A NOTE

ON WEIERSTRASS'

AUXILIARY

THEOREM

V. I . A r n o l l d

Let t h e r e be given an analytic mapping, y = f ( x ) , f ( o ) = o, in the neighborhood of zero in the n n n dimensional c o m p l e x s p a c e Cx to another c o m p l e x s p a c e , Cy. Then e a c h function ~0(y), analytic in the neighborhood of the point y = 0, can he c o n s i d e r e d as a function of x, ~o(f(x)). In g e n e r a l , t h e s e functions do not exhaust the ring of all functions of x which a r e analytic in the neighborhood of x = 0. The a u x i l i a r y t h e o r e m of W e i e r s t r a s s a s s e r t s that, if the m a p p i n g f at 0 h a s finite multiplicity (i.e., t h e r e a r e only a finite n u m b e r of p r o i m a g e s of points close to 0), then the r i n g of analytic functions df x f o r m s a module with a finite n u m b e r of g e n e r a t o r s on the ring of functions ~o(f(x)) (see below, T h e o r e m 1). Thus, if we denote the g e n e r a t o r s of this module by ei(x), then e a c h function, ~o(r0, p e r m i t s a nWeiers t r a s s expansion" r

(x) = y ~, (y) e, (x),

w h e r e y = f ( x ) and the ~0i(y) a r e analytic functions in the neighborhood of y = 0. More than that, this is a f r e e module, i.e., the g e n e r a t o r s ei(x) can he so chosen that the expansion is unique. T h e r e thus a p p e a r s a l i n e a r o p e r a t o r which puts into c o r r e s p o n d e n c e with e a c h function ~o(x) its collection of coefficients (~oi(y)). F o r this o p e r a t o r , we shall obtain, in t h i s n o t e , a "bound without l o s s of domain," showing that its analytic p r o p e r t i e s a r e no w o r s e than those of the o p e r a t o r of ( r - D - f o l d d i f f e r entiation. Such a bound is useful, f o r e x a m p l e , in studying the d e f o r m a t i o n s of mappings: it is e a s y to d e r i v e f r o m it the analytical stability of e a c h infinitesimally stable shoot.* § 1.

Notation Let C n = {x} be an n - d i m e n s i o n a i c o m p l e x space, x = xi, . . . , Xn, U C C~,

f:u-~c~, y =fCx). a mapping which is analytic in neighborhood V of the point x = O. Let f(O) = 0 and let 0 be an isolated point, the p r e i m a g e f - t ( O ) . By the multiplicity, r, of m a p p i n g f at O, we m e a n r = lim max r (y, v), U--,O

y

w h e r e r(y, U) is the n u m b e r of points in the p r e i m a g e , f - t ( y ) in the neighborhood U of point x = O. It is known that the limit, r, e x i s t s and is finite, w h e r e b y , f o r sufficiently s m a l l U, a l m o s t e v e r y point y s u f ficiently close to 0 has, i n U, e x a c t l y r p r e i m a g e s (see [1]). By the local ring, Q, o r m a p p i n g f at O, we m e a n

An analogous t h e o r e m of stability in the infinitely differentiable c a s e w a s announced r e c e n t l y by Dzh. M e z e r (but not yet published). Moscow State U n i v e r s i t y . T r a n s l a t e d f r o m Funktsional,nyi Anaiiz i Ego Prilozheniya, Vol. 1, No. 3, pp. 1-8, J u l y - S e p t e m b e r , 1967. Original a r t i c l e submitted J a n u a r y 25, 1967.

173

w h e r e Hx (H~J a r e r i n g s of shoots of analytic functions of x (of y) at 0, ~ y is a m a x i m a l ideal of Hy, and f * is the mapping induced by f : f * ( ~ ( y ) ) (x) = 7(f(x)). O n s o f the m o d e r n f o r m u l a t i o n s of the s o - c a l l e d a u x i l i a r y t h e o r e m of W e i e r s t r a s s is as follows: THEOREM 1. As a C - l i n e a r space, r i n g Q has dimensionality r . If ei E Hx (i = 1 . . . . . r) a r e s o m e r e p r e s e n t a t i v e s of the g e n e r a t o r s of Q as a C - l i n e a r space, then e a c h shoot, ~0(x) E Hx, is r e p r e sented, in a s i n g i s - v a l u e d m a n n e r , in the f o r m

Cx) = ~ ~ (y) e+(x),

(1)

w h e r e ~oi E Hy, and y --f(x) (i.e., ~0i(y) m e a n s ( f * ~i) (x)). The existence of Ex't~nmion (1) w a s p r o v e n in [2], its uniqueness in [3]* (see also, [4]). Expansion (1) defines the l i n e a r " W e i e r s t r a s s o p e r a t o r " IV: H,--~ (Hv)r, Example.

