THE CLASSIFICATION OF ELEMENTARY CATASTROPHES: Christopher Zeeman PDF

THE CLASSIFICATION OF ELEMENTARY CATASTROPHES OF CODIMENSION* ~ 5. by Christopher Zeeman University of Warwick...

THE CLASSIFICATION OF ELEMENTARY CATASTROPHES OF CODIMENSION* ~ 5. by Christopher Zeeman University of Warwick Coventry, England (Notes written and revised by David Trotman) INTRODUCTION. These lecture notes are an attempt to give a minimal complete proof of the classification theorem from first principles. All results which are not standard theorems of differential topology are proved. The theorem is stated in Chapter 1 in a form that is useful for applications. The elementary catastrophes are certain singularities of smooth maps R r + R r. They arise generically from considering the stationary values of r-dlmensional families of functions on a manifold, or from considering the fixed points of r-dimensional families of gradient dynamical systems on a manifold. Therefore they are of central importance in the bifurcation theory of ordinary differential equations. In particular the case r - 4 is important for applications parametrised by space-time. The concept of elementary catastrophes, and the recognition of their importance, is due to Ren@ Thom. He realized as early as about 1963 that they could be finitely classified for r ~ 4, by unfolding certain polynomial germs (x3,x4,x5,x6,x3_+xy2,x2y+y4). Thom's sources of inspiration were four- fold: firstly Whitney's paper [ii] on stable-singularitles for r - 2, secondly his own work extending these results to r > 2, thirdly light caustics, and fourthly biological morphogenesis. *This paper, giving a complete proof of Thom's classification theorem, seems not to be readily available. In response to many requests from conference participants, Zeeman and his collaborator, David Trotman, agreed to make a revised version of the paper (July, 1875) available for the conference proceedings. I would like to express my appreciation to both Christopher Zeem~n and David Trotman. Peter Hilton  264 However although Thom had conjectured the classification, it was some years before the conjecture could be proved, because several branches of mathematics had to be developed in order to provide the necessary tools. Indeed the greatest achievement of catastrophe theory to date is to have stimulated these developments in mathematics, notably in the areas of bifurcation, singularities, unfoldings and stratifications. In particular the heart of the proof lies in the concept of unfoldings, which is due to Thom. The key result is that two transversal unfoldings are isomorphic, and for this Thom needed a C= version of the Weierstrass preparation theorem. He persuaded Malgrange to prove this around 1965. Since then several mathematicians, notably Mather, have contributed to giving simpler alternative proofs [4,5,7,8] andthe proof we give in Chapter 5 is mainly taken from [i]. The preparation theorem is a way of synthesising the analysis into an algebraic tool; then with this algebraic tool it is possible to construct the geometric diffeomorphism required to prove two unfoldings equivalent. The first person to write down an explicit construction, and therefore a rigorous proof of the classification theorem, was John Mather, in about 1967. The essence of the proof is contained in his published papers [4,5] about more general singularities. However the particular theorem that we need is somewhat buried in these papers, and so in 1967 Mather wrote a delightful unpublished manuscript giving an explicit minimal proof of the classification of the germs of functions that give rise to the elementary catastrophes. The basic idea is to localise functions to germs, and then by determinacy reduce germs to Jets, thereby reducing the | problem in analysis to a finite dimensional problem in algebraic geometry. Regrettably Mather's