Colloquium publications.

182

THE MADISON COLLOQUIUM.

identically, but does vanish at (a); and if, in the neighorhood of
(a), a relation of the form
F(ZI, * * *, zn) = Q(z, * zn);4i(Z1, Zn)
holds, Q being analytic at (a), then F is said to be divisible by
1 in the point (a). If G(zi, * -, Zn) is analytic in the point
(a) = (a,, * * *, an) and vanishes there, then G is said to be irreducible at (a) if no equation of the form exists:
G(zi, * -, Zn) = Gl(zi, * -, Zn) 02(Z1, ***, )
where G1 and G2 are both analytic at (a) and both vanish there.
Two irreducible factors are equivalent if their quotient, taken
either way, presents at most removable singularities.
A function G(z1, *., Zn) analytic at (a) and vanishing there, but
not vanishing identically, can be written in one, and essentially
in only one, way as the product of factors each irreducible in (a).
A factor which is irreducible at a given point is not necessarily
irreducible at every one of its vanishing points which lies in a certain neighborhood of the point. Hence the expression of a function at a given point as a product of factors each irreducible at
that point does not always retain this character when that point
is replaced by a second root of the function that lies in the
neighborhood of that point.
The theorem of algebraic geometry that two curves or surfaces
which have ever so short an arc or small a region in common,
must necessarily have a whole irreducible piece in common,
finds its counterpart here. Let F(z1, * * *, Zn) and i(z1, * * a, Zn)
both be analytic at the origin and vanish there, and let $ be
irreducible there. If F vanishes at all points in the neighborhood
of the origin at which 4 vanishes, then F is divisible by (D.
The Roots of an Analytic Function of Several Variables. In
the case of analytic functions of a single variable the roots are
isolated. This theorem appears to be lost for functions of several
variables, since such a function which vanishes at all has an
infinite number of roots clustering about any given root. The
theorem admits, nevertheless, a perfectly good generalization.