A PROOF OF THE ~K~LGRAN,GE, ~ P ~ T I O N THEOREM

L. Nirenb erg

Malgrange's preparation theorem [3], [4] is an extension to real C ~ functions

of the Weirstrass preparation theorem. In this paper we give a short, rather

direct proof of the theorem which applies also to complex functions: t

Theorem 1. Let f(t, x) be a C ~ complex-valued function defined in a

nei~hourhood of the ori6in in R n+l, here t E R I, x E R n, and let p > 0 be the

first integer such that

(~)P f(O, 0) / 0 . (i)

Then in a nei~hbourhood of the ori6in one has the fact orization

(2)

where

(3)

and q

If f

f(t,x) : Q(t, x)P ,

P P = t p + ~ kj(x)t p-j

i

and ~j are ~ complex-valued functions with Q(0, 0) / 0.

is real there is such a factorization with Q and P real .

This paper is a slightly expanded version of a letter written to B. Ma!grange in

November 1969. The work was supported by the U.S. Air Force Contract AF-49(638), 1719. Reproduction in whole or in part is permitted for any purpose of the U.S. government.

~Nalgrange informed me that the complex case can be derived from the results in [4]. In addition, G.I. Eskin, using the results of [4], observed independently that for each positive integer k the complex function f admits the factori~ation (2) with Q and ~i of class C k . See a&so the papers in this volume by S. Lojasiewicz ~and G. Glaeser.

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af (0, O) # 0 then one For p = i the theorem states that if f(O, O) = 0 ,

has the C °o factorization

(4) f(t, x) = Q(t, x)(t + kl(X)) , Q(0, 0) # 0 .

If f is real this follows easily from the implicit function theorem, and the

factorization is then unique. However for complex f this simple looking result

seems to be nontrivial. Furthermore, the factorization is not unique, for the

function f t + i(x + e -l/x2 = ) , here n = 1 , admits the factorization f = 1 .f

and f = Q(t, x)(t + ix) - one easily sees that Q belongs to C ~ , and

Q(o, o) : i .

The preparation theorem is based on the Maigrange division theorem which we

state for complex functions :

C ~ , ,,,,,, ...... ,,,,,,, ,, Theorem 2. ~ Let f(t, x) be a complex-valued function in a nei~hbour-

hood of the orisin in R n+l and let

tP-J x : (h ..... ~) (5) P(t, A) = t p + Aj ,

1

be a generic monic polynomial in t with constant complex coefficients

in a nei~hbourhood of the origin one has division by P(t, %) :

(6) f(t, x) : q(t, x, ~) P(t, %) + r(t, x, %) ,

%. . Then

where q and r are complex-vaiued C ~ functions in a n eighbourhood of the origin

in R n+l × ~P , an d r is a polynomial in t of desTee less than p . A~ain if

f and the ~. are real one can obtain this with q and r real. J

tThe functions q and r are not unique. In the real case a stronger form of Theorem 2 was proved by J. Mather [5]. He showed that for every f one can find q and r in (6) in such a way as to depend linearly on f . In addition he established a global form of the theorem. Our proof does not yield q and r depending linearly on f . In a paper in this volume Mather shows how to modify

our proof (in particular he presents an improved version of our Lemma l) so as to obtain q and r depending linearly on f .

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Proof of Theorem ! : This follows [3] • Let

consider the division (6) by the generic polynomial

remainder r has the form

r = rj(x, A) t p-j

1

We shall solve the p complex equations

(7) rj(x,x) = o, j =l ..... p

for k = k(x) , regarding this system as 2p real equations for

Re lj , Im Aj , j = 1 ,..., p .

f(t, x) =

f be as in Theorem I and

P( t , ~) of (5) . ~he

2p real unknown

We then obtain the desired factorization

q(t , x, X(x)) P( t , X(x)) .

From the definition of p we see that rj(O, 0) = O, j = 1,..., p and

q(O,0,0) / 0 hence to solve (7) it suffices to show that the Jacobian matrix

a(Re r j , Im rj) is nonsingular at x = O , ~ = 0 . In fact we shall show that

a(~e ~k' ~m ~k) Or. Or.

the (complex) p xp matrix --/ (0, O) is nonsingular while 0 (0,0) = 0 ,

from which the desired result follows easily.

a 1 (__a a a~ 2 O~k+ i~vk ) , for ~ = ~k + irk .