manuscript was never quite finished, ahtough copies of it have circulated widely. We base Chapters 2, 3, 4, 6 primarily upon his exposition. Mather's paper is confined to the local problem of classifying germs of functions. To put the theory in a usable form for applications three further steps are necessary. Firstly we need to globalise from germs back to  265 functions again, in order to obtain an open-dense set of functions, that can be used for modeling. For this we need the Thom transversallty lemma, and Chapter 8 is based on Levine's exposition. Secondly we have to relate the function germs, as classified by Mather, to the induced elementary catastrophes, which are needed for the applications. For instance the elliptic umbilic starts as an unstable germ ~2 + R, which then unfolds to a stable-germ R 2 x ~3 + ~ ~3, or equivalently to a germ f: R 2 x R 3 + ~, and eventually induces the elementary catastrophe germ Xf: R 3 + R 3. The relation between these is explained in Chapter 7. Finally in Chapter 9 we verify the stability of the elementary catastrophes, in other words the stability of Xf under perturbations of f. A word of warning here: although the elementary catastrophes are singularities, and are stable, they are different from the classical stable-singularities [1,2,4,5,11]. The unfolded germ is indeed a stable-singularity, but the induced catastrophe germ may not be. The difference can be explained as follows. Let M denote the space of all C~ maps ~r + ~r, and C the subspace of catastrophe maps. Then C # M because not all maps can be induced by a function. Therefore a stable-singularity, such as Z2' may appear in M, but not in C, and therefore will not occur as an elementary catastrophe. Conversely an elementary catastrophe, such as an umbilic, may appear in C, and be stable in C, but become unstable if perturbations in M are allowed, and therefore will not occur as a stable-singularity.. For r = 2 the two concepts accidentally coincide, because Whitney [ii] showed that the only two stable- singularities were the fold and cusp, and these are the two elementary catastrophes. However for r = 3 the concepts diverge, and for r = 4, for instance, there are 6 stable-singularities and 7 elementary catastrophes, as follows:  266 ) stable-singularit ies elementary catastrophes We are grateful to Mario De Oliveira and Peter Stefan for their helpful comments: these have led to several corrections in the text.* *As editor, I w o u ~ also like to express my gratitude to Sandra Smith for adapting the original manuscript to a form suitable for the Lecture Notes, and to Sarah Rosenberg for her skillful reproduction of the diagrams. Peter Hilton  267 CONTENTS Chapter i. Stating Thom's Theorem Chapter 2. Determinacy Chapter 3. Codlmenslon Chapter 4. Classification Chapter 5. The Preparation Theorem Chapter 6. Unfoldlngs Chapter 7. Catastrophe Germs Chapter 8. Globallsatlon Chapter 9. Stability  268 CHAPTER i. STATING THOMIs THEOREM. Let f: ~ n M r + ~ be a smooth function. Define Mf c ~ n+r to be given by ~(~f-~''''' ~f ~x ) = gradxf = 0, where Xl,..., x n are coordinates for A n, and w I n YI' "''' Yr are coordinates for M r. Generically Mf is an r-manifold because it is codimenslon n, given by n equations. Let Xf: Mfr § be the map induced by the projection ~n+r + Mr. We call Xf the catastrophe map of f. Let F denote the space of C~-functions on ~n+r, with the Whitney C=-topology. We can now state Thom's theorem. Theorem. If r S 5, there is an open dense set F, r F which we call senerlc functions. If f is generic, then (i) Mf is an r-manifold. (2) Any singularity of Xf is equivalent to one of a finite number of types called elementary catastrophes. (c) Xf is locally stable at all points of Mf with respect to small perturbations of f. The number of elementary catastrophes depends only upon r, as follows: r i 2 3 4 5 6 7 elem. cats. 1 2 5 7 Ii ~ Here equivalence means the following: two maps X: M ~ N and X': M' ~ N' are equivalent if 3 diffeomorphisms h, k such that the following diagram commutes: M N Xv Now suppose the maps X, X' have singularities at x, x' respectively. Then the singularities are equivalent if the above definition holds locally, with hx = x'.  