J . Math. Kyoto Univ. (J MKYAZ)16-3 (1976) 683-696

Zero sets of certain ideals ofdifferentiable functions

By

Shuzo lzumi

(Communicated by Prof. Mizohata, Aug. 20, 1975;Received Feb. 19, 1976)

Introduction

T h is is an a ttem pt to define a nice family o f closed subsets on am anifold using a Cs°-structure. Thom has proved that the zero set ofa Lojasiewicz id ea l is th e c lo su re o f a m a n if o ld . Bochnak has ob-ta ined som e resu lts o n ideals associated to closed subsets assum ingth a t t h e y a r e finitely generated ( t h e n t h e y a r e Lojasiewicz ideals).B u t it seem s that there is little know n about dimensions of zero sets.I f a germ X„, o f a closed subset is real analytic, its geometric dimensioncoincides with the algebraic one(t) (defined by th e form al ideal o f X.).W e define a fam ily O . o f closed subsets o f R " w hich h av e similarstructure to re a l a n a ly t ic s e t s a n d w h o se tw o k in d s of dimensionscoincide at a. O a is verified to be a generalization o f coherent analyticsets. B u t its elements d o not always adm it a stratification.

Terminology and notation

S ( U ) ; the ring of C ° functions on an open set U cR".

S nu= e a ; the ring of germs o f Cc° functions at a e R".i n ; the maximal ideal of

The author w ishes to express thanks to Professors: S . N akano, M . Shiota andM . Adachi for helpful discussions and encouragement.

( t ) This can be verified by the last argument in [7].

684 S huz o /z um i

= a; th e r in g o f form al pow er series in th e n coordinate vari-ables centered at a.n ; the maximal ideal o f ..Fa.

: a F a (Ta: S (U)— +g-a); the formal Taylor expansion a t a.f E S (U ) is f lat o n A ; T x f =0 for any x e A.A closed subset A c X is a z ero of inf inite order o f a real valued func-tion f defined o n X c 12''; Ix e I f (x )1 . d(x , A)k} a re neighbourhoodsof A in X for any k = 1, 2,....X a ; the germ o f a closed subset X c U a t a.J(X , U ) (the ideal o f X ) ; { f e e (U ): f= 0 on X ) .Ja(X )=J(X „) (the ideal of X a (of X a t a)); { f E f= 0 o n Xa}

K a(X )=K (X a) (the f orm al ideal of X a ) ; { T ie fe Ja(X)} •J'a(X )=J'(X a) (the w eak ideal o f X a} : { f e S a: a is a z e ro o f infiniteorder of the restriction f I X} .

1<:,(X )= K '(X ) (the w eak form al ideal of X a) ; { T a f .9 7 f e JJX ) }

Z (f ,, ; the intersection of the zero sets of fp.

Dimensions of zero sets

L e t Xa b e th e germ o f a closed subset X c R " a t a. We defineth e geom etric dim ension o f X a by dim Xa = m in dim (X n U ), where

aeU:opendim denotes th e inductive ( = covering) d im ension . T he algebraic di-m ension adim X a is d e f in e d b y t h e Krull d im e n sio n o f .FalKa(X ),

i.e. t h e largest number k fo r which there exists a sequence (K (X )c :)

= (the maximal ideal) of prime ideals.

1. Theorem. dim X a a n d adim X „ are upper sem icontinuous withrespect to a and dim X 0 ad im X 0 .

Pro o f . T h e semicontinuity o f d im is o b v io u s . A s in th e analyticc a se , th e re e x is ts a n affine coord inate system (x , y )=(x 1, x 2 ,. . . , x,„

Y i' y2, ••• , Y i) ( k + 1 = n ) centered at a su ch th a t th e canonical map—>.F„/Toa (a =./a(X ) ) is in je c tiv e a n d finite, w h e re k deno tes theform al power series i n x1, x2,..., x k (cf. [4; p. 49]). Then th e m ap Sk