IV~ = (~1. . . . . ~,).

(2)

Let n -- 1, f ( x ) = x ~, e 1 = 1, e 2 = xo then

2

'

2x

"

The a i m of the p r e s e n t note is to study the analytic p r o p e r t i e s of the W e i e r s t r a s s o p e r a t o r . § 2.

Bounds

Without

Loss

of Domain

We fix m a p p i n g f and g e n e r a t o r s ei. Let V be a sufficiently s m a l l neighborhood of point y = 0 in C~, let U ~---f - l ( V ) be a connected component of the null p r e i m a g e of V in C n, and let d 0 > 0 be a sufficiently s m a l l positive n u m b e r . Then, T h e o r e m s 2 ~nd 3 following a r e valid. THEOREM 2. If ~o(x) is an Analytic function in the neighborhood of region U, then the v e c t o r - f l m c tion W~o is analyUc in the c l o s u r e of V, And

U(IV+)(~)IIv< q

I1~ (x) Ilv++

(3)

f o r any 3, 0 < 3 < 30, w h e r e the constant C l > 0 does not depend on 9, #, U o r V, but depends only on the original mapping, f , and the g e n e r a t o r s , el. H e r e , I I ~ ( z ) [ I o = s ~ l ~ ( z ) [ , [#~. . . . . , h i =

maxl~ki ,

a n d U + 3 i s a S-neighborhood of U i n t h e m e t r i c

~l = max Ixkl. Note. It is i m p o s s i b l e to d e c r e a s e the exponent r = 1 in (3); f o r example, in the c a s e n = 1 , f ( x ) = x r ; e I = 1. . . . . e r = x r - l | ~(x) = x r - t . T h e p r o o f of T h e o r e m 2 is b a s e d on the construction of a c e r t a i n r e p r e s e n t a t i o n of the r i n g of a n a l y tic functions of x in the r i n g of analytic m a t r i c e s of y. We denote by Q, the r - d i m e n s i o n a l s p a c e spanned by the ei (x):

We make c o r r e s p o n d to each function ~o(x), analytic in a neighborhood of region U, the e n d o m o r p b t s m @(y): Q ' - - Q ' , analytically depending on y and acting a s multiplication by ¢(x): (® (y) ~) (x) = ~ (x) ~ (x), if

*See [31, p. 299.

174

y = [ (x), m~ Q'.

(4)

THEOREM 3. The e n d o m o r p h i s m ~(y) : Q' --~ Q', which s a t i s f i e s Condition (4) and is continuous f o r y E V, exists, is unique, is analytic in y in the c l o s u r e of V, and a d m i t s the bound

II • (y)try < c~ tl~ (~)tl,j++

(5)

(~r--1

for any 6, 0 < 8 < 80, w h e r e the c o n s t a n t C 2 > 0 does not depend on @, 8, U o r V, but depends only o n f and the el. We note that T h e o r e m 2 is a consequence of T h e o r e m 3. Indeed, a c c o r d i n g to T h e o r e m 1, in s o m e neighborhood of the point x = 0 (it can be c o n s i d e r e d that this neighborhood contains U + 50), t h e r e e x i s t s the expansion 1 = ~.~Jf(x))et(x)

. Consider the v e c t o r , depending o n y e V, l y =

i=l

~, ~i(y)e~(x)EQ'

• If

i=1

¢(Y) lu = ~, ~i(y)et(x)EQ',then,

by virtue of (1) and (4), (W~o) (y) = (¢1 (Y) . . . . .

Cr(Y)) and, f r o m the bound

i=l

in (5) the bound of (3) follows. Thus, for the p r o o f of T h e o r e m 2, it suffices to c o n s t r u c t the e n d o m o r p h i s m @(y) s a t i s f y i n g T h e o r e m 3, which we shall do in the following sections. § 3.