269 Remarks. The reason for keeping r ~ 5 is that for r > 5 the classification becomes infinite, because there are equivalence classes of singularities depending upon a continuous parameter. One can obtain a finite classification under topological equivalence, but for applications the smooth classification in low dimensions is more important. The theorem remains true when ~n+r is replaced by a bundle over an arbitrary r-manifold, with fibre an arbitrary n-manifold. The theorem stated above is a classification theorem: we classify the types of singularity that 'most' Xf can have. We find that if Xf has a singularity at (x,y) ~ n + r n Mf, and if n is the germ at (x,y) of fl~nx y, then the equivalence class of Xf at (x,y) depends only upon the (right) equivalence class of n (Theorem 7.8). This result is hard and requires an application of the Malgrange Preparation Theorem, itself a consequence of the Division Theorem (Chapter 5), and study of the category of unfoldings of a germ ~ (Chapter 6). To use it we have first to classify germs n of C~ functions ~n,o ~ ~,0. We use two related integer invariants, determinacy and codimension, and the jacobian ideal A(~) (the ideal spanned by ~n..., ~n_~_ in the ~x I ' ~x n local ring ~ of germs at 0 of C~ functions ] i n § R). The determinacy of a germ n is the least integer k such that if any germ $ has the same k-jet as n then E is right equivalent to n. Theorem 2.9 gives necessary and sufficient conditions for k-determinacy in terms of A. Defining the codimension of n as the dimension of m/t, where m is the unique maximal ideal of E, we use this theorem to show that det ~ - 2 ~ cod n in Lem~na 3.1. If r ~ 5 and f E F, then if n = flRnxy , for any y ERr, we have cod n ~ r. Hence since we can restrict to cod n ~ 5 we need only look at 7-determined germs in the vector space j7 of 7-jets. We must restrict to r ~ 5, for if cod n ~ 7 there are equivalence classes depending upon a continuous parameter, and the definition of F, ensures that if r = 6 then each of these equivalence classes contains an fI~nx y for some y ERr and f E F,.  270 The 7-jets of codimension ~ 6 form a closed algebraic variety Z in j7 and the partition by codimension of J7-Z forms a regular stratification (Chapters 3 and 8). We in fact use a condition implied by a-regularity (Definition 8.2). This is necessary to show that F, is open in F. That it is dense follows from Thom's transversality lemma; and transversality gives that Mf is an r-manlfold for f E F, (Chapter 8). The classification of germs of codimension ~ 5 is completed in Chapter 4 and in Chapter 7 the connection is made with catastrophe germs. Finally in Chapter 9 we show the local stability of Xf. CHAPTER 2. DETERMINACY. Definition. Suppose C'(M,Q) is the space of C" maps M ~ Q, where M and Q are C = manifolds. If x ~ M and f and g ~ C'(M,Q) let f ~ g if a neighborhood N of x such that fin = giN. The equivalence class If] is called a germ, the germ of f at x. Let En be the set of germs at 0 of C" functions Rn§ It is a real vector space of infinite dimension, and a ring with a i, the I being the germ at 0 of the constant function taking the value i ~R. Addition, multiplication, and scalar multiplication are induced pointwise from the structure in ~. Definition. A local ring is a commutative ring with a i with a unique maximal ideal. We shall show that E is a local ring with maximal ideal m n n being the set of germs at 0 of C| functions vanishing at 0 (written as functions Rn,0 ~ R,O). Lemma 2.1. m is a