—>d'a/a is a lso fin ite by Malgrange's preparation theorem . H ence d'n/a

Ideals o f differentiable functions 685

is in te g r a l o v e r d'k/Sk n a ; th e r e e x is t au(x) e e , s u c h th a t f i(x , y)

y 'p + E o-ii(x)yi; e a. T h e n t h e r e e x i s t s a product neighbourhoodi=o

U = Vx W ( VcRk, WŒR"-k) o f a su ch th a t th e representatives f i(x , y)yip + Ea ii(x)yi; b e lo n g to J(X , U). Since 9.„,/(T, f,,..., T b f i ) is finite

over "-kb its K rull dimension is le ss than k b y th e theorem of Cohena n d Seidenberg. Hence, adim X b k and ad im is upper semicontinuous.L et n: 12" —>ilk denotes the projection (x, y)—>(x). I f W is chosen suffi-ciently sm all the re str ic tion irk- ' (U) n X is finite-to-one and proper.T h e n b y dimension theory dim X dim U= k (c f . [5 ; p. 63]). q. e. d.

2. C oro lla ry . Suppose th a t adim X a= k , K a (X ) is p r im e a n d X

contains Y with dim Y = k . I f fe d'a vanishes o n "a, Taf E K„(X).

T h is i s a n a n a lo g u e o f th e theorem o f identity in t h e complexfunction theory.

T o ob ta in the opposite estimate fo r dim X„ we assume th a t J(X „)

is finitely generated. L e t g ( ) denotes th e number o f th e elements ofa m in im al basis o f a n ideal, which is uniquely determ ined f o r quasi-local ring (cf. [6]).

3. P roposition . Let X cU b e c lo s e d a n d .1 a(X ) be finitely gener-

ated. Then we have:

( i) g ( f(X ) ) is upper senticontinuous with respect to x e U.

(ii) d im X „ n — g(J(X„)).

(iii) dim, .1(X )linf(X a)= 9(.1(X,))= g(K(X „))= dim, K(X. „)IIIK(X a).

Proof. ( i ) Immediate from (17).(ii) X is the closure of a m anifold Y b y (1 8 ) . If xe Y n V (V in (17)),J ( X ) includes the equation of Y . H e n c e

dim X a dim Yx_„n — g(J(X.)).

(iii) The second equality follows from (19). The others are well known(cf. [6; p. 13]). q. e. d.

The equality does not always hold in (ii) (cf. (12)).

686 Shuzo lzum i

The main theorem

I n th is section w e give a family 012 o f closed se ts which have asimilar property to irreducible real analytic sets.

4. Definition. L e t a be a point of 12". cbg(R ") denotes the fam ilyo f all c lo se d su b se ts X o f R " w hich satisf y the following:

(a) J ( X ) is f initely generated.(b) K (X ) is prime.

5. Theorem. If X e 08(R n) and if adimXo=k, then f o r any givenneighbourhood U0 of 0 there ex ist a linear subspace R ' R ", a productneighbourhood U V x (V c R" - k) an d 3 e (R n) satisfying the following:

( i ) K ( X n E) n 0 where E={ x e U: 3(x)= 0}(ii) T he natural projection 7E : X n U - ±11" is proper. The number

of points of a point inv erse of it i f uniform ly bounded on V .(iii) Z X n U n Ec is a nonem pty k -dim ensional m anif old and

Z ut i s a local dif feomorphism .( iv ) adim (X n E ) 0 k — 1.W e prove this in a m ore detailed form:

5'. Lem m a. L et X e 08(R n) and adim X o = k . Then f o r an y openneighbourhood U0 of 0 there ex ist the following:

(1) af f ine coordinates (x , y , z )=(x ,,..., x k , y , z ,,..., z 1) centered at0 (k +1 +I=n );

(2) a product neighbourhood U = V x W U 0 (V cR , W c R 1 + 1 );P - 1

(3) f (y ; x , yP + E crp_,(x, z)yi eJ(X , U) w ith discrim inant 5(x , z )i=o

such t h a t T oo-p_ iE .F",, an d T o f i s p rim e inP - 1

(4) g i(x, y, z)- (5(x, z)z; — E T'

i p i( x , z)y, E J(X , U ) (j=1,..., 1) suchi=o

that Toti„,_,E.Fk;(5) cp(x, y , z )e S (U) such that T0p=0;(6) a natural num ber N.

