Topological triviality of deformations of functions and Newton filtrations

Invent. math. 72, 335-358 (1983) ln~;e~ltloYle$ mathematicae �9 Springer-Verlag 1983

Topological Triviality of Deformations of Functions and Newton Filtrations

James D a m o n a* a n d Te rence Gaf fney 2 .

1 Department of Mathematics, University of North Carolina, Chapel Hill, NC 27514, USA 2 Department of Mathematics, Northeastern University, Boston, MA 02115, USA

Table of Contents

Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 335 w 0. Notation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 336 w 1. The Newton Filtration . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 337 w 2. A Sufficient Filtration Condition for Topological Triviality . . . . . . . . . . . . . 339 w 3. Topological Triviality via Controlled Vector Fields . . . . . . . . . . . . . . . . 341 w 4. Patching Data and Non-degeneracy . . . . . . . . . . . . . . . . . . . . . . . 344 w 5. Jump Conditions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 345 w 6. Sufficient Conditions via Local Patching Data . . . . . . . . . . . . . . . . . . 349 w 7. Examples . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 350 w 8. The Push-off Lemma and a First Consequence . . . . . . . . . . . . . . . . . . 352 w 9. Deriving Filtration Conditions from Jump Conditions . . . . . . . . . . . . . . . 355 w 10. # Constant Deformations of Real Singularities . . . . . . . . . . . . . . . . . . 357

Introduction

W e cons ide r a d e f o r m a t i o n f o f a g e r m f0 = k", 0 ~ k , 0 which has a n (a lgebrai - cally) i so la ted s ingu la r i t y at 0 (here k = ~ or IE a n d the germs are co r r e spond - ing ly smooth , ana ly t i c or ho lomorph ic ) . A f u n d a m e n t a l p r o b l e m is to de- t e r m i n e w h e n such a d e f o r m a t i o n is t opo log ica l ly t r ivial (i.e. t opo log ica l ly r ight e q u i v a l e n t as a d e f o r m a t i o n to the c o n s t a n t de format ion) . W h e n k = C , the bas ic a p p r o a c h to this p r o b l e m is to p rove the M i l n o r n u m b e r is c o n s t a n t a n d use the resul ts of L & R a m a n u j a m a n d T i m o u r i a n [10], [14] to c o n c l u d e tha t the d e f o r m a t i o n is t opo log ica l ly tr ivial . However , this m e t h o d does n o t app ly w h e n k = ~ (nor w h e n fo def ines a complex surface s ingular i ty) . Also, the c o m p u t a t i o n of the M i l n o r n u m b e r c an be difficult (see e.g. K o u c h n i r e n k o [8]).

* Partially supported by grants from the National Science Foundation and the British Science Research Council

336 J. Damon and T. Gaffney

An alternate approach is to use Teisser's p,-constant condition [13] to prove that the Whitney conditions hold and then obtain topological triviality as a corollary.

Here we describe an approach based on the Newton filtration of f0. Kouchnirenko [8] made use of the Newton filtration in his computation of the Milnor number for non-degenerate functions. Our method uses certain filtration preserving properties of the deformation f to prove directly that f is topologically trivial without requiring non-degeneracy. This method gives a reasonably practical algorithm for proving topological triviality which also works for real germs and the germs defining complex surface singularities.

The general form of this result is given by a filtration condition involving the deformed terms ~f/~u~ and the deformation f. We also describe how this general form can be reduced to a useable criterion involving the deformed terms and the original germ fo. For this, we consider the polyhedral structure of the Newton polyhedron of fo. We construct local patching data for the faces of the polyhedron. When the filtration properties of the deformation behave well with respect to the local data, then the local conditions "patch together" to give the "global" filtration properties. Then, topological triviality is proven along the lines of [5] or [11] by solving the localized equation for triviality (with respect to right equivalence) using controlled vector fields.

In the case of non-degenerate functions, the filtration criteria reduces to the requirement that the deformation does not reduce the Newton filtration. Thus, we obtain as a corollary, a topological proof of the result of Kouchnirenko that for non-degenerate functions, the Milnor number only depends on the Newton polyhedron. Since our original announcement [6], we have learned of several other geometric proofs of this same fact; both Merle and Brian~on have described distinct ways of obtaining Teisser's/~.-constant condition from non-degeneracy (in fact, our method also implies the analytic version of Teissier's condition (c) which is equivalent to ~t.-constant). Also, Oka [12] has used the non-degeneracy condition to bound the Milnor radius and obtain topological triviality directly.

We wish to express our gratitude to the British Science Research Council for its support and to the Department of Mathematics of the University of Liverpool for its warm hospitality during the completion of this research.

w O. Notation

Throughout we consider germs f: k", 0 ~ k p which are either C OO or real analytic if k = N or holomorphic if k- -~ . We will use local coordinates x for k" and denote the ring of germs k",O~k (in the appropriate category) by cg x with maximal ideal ~ , . For a germ f: k",O~kP, O we let O(f) denote the module of germs of vector fields (in the appropriate category) ~: k " ,O~Tk p such that npo = f for np: TkP~k p. We denote O(idk, ) by 0,; this is a free ~fx-module generated by O/~?x i i=1 . . . . ,n. For a deformation f:k"+r,O-~k,O of a germ fo: k",O~k,O we let u denote local coordinates for U, the space of deformation parameters. Also, we let cgx. u denote the ring of germs k"+r,O~k. If

Topological Triviality of Deformations of Functions and Newton Filtrations 337

~,,r: k"+r, 0-~kr, 0 is the projection, then 0(~,.r) is also a free ~x,,-module generated by the ~/Ox i.

Also, we denote the polynomial ring k[x~,.. . ,x,] and the formal power series ring k[[x~,.. . ,x,]] by k[x,] and k[[x,]] respectively. For any ring R and elements u~ ... . ,u~, we let R{ui} denote the R-module generated by the u~ (the number r will be clear from the context).

w 1. The Newton Filtration

In this section, we recall [8] the definition of the Newton polyhedron and Newton filtration associated to a germ f0: k", 0~k, 0. In fact, we give a slightly more general definition which is defined even when the usual Newton filtration may not be.

Let ~c~x ~ denote the Taylor series of fo- Also, let I(+ ={(Xl,...,x,)~lR", each x i > 0}. By a defining half-space for )Co, we mean a closed half-space H of N" such that i) if c,:t=0, then eeH; ii) there are n+ 1 independent ~ with c.4=0 which lie in the hyperplane boundary 0H for H, and iii) OH intersects each axis of IR+ (at some point other than the origin).

If fo has at least one defining half-space, we will call fo semi-fit. For a semi- fit f0, we let F+ (fo) denote the intersection in ~ + of all defining half-spaces for f0. It is called the Newton polyhedron for fo. It is a convex polyhedron in ~ + and so has a polyhedral decomposition. We let F(fo) denote the union of the compact faces of F+(fo ) and call it the Newton boundary. It also has a polyhedral decomposition.