The

Basis

Endomorphisms

Xi(y)

Turning now to the construction of the r e p r e s e n t a t i o n of function ¢(x) by the e n d o m o r p h i s m s ~(y) in the entire domain V = f ( U ) : the e x i s t e n c e of @(y) in s o m e neighborhood of point y = 0 f o r e a c h fixed a n a l y tic function ~o(y) follows d i r e c t l y f r o m T h e o r e m 1. In p a r t i c u l a r , f o r e a c h of the coordinate functions xi (1 -< i -< n), we define the e n d o m o r p h i s m Xi(y) : Qr --. QT of its action on b a s i s QW: r X~ (y) e~ = ~, Zi~ (Y)et,

1 ~ k ~ r,

l=1

w h e r e the coefficients ~ikl (y) a r e the s a m e as in the rn expansions of the f o r m of (1) : r

x~ek = ~ x~kt(~) e~ (x),

y = / (x),

1 < k ~< r,

1 ~< i ~< n.

l=l

Then, Xi(f(x)) a c t s in QT as multiplication by xi, i.e., (X,. q (x)) ~) (x) = x,.,~ (x),

W z ~ Q'.

(6)

In a sufficiently s m a l l neighborhood, V0, of the point y = 0, all the e n d o m o r p h i s m s Xi(y) a r e analytic and bounded:

[t Xi (y) [[Vo~ C3, § 4.

1 ~ i ~ n.

(7)

Eigenvalues

We now find the eigenvalues of the e n d o m o r p h i s m s Xi(y). We denote by U 0 the connected component of zero in the p r e i m a g e f - l ( V 0 ) . If region V 0 is sufficiently small, then a l m o s t e v e r y point y E V0 h a s exactly r p r e i m a g e s in U 0. We shall call such points y points of g e n e r a l position. Let ~l . . . . .

~r E f - l ( y } N U 0 be p r e i m a g e s of point of general position y E V0.

L E M M A 1. If e n d o m o r p h i s m @(y): Q T - . ~o' s a t i s f i e s Relationship (4), then each of its r eigenvalues equals the value of ~0 at one of the points ~i, and the c o r r e s p o n d i n g e i g e n v e c t o r is that function, 8 ~i(x), whose value at point ~i equals 1 and, at all o t h e r points ~ k equals z e r o :

(Y) Sd = ~ (~)~" 8d, 1 < i ~< r.

* 8ik=

(9)

t 1 ifk =i, 0 ffk~i.

175

P r o o f . Relationship (95 follows i m m e d i a t e l y f r o m (45 and (85. If, now, y is a point of g e n e r a l p o s i tion, then the r v e c t o r s 5~i(1 _< i ~ r5 f o r m a b a s i s in Qf, which also p r o v e s the L e m m a . COROLLARY 1: F o r .each point yfiV 0, the e n d o m o r p h i s m s Xi(y ) (1 -< i - n) c o m m u t e i n p a i r s , andthe eigenvalues of Xi(y) equal the values of the coordinates xi at the points f - i ( y ) . Indeed, in a c c o r d a n c e w i t h R e l a t i o n s h i p (6), Condition (4) is m e t (where @ : Xi, q~ = xi). F o r a point y of general position, the a s s e r t i o n follows f r o m (9). But then, b y the continuity of Xi(y), it is p r o v e n f o r a l l y 6 : V o. COROLLARY 2. E n d o m o r p h i s m s ~I(Y) and ~z(y), continuously depending o n y and satisfying Condition (4), coincide. Indeed, they coincide, a c c o r d i n g to (95, on the e v e r y w h e r e dense set of points y of general position. Thus, the uniqueness of ~ y ) is proven. In o r d e r to construct q,(y), we consider function q~and the e n d o m o r p h i s m s Xi. § 5.

Construction

o f @(y)

Let V be an a r b i t r a r i l y s m a l l neighborhood of the point y = O, so s m a l l that its connected component in the p r e i m a g e of V e n t e r s into U 0 in a 60-neighborhood. Let ¢p(x) be an analytic function in U + 5, w h e r e 0 < 6 < 50. We fix point Y0 E V. Starting out f r o m the b a s i s e n d o m o r p h i s m s Xi(y), we provide o u r s e l v e s with the e n d o m o r p h i s m @(YS: Qt--. QV in the neighborhood of point Y0 by the Cauchy integral m (u) :

(')" I ~°(x,~A'' Adx~n ~7

"~lyo)

H

(i0)

(xkE -- Xk (Y))

k=l

The contour of integration, 3' (Y0), depends both on Y0 and on 5; it will he d e s c r i b e d in §6. We so choose it that 6(y5 will act on Q ' a s multiplication by q~: Relationship (95 will hold for each point of general position y E V. F o r the e n d o m o r p h i s m of (10), Relationship (95 is p r o v e n in §8, while the bound of (5) is obtainedin §7. § 6.