W ith that if w e put

Ideals o f differentiable functions 687

E={(x, y , z ) U: (SN —(p)(0(f, g , . . . , g ,)1a(y, z))=0} ,

Y= {(x, y, z) e U: f=g1=•••=g,=0},

we have the fo llow ing:

(7) The canonical p ro jec tion IT: X — >Rk is fin ite -to -one and p rope r;

(8) (X n U) n Ec is a k-dimensional m an ifo ld w ith nonempty germ

a t 0 a n d th e re s tr ic t io n rci(X n U)n E , is a lo c a l diffeomorphism into

Rk;

(9) adim ((X n u) n E )0 k-1;

(10) (5(x, z) 4 ( X n E).

P ro o f. L et's p u t a =Jo(X ), 91= K o (X ) . B y th e sam e argument asin analytic case (c f . [4 ] ), w e can choose affine coordinates (x, y, z)=

(x1,..., xk, y , z,,..., z1) o f R" (k +1 +1= n ) such that:

( i ) The canonical map ,Fk--■,F„/V1 is injective and finite.

(ii) (,-„/% )- is generated by th e class y o f y over k. Here k

denotes th e form al pow er series r in g in (x1,..., xk) a n d - denotes thequotient field.

P-L e t F'(t; x).--111+

1E s i ( x ) t i ( s , ( x ) e k, S AO) = 0 ) be the minimali=o

polynomial o f y over Tk. Then there exists

P-'(y; x,

1E 0-;_i(x)yi +0(x, y , z) E ai=o

s u c h th a t Toc_,(x)= S"p-i(x), T00(x, y, z ) = 0 . B y Malgrange's prepara-tion theorem we may assume:

P- 1(iii) f(y; x, z)-yP+ Ea _1(x, z)yi e a

i=o

su c h th a t Toup_,(x, z)= Sp _1(x), S , ( 0 ) = 0 . N o t e th a t z does no t ap-pear in Too- ,(x, z) because o f th e uniqueness o f th e remainder in theform al p repara tion theorem . (T here ex ists even a monic f(y; x)E aw ith coefficients in S k (c f . th e p ro o f o f (1)). B u t it is n o t a lw a y sformally m inim al.) Obviously

(iii)' F(t; x).; fP + E S p _ i(X ) i i

688 Shuzo lzurni

is th e minimal polynomial of 37. L et S(x, z) a n d t l(x ) b e the discrimi-nan ts o f f (y ; x , z ) a n d F (t ; x ) respectively. Since <?.k is characteristic0, T06(x, 4 ( x ) 0 O. I t i s a l s o the same with analytic case that:

P - IGi(x, y , z)=- .4(x)zi— E Ti _.(x)yi e‘21 (j= 1 ,..., I) •

i=o

Ti,p-i(X)Gg-k. Then there exist

g;(x, y , z)-a S(x, +0i(x, y , z)e

such that Tofi,p_i(x)— T003(x, y , z)= O. B y (iii) and the prepara-tion theorem there exist

P - 1(iv)' gi(x, y , z) S (x , z )z i— i; p i(x, z)yi e a ( j= 1 ,. . . , I)

w ith Togi(x, y , z)=G i(x , y , z ) a n d To-ci,_i(x, z)= 7 , , _ ( x ) . A s in th eproof of (1) there exist