We also say that fo is f i t (commode in the terminology of Kouchnirenko [8]) if for each j there is an x~'~ with non-zero coefficient in the Taylor expansion of fo. Note that Kouchnirenko only carries out his analysis for fit germs.

It is easy to see that if fo is fit, it is semi-fit; and then the Newton polyhedron defined here agrees with that defined in [8]. However, the converse is not true if f0 is only semi-fit.

Example. (Briangon-Speder example [4]).

fo(x,y,z)= x 5 + yV z + z15

Then, fo is semi-fit but not fit. It is possible to change coordinates ( z~z+y) to make fo fit; however, the complicates the problem of determining the topologically trivial deformations (see w 7).

From the Newton polyhedron, we construct the Newton filtration. For each face A ~ F(fo) (or closed face A), C(A) (respectively C(zl)) denotes the cone of half-rays emanating from 0 and passing through A (or z/). The {C(A)} gives a polyhedral decomposition of I(+ which we call the Newton decomposition. The Newton filtration is then defined via a piecewise-linear map ~b: N + ~ N satisfying:

i) 4~ is linear on each C(zl).

ii) q5 takes positive integer values on the lattice points of I(+\{0}.

iii) q~IF(fo)=m (for some positive integer m).


Fig. 1

For any monomial x ", we define fil(x ~) = q~(~t). This extends to a filtration on k[x,], k[[x,]] , and cg x (via Taylor expansions) by defining

fil(~ c.x ~) = min {~b(~t): e, 4: 0}.

It also extends to Cgx, . by defning fil(x'u~)=fil(x'). We denote the set of g with fil(g)>l in k[x,], k[[x.]] , cgx, or cg~,, by A t, ~ , ~r or ~ .. . . . respectively.

By its definition, this filtration has the property that on any cone C(A), it is given by assigning appropriate weights to the x i. Furthermore, it has the additional property (see [81) that fil(x'xt~)=fil(x')+ fil(x ~) if and only if �9 and fl are contained in a common closed cone.

Because of the identification of a monomial x" with ateN+, we will fre- quently abuse notation and speak of x" belonging to C(A) when we mean that the associated ~t~C(A). Also, we can define rings k[zt], k[[z~]], etc., to consist of polynomials, power series, etc., with non-zero monomials in C(A). For these rings, the associated Newton filtration is the restriction of {At}, {~r etc. For these rings, it corresponds exactly to assigning appropriate weights to the x i.

The rings k[[x.]] , k[[z~]], ~x, and ~ , . are all local rings and also have filtrations associated to the powers of the ideals ~ . , ~n, ~ , and ~xCgx,,. In each case, the Newton fltration is equivalent to the filtration by powers of these ideals (i.e. for example in k[[x,]] , for each l there are k i so that

~ k~ and /~/gnl ~ ~k2)"

A useful tool in working with these filtered rings is the following simple extension of Nakayama's lemma.

Lemma 1.1 (Filtered version of Nakayama's lemma). Let (R, {I~}) be a filtered local ring (with say I I ~ R ) and let (M, {Mj}) be a finitely generated filtered R- module (I l Mk c Mj+k) with {Mj} equivalent to the filtration {I~ . M}. I f (N, {Ni} ) is a submodule of (M, {Mj}) (so Nj~_ Mj all j), then

M j = N j + M j + i all j ~ O implies M = N .


Pro(~

Thus, by induction

M = M o = N o + M I = N + M 1

= N + N 1 + M 2 = N + M 2.

M = N + M k for all k>0.

By the equivalence of {M,} and {1 ]- M} there is a k so that 11 �9 M ~ M k. Hence

M = N + I 1 M .

Since 11 c ~, the maximal ideal of R, Nakayama's lemma implies M = N. D

As with any filtration, we can define for a germ g in any of these rings, its initial part, in(g), to be that part of the Taylor expansion of g whose terms have filtration= ill(g) and so that i l l (g- in(g))> ill(g). For a face A eF(fo), we let in(g)la denote the terms remaining after further removing from in(g) terms not in C(A).

Finally, we also mention the special case of semi-weighted homogeneous germs. For such germs fo: k",O~k,O we can assign weights wt(x~)=a~ and obtain the associated weight filtration where fil(x')=~cqa~. Then, fo is semi- weighted homogeneous if for some such weighting in(fo ) still defines an isolated singularity. For such semi-weighted homogeneous germs, we can replace the Newton filtration by the weight filtration and the results we shall describe will also follow for it.

w 2. A Sufficient Filtration Condition for Topological Triviality

In preparation for the specific conditions which we will describe in later sections, we begin by giving general sufficient conditions that a deformation be topologically trivial. We consider a germ f0: k",O~k,O which is semi-fit. Then, the vertices of the Newton boundary F(fo) will in general only have rational coordinates. We say that the level ~ of the Newton filtration is f i t if all of the vertices of 4>- 1(/) are lattice points of IR+ (here ~b is the function used to define the Newton filtration). If ~t is fit, then by the linearity of the Newton filtration on cones, ,~r is also fit for integers r > l . Also, i f fo is already fit, then S~Ctm is fit for all integers l> 1. For ~ which is fit, we let

ver(sCt) = {x': a is a vertex of q~-1(I)}.

We also define a filtration on 0, associated to that defined on cg~. Specifically (with sJ~+l{t3/t?xl} denoting the ~r generated by the O/Oxi, i = 1 .. . . , n), we let

Then {~//~} is a filtration on 0, which makes 0, a filtered cgx-module. We can analogously define a filtration { ~ . . . . } on 0(ft,,r) which makes it a filtered ~x,. module.


For example, we have for the Newton filtration

If {s~,x } is the weight filtration for a semi-weighted homogeneous germ fo, then

i= 1 ~ /+wt (x i ) ~X i

We let r ~ r By a deformation f: k "+', 0-*k, 0 of f0 : k", 0~k , 0 being topologically trivial,

we mean that there is a germ of a homeomorphism ~,: k"+r,O--k"+',O of the form qJ(x, u) = (~(x, u), u) such that

f o O(x, u) = fo(X).

Then, we have the sufficient condition for topological triviality.

Theorem 1. Suppose that f: k"+r,O~k,O is a deformation of a semi-fit germ fo : k",Ook, O. .Then a sufficient condition that f be a topologically trivial deformation is that there exists a f i t ~ so that

ver(~Cz) �9 ~8~f c ~//~l x ,(f), 1 <i<r (*) G b l i ' ,

Remark. Even if fo is not semi-fit but is semi-weighted homogeneous, then the result holds using the weight filtration.

An important special case is given by the following result.