Choice

of the

Contour

of IntesrationT

We begin with c e r t a i n notation r e l a t e d to the Cauchy integral. We call the c r o s s , K~, of a point ~ ~ Cn, the union of all the coordinate planes p a s s i n g through ~: K~=

x:

(xk--h)=0 . R-~--I

We call a p o l y c y l i n d e r in C n a d i r e c t product of c i r c l e s , o~= {x:Jx~--x°kl~l~k,

k=l .....

n}.

0 We call the hull of a polycylinder the d i r e c t product of neighborhoods T = {x:[ x~ - - X k! = Rk, k = 1, . . o r i e n t e d in the usual fashion.

•

n},

Let g(x) be an analytic function in polycylinder w. If the c r o s s of point ~ does not i n t e r s e c t hull 7 of polycylinder ~, we then define the Cauchy i n t e g r a l 6(~) =/_..t_t ~nC ~(x)dx, A . . . A dxn

k~tJ~

n

H (xk-- ~) ~ml

With this, ff ~ lies within ¢o, then G(~) = g(~), whfl'e i f ~ i s o u t s i d e ¢o, then G(~) = 0. LEMMA 2. Let ~I . . . . , ~r b e points in C n (not n e c e s s a r i l y different), and let r > 0. T h e r e then e x i s t p < r p a i r w i s e n o n - i n t e r s e c t i n g polycylinders wJ(1 -< j ~ < p) such that:

1) each point ~ l i e s in one and only one of the polycylinders wJ;

176

(11)

2) the hull of each of the p o l y c y l i n d e r s ¢oJ is at a distance of not l e s s than r f r o m e a c h of the r c r o s s e s K~i; 3) the radius of none of the p o l y c y l i n d e r s ¢0J e x c e e d s r r . The proof of L e m m a 2 is given in §9. T u r n i n g now to the c o n s t r u c t i o n of the contour 9 f i n t e g r a t i o n in Cauchy I n t e g r a l (10), we apply L e m m a 2 to the r (not n e c e s s a r i l y different) p r e i m a g e s ~ E f-t(y0) [-1 U of point Y0 IE V, setting r = 6 / 2 r . Then, all the polycylinders 60J lie in U + 6. We f o r m the contour of integration, 2/(Y0), in (10) f r o m the hulls of the polycylinders 60J(1-< j - p). § 7.

Bound

on the

Cauchy

Integral

The bound on I n t e g r a l (10) is b a s e d on the following L e m m a , p r o v e n in §10. LEMMA 3. Let Ak: C~--~ £~ (/e = 1. . . . . n) be c o m m u t i n g e n d o m o r p h i s m s of C r . If [IAkl[ --- a and all eigenvalues of all Ak a r e not l e s s than ~ in modulus, then kI~__A~- 1 % C,e -(~+~-t~, where Ca (a. r. n) = r (ran)~-' (the n o r m is Hermitian). Let y E V be a point of V sufficiently close to Y0. We denote its p r e i m a g e s (not n e c e s s a r i l y d i f f e r ent) by ~J E f - i ( y ) [q U(1 --- j -< r). Let x = (x 1. . . . . xn) E 2/. We set Ak = x k E - X k ( Y ) . It follows f r o m (7) that [IAkll ___2C 3. According to §4, the e n d o m o r p h i s m s Ak c o m m u t e , and t h e i r eigenvalues a r e Xl~Xk(~J) (1 -< j - r). According to the choice of 2/, and by virtue of a s s e r t i o n 2) of L e m m a 2, IXk-Xk(~)[ - T SO that, for a point y sufficiently close to Y0, we have

I xk -- x~ (~i) I > 8 = x/2 = 8/4r. Applying L e m m a 3 to the Ak, we obtain

i[ ~k~__(xkg-- X, (y))-~ [l ~ C66-(~+~-',, w h e r e C 5 = C4(2C 3, r, n) (5r~a+r-1. It follows f r o m this that Integral (10) e x i s t s , is analytic with r e s p e c t to y, and s a t i s f i e s the bound

II (P (y) II < c~8-~-'~ II q~(x) IIv+o,

(12)

w h e r e C e = Csr because, a c c o r d i n g to a s s e r t i o n 3) of L e m m a 2, the m e a s u r e of 2/ does not exceed r(2rS) n. § 8.