(y) hi(x, zip + E n a ,i=o

Pi - 1(v)' II Z IP E S

'j p X ) Z E 9f

i=o

w ith Tohi(x, z)= z), Tocr _i(x )= S j,p_i(x ). I f w e p u t a=(ei,..., em)a n d b= (f, g 1 ..... g1), w e have o p b . B y th e preparation theorem, thereexist cr, j(x, y , z )e g „ and p ,111 11(x , y ) e + 1 such that

eo(x, y , z )= E1

1 ami(x, y , z)hi(x, z)=

y)zil•••zji .i l < P i ...... i i < P i

If we pu t L = max { ( p i - 1 ) , p i } , we haveJ=

P -Z ) - I E z)yi e h,

i=o

P - 1

( 5 L ( X , z ) E E Pi(x, z).Yle h1=0

(iv)

fo r some

Ideals of dif f erentiable functions 689

Top'ji(x, z) E Top; i(x, z) E ,F k

b y (iii) and (iv). Since h i a n d Epo,...iszi, ••zji a r e contained in a ,Top"ii=-Topp." i = 0 by (iii . Thus there exist

(vi) tip(x, y, (51-(x, z)e,i(x, y, z)— p,,(x, y, z) a b

where p,1 is flat a t 0.N o w , le t U = V x W (V cR , Wc121+') b e a product neighbourhood

o f 0 s u c h th a t a , e o a n d pi, h a v e representatives inS(U). Hence f(y; x, z), 6(x, z), gi(x, z), hi(x, y, z ) a n d u,,(x, y, z ) aredefined naturally a s elements o f g ( U ) . Again we p u t a = (e ,(x , y, z),...,e,„(x, y, z )), b -=(f(y; x, z), g1(x, y, z),..., g i(x, y, z ) ) a s idea ls o f g( U).N arrow ing U i f necessary , w e m ay assum e that o b , u ,1e 6 a n d a= J (X , U ) by (17). W e put

Y= {(x, y, z )e U : f(y ; x, z)-=gi(x, y, z)= • •• =gi(x, y, z )=0} .

I t i s o b v io u s th a t X n LI= Y. B y (20 ) th e re e x is t tii(x, y, z) a g(U ),if,(x, y, z ) ( U ) su c h th a t po=/3,1(x, y, z) • (11(x, y, z), = 0 a n d 1/f(x,y, z )> 0 i f (x, y, z)0 (0, 0, 0). Since p i, vanishes o n X n U b y th e factu,,eb, we have flu E a excepting the trivial case (0, 0, 0) is isolated in X .Hence there exist f3,,,,(x, y, z )e S (U ) such that

SLe„—tk vitiq,ve,=0 (i1=1,..., in)

o n Y. S in c e th is sy s te m o f linear equations ( fo r fixed (x, y, z ) ) hasnontrivial solution on Y n Xc, w e have

det [SL(x, z)I„,—tli(x, y, z) 013,,„(x, y, z )))]= 6 L '" (x , z )-0 (x , y, z)I3(x,y, z )=0

o n Y n 'c c , where denotes th e u n it m atrix a n d fl e S (U ) . W e put

30c, y, z)---( m—tPig)•(aU, g g1)/(y , z )),

E = {(x, y, z) E u : =0 } .

T h e n Y n X c = E and Y n E c is a k-dimensional manifold a n d it: Y n Ec

690 Shuzo lzum i

--÷11k is a local di ffeomorphism. Since

(X n U) n Ecc Y n E c c Y n E cn (Y n xc)----(xn u) n Ec,

we have Y n =(X n U) n E c . It is easy to see that

O(f, 91, 91)/a(Y , z)