Theorem 2. Suppose that f: k"+r,O~k,O is a deformation of the f i t germ fo: k",O~k,O. Then, a sufficient condition that f is a topologically trivial deformation of fo is that there exists an I so that

ver(.~z,.) �9 ~u~u / d , . . . . . {xj ~ } , l<_i<_r (**)

again recall txj ~--f-f t denotes -module generated by the that ~r the s~t

Remark. We will also show that condition (**), in fact, implies Teissier's condition (c) [13]; hence, these deformations will, in fact, satisfy the Whitney conditions.

We will call the conditions (*) or (**) the filtration conditions. In the sections that follow, one of our principal goals will be to derive simple conditions in terms of the deformed terms 8f/~u i and the Newton filtration of fo which will imply the filtration conditions. However, before doing so, we first give the proofs of these results in the next section.


w 3. Topological Triviality via Controlled Vector Fields

Here we use the method of solving the localized equations for triviality using controlled vector fields to obtain topological triviality (see [5] and [11]). Specifically, we use the terminology and notation of [5]. Although in these papers triviality of unfoldings was considered, the method likewise applies to deformations f: k "+r, 0 ~ k , 0 of a germ fo:k", O~k,O.

We seek to solve the localized equations

(3.1) - P ~ = ~ ' i ( f ) . l<i<_r

where p: k"+r, 0-~lR+ is a (C ~) control function, so that p-l(0)={0} xk ' and ~'i: k"+r, O--*Tk" is a smooth germ of a vector field. Also, we require that p-1 ~'i be controlled by p along c~/Oui. This requires that:

i) p- 1 ~'~ extends continuously in a neigborhood of 0 to be zero on {0} x U

ii) in a neighborhood of 0 there are constants C~ so that

Ip- l ~'i(p)l <= C1. p and O P < c c~ui = 2" P"

If p is independent of u, then ii) is equivalent to ]~'i(P)l < C1" p2. If we can solve (3.1) with vector fields ~'i controlled by p, then f is a

topologically trivial deformation. Thus, to prove theorem 1, we will show that the filtration condition implies that we can solve the localized equations (3.1) using controlled vector fields. If ver(~) = {x'} then we may write

c~u~ j ~

Multiplying by x', or its conjugate if k = ~ , and summing over x'Ever(.~) we obtain

(3.2) - ( Z IXe~12) ~ / / = 2 ~ij ~f �9 0x~"

We extend the filtration so that ill(if')= fil(x') in the complex case. Then, we let

0 __ ] ~2 P - Z x I and ~'g=Z~'iJc~x/

From the filtration properties of the (I~), we obtain

fil(~'~) > fil({l~ ~) + fil(x') > 21 + 1.

Also, since ~'i = ~ ~" ~I ")

(3.3) fil(~'~(p))> min {fil(ff'.~l')(p))}. x~Ever(~tl)

Then fil(~l~)(p))> min {fil(~-~l~(x~)}.

xi~ ver 0:r


By the properties of ~I ~), fil(~l')(xtJ))>2l. Thus,

fil(~l~)(p)) > l + 21 = 3 I.

F rom (3.3) we obtain

fil(~'~(p)) > l + 3 l= 41.

As p is independent of u, to establish i) and ii) it is sufficient to show:

i) ][~'iJl~Cp 1+~ in a neighborhood of 0 for some 6>0 . Then lip -~ ~'il[ < Cp ~ on the neighborhood minus {0} x k r. This implies that it extends continuously to be zero on {0} • U.

ii) Ir < C~. p2. Both conditions will follow from the following lemmas guaranteeing the bound- edness of functions satisfying filtration conditions.

For the first lemma, we consider the case where p: k",0--+~§ is any continuous function. Let I ~ = ( l ~ " c ~ . We define

B = { ~ O / ~ : 3 a constant C > 0 and a neighborhood U of 0 (which depends on a 0 such that Ix'l__< C.p on U}.

Note that x ~ may only be defined as a complex number even though k = N ; however, this is allowed for the definition of B.

The first lemma describes the structure of B.

Lemma 3.4. Given p and B as above, then B is a convex set (in II~"+) closed under addition by elements o f q)"+.

Proof For convexity, consider ~, f l~B so that Ix'l < Cx.p on U 1 and IxPl__< C2" p on U 2. Let 2eQ, with 0 < 2 < 1 , and define ~ = 2 a t + ( 1 - 2 ) f l . Then, on U = U 1 c~ U z we have

IX~,l _____ IXZ~*+ (i -Z)/I I = IX,IZ ixtll 1 - ;~ 5 ( C 1 p)Z(C2, p) l -,t.

Hence, on U, IxVl < C . p where C = C] C, 1-z. Secondly, if a~B and f l~( l~, then Ix~l ~ C~.p on U 1 and IxPl < C 2 on some

neighborhood U 2 of 0. Thus

Ix~+Pl=lx'llx~t~flfz.p on Ulc~U2. D

Next, we return to the situation of interest to us, namely p = ~ l x ' l 2 where x 'ever(a/ l) for a fit ar

Lemma 3.5. I f geag,,,x,,, there is a neighborhood U of 0 in k "+r and a constant C > 0 so that

Igl < C. pm/2l on U.

Proo f Raising the desired inequality to the 2l-th power, we see it is sufficient to assume g~ar . . . . . and show

lgl=<C'pm on U.

Topo log ica l Tr iv ia l i ty of Defo rmat ions of Func t ions and N e w t o n F i l t ra t ions 343

Since ~r as an ideal is generated by a finite number of monomials x r in k[[x . ] ] , it follows by Taylor's theorem that ~'21 . . . . . is also generated by a finite number of monomials x r in ~ , , . It is enough to verify the result for the monomials x r for then it is also true for any finite linear combination ~g~x r with gTECs by just bounding each gr on some neighborhood by a constant.

Applying Lemma 3.4 to p ' , we see that B contains (x~') z" for x'ever(~r By the linearity of the Newton filtration on cones, it follows that

ver (alE,.,) = {(x.)2 m: x ' eve r (~ )} .

Hence, the xr~B. The result follows. [3

Lastly, we complete the verification of the conditions i) and ii). We have , > fil(~ij)=2/+ 1. Thus, by Lemma 3.5,

21+1 I~'J<fi.p 2, = C , p 1+~ with 6=(2/) - t on U/.