Here a~ = 1 (10 . i a 2 % )'

Applying a/a~ to (6) we find

@r~ (0,0) t p-j = - t p 8

j = l aXk aik

for all t , from which it follows that

(6) we find

q(t, 0, O) , k = i,..., p ,

~r. (0, O)= 0 . Applying a//O~ to

q(t, O, 0)t p-k + t p 0 ~ ar.

OZ]< q(t , O, O) + a2~ (0,0) t p-j = 0 j = l

k = i,..., p .

Suppose that for some complex vector ~ = (~i'''" ~p) '

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at. c00) k

k

= 0 , j = i,..., p ;

then we find

q(t,o, o) ~t p-k ~k:°(tP)' as t*O. 1

Since q(0,O,O) ~ 0 this implies that ~ tP-k ~k ~ 0 for all

1

~ 0 . Hence arj/0~ k (O, O) is nonsingular. Q.e.d.

t which means

Our proof of Theorem 2 is an adaptation of one of the well known proofs of

the Weierstrass Division Theorem (Theorem 2 in the analytic case), which we repeat

here. If f is analytic we have, by Cauchy's integral theorem

(8) f ( t , x ) = 2~il ~ f(~,x),~_t d~ aD

where D is a disc about the origin in the complex z plane containing t , and

all the roots of P(z, ~t) for [%[ small, in its interior. The function

P(~, x) - P(t, x) R(t, z, ~) =

z - t

is analytic, and a polynomial in t and z of degree less than p . Replacing

1/(z - t) in the proceding integral by

(9) l P( t , X) R(t, z, X) z-t = (z-t) P(z,A) + P(z,k) '

we find

f(t,x) = P(t,~) i f(z,x) l f (s- t )P(z ,X) ds + 2~i f R(t,

aD aD

z, ~) dz .

This is the desired form (6) with

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q (, - t) P(~, ~)

aD aD

This proof was also used by Mather in [5] •

We shall make use of the following extension lemma.

Lemma i. t Let f(t, x) beta C °o complex-valued function in a nei~hbourhood

• C °° , of the origin in R n+l There exists a complex function F(z, x, %) defined

in a nei~hbourhood of the origi n for z ~ ~I, x e R n ~P . , ~ E , satisfying

(i) F(t, x, %) = f(t, x) for t real

(ii) F z vanishes of infinite order on Im z = 0 and on P(z,2~) = 0.

Proof of Theorem 2 : We wish to apply the analogue of (8) to our extension

F . This is the well known generalization of the Cauchy integral theorem for a

smooth , but non-holomorphic function of a complexvariable z (see for instance

Theorem 1.2.1. in [i] ):

f(t,x):F(t,x,~) : 1 ~ F<z,x,~l dz+ 1 ~j dz dz ( ) ^ 2~i z-t ~ F z z,x,% z - t

aD D

where the area of integration is a disc D about the origin containing in its

interior the point t and all roots z of P(z, ~) for IA I small. Substituting

(9) for 1/(z- t) in these integrals we obtain (6) with :

1 r F(z,x,%) 1 r r F~(z,x,A) dz + dz dz q 2~i J (~-t~P(z,~) ~ JJ (z-t~P(z,~ ^

~D, D

r - 2~i f F(z,x,%) P(z,k) dz + ~'~ If Fz (z,x,%) R(t.z,2~)p(z,;t) dz ^ d~

~D D

In his paper in this issue Mather proves a global version of this and obtains F depending linearly on f ; the F constructed here depends on f in a nonlinear way.

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The denominators in the double integrals are zero when z - t or P(z, ~) vanishes

- but the numerators vanish of infinite order there, by Lemma 1. Hence those

integrals converge absolutely. Furthermore one may differentiate under the integral

signs with respect to t, x, and k ; the resulting integrals are all absolutely

convergent. Thus q and r are of class C ~ and Theorem 2 is proved for the

complex case. In case f is real we may use ~F(z,x,X) + F(~,x,k)) in place of

F in the preceding; it is then readily verified that q and r are real for

real.

We remark that H~rmander [2], in quite a different connection, has used the

device of extending smooth functions of real variables to complex variables as

functions satisfying the Cauchy-Riemann equations to infinite order on the real

subvariety.