= (61+ A)(a.f lay)— (6`-'6.i.r+Aft)(agilay)(af lazy),i,F=1

where A, A ir eS (U ) a re fla t a t 0 . T h e n , if U is sufficiently small, wehave

or 1" ..1,11•1611/2 16" (51f + • lagi/aYI • M• max leudazilJ,F =

f o r any (x , y , z )e Y satisfying 0(f, g1,..., gt)10(y, z)= 0 b y (2 2 ) . S in c eao-dazi and are fla t a t 0 , 0 is a ze ro o f infinite order o f the restric-

t io n Si Y n E . Since i s p r im e a n d F is m in im a l T0 =4 L m + 1 ( 8 F 1

at)It= y 21. T h i s im p lie s th a t 3 a . T h e n X n Un EcO 4) and adim (xn. 0 n E)o k — 1 . y a n d z ,,..., z i satisfy monic equations over d"

a s ( v ) ) . Hence i f V is sufficiently small ir is fin ite to one and proper.q.e. d.

6. Corollary. In (5) w e have

dim (X n U)= ditn (X n U )0 = k , dim (x nu n k — 1 ,

x n u n E { (x , y, z ): 13(x, y, z)151//(x, y, z)}

f o r some 1,1r E S ( U ) with Totlf =O.

Pro o f . T he first and the second fo llow from (1 ). T h e last asser-tion follows from (21).

Reducible case and analytic case

L e t a b e a n idea l o f E a su c h th a t 7 (a )= ,/7 (a ) ( r a d ic a l) . Wec a l l i t uniquely decom posable i f it sa tis f ie s t h e following condition :

Ideals of dif ferentiable functions 691

If Ta(a)= 2t, n • • • n 91. and 91, = 13,•J such that Ta(a)=-- n P i ; givesP th e irredundant decomposition into minimal prime devisors, then there

exist ideals a,,..., a p uniquely such that

a= n , n ••• nap , Ta(cii) = 91, .

7. Definition. L e t a b e a point of R " . O a denotes the f am ily ofall closed subsets X c R " satisfy ing the following:

(a) J(X a) is f initely generated.( c ) K (X a)= K (X a).(b) J(X (,) is uniquely decomposable.

8. Proposition. I f X e 0 0 , Ka(X )= 911 n : 4 = n P i, 93= n Q i a n di=1 j 1

K a(X )=(n P ) n ( n .21) g iv e s th e irre d u n d an t prime decom position ofK a(X ), then the uniquely determ ined ideals a, b w ith Tan= 9E, Tab=

and J a ( X ) = a ro are .f in ite ly g e n e rate d . H e n c e th e ir zero sets Ya

and Z a are germ s of elements of Oa satisfy ing X 0= 'a U Za.

P ro o f . L e t Ja(X )= (fi • • • •• fr) a n d 91= (Tag i,• • •, Tag s), = (Tail 1,• • •,T h ) (gk E a, h, E b). Then there exist am, bk, e S a a n d q)„ e a , 1//k E b such

that

.fk E a t + (Pk= ± b kill 1+ th ,1=1 1= I

Ta9k= Tak = 0.

Putting a' =(g gs, (Pr), b' =(h ,,.. . , he, , , . . . , 0 0 w e have J0 (X )

=a' n b', Ta(d)= 91, Ta(b)= O. Hence a = a ' a n d b = b ' b y t h e unique-ness. q. e. d.

Thus a n element o f O a can be locally decomposed into a finite

number o f irreducible com ponents (E OD uniquely. Then by ( 6 ) wehave the following:

9. Corollary. I f X E Oa, dim X a= adim X . •

10. Lem m a. L e t X ,,..., X pe(I),? an d e ac h X , b e th e c lo s u re o f

692 Shuzo /zu i

k ,-d im e n s io n a l m a n ifo ld in a ne ighbourhood o f a . P u t X =U

and suppose th a t Ka(X) is a ra d ic a l a n d 11<„(Xi)1 a re a ll o f i ts m in i-

m a l p r im e d e v is o rs . I f Ja( U X i ) a re f in ite ly g e n e ra te d fo r a ll s u b -

fa m ilie s {X X ; 0 } OE ,,..., X1,}, Ja(X) is un ique ly decom posab le andhence X e Oa.