Hence t - - p l I1r162 +~ on U

J

where C = ~ C, and U = ~ U i. Also, fil(~'~(p))>4/, so

41 Ir < C1" p~5= C 1 �9 19 2 o n some U.

This completes the proof of the topological triviality. Theorem 2 is then a special case of Theorem 1. However, conversations with Andrew Duplessis led us to realize that the form the localized equations take for Theorem 2, in fact, implies that the deformation satisfies Teissier's condition (c). We obtain the localized equation

. . . . and fil(gi~)>2/. Thus where ~t i -'~- 2 g,jXj ~Xj

t 3 f < n . max Igljl �9 max Ix~[" max i f . P' c~u i = ~ j ~ ~xj

By Lemma 3.5, there is a common constant C~ >0 so that

[gijl~Clp o n s o m e U

Also, max Ixjl < IIxtl, maxj ~ ~ Ilgrad x fll. Thus

l < j < n .

p. <=C.p. Ilxli IIgrad~ fl[


dividing by p for x 4= 0, we obtain the inequality which is also valid for x = 0

~ <Cllxll l lgrad~fll on U.

This is exactly the analytic version of Teissier's condition (c).

w 4. Patching Data, and Non-Degeneracy

The local patching data which we define give us a method for determining the effects of a deformation on the various cones in the Newton polyhedron. Let )Co: k",O~k,O be a semi-fit germ.

Definition 4.1. A set of local patching data for fo consists of: for each face A cF(fo), a set {~a,i} of germs of polynomial vector fields in 0, such that

i) ~a,,=~,g~,i,jXJ axj

ii) {in (~a,i(f0))ta} generates an ideal of finite codimension in k[[A]].

Remark 1. The {C(A)} give a polyhedral decomposition of IR+ which is the cone on the polyhedral decomposition of F(fo). The local patching data can be viewed as being defined via {C(A)} or {d}.

Remark 2. It is sufficient to construct local patching data for the top dimensional faces and restrict these to obtain data for the lower dimensional faces.

The existence of local patching data is guaranteed by the following

( afo] Proposition 4.2. If ~x i ~xi; generates an ideal of finite codimension in ~x, then

there exists a set of local patching data for fo.

( afo] ( afo~ Proof If ].x i ~xl; generates an ideal of finite codimension in c~x, then lxi ~xi;

generates an ideal of finite codimension in k[-[x.]], where fo denotes the Taylor expansion of f0, If ~ . denotes the maximal ideal of k [ [ x J ] , then by

t a?o~ t a~o~ assumption there is ~ r so that < = ? , ~ I , ~where V' ~ t denotes the ideal a~0~ generated by the x i ~ixl ].

Then, ~ c ~ k [ [ z l ] ] is an ideal of finite codimension in k[[zl]]. As k[[zl]] is

Noetherian, there exists a finite set of monomials ha,ie ~xi~x~] which generate J , ~ k [[z]]]. We can write

~o ha,i= ~',g,a,~jx~ c3xj"

Let ga,~,~ denote ga, i, j truncated above filtration level = fil(ha, 3, and define

~A,i = Zj ga,l,j Xj C~Xj.


Then, h~,i=in(h~,i)l~:in((a,i(fo))[a. [-1

If fo has an (algebraically) isolated singularity, then {Ofo/OX~} generate an

f afo) ideal of finite codimension. Hence, for ].x i ~ixi ~ to fail to generate an ideal of

finite codimension, the restriction of fo to some coordinate hyperplane x i=0 must fail to have an algebraically isolated singularity. However, for almost all hyperplanes OeYIck",folH has an algebraically isolated singularity. Thus, for

{ almost all choices of coordinates x, X~axij will generate an ideal of finite codimension.

Thus, by a generic choice of coordinates, we can ensure that fo has local patching data. At the same time we point out that a generic choice of coordinates gives the least interesting form for the Newton polyhedron. The

f afo] most desirable course is to choose coordinates so that ~x i~xi~ generates an

ideal of finite codimension, yet the Newton polyhedron "presents the richest picture of the singularity".

f #fo] When ~x i~xi( " generates an ideal of finite codimension, we can algebrai-

cally construct a canonical choice for local patching data. If we construct the graded algebra associated to {k [[zl]], ~[~}, we obtain k [zJ]-~gr(k [[z]]]).

Given the ideal I=(xiCS~~ we associate the ideal 14 in k[-z]] consisting of

the images of ge l in gr(k[[A]]) by the map g~-~in(g)]~. As k[z]] is Noetherian, I~ is generated by a finite number of images {gi=r These {Ca, i} are the canonical patching data associated to 14. It is not necessary to find the canonical patching data to apply our results.

We recall the definition of a non-degenerate germ fo [8] as one which is fit

and f~ which {in ( x i~ ) } generate an ideal ~ finite c~176 in k[[~]]

for each closed face AcF(fo). It is immediate that ~ x i is a set of local

patching data (in fact, canonical data) for each A c F(fo). We shall also say

thatagermfoisnon-degenerateonafaceAcF(fo) i f { i n ( x i~ ) ~}generates an ideal of finite codimension in k[[zl]]. More generally we shall see how local patching data also allow us to handle the situation where fo is degen-

I f0xL erate. In fact, the data {(~,i} arise from the relations between the in ~xi Jx~xi ~ .

w 5. Jump Conditions

In this section, we fix a set of local patching data. Then, to measure the effect of local patching data applied to the filtration, we use the notion of jumps. For

346 J. D a m o n and T. Gaffney

local patching data {~d.~} for fo, we define

jump(~d,i)=fil(~,i(fo))--rrfin{fil(g~,i,jxj~o~j} �9

Also, for each face A ~ F(fo) we define

jump(A) = max {jump((a, i)}. i

Lastly, for each vertex veF(fo) we define

jump(v) = max {jump(A): A = star(v)}.

Here recall that for any face A ~F(fo), star(A) is defined as for any polyhedral decomposition star(A)= U {A'EF(fo): A czi'}. In particular, star(A) is an open subset of F(fo). In the non-degenerate case, with local patching data

~ x i ~ , all jumps are zero. k v ~ i )

Lastly, for each vertex veF(fo), we wish to be able to "pull" any polynomial germ in x, 0ecgu[x,] into star(v) and measure the filtration of its image. For this we let ~eC(v). Given any qSe~x, . we write its Taylor expansion with respect to x as ~ bp(u)x I~ with btj(u)eCg .. We define

supp(ck)={AeF(fo): there is a •eC(A) with ha+0 }.

Then, we have the following lemma which allows us to contract polynomials into stars of faces.

Lemma 5.1. (contraction lemma). Given any polynomial germ 0e~ , [x , ] , there is , > an l so that for l = l

i) supp((x') r. O) c star(v) '_> ii) fil((x') r. O) - fil((x*) r) is independent of I _ I.

We then define for the vertex v of F(fo ),

filv(~k) = fil((x') r" 0) - fil((x') r) l'_>_ l.

Proof It is clearly enough to establish the result for a monomial in x, say x ~. For i), if we consider the plane P spanned by C(v) and x a, then star(v) will

intersect P in an open cone C containing C(v) in its interior. See Fig. 2. Then, by the "parallelogram law for multiplication", (x')t .x Ij lies on the ray parallel to C(v) through x ~. Thus, for all large l if will lie in C.