In proving Lemaa 1 we shall make use of a special, simple case of the Whitney

extension theorem for which we include a proof :

Lemma 2. Let u(y), v(y) be C ~ complex-valued functions in a nei~hbourhood

of the origin in R N

t~J)

D°(u-v)

Then there is a C ~

D~(F - u) : o

~=(F - v) : o

sat%s.f~in g (here D ~ represents any derivative of order

= 0 o~n Yl = "'" = Yk = 0 for all

function F(y) satisfying

on Yl = "'" = Yj = 0 forall ~ ,

o_n Yj+l .... = Yk = 0 for all

proof : The function

where the sum is over

where ~(s) is a

for s > l, and where

The choice of the t. J

(z .

D~(u-v)(O ..... O,yj+I,---,Y N) • ¢(ti~ I

J

1

: (~i''''' ~j' 0,..., 0) , has the desired property,

function on s ~> 0 which is i on 0 ~ s ~< 1/2 and zero

t..~ co is a rapidly increasing sequence tending to infinity.

depends on the functions u and v , and the resulting F

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therefore does not@epend linearly on u, v .

Proof of Lemma 1 :

it has been proved for

variable in (z, ~) space to straighten out the variety P(z, l) =

l' = (h i ..... kp_ I) alone and introduce a new variable

= P(z, ~) .

Then in (z, l' , N) space the variety P = 0 becomes

@/aT now becomes , in the new coordinates,

(io) L a + ~ _t a~ a~

P' represents the derivative of P where

independent of A P

We shall construct two

Induction on p ; Lemma I is true for p = 0 . Assuming

p-i we wish to prove it for p . First we make a change of

0 . Leave z and

to replace ;t : P

= 0 , and the operator

with respect to z - note that P' is

C ~ functions v(z, x, A') and u(z, x, A', ~) such

that

(ii)

(12)

and

(i3)

(l~)

(15)

v(t, X, ~') = f(t, X) for t real ,'

V vanishes of infinite order on Im z = 0 Z

u=v on fl = 0

Lu vanishes of infinite order on ~ = 0

D~(u - v) = 0 on Im z = Re ~ = Im ~ = 0 for all ~ .

Using Lemma 2 we can then put these functions together to obtain the desired

function F(z, x, ~) .

The induction hypothesis is used to construct v , namely v(z, x, ~')

C ~ function satisfying (ll), (12) and a/so satisfying :

(16) v_ vanishes of infinite order on Z

P,(z,~'):p2 -l+(p-l)~s p-2+...÷~p_l = o.

is a

Then we set

- l O ~ -

l a )J v(z,X') ~ ¢(1~12 tj) (- P~ a~

u

0

where

for

Next

t./~ oo is rapidly increasing (depending on the function v) . J

Because v vanishes of infinite order on the set where P' = 0 one sees that Z

tj rapidly increasing the function u belongs to C °O and satisfies (13).

Lu = ~ ( 1 a + !)u

p--r a~ a~

= p' ( 1 a )j+l v(., x,) £ [~(1.t 2 tj÷ l) - ¢(l~12tj)] P-~ a~ J~ 0

_l ~ )J v(z, k') ~j ~'(i~i2tj + P' ( - p-T a~ j'T t j . ) ;

0

and one sees that Lu vanishes of infinite order on

to verify (15) we see from (12) that

u - v (s , x ' ) ~(I.12to )

= 0 , i.e. (14) holds. Final]z,

vanishes of infinite order on Imz : 0 . Since v(z,A')(~(I~I2t0' ' -

infinite order on ~ = O , w@ infer that (15) holds.

!) vanishes of

REFERZNC E S

[1] L. H~rmander,

[2] L. H~rmander,

[3] B. Malgrange,

[4] B. ~algrange,

An Introduction to Complex Analysis in Several Variables.

D. Van Nostrand Co., Inc., Princeton, 1966

On the singularities of solutions of partial differenti-

-al equations. Conf. on Functional Analysis and Related

Topics, Tokyo, 1969, 31-40.

The preparation theorem for differentiable functions.

Differential Analysis, Oxford Univ. Press, 1964, 203-208.

Ideals of differentiable functions, Oxford Univ.Press,

1966.

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[5] J.N. Nather, Stability of C °o mappings : I. The division theory,

Annals of Math. 87, 1968, 89-104.