Proof. L e t J„(X)= Ta(ni)= Kcxx ;,) su c h th a t ,,...,i=1

{X ...... X } . Suppose that l e n J( kJ X „a)c. Then f does noti=

vanish on som e open subset V o f som e X„„0 w hich is adherent to a.

Let k b e the K a lil dimension of ,F Ko(X „„„). By our assumptiona n d (5 ) w e m ay assume th a t V i s a k-dimensional manifold. On the

other hand there exists g e Ka(X„,10)c fl ( K„( X11)) (cf. e .g . [6 ; p. 6]).j 1=1

H ence there ex ist g1 E ni (j 0 i n ) s u c h th a t Tag i= g . Since h g1

ap J0(X) and h vanishes on V. This im plies that g e K„(X„,10)

b y (2 ), a con trad ic tion . T hus w e have p ro v e d th a t a„,cJa( J X,„/).i=1

The converse inclusion follows from (19). These prove the uniquenessof a1,..., a,..

11. Proposition. T h e fo llo w in g c o n d it io n s a r e e q u iva le n t f o r a

re a l a n a ly tic set X :

(i) X „ is coherent.

(ii) Ja(X) is fin ite ly generated.

(iii) X E Oa.

Proof. ( i) ..( ii) is proved in [3 ]. (iii)= .(ii) is trivial. Suppose thatX a is c o h e r e n t . Let ( 9 „ c , „ deno te the ring of convergent powerseries a t a E R" and /a( Y) denote the ideal o f a ll f e (9„ vanishing on Y.

I„(Y) is of course a radical. I f X = J X ; is the irreducible decomposi-1= 1

tion as an analytic germ at a, I a(X)= I0(X,) is the irredundant primei = 1

decom position. Then ,"'„ • /„(X)= „ • 10(X,) is the irredundant prime

decomposition by the theorem of Zariski and Nagata (cf. [6] or [4; p. 89]).

Malgrange [4; p. 90] has shown that F„ • la(Y)= Y ) for analytic Y. H.

Id e a ls of differentiable functions 693

Cartan- [2 ; (1 3 )] h as sh o w n th a t i f analytic Y= Yi is coherent anyi=1

partial union n Y, is coherent, where Y1 ..... Y a r e irreducible componentspi= I

of Y as an analytic set. Hence K a (X )= r K a (X i) is the irredundant primei=1

decomposition and fa( X1 ) is finitely generated fo r any partial unioni=

P PU X i o f U X . T his im plies that X,,..., Xp e 0,?. It is a lso know n

th a t t h e K ru ll d im e n s io n o f a coherent analytic se t is constant nearits irreducible p o in t . Then X e l9 . b y the previous lemma. q. e. d.

Reviews and examples

Elem ents o f O a h a v e still m a n y b a d p ro p e rtie s . E sp e c ia lly thes e t o f singular p o in ts o f X does not a lw ays belongs to O a, not evenlocally, a n d so m e X have locally infinite topological ty p e s . Henceso m e X adm it no stra tifica tion a t a. T h e au th o r d o es n o t k n o wwhether the following hold o r not:

( i ) dim X x is constant near a for X e (1)„°.( ii) The strong condition uniqueness in (d) of (7) is removable.(iii) { x e X : X e Ox} is open in X.

( i ) is a ffirm ab le i f X e O a is a n a ly tic . A s f o r ( i i ) there ex ists aclosed se t X whose ideal is decomposable in infinitely many ways andw hose form al ideal is a rad ica l (c f. (15)). A s fo r ( iii) w e know thatth e property (a ) is a n o p e n property by (17). I f X is locally analytic(iii) holds by (1 l).

N ow le t 9(x) E g (R ) b e z e ro o n ( — co, 0 ] an d p o sitiv e elsewherea n d 0(X) E g(R) b e z e ro o n ( — co, 0 ] , flat a t 1, 1/2, 1/3,... a n d positiveelsewhere.