For ii), note that once supp((x') ~- O) ~ star(v), then by the property of New- ton filtrations described in w 1

fil((x') k. (x ' ) t. ~0) = fi l((x') ~) + fi l((x ') ~. 0).

Then, ii) follows immediately. [q


C(v) C /

Fig. 2

For any germ ~O~cgx,,, we can also define filv(~b ). If ~k(L ) denotes the Taylor expansion of ff in x up through filtration < L, then filv(qJ(L)) is independent of L for all sufficiently large L. Then, filv(~) is defined to be this common value.

Remark. fil,( ) can also be defined as the piecewise linear extension to R+ of filvlstar(v).

We are now ready to define the jump conditions which will be sufficient to establish the filtration conditions of w 2.

Definition 5.3. A germ ~, satisfies a simple jump condition for fo if

fil~(qs)>m+jump(v) for all vertices vEF(fo).

To illustrate the simple jump condition, consider a semi-fit germ f0 with F(fo) as shown in Fig. 3 so that fo is non-degenerate on each face except A and jump(A)=/. Then, monomials satisfying the simple jump condition are those in the shaded region.

This is a purely numerical condition. It must be supplemented, in general, with a more specific higher order jump condition.

Definition 5.5. A germ ~k~x,, satisfes a general jump condition for a deformation f offo if for each vertex v~F(fo), there is an x'~C(v) of filtration =m~

f / level m+s

? ( f o ) ~ Fig. 3


and a polynomial germ O ' e d . . . . . . x~ so that

fil,,(x" �9 ~ - ~') > m~ + m +jump(v).

There is an inductive procedure for establishing the general jump condition. If fil,(~b)<m+jump(v), then we inductively seek to "stably" reduce the in-

equality. First seek a x~"EC(v) and a ~1 = ~ ~l,ixi~zj with ~1,i~r ..... where m I = fil(x ' ' ) so that tJ~ i

filv(x" �9 0 - ~ 1 (fo)) > filv(O) + m,.

If the remainder also satisfies

filv (~ l ( f ) -- ~ 1 (f0)) > filv (0) + m,,

then we replace ~ by O l = x ~1 ~9_ ~l(f) . Then,

fil,,(ff 1) - - m I > fil,,(O).

Inductively, we repeat the construction finding x "', ~i, and Oi so

f i lo(q, ,)- m, > fil~(q,,_ ,). Eventually,

fil~ (Ok) -- mk> m +jump (v).

k

Then, we let ~ = ~ i , mv=~m~, and ~=~x~,r where fl~= ~ ~j; ~(f) is our desired t)'. j= i + 1

In what follows, we will generically refer to either form of jump condition as simply a jump condition.

To place the ideas of local patching data and jumps into perspective, we indicate how they naturally arise in a spectral sequence which is a slight

variation of the one used by Arnold [1]. Let U = d o xl and I be the

ideal x i ~?xi ! as defined earlier. Define d: U ~ I by d(~)=~(fo). Then, U has a

filtration with U p = d p { x i ~ } , and I has a natural filtration which can be

renumbered so that d preserves filtration. We consider the spectral sequence of the complex

0--~ U~I--~O, {d;: Sp~I'p+,} ( So= Up/Up+r I~ =(I~dp+,,)/(Ic~dp+m+ l), and d~

Then, S~ is generated by vector fields of filtration p, which when applied to fo jump filtration r. Thus, if d~=~_d~, Im(d*) is generated by ~(fo) with jump(~)

P

=r. Also, if r o is the minimum integer such that I~ ~ is non-zero for only finitely many p, then there exists a set of local patching data {~a,i} with max {jump (~a, i)} = ro.


w 6. Sufficient Conditions via Local Patching Data

We are now ready to describe the principal result which states how the jump conditions for the local patching data for a semifit germ fo allow us to derive the global filtration condition.

Theorem 3. Let f: k "+r, 0--* k, 0 be a deformation o f f o. Suppose that fo has local patching data {~A,i} SO that:

i) fil(~+~(f))=fil(~A,,(fo)) for all A, i ii) the ~f/Ou i satisfy a jump condition.

Then, f satisfies the filtration condition (**) and hence is a topologically trivial deformation.

As an immediate corollary we have

Corollary 1. Suppose that fo: k", O--+k, 0 is a non-degenerate germ with an isolated singularity. Any deformation o f f o of non-decreasing Newton filtration (i l l(f) = fil (fo)) is a topologically trivial deformation.

As a corollary of this, we have a topological proof of a result of Kouchni- renko.

Corollary 2. I f fo: I12", O--+(U, 0 is a non-degenerate function with isolated singularity then the topological type o f f o and hence its Milnor number only depends on its Newton polyhedron.

Over the reals we can still say

Corollary 3. For non-degenerate germs fo: IR", 0 ~ I R , 0 the topological type o f f o only depends on its connected component in the subspace of all such germs with the same Newton polyhedron.

Also, as a corollary, we have an estimate of the order of topological determinacy in terms of the filtration.

Corollary 4. Let fo be a germ with local patching data {~A.~}- I f s = m +max{jump(v) : vsF(fo)}, then fo is s-topologically right determined with respect to the filtration {d}. Specifically, if f t -)Co m~ then f; and fo are topologically right equivalent.

In the semi-weighted homogeneous case, we obtain

Corollary 5. I f fo is a semi-weighted homogeneous germ, then any deformation which does not decrease weight is topologically trivial.

Before turning to the proofs of the results, we first illustrate these results with several examples.

w 7. Examples

By Kouchnirenko, we expect almost all germs with a given Newton polyhedron to be non-degenerate. For example, if we examine Laufer's list of mi-


nimally elliptic surface singularities [9] (or Arnold's list of unimodal and bimodal singularities [2]), we see that with the exception of eight families, all other singularities are either weighted homogeneous or non-degenerate (possi- bly after a change of coordinates to make them fit). Thus, we can conclude that, excluding these eight families, deformations of the other minimally elliptic surface singularities of non-decreasing Newton Filtration (using new coordinates where necessary) are topologically trivial.

For the eight families, we must construct local patching data. We consider the first such family where this occurs.

Example 7.1. fo(x, y, z)=(xZ + y3)2-z2 + xay b where 3a+ 2b=n+ 11 (n>2). The Newton polyhedron has just one face.

I Z2

Fig. 4

This is degenerate on the top dimensional face and on the edge {x 4, y6}. ( O haveth commo a to We see that on these faces, in x 0 x ! and in y Oy!

x2+ y3. There is a relation between these initial parts obtained from

Then,

f_~ 0 ( = 3 y 3 . x - 2 x 2 . y ~ y .