maximal ideal of E. n n Proof. Suppose n ~ E and n ~ m. We claim that the ideal generated by m n n n and n, (mn,n) E , is equal to En. n  271 Let the function e 6 n, i,e, n is the germ at 0 of e, and choose a neighborhood U of 0 in ~n such that e ~ 0 on U. Then i/e exists on U. Let ~ be the germ [l/e], then En = [i/e]'[e] = [i/e'e] = [i] = i. Also ~n ~ (mn,n) E 9 Thus (mn,n) E = E. n n n Lemma 2.2. m is the unique maximal ideal of E. n n Proof. Given I ~W En, we claim I c mn. If not 3n E I - n m ' and then as in Lemma 2.1 an inverse exists in E. 1 = i/n-~ E I, and so I = E. n n Lemma 2.1 and Lemma 2.2 show that E is a local ring. n Let G be the set of germs at 0 of C~ dtffeomorphisms IRn,o+Rn,o. n Gn is a group with multiplication induced b y composition. We shall drop suffices and use E, m and G, when referring to En, m n and G rather than E n s when n # s, etc. Given ~l' "''' ar ( E, we let (al,...,ar) E be the ideal r generated b y {ai } = { Z ~.~. : gi s E}, and drop the suffix if there is no risk i=l z l of confusion. Choose coordinates Xl,..., x n in IRn (linear or curvilinear). The s y m b o l 'x t' will be used ambiguously as: (i) coordinate of x = (Xl,...,Xn) , x i (I~. (ii) function x. : l~n,o -+ ]R,O. i (iii) the germ at 0 of this function in m c E. (iv) the k-jet of that germ (see below). Lemma 2.3. m = (x I..... Xn) E = ideal of E generated by the germs x i. Proof. Given n E m, represent n by e: ~n,o.-----~,0. VxE~ n, ae e(x) = f~ ~ (tx)dt n ~e - (tx)xi(x)dt 1 n = z e. ( x ) x _ ( x ). i=l i i n n e = Z e~x. as functions and so n = Z e,x, as germs. Thus m c (x 1..... Xn). i=l x x i =I i i (x I..... Xn) C m because each x i ( m.  272 k Corollary 2.4. m is the ideal generated by all monomials in x. of degree k. i k Corollary 2.5. m is a finitely generated E-module. We let 7k be the q u o t i e n t E/mk+l, and l e t jk be m/mk+l, jk denotes the canonical projection E § 7 k. Lemma 2.6. jk is i) a local ring with maximal ideal jk, 2) a finite-dimensional real vector space (generated by monomials in {xi} , of degree ~ k). Proof. i) jk is a quotient ring of E and thus is a commutative ring with a i. There is a i-i correspondence between ideals: E E/mk+l=J k U U 9 I/m k+l U k+l m So 7k is a local ring. 2) jk is a quotient vector space of E and is finite-dimensional. For given n ~ E, the Taylor expansion at 0 is, n = n o + nl + " "" + nk + Pk+l' where n~ is a homogeneous polynomial in {x i} of degree j, with coefficients k+l. the corresponding partial derivatives at 0, and Pk+l ~ m.k Definition. The k-jet of n = 3 n = n o +... + n k = Taylor series cut off at k. jk and jk are spaces of k-jets, or jet spaces. Definition. If n, ~ ~ E we say they are right equivalent (~) if they belong to the same G-orbit. n ~ ~ ~ 3~ ~ G such that ~ = ~y.  273 Definition. If n, ~ E E we say they are k-equivalent (~) if they have the same k-jet, n ~k ~ ~ jkq = jk~. Definition. n ( E is k-determinate if V~ E E, n ~ $ = n ~ $. Clearly n k-determinate = n i-determinate Vi ~ k. The determinacy of q is the least k such that q is k-determinate. We write det D. Lemma 2.7. If q is k-determinate then i) q ~ $ = ~ k-determinate, 2) q ~ $ = ~ k-determinate. Proof. i) follows at once from 2), which we shall prove. Assume n ~ $, i.e. D = ~YI' some 71 E G. Suppose ~ ~k v, i.e. jk$ = jkv, i.e. jk(qy;l) = jkv. Then jkq = jk(qy;iyl ) = jk(qy~l).jk(yl) = jkv.jkyl = jk(vYl). So k q ~ ~Yl' which = n ~ vy I, i.e. n = ~yiy 2 some y2 E G. Then ~Yl = v ~ Y 2 ' -i and ~ = v~172Y I , i.e. ~ ~ v. So 2) is proved. Definition. If n E E, choose coordinates {x i} for A n, and let A = A(q) = (~x~ Bq )E" A is independent of the choice of coordinates. For J. ,''',~x n ~n ~_) ~n ~ ~n ~xi if A x = (~-~) and A = ( = ---- E ~ and so A c A. l Y J ' ~Yj i=l ~x i ~Yj x y x @n ~xi (~-~i 6 Ax, each i, and --~yj E E, each i, j). Similarly Ax c &y, so &x = Ay. Lemmm 2.8. If n E