in an
arbitrarily small neighbourhood of the point For algebraic functions of
one variable this local group is commutative; as a consequence the local
monodromy group of an algebraic function which is expressed by means
of sums and differences of integer functions must be soluble. But the local
monodromy group of the function
near the point (0,0) is the group S(5) of all permutations of five elements,
which is not soluble. This explains the statement of the theorem.
Observe that if the operation of division is allowed, then the above
argument no longer holds. Indeed, the division is an operation killing the
continuity and its application destroys the locality. In fact, the function
satisfying
can be expressed by means of division in terms of a function of one
variable, defined by the equation
and of the function of one variable It is not difficult to see
that
the study of compositions of algebraic functions, were afterwards applied by Vassiliev
and Smale [21],[22],[23] to the problem of finding the topologically necessary number
of ramifications in the numeric algorithms for the approximate calculation of roots of
polynomials. (The number of ramifications is of the order of for a polynomial of
degree (Translator’s note.)