~(fo) =(3ay 3 - 2bx2) x"Y b.

0 8 z 8 This [ together with x - y - and forms local patching data for these two faces. Then ~x' Oy' Oz

f i l ( ( ( f o ) ) = n + l l + 6 = n + 1 7 and j u m p ( ~ ) = ( n + 1 7 ) - 1 8 = n - 1 .

Hence, a deformation f must satisfy

(1) fil (((f)) = n + 17 and

(2) fil O(~-~u~)>__n-l+12=n+ll.

In fact, in this case (2) implies (1); hence, we conclude that any deformation off0 by terms of filtration > n + 11 is topologically trivial.

As a second example, we consider

Example 7.2. fo(X, y)= y5 +2x2ya +X4y..FX6.

Topological Triviality of Deformations of Functions and Newton Filtrations

/wt(x'Y)=(6'6)

[ yS

~ y ) = (5,10)

x 6

Fig. 5

351

This has the Newton polygon shown in Fig. 5 where the Newton filtration on each cone can be described using the indicated weights for x and y. Then m=30.

Also, x c~f~ = 4x 2 y(x 2 + y2) + 6x6 0x

Y ~ = y ( 5 y 2 + X2)(y 2 -.-~ X2).

On A l and the vertices, fo is non-degenerate; thus, for these faces we may use

x ? ~ , ~yy for our patching data. On A2, fo is degenerate; in fact,

in x OxIla2 and in Y a2 have the common factor y2+x2. We obtain a

relation between these using

Then,

~ = ye ( 5 y2 + xZ) " x -- 4 yZ x2 " y o~.

~(f0) = 30y 4x6 + 6Y 2x8"

In this case, in(~(fo))=~(fo), fil(~(fo))=60 and j u m p ( ~ ) = 6 0 - 5 4 = 6 . Thus, for

A 2 we use local patching data x , y • y , ~ . We must have fil(~(f))>60.

This only requires in C(At) that we must use terms of filtration >30; and in C(A2), we must use terms of filtration >36. Also, it is readily checked that such terms satisfy filr,( )>36. It remains to check the jump condition at x4y . Since x 4 y is a vertex of both faces, filx4y(q~)=fil(q~ ). Thus, we must have fil(q~)>36. Thus, deformations involving terms except x 6 and x 7 satisfy the hypothesis of Theorem 3. More generally, for a deformation

f = f o + u i X6 + U2 X7 + h(x, y, u 3 . . . . )


with fil(h)>36, we claim that both X 6 and x 7 satisfy a general jump condition. 0f As Y ~YY - Y OJo mod ~ag36,,x,, both x4 y . x6 - x6 . y ~y and x4 y . x 7

7 Of - x .y~ves466 . . . . . Thus, the general deformation f and hence any defor-

marion involving x 6, x 7 and terms of filtration > 36 is topologically trivial.

Example 7.3. (Modified Brian~on-Speder example [4]).

fo(X ' y, x ) = x 5 + y7 2+y8 +715.

This fo is non-degenerate with Newton polyhedron shown in Fig. 6.

Z 15

y7 Z

yS

X 5

The term x y 6 lies below the Newton polyhedron. However, we still claim the deformation fo--l-13xy 6 is topologically trivial. In fact, fo is semi-weighted homogeneous, and with respect to the weighting wt(x ,y , z)=(3, 2, 1), w t ( x y 6) = 15. Thus, by Corollary 5, the deformation is topologically trivial. A similar situation can arise for other deformations under the Newton polyhedron, where it is necessary to replace a non-degenerate function by a semi-fit part which still has an isolated singularity and apply Theorem 1 to prove topological triviality.

w 8. The Push-off Lemma and a First Consequence

Using the polyhedral structure of F(fo ), we use an inductive procedure to establish the filtration conditions from the jump conditions. We inductively show that, modulo higher filtration, we can push the support of germs for which the filtration condition fails onto higher dimensional faces. After n-steps, we have pushed it into a higher filtration and then we can use the filtered version of Nakayama's lemma (Lemma 1.1). In the most general form, we have to be able to do this "stably", i.e., after first multiplying by sufficiently high powers of vertices.

Lemma 8.1 (push-off lemma). Let f: k "+', O ~ k , 0 be a deformation of the germ fo:kn, O ~ k , O so that fo has local patching data {~ , i } and fil(~A,i(f)) =fil(~a.i(fo)) for all AeC( fo ) and all i. Then, there exists an N so that if O S ~ , u


has f i l (O)=q>N and supp(O)cA then there is a O 1 E ~ q . . . . with supp(Ol) c s t a r (A) -A and

~ - t p , e ~ dq ... . . ~A,i(f)+~Cq+ 1 . . . . i

where ql =q - fil (~a, i(fo)).

Proof Let .~t be a fit level for the Newton filtration offo , then -~"~r is fit for all integers r > 1. From the definition of local patching data, it follows that there is r so that the ideal Ia generated by {in(~A,i(fo))[3 } contains ~4,tc~k[[zl]] for all faces A6F(fo). Then, there are two main steps.

Step 1. We let {~r denote the Newton Filtration on (~,[[x,]] analogous to that defined on cgx. u. We first wish to give for each face A~F(fo) a set of generators for ~/rt,,~cg,[[zl]] which restrict to a set of monomial generators for ,4~, when u=0.

We note that ,~,tc~k[[A]] is generated as an ideal in k[[z]]] by a finite number of monomials. Then, it is easily seen that ~q/,,.c~c-g,[[z/]] is also generated as an ideal in ~,[[zl]] by the same set of monomials. Thus, by Nakayama's lemma it is sufficient to give a set of elements in ,~J,l,,c~Cg,[[J]] which restrict to the set of monomial generators of .~,c~k[[zl]] when u=0.

We may represent the monomial generators of . ~ c ~ k [[A]] in the form

x " = ~ hij in (r j(fo))[a" d

As ill( )]k[[zl]] comes from a weighting and in( )]a is weighted homogeneous with respect to this weighting, we may also assume the h~2 are weighted homogeneous with respect to this weighting. Thus, in(h~)=h~_and fil(h~) =fil(x")-fil((a,i(fo)). Then, the set of generators for ~'rl,~c~%[[A]] is given by {~i} where

~i = ~', hij in(ff~, i(f))l~. J

Note the assumption fil(~z,i(f))= fil(ffz,i(fo)) ensures that ffir162

Step 2. By Step 1, x'~ver(~Crt)~k[[A]] has a representation

x ~ = ~ hl ~) in ((a. ~(f))13 (rood ,~r + t,,,) (8.2)

with hl')~k[A]. Then, by the contraction lemma, we may suppose

x~ - ~ hl ~ ~ , i(f) = g.(mod ,q/,, + ~,,,)

with supp(g,) star(x ) - A (otherwise, we could multiply (8.2) by (x~) L for some L and replace rl by a larger R l). Denote ~ hl ") ~a, i(f) by qS~a ").