E - m, and q' = n - n(O) E m, then A(n) = A(q'), and n is k-determinate n' is k-determinate. Proof. = is trivial trivially. t n( 0) = ~(o). Also n = ~y ~, ~q' = ~ ' y , y ( G n(O) = ~ ( 0 ). Thus n ~ ~ "~ ~n' ~ ~' n(O) = ~(o). So from now on we shall suppose q ( m.  274 Theorem 2.9. If n E m and A = A(n), then m k+l m2A ~ k+l C is k-determinate = m 9 mA. Proof. We shall use the following form of Nakayama's Lemma: Lemma 2.10. If A is a local ring, d its maximal ideal, and M, N are A-modules (contained in some larger A-module) with M finitely generated over A, then M c N +aM = M c N. Sublemma. t E A, ~ ~ a = l-I E A. Proof. IA is an ideal ~ a. So XA = A 9 i, 3~ such that X~ = i. Proof of Lemma 2.10. We shall first prove the special case of N = 0, i.e., M 9 a M = M = 0. Let Vl,..., v r generate M. v i E aM by hypothesis, r so v. = E X..v. (Xij6~) z j=l 13 3 r or j=IZ(6..13 - Aij)vj = 0, i.e. (l-^)v = O, where A is an (r (llj), and v =~Vl~. The determinant II-^] = i + X, some I E a. Now 1 + I ~ a, else -J\Vr, 1 E a and a = A. So (l+l) -I exists by the sublemma. Then (I-A) -I exists, giving v = 0 and M = 0. To prove the general case consider the quotient by N~(M+N)/N c N/N + (aM+N)/N. We claim the R.H.S. =a(M+N)/N. (*) Then by the special case, (M+N)/N = O, giving Mc N. Q.E.D. The A-module structure on (M+N)/N is induced by that on M + N by %(v+N) = %v + N. a(M+N)/N = {X(v+N): X ( a, v ( M } = {~v+N: X E a, v E M ) = (aM+N)/N, proving (*). Continuing the proof of Theorem 2.9, we assume m k+l c m2A, and must show that k n ~ ~ = n ~ ~. The idea of the proof is to change n into ~ continuously with the assumption ~ ~ 6. Let ~ denote the germ at 0 x ~ of a function ~n x ~ +~ given by @(x,t) = (l-t)n(x) + t~(x), x E ~ n, t E 2. Let  275 ~t(x) = ~(x,t) = ~n(x) t = 0 (~(x) t = i. Lemma i. Fixing to, 0 ~ t o ~ i, 3 a family Ft ~ G defined for t in a neighborhood of to in such that i) rt0 = identity 2) #tyt to Lemma i will give n ~ ~: Using compactness and connectedness of [0,i], cover by a finite number of neighborhoods as in Lemma i, then pick {t.} in i the overlaps, and construct V satisfying n = ~Y by a finite composition o f {rti @0. ~ ~i {. }, i.e. ~ = ~.. = Lemma 2. For 0 ~ t ~ i, 3 a germ r at (p,t o) of C" maps IRn x I{ -+I~n o satisfying (a) r(X,to) = x, (b) r(o,t) = 0, (c) ~(r(x,t),t) = ~(X,to), for all (x,t) in some neighborhood of (0,to). Lemma 2 will give Lemma i: Define rt(x) = r(x,t) from a neighbor- hood of 0 in ~n to ~n; rt is a germ of C" maps ~n,0 ~ R n , 0 by (b); r ~~ is the identity by (a), C" dlffeomorphisms are open in the space of C" maps ]Rn,O ~ n , o (because they correspond to maps with Jacoblan of maximal rank, i.e. to the non-vanishing of a certain determinant), and so 3 a neighborhood of t such that rt is a germ of dlffeomorphlsms for t in that o neighborhood, i.e. st s Lemma 3. (c) in Lemma 2 is equivalent to, Ss i (e') ~ (r (x,t) ,t) (x,t) + ~ ( r ( x , t ) , t ) -- O. (c) = (c'): by differentiation with respect to t. (c') ~ (c): 0 = ft (c')dt = 0(r(x,t),t) = ~(r(X,to),t o) t o = ~(r(x,t),t) - ~(x,t o) by (a) in Lerma 2. Thus we have (c). Lemma 4. For 0 _< t o _< i, 3 a germ ~ at (0,t o) of a C" map I~n 1~-~1~ n  276 satisfying (d) ~(0,t) = O, (e) ~ 8~ (x,t)~i(x,t) + 3~ (x,t) = 0, i=l ~ for all (x,t) in some neighborhood of (0,to). Lemma 4 = Lemmas 3 and 2: The existence theorem for ordinary ~F differential equations gives a solution F(x,t) of ~ = ~(F,t), with initial condition r(x,t o) = x (i.e. (a) of Lemma 2). In (e) put x = r(x,t) to give (c'). (d) = r = 0 is a solution, i.e. F(0,t) = 0 for all t in some neighborhood of to, which is (b). Let A denote the ring of germs at (0,to) of C~ functions A n A § A. Projection A n A +A n induces an embedding E c A by composition. Let ~ = 8(~_~ 3~.... '~x )A" ~i n Le~ma 5. m k+l c m2A = m k+l c m2n. Lemma 5 = Lemma 4 as follows: 8--~ = ~ - n ( m k+l c m2~. Thus ~8~ = jZN.m., J 3 Uj ( m 2, mj ( ~. (finite sum) 8r 8~ = Zij ~jaij 8-~i, where ~.