12. E xam ple . (A sa m i) . Let X = Z(x2 — yz , y 3 — x z , z2 — xy2) c R3.

T h e n X is c o h e re n t a n d J(X0)=(x— yz, y3 — xz, z2 — xy2) b u t dim X ,= 1> 3 —g(J(X0))=- O.

13. E xam p le . Let X = Z (y ) and Y = Z ( y q ) ( x ) ) . T hen X , Ye (1),(R2)

but .10(X u Y) and Jo(x n Y) are not finitely generated.

694 S huz o lz um i

14.Example. T h e principal ideal (xy+ (p(z))cg(R 3) is not decom-posable.

15.Example. Suppose th a t ( i ) J ( X ) is finitely generated; (ii) a,

a t, a2,•••, o r a r e ideals o f e a ; ( ) a c Ja(X ); (iv) n T a c T aa; (v ) Tanii=

a r e p r im e ; ( v i ) dim (Z(ai) n x)= dim ,F a /T aa i. T h e n Ja(X )= a b y (2)a n d (19). F r o m th is f a c t t h e id e a l J „ (Y ) o f Y=Z{(x2—zy2)(y2—z(x— tp(— z))2)1 is n o t f in ite ly g e n e ra te d . W e c a n e a s ily sh o w thatK(Y)= N/Ka(Y) and Ja(Y ) is decomposable in infinitely many ways.

16. Example. Let's put f= y2 — u(u+ y2 +1/, (x)), X = Z ( f ) R4.Since th e firs t Jaco b ian extension ( f , Of 10v ) o f ( f ) i s (u2, y,ut//(x), tty , 2u + y2 + tP(x)), "the critical set" of ( f ) is

Y = (x , 0, 0, 0): n t(l, o, 0, 0), (1/2, 0, 0, 0), (1/3, 0, 0, 0),...}

(cf. [1]).(i) If p a X — Y , K p(X )=(T pf ). I f p e Y a n y g e g 4 a can be ex-

p re sse d as g =cif+ 11(x, y , tt)v+ k (x , y, u ) (g E or,p,; h , k e e 3 p ) . If uo(u0+y,2)+1/1(x0))>0 there a re tw o y satisfying (xo, yo, uo, it) e X . Henceh, k vanishes tt(u + y 2 + k x ) )> 0 fo r a n y g e J p(X ). T h u s Tph = Tpk = 0a n d I< p ( X ) =( T p l) . T hese m ean tha t J(X , R4) =Q5 (the closure w ithrespect to el'-topology).

(ii) It i s e a s y t o s e e t h a t LI E \/( f , O f lav ) a n d T u i sn o t a z e r o d ivisor i n <F„/(T i,f ). T h e n b y th e th e o re m o f Tougerona n d Merrien [ I l ] , ( f ) is c lo sed a s w ell a s t h e analytic ideal (f , u)=(u , y2).

B y ( i) and ( ii) , J( X , R 4 ) = ( f ) . It is easy to see tha t X e oon- '

But its singular se t Y is locally infinite.

Lemmas on funtions, ideals and sets

Finally we list the important lemmas used in this paper.

17. Lemma. (T ougeron [ 9 ] , c f . [3]). L et ac J( X , U ) be an idealsuch that the restriction 0 a=.1 a(X ). T hen there ex ists a neighbourhood

Ideals of dif ferentiable functions 695

V of a such that the restriction al V =A X , V).

18. Lem m a. (Bochnak [1]). 1 f J (X , U ) is f in ite ly generated , Xis the closure o f a m anif o ld Y.

19. Lem m a. ([3]).(1 ) S u p p o s e th at Jp(X, U) is f initely generated.I f an ideal et Ja(X, U) satisfies Tao= Ka(X, U), we have a=J„(X, U).

