Colloquium publications.

FUNCTIONS OF SEVERAL COMPLEX VARIABLES.

193

TF1(w., *, m, ZUm+1) = ZO+l + A1 iu+l + ' + AN = 0,
(3) {Fl, = Fj(.Wi, *, Wm, Wm+l)
WmA-j Fl(Wl, ' W in, Wn- +l)' -- 72 '' ', n Zm,
Fi(w1,..., wy, wm+i)
where Ak is analytic in Wi, *, wm at the origin and vanishes
there, and Fj is analytic in wI, * *, Wm, Wm+l at the origin and
vanishes there. Fi(wz, * *, wI,, wm+l) is irreducible at the origin,
and
awm+l
To each root (z) = (0) of (1) lying in the neighborhood in
question corresponds at least one system (3) such that F[ does
not vanish in the point (w) which is the image of (z). Conversely, each system of values (w) lying in a certain neighborhood of the origin and satisfying (3) yields a root (z) of (1) lying
in the neighborhood of (z) = (0).
When the conditions of the problem are such that all n variables
Z1, *..,,n are coordinate, so that the transformation (2) is
available, this theorem yields complete and satisfactory information regarding the solution of equations (1) im Kleinen.
The proof of the theorem is direct, and is given by means of
the factor theorem and the algorithm of the greatest common
divisor.
~ 7. CONTINUATION. A GENERAL THEOREM
It may happen that the variables with respect to which it is
desired to solve may not be interchanged with the remaining
variables, so that the factor theorem is not available. In this
case the following theorem may be useful.* The proof is closely
allied to that of Weierstrass's theorem, ~ 6. The case n = 2 is
covered by a theorem of Bliss's.t
* The theorem is suggested by a theorem of Poincare's, These, 1879,
Lemma IV, p. 14. It is not clear what Poincare meant by the words: "... si
les equations p01 = 2 = * = pp = 0 restent distinctes quand on annule
tous les x...."
f Princeton Colloquium, 1909, published, 1913, p. 71.