= j ~aij 8-~i, aij ( A. = -Z~ii --Sxi, setting ~i = - ~ j a i j E A. This gives (e). Now ~j = ~j(x) and aij = aij(x,t). T = {%} is a germ at (O,t o) of a map ~n x A § n, and ~i(0,t) = 0 as each ~j(O) = O, so (d) holds for ~. Proof of Lemma 5. (and hence the completion of the proof of a sufficient condition for k-determinacy) ~ ~n + (~-n) 8x i ~x i ~x i ~n Ak (t ~A, ~ n (k+l) ( 8--~i+ i.e. ~n__q_ ( ~ + Am k c ~ + Am k. 3x i ~x i  277 So A c ~ + A m k. Denote the maximal ideal of A by a, i.e. those germs vanishing at (0,to). Then m c a. Now Am k+l e Am 2 (hypothesis) c Am 2 (~+Am k) = m2~ + Am k+2 c m2~ + ~Am k+l. Now apply Nakayama's Lemma 2.10 for A, a, M, N where M = Am k+l is finitely generated by monomials in {x i} of degree k + i By Corollary 2.4, and k+l N = m2~. This gives Am k+l c m 2 ~. In particular m e m2~, completing Lemma 5. Now we prove that m k+l c m& is a necessary condition of k-determinacy. a natural map m ~ jk+l ~ jk, ~ = jk+i/m" jk+l +__~.k nl 9 3 Let P = {$ ~ m: ~ ~k ~), and Q = {~ ~ m: ~ ~ ~} = orbit nG. Assuming that n is k-determinate then p c Q, so that ~p c ~Q. (*) P = n + m k+l, so ~P = z + +i /k +2 = z + ~m k+l (Letting z = jk+In). The k+l tangent plane to ~P at z, Tz(~P) = ~m Let Gk denote the k-jets of germs belonging to G; 0 k is a finite- dimensional Lie group. Now jk+l(ny) = jk+l(n)jk+l(y) for Y E G, i.e. ~ is equivariant with respect to G, G k+l. So ~Q = ~(~G) = zG k+l, an orbit under a Lie group, and hence is a manifold. In particular Tz(ZQ) exists. Lemma 2.11. Tz(~Q) = ~(mA). k+l Now (*) gives Tz(~P) CTz(~Q). Then Lemma 2.11 gives ~m c ~(mA), i.e. m k+l c mA + m k+2. Apply Nakayama's Lemma 2.10 with A = E, a = m, M = m k+l, N = mA, using Lem~as 2.1, 2.2 and Corollary 2.4, to yield m k+l r md. Proof (of 2.11). Suppose V E G. As Rn is additive we can write Y = i + ~, where I is the germ of the identity map, and ~ is the germ at 0 of a C~ map IRn,o § ]Rn,0. Join i to V by a continuous path of map-germs, t = i + t~,  278 t 0 -< t -< I. When t = 0 or i, y is a dlffeomorphlsm-germ. Diffeomorphisms are open in the space of C maps, and so 3 t > 0 such that ~t E G, o o_ Hence cod q = dim m/A ~ k - 2 ~ det n - 2, as required. Case (ii): If det n is finite, then mk c A for some k (Corollary 2.12). Then k + A = A = m k+l + A, and we are in Case (i). So det n is infinite. m/A D (m2+A)/A 9... is a strictly decreasing sequence and so cod n (= dim m/A) is infinity. Let F = {n ~ m2: cod q = c} (a 'c-stratum' of m2), and let c = {N ( m2: cod n ~ c , and Z = {~ ( m2: cod ~ ~ c}, so that c c 2 m = r 0 U r I U r 2 U... U pc U... U p. (disjoint union) Let Fkc' ~kc, Ekc be the images of Fc, ~e' Ec under the map ~: m 2 + I k (~ = j k l m 2 ) , where Ik is defined as m2/m k+l Just as jk is m/mk+l. Theorem 3.3. If 0 ~ c ~ k - 2, then i k = ~k U Z k (disjoint union), and c c+l Ek is a (closed) real algebraic variety. c+l Remark. Both statements are false for e > k - 2. Lemma 3.4. Dim E/m k+l (n+k)! n!k! , V n, k >_ O.  281 Proof. If n ffi 0, ~ = ~ , m = 0; L.H.S. = i = R.H.S. V k. If k = 0, E/m = ~ ; L.H.S. ffi i = R.H.S. V n. Use induction on n + k. Then ~/k+l = polynomials of degree E k in Xl,..., x n = (polynomials of degree k in x I..... Xn_ I) + x (polynomials of degree k - i in Xl,..., x n) n So dim E/mk + l (n+k-l)! (n+k-l)! (n-k)!k! + n!