Proof

of Theorem

3

Let y be a point of g e n e r a l position, sufficiently close to Y0. We shall p r o v e Relationship (9), We apply 4S(y) of (10) to v e c t o r 8 of (8). Since, a c c o r d i n g to § 4, Xk(y)6~j = ~ 8 ~ j , then, using Cauchy F o r m u l a (11), we find that

~P(v) 8~ = ~,, ~

J

~-

8~,, = ~,, ~(~)svx, (~),

w h e r e 2/1 is the hull of polycylinder 60/, and XI is the c h a r a c t e r i s t i c function of coI. In a c c o r d a n c e with the choice of contour 2/, and by v i r t u e of a s s e r t i o n 1) of L e m m a 2, each point ~J P

lies in exactly one of the polycylinders w l . T h e r e f o r e , ~ ~ (~J) ~ 1, and Relationship (9) is proven. It was shown in §5 that (4) follows f r o m Relationship (9). Since the bound of (12) h a s the f o r m of (5), T h e o r e m 3 is proven.

177

§ 9.

Proof

of

Lemma

2

Initially, let n = 1. We call a mapping, ~, of a finite set, {~ J~, of points of C 1 into the set of c i r c l e s in C l a ~- - a d m i s s i b l e s y s t e m o f c i r c l e s if 1) the point ~i lies in a r - n e i g h b o r h o o d of c i r c l e ~(~i) ; 2) the r a d i u s of a(~i) does not e x c e e d v r , w h e r e v is the n u m b e r of points ~J f o r which a(~i) = a(~j). F o r example, the m a p p i n g ~0, m a k i n g c o r r e s p o n d to e a c h point ~i, 1 -< i ~ r the c l o s u r e of its 1neighborhood, is a I - - a d m i s s i b l e s y s t e m of r c i r c l e s . If the c i r c l e s of a T - a d m i s s i b l e s y s t e m a r e p a i r w i s e n o n - i n t e r s e c t i n g , the L e m m a is then p r o v e n . If, now, two c i r c l e s , oJ1 and co2, of s y s t e m ~ i n t e r s e c t , then the n u m b e r of c i r c l e s c a n be d e c r e a s e d , r e p l a c i n g off and ~2 by the c i r c u m s c r i b e d c i r c l e o:, i.e., s e t t i n g

It is e a s i l y v e r i f i e d that the s y s t e m ~' thus obtained will be ~- - a d m i s s i b l e s y s t e m c o n s i s t i n g of the l e a s t n u m b e r of c i r c l e s , the c i r c l e s do not i n t e r s e c t and, f o r n = 1, L e m m a 2 is p r o v e n . Now, let n > 1. F o r given i (1 ~ i - n), we c o n s i d e r the iVth c o o r d i n a t e of our r points ~ . . . . ~ C 1. Since the L e m m a is p r o v e n f o r n = 1, t h e r e e x i s t s a ~--admissible s y s t e m of p(i) c i r c l e s f~i(~) (1 ~ j -< r), p a i r w i s e n o n - i n t e r s e c t i n g . We c o v e r e a c h point ~J E C n b y the p o l y c y l i n d e r c o r r e s p o n d i n g to the c i r c l e c o v e r i n g its p r o j e c t i o n :

T h e s e p o l y c y l i n d e r s s a t i s f y all the r e q u i r e m e n t s of L e m m a 2 ( r e q u i r e m e n t 2) is s a t i s f i e d b e c a u s e • f o r any i, e a c h of the p(i) n e i g h b o r h o o d s a 9,i(~i) is at a d i s t a n c e no l e s s than ~ f r o m e a c h of the r points ~ E c i ) . L e m m a 2 is p r o v e n .

w h e r e A is a nilpotent u p p e r - t r i a n g u l a r m a t r i x all of w h o s e r ( r - 1 ) / 2 e l e m e n t s equal ha." P r o o f . We expand T into a diagonal and, a s u p e r d i a g o n a l p a r t : T = A + A = A(E + A - t ~ .

Then,

P r o o f of L e m m a 3. Since the Ak c o m m u t e , t h e i r m a t r i c e s have the t r i a n g u l a r f o r m s Tk in one and the s a m e H e r m i t i a n o r t h o n o r m a t b a s i s . With this, e a c h e l e m e n t of m a t r i x Tk is no l a r g e r than "a, w and the d i a g o n a l e l e m e n t s a r e no s m a l l e r than ~. By L e m m a 4, the m a t r i c e s T~ 1 have a c o m m o n m a j o r a n t , given in (13). Since A r = 0,

LITERATURE 1o

2. 3. 4.

CITED

M. Erve, Functions of Many Complex Variables [in Russian], Moscow, "Mir" (1965). S~minaire H. Cartan, Paris, No. 18 (1960/1961). G. Grauert, Complex Spaces [in Russian], Moscow, "Mir" (1965). V. P. Palamodov, "On the multiplicity of holomorphic mappings," Funkts, Analiz, 1_, No. 3, 54-65 (1967)

179