20. Lem m a. (Tougeron [10; p. 9 3 ]) . L et fi, f2,... e S(U) be f lat ona closed subset X OE U. T h e n th e re e x is t s g e d'(U) w hich is strictlypositiv e on U — X , f lat on X and (fi, f2,...)OEg • rnx. W here mx denotesthe ideal o f all h e ‘ (U ) fla t on X .

The proof o f this lemma applies to the following:

21. Lem m a. L et f b e a re al v alu e d f u n c tio n o n U . I f a closedse t X OEU i s a z ero of in f in ite order o f f , there ex ists g g (U ) whichis strictly positiv e on U— X , f lat on X an d If(x)1 g(x ) o n U.

L e t U b e a n o p e n neighbourhood o f 0 E R k = { (X „ X 2 ,..., x,)}.Suppose th a t cr,(x), o-2(x),..., ,(x )e E (U ) a n d (5(x) is the discriminant

P- 1o f f(y; x)= yP + E o-p_i(x)yi. Then f(y ; x ) = 0 h a s p distinct solutions

1=i)y = ,(x), 2 (x ), o p(x) in {(x, y) e U x C: S(x) 0 01.

22. Lm m a. There ex ists a constant M depending only upon f andupon com pact set K c U such that

,N/16(x)I 1( 9OE/Oxi)(x)i M • max {i(ecri/ax,i)(x)11

on {(x) K 5(x)0 0} ,

N/ 0001' l((3f lax 1)(y ; x)I M • max l(8ci/axi)(x)1 l(af la y ) ( y ; x ) i

on {(x, y)e K xC: f(y; x)=01.

P ro o f . I t is easy to see

( t) A m o re general proposition is found i n Bochnak-Risler; Quelques questionsouvertes (preprint).

0u2/0X,

696 Shuzo lzum i

E 9.9pa,I3* I

ct <II

( (p 2 . ( p p ) ......................... (foci — 4 9 p - 1 )

a(p3/axta<I3

E (11.(Ppa,f l* p

E (P.oLo p

a(P2/axi

T h is m atrix h as d e te rm inan t o f abso lu te v a lu e ,./161 a n d its elementsa r e b o u n d e d o n {(x)e K : 6(x)00 }. T h e n t h e first inequality follow sfro m C ram er 's f o r m u la . T h e s e c o n d h o ld s f ro m t h e f a c t th a t aflax,,=—(0(pc10x,)(0flay) o n th e z e r o s e t o f f q . e . d .

FACULTY OF SCIENCE AND

TECHNOLOGY, KINK! UNIVERSITY

References

J. B ochnak , S u r le théo rém des zé ro de H ilbert "D iffé ren tiab le" TopologyVol. 12 (1973) 417-424.H . C artan, V arié tés analytiques réelles e t varié tés analytiques com plexes,Bull. Soc. Math. France, 85 (1957) 77-99.S . Izu m i, Zeros o f ideals o f C r functions, J. M ath. K yoto Univ., to appear.B . M algrange, Ideals o f differentiable functions, O xford U n iv . Press (1966).J . N a g a ta , Modern dimension th e o ry , Amsterdam, North-Holland (1965).M . N agata, L ocal rings, N ew Y ork, John Wiley & Sons (1962).J. J . R isle r , U n theo rem d es zé ro s en g éo m é trie an a ly tiq u e rée lle , C. R.Acad. Sci. Paris 274, 1488-1490 (1972).R . T h o m , O n som e ideals o f differentiable functions, J. M ath . S oc . Japan19 (1967) 255-259.J. Cl. Tougeron, Faisceaux différentiables quasi-flasque. C. R . Acad. Sci. Paris260, 2971-2973 (1965).J. Cl. Tougeron, Idéaux de fonctions différentiables, Springer (1972).J. C l. Tougeron, J . M errien , Id éaux de fonctions différentiables, I I , Ann.Inst. Fourier 20, 179-233 (1970).