(k-1)! (by induction) (n+k)! nlk! Proof of Theorem 3.3. We define an invariant z(z) for z ( Ik = m 2 / k + l , Choose n E ~-iz. n ( m 2 , so A(n) = A c m. Define T(Z) = dim m/ (A+mk). We claim that T(z) is independent of the choice of n. Let n' be another choice, A(~') = A'. Then D - n' E m k+l, so a~ ax i ~n' ~xi E m k, and an__n_ ( A' + k. Hence A c 4' + m k and A + m k c A' + m k. ax i A' + m k c A + m k by symmetry. Hence A + k = A' + m k and T(z) is well defined. We claim that, (i) T(z) -< c = cod q = T(z), so z E 9 k. (3.5) (ii) T(z) > c = cod n > c, so z E Ik (cod n perhaps + T(z)) c+l" Because (i) and (ii) are disjoint, Ik is the disjoint union of flk and Zk c c+l' once we have shown (i) and (ii) hold. W e have E (Lemma 3.4) / Note that T(z) is finite, +i (n+k-l) ! ~ / although cod q may be infinite. n! (k-i) '/ m A-~k mk ~(z)+ ~ A  282 Case (ii): cod n >- ~(z) (from the diagram Thus (ii) holds. > C (hypothesis of (ii)) Case (i): k - 2 >- c (hypothesos of the theorem) >_ ~ ( z ) (hypothesis of (i)) We have a sequence, 0 = m/m = m/A + m < - - - - m / A + m 2 <... < m/A + m k < k-i steps > k - 2 > T(z) = dim m/A + k , so one step must collapse, i.e. A + m i-I = A + m i for some i ~ k, i.e. m i-I c A + m i. Nakayama's Lemma 2.10 = (mk c} nl(k-l)! c+l = {z ~ I k : ~(z) > c} (c ~ k - 2) = {z s Ik: o(z) < K}, w h i c h we shall show is an algebraic variety (real). If Xl,..., x n are coordinates for ~n, let the monomials of degree k in {xi} be {Xj} as below: (n+k)!l X1 X2 X3... Xn+ I Xn+ 2 Xn+ 3... X~ (8 = nlk! " 1 x1 x2... xn x~ XlX2 "'" xkn Now jk is the space of polynomials in {x i} of degree ~ k with coefficients in R and no constant term. z E Ik can be w r i t t e n z = E a.X. (aj E ~ ) o j=n+2 3 3 ~z Because ~ is a polynomial of degree k - i with no constant term it belongs w i ~z B (n+k-l)' to jk-l, so ~x i = j=2 Z aijXj' (~ = n!(k-l)!"), w h e r e each aij is an integer multiplied by some ak. Just as A is the ideal of E generated by ~n {~-~.}, so "-'(L+mk)/m k 1 is the ideal of 3 k-I generated by {~z__~_}. Now jk as a vector space has a 0x i basis X 2,..., X~. (A+mk)/m k is now the vector subspaee of jk-i spanned by  283 {~z Xj}. ~x i Let each ~z ~-~i Xj = k ~ 2 a i j , k ~ , where each aij,k is some am. We put M = the matrix (alj,k) = the coordinates of vectors spanning (A+mk)/m k. Now o(z) < K -- dim ( A + m k ) / k < K rank of M < K all K-minors of M vanish. And so Ek is given by polynomials in the {aij,k} , k.e. by polynomials in c+l the {al} , each a i E ~. Hence Ek is a real algebraic variety in the real c+l (n+k)! vector space Ik of dimension n!k! n - i, itself a suhspace of jk ((n+k)! which is " n!k! - l)-dimensional. Corollary. Ik is the disjoint union POk U s k U... U Fk-2 k U E kk-l' and each sk is the difference ~k _ Ek between 2 algebraic varieties. c c c+l Recall that the map ~: m 2 -~ i k is equivariant with respect to G, Gk; q. 9 Z also the image of the orbit nG is zG k, a submanlfold of I k, as in the proof of Theorem 2.9. Theorem 3.7. Let n ( m2 and cod n = c where 0 ~ e ~ k - 2. Then zG k is a submanlfold of Ik of codimension c. Proof. By Lemma 2.11, T (zG k) = ~(mA). (A = A(n)) z By Lemma 3.1, det n - 2 ~ cod ~ = c S k - 2, by the hypotheses. So det N ~ k, i.e. q is k-determlnate. By Theorem 2.9, m k+l c mA. The codlmension of zG k in I k = dim I k - dim ~(mA) = dim m2/m k+l - dim m&/k+l = dim m2/mA. Now m/mA = m/m 2 + m2/mA, so dim m2/mA = dim m/mA - dim m/m 2. So the codlmension of zG k in I k = dim m/mA - dim m/m 2 = dim m/A + dim A/mA - dim m/m 2 = C 4- n -- n  284 using the following lemma. Lemma 3.8. If n ~ m2 and cod ~ < ~, then dim A/md = n. This completes the proof of the theorem. Proof of Lemma 3.8. Since A is the ideal of E generated by ~n {~-~--}, every n ~D i ( A can be written as ~ = i~lai ~-~i where a i ( E, a i = a i + Pi' Pi E m, n ~ ~n a i E ~. Then ~ = i=l I Z a. ~ - ~ m o d n~. So {~-~} span A over ~, mod md, and dim A/mA ~ n. It remains to prove dim A/mA ~ n. Suppose not, i.e. that dim A/mA < n. Then ~ are linearly ~ L dependent mod mA. 3 al,..., a n ( ~, not all zero, such that n 8n n 8n Z ai ~ = Z pi ~ ~ mA, some {pi } E m. i=l i=l n ~ n Then XD = i=IZ (ai-~i) ~ = 0 where X = i=l ~ (a~-p~) Z~ ~ is a vector field on a neighborhood of 0 in R n. X is nonzero at 0 because {pi } ( m and so vanish at 0 and {a i} are not all zero. Change local coordinates so that X = where {yi } are the new ~Yl coordinates. Then ~ = 0. So n = ~(y2,...,yn ). Ess n = ~ with respect ~Yl to {yi }. But det ~ ~ ess ~, by Corollary 2.14. By Lemma 3.1., cod ~ = ~,. We have shown that dim A/mA = n. Theorem 3.7 Justifies the notation cod n, as an abbreviation for codlmension.  