Then, ver(~'~3~ C(z]) generates an ideal of finite codimension in k[[zl]] for each face AcF(fo) . We suppose that N is chosen so that _~uc~k[[A]] is contained in the ideal generated by ~za" (ver (~r C(z~)) in k[[A]].

To prove the result, it is enough to verify it for monomials x ~ with ~ C ( A ) and fil(x~)=q>N. We may write x~=xt~.x" with x'ever(s~',3nC(zl), and fleC(A). As 7eC(A), fl and �9 are not both contained in a closed subface of z~

354 J. D a m o n and T, Gaffney

(hence star(xtJ)c~star(x')=star(A), where for x ~ not a vertex, star(x ~) denotes star(A') for A' the smallest face with C(A') containing fl). Thus,

and xt~(x ~ - r )) =- x tj g'. (mod ~'q+ x, ,)

supp (x p g'.) c star (x ~) c~ star (x ') - A = star (A) - A.

Here g', consists of the terms of g. of filtration q and supp c star(xr otherwise by the properties of the Newton filtration, the filtration of x a times such a term will have filtration > q.

Lastly, xP q~') = }-'1 xP hl ") ~A,,(f);

and by the properties of ill( )l C(A)

ill(h} ")) = ill(x*) - fil (~a, , ( f)) s o

ill (x ~ h} ")) = ill (x ~) + fil (x') - ill ({a,~ (f))

= fil (x') - fil (~a,,(f)). Hence,

x ~ = x tl g ' ( m o d ~ dq , , , Q~, , ( f ) + ~r + ,,.).

Since x ~ hl')ecg~,., this also yields

x '=-xPg ' , (mod~s~q .... , ~A , , ( f )+~ 'q+ , . . . . ). [7

As a consequence of the push-off lemma we have

Proposition 8.3. Let fo have local patching data {~ , i} and let f be a deformation o f f o so that f i l (~n , i ( f ) )=f i l (~ , i ( fo ) ) fo r all A c F(fo) and all i. Then, there is an N so that if qJecgx, . with f i l (~9)=q>N then

A,i

where qa, ~ = q - fil (~ , i(f)) > O.

Proof Let N be as in the push-off lemma and so that N>fi l (~a,~(f)) all A, i. Fo r q > N , we have the cgx,.-modules dq+ j . . . . and ~- 'dj . . . . . . . �9 ~a,i(f) (summed over all A, j) where Ja, i = q + J - fil (~a, i(f)).

By inductively applying the push-off lemma to faces of F(fo) of increasing dimensions, we obtain

dq+j . . . . = ~ ~ j . . . . . . . �9 ~ , , ( f ) + s l q + j + t , , , . j>O. (8.4)

Since the filtration {dq+j . . . . } is equivalent to that of (~l,f'~q+Jx, ul~--t~q+J'(~-- I.""x ~x, u}' we can apply the filtered version of Nakayama ' s lemma to (8.4) to conclude


w 9. Deriving the Filtration Conditions from Jump Conditions

We need a "stable version" of the push-off lemma.

Lemma 9.1 (stable push-off lemma). Let fo be a germ with local patching data {(a,i} and let f be a deformation with fil((A,i(f))=fil((a,i(fo)) for all A c F ( f o ) and all i. Let ~Ecg,[x,] have f i l (~)=q and s u p p ( ~ ) c s t a r ( v ) f o r v a vertex of F(fo). Also, let x '~C(v) with f i l (x ' )=l . Then, there is a r > 0 and a polynomial

OIEY',~4q . . . . . . . ' ( a , i ( f ) where qa, i=q+rl - f i l ( (a , i ( fo) )

(summed over A c star(v)) so that

i) (x')" 0 - 0 , (mod ~/q+,l +1 . . . . ) ii) supp ((x') ' �9 0 - 01) c star (v).

Proof We will inductively apply the push-off lemma for a r~ and 0 ti) so that

supp ((x'F ' . 0 - O~li~)(q + ,,t~) c star (v) - U {A of dim < i}.

Here gm denotes the terms of g of filtration __<r. Then, r, and 0(~ ") will be our desired r and 01.

We begin the induct ion with r o = 0 and 0~ ~ Suppose the result is true for i<j (where j<n). Let 0 ' = ( x ' ) ' , - ' t b - 0 ~ j - i ) . There is an s so that fil(~k') + s l > N , where N is that obtained from the push-off lemma. Then, by the push-off lemma, combined with Proposi t ion 8.3, there is, in fact, a polynomial

with

so that PA, i = fil (O') + s l - fil (~a, , (f))

supp ((x') s. ~' - I]/")(fil(O,)+sl)) CZ star (x ~) -- [.) {A of dim _-<j}.

However, s u p p ( ( x ' ) * . 0 ' - O " ) may not be in star(v). Then, by the con- t ract ion lemma there is an s' so that

" f ; CZ supp((x')*'((x=) ~. ~ - ~ )) star(v).

We let rj = r j_ ~ + s + s' and

O~'=(x'r+s'O~-"+(x') ~'' ~". H

Corollary 9.2. Let fo be a germ with local patching data {(A,i} and let f be a deformation with fil((a,i(f))--fil((a,i(fo)) for all d c F ( f o ) and all i. Let ~ecg.[-x.] with q v = f i l . ( ~ ) f o r all vEF(fo). Then, for x~eC(v) with f i l(x~)=l and r > 0, there exists an s > 0 so that

(x')*. ~ e ~ dq., , . . . . �9 {a,i(f)+ dq.+s,+ . . . . .

with qa, i=qv+ s l - f i l ( (a , i ( f ) ) and the sum is over d c s t a r (v ) and the i's for these d's.


Proof We repeatedly apply the stable push-off lemma beginning with (x') s'. ~p so that supp((x')~' .@)=star(v) (by the contraction lemma). After r appli- cations, we have the desired result. VI

We are now in a position to prove Theorem 3.

Proof of Theorem 3

Let d t be fit and let ver(d~)={x '} . Next, let N be as in proposition 8.3. We consider a germ ~O~cg~,u which satisfies a jump condition. We let qJl denote the terms of filtration < N in the Taylor expansion of ~ (in the x-coordinates). Then qJ2=@-~91 has filtration fil(I//z)~N. First, consider the case of the simple jump condition.

