Colloquium publications.

188

THE MADISON COLLOQUIUM.

~ 4. CONTINUATION. THE BRANCH POINTS OF THE
DISCRIMINANT
It is important to notice how the dependent variable behaves
in the points of a (2n - 2)-dimensional manifold of branch
points. If we are at liberty to make, if necessary, a non-singular
linear transformation of the x's, we may assume that
A(O,., ) 0, Xn) + o,
and hence replace the equation A = 0 by an algebroid equation
in xn. Let
(3) D = Xn + BXnl-l + * + B == 0,
where D is an irreducible factor of A; and let D1(x1,...,,n-l)
be the discriminant of D. Then D,1 0. For simplicity in the
presentation, we confine ourselves to the case that A has no
further irreducible factor.
Consider a point Po: (xl~, "' xn-l~) in which D1 = 0. In
the neighborhood of this point the roots of (3) can be grouped
to I functions xn, xn, * *, x (l) each analytic in the above point
and all elements of the same monogenic analytic function.
If we substitute one of these elements, x,', in the coefficients
of (1), the new polynomial,
F=um"+ Alum-1+. + A, = 0
-where Ak(xl, * *, x,_n-) is analytic in the point (xl~, * *, Xn-1~)
but does not necessarily vanish there,-will have a common
factor with its allied polynomial
aF
mu
Consider the greatest common divisor of F and F'. Let its
irreducible factors be
Gk(u, X1, *.., n-1) =,. 1, *,.
In general, v = 1 and G1 is linear in u.