Let x'~ver(~Ct) with say x '~C(v) for v~F(fo). By Corollary 9.2, there is an s > 0 so that

(x') s" I/I I E Z ~q . . . . . . . " (~, i( f ) + ds~ +q~ + N (9.3)

where the sum is over A=star(v) and all i for each such A and qA, i=qv+Sl -- ill(Ca, ~(f)). Then

fil (Ca, i ( f ) )= fil (r (fo))

= m i n { f i l ( g ~ , i , j ) + f i l ( x j ~ ) } + j u m p ( ( ~ , i )

because fo being semi-fit guarantees that xj ~xj wilt have a non-zero term of

filtration = m in every maximal dimensional closed face. Thus,

fil (~ , i(f)) < min { fil (g~, i, j)} + m + jump (v). Hence,

qa, i > q~ + s I - m - j u m p (v) - min { fil (g~, i, ~)}. (9.4)

If from 9.3, we write

then,

Also,

(x~) ~' ~ , = ~ hA,i ~ , i ( f ) + ~P'

(x~)~ " ~ ' = ~-'j (~,i h~'ig'J'i'J) XJ ff~fxj + tp'.

fil (ha, ~ ga, ,, j) > fil (ha, ~) + fil (ga, ,, j)

> qa, i + fil (ga, i,j) > sl + q v - m - j u m p ( v )

>s l

by (9.4) and the simple jump condit ion on ~b (recall qv=filv(@0). Thus,


Then,

Hence,

Also,

(x') ~" 0n - 0' mod d~, . . . . xj ~ .

(x ' ) s O = ( x ' ) s O ~ + (x') s 02.

(x~)S'~=llt'+(x*)S'O2mod~4st .... {xJ~fxj }"

fil (O' +(x~)Stp 2)>=sl + L > sl + N.

Thus, by Proposition 8.3

tP' +(x') s" 0 2 ~ , d , ....... ~, i( f)

where r~,i>sl+N-fil(~a,i(f) ). By Proposition 8.3 rA, i>=sl. Hence,

, s 0f

By (9.5), we conclude

(9.5)

N > fi l(~,i(f)) , thus,

(x')S. 0e~'s, .... {xj ~@j}.

For the general jump condition, the argument is similar. We let (x*) �9 01

- 0 '=0" , where q*e~ ' m . . . . . xj for mv=fil(x ") and filv(O")>m,,+m

+jump(v). By an argument similar to that for the simple jump condition, there is an s so that

(x~) s" O " e d ~ s + ,~ . . . . . . x j .

(x,) ,+ ~. ~, =(x,)S+ n 0n + ( x ' ) ~+' ~'2

= ( x ' ) s ~" + (x') ~" 0" + (x') s + n ~2.

Then,

Again

This together with (9.6) gives the result. D

w I0. t~-constant Deformations of Real Singularities

The structure of real isolated singularities fo: 11", 0 ~ I ( , 0 barely begins to approach the rich structure for complex isolated singularities which results from the Milnor fibration. Nonetheless, we have seen that even without the richer topological structure, deformations of real germs are topologically trivial under conditions analogous to when complex germs are (however, not con- versely, for example, any Pham-Brieskorn polynomial e 1 x~' + . . . + e, x~," over IR defnes a germ topologically equivalent to either the germ of a submersion or a


Morse singularity). This suggests that there may be an analogue of the L6- R a m a n u j a m result for real germs. We end our discussion by indicating how results of King [7] yield analogues for real plane curve singularities and surface singularities in R 3.

Theorem 10.1. Let f (x ,u) : ~ s + ~ , O ~ , O be a deformation of fo:RS, O--*~,O, s = 2 or 3, so that the Milnor number # ( f ( ' , u)) is constant in a neighborhood of O. Then, f is a topologically trivial deformation.

cgx/[Of( ' ,u) , 3f(',_u)~, which is the dimen- Proof Since # ( f ( - , u ) ) = d i n ~ / \ 0x 1 " " ' 0x~ ]

sion of the local algebra of g radx f : R~, 0 ~ R ~, 0, it follows by the upper semi- continuity of the algebraic multiplicity that in a ne ighborhood of 0 in N~+r, the singular set o f f is {0} x R r. Thus, g radxf : ~ , ? - { 0 } - - * R s - { 0 } is a ho- mo topy for small u ~ R ' . Hence, d ( f ( . , u ) ) = d e g ( g r a d j ( . , u)) is independent of u for small u. In the case of plane curve singularities, this is sufficient to determine the structure of the curve singularity. For by a result of Arnold [3], the Euler class o f f ( . , u)-l(O)c~(S~ x {u}) (which is a finite set of points) is given by 2 ( 1 - d ( f ( . , u))). Thus, locally f ( . , u) - l (0) is a cone on this number of points, which is independent of u for u near 0.

However, more generally since the singular set of f ( x , u)={0} • in a ne ighborhood of (0, 0), by the results of Henry King (Theorem 2 and Corol- lary 1 of [7]) these deformations for the case of curves and surfaces are topologically trivial. D

References

1. Arnold, V.I.: Spectral sequence for reduction of functions to normal form. Funct. Anal. and Appl. 9, 251-253 (1975)

2. Arnold, V.I.: Local normal forms of functions. Invent. Math. 35, 87-109 (1976) 3. Arnold, V.I.: Index of a singular point of a vector field, the Petrovski-Oleinik inequality, and

mixed hodge structures. Funct. Anal. and Appl. 12, 1-11 (1978) 4. Brianqon, J., Speder, J.P.: La trivialite topologigue n'implique pas les conditions de Whitney.

Comp. Rendus t280, (series A) 365 (1975) 5. Damon, J.: Finite determinacy and topological triviality I. Invent. Math. 62, pp. 299-324 (1980) 6. Damon, J., Gaffney, T.: Topological triviality of deformations of functions and newton fil-

trations, preliminary announcement 7. King, H.: Topological type in families of singularities. Invent. Math. 62, 1-13 (1980) 8. Kouchnirenko, A.G.: Polyedres de Newton et Nombres de Milnor. Invent. Math. 32, 1-31

(1976) 9. Laufer, H.: On minimally elliptic singularities. Amer. Jour. Math, 99, 1257-t295 (1977)

10. L6 D~ng Trfi.ng, Ramanujam, C.P.: The invariance of Milnor's number implies the invariance of topological type. Amer. Jour. Math. 98, 67-78 (1976)

11. Looijenga, E.: Semi-universal deformation of a simple elliptic singularity: Part I unimodularity. Topology 16, 257-262 (1977)

12. Oka, M.: On the bifurcation of the multiplicity and topology of the newton boundary. J. Math. Soc. Japan 31, 435-450 (1979)

13. Teissier, B.: Cycles evanescents, sections planes, et conditions de Whitney. Singularities a Cargese. Asterisque 7 and $, 285-362 (1973)

14. Timourian, J.G.: Invariance of Milnor's number implies topological triviality. Amer. Jour. Math. 99, 437-446 (1977)

Oblatum 39-I & 25-X-1982

Topological triviality of deformations of functions and Newton filtrations

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