285 CHAPTER 4, CLASSIFICATION 7 Key: k vanish in Ik. So jkq, = y~ +... _ y~ + P(Yo+I..... Yn ) + PI(Yo+I..... Yn )' completing the lemsna. Addendum 4.10. The function q ~--~p is well-deflned because the construction is explicit.  289 Lemma 4.11. If rank n ~ n - 3, then cod n ~ 6. Proof. Either n is not finitely determinate, in which case cod n = ~ (Lemma 3.1), or n is k-determinate, some k, i.e. n ~ jk n, and jkn ~ q + p (essentials), by Lemma 4.9. Then cod B = cod (q+p). A(q+p) = 3p ap ). (2Xl""'-2Xp'ax '""ax p+l n So cod ~ = dim m/&(q+p) e dim m / ( A ( q + p ) + m 3) = number of the missing linear and quadratic terms in the essentials. If n-rank n = ~, all ~ linear terms are missing, as too are at least all but ~ of the 89 quadratic terms. So cod n ~ ~ + 89 - A= 89 If rank n ~ n - 3, then ~ ~ 3 and cod n ~ 6. We have that U {Qo x m3/m 8} consists of n with cod ~ ~ 6. p~m-3 By Lemma 4.8., this subspace has codimension 6 in 17. It remains to investigate Qn-i x m3/m 8 and ~-2 x m 3 / m 8. k Lem~na 4.12. (Classifying cuspoids) If rank n = n - I, then n ~ q + x n, 3 - 6. Proof. By the reduction lemma 4.9., n~n' where j7n' = q and p is a polynomial P(Xn) with 3 ~ degree of monomlals of p ~ 7. Let k be the k least degree appearing, so that p = akx n +.... Then jkn' is k-determinate, because A(jkn') = (Xl'.. "'Xn-l' x kn - l ) and so m2A ~ mk + l a n d we c a n u s e Theorem 2.9. Thus n' ~ jk n ' = q + akx kn = q + y~, changing coordinates so that lakIl/kxn=Yn 9 k k If k is odd changing coordinates Yn § -Yn makes n' = q - Yn ~ q + Yn" 3 4 5 6 7 Classify as q + p where p = x + x x + x x 0 n -- n n -- n n and cod(q+p) = i 2 3 4 5 Lemma 4.13. The cuspoids n with cod n ~ 6 form a submanifold of 17 of codimension 6. p 2 Proof. If n is a cuspoid, j2n = q = x I +... - Xn_ I. Write m 3 / m 8 = R x S, w h e r e R is the set of polynomials involving  290 one of Xl,..., Xn_ I and such that 3 ~ degree of monomials in r E R E 7, and S is the set of polynomials in x only, so that S ~5. Then n j7n = q + r + s, r E R, s 6 S. The reduction lemma 4.9. gave a (unique algebraic) map O: R + S such that n ~ n', and j7 , = q + 0 + (Sr+s). cod n > 6 ~ cod n' -> 6 ~'Sr + s = 0 " s = -Sr (r,s) ( Me, where Mo is the graph of -8, and is a submanifold of R x S of codimension 5. (8 is algebraic and so graph e ~ source of e.) As q varies through Qn-i we find that the required set of cuspoids ~ with cod n ~ 6 form a bundle over Qn-i (of codimension i in m2/m 3 by Lemma 4.8) with fibre M8 which has codimension 5 in m3/m 8. Thus the bundle has codimension 6 in m2/m 8 = 17. Now we classify the umbilics, Qn-2 m3/mS" Let n ~ m2 be such 2 that j2n = q, and q = x~ +... - Xn_ 2. By the reduction lemma 4.9., ~ ~' where j3n' = q + p and p is a homogeneous cubic in Xn_l, x n. In place of Xn_l, xn we shall use x, y respectively, for clarity. Note that Lemma 4.12., which classifies cuspoids, has been interpreted in this way in Diagram 4.1 with x replacing x. n Let (x,y) E R 2. The space of cubic forms in x, y is, 3 2 2 3 ~R} = R4. B2 {(alx +a2x Y+a3xY +a4Y ): al,a2,a3,a 4 The action of GL(2~R) on induces an action on ~4. Le~na 4.14. There are 5 GL(2~R)-orbits in R4, and so each P ~R4 is equivalent to one of 5 forms: dimension codimension (I) x 3 + y3 hyperbolic umbilic 4 0 3 2 (2) x - xy elliptic umbilic 4 0 2 (3) x y parabolic umbilic 3 i 3 (4) x symbolic umbilic 2 2 (5) 0 0 4  291 Proof. Consider the roots x, y of p(x,y) = 0, p E ~ 4. There are 5 cases (i) 2 complex, i real (2) 3 real distinct (3) 3 real, 2 same (4) 3 real equal (5) 3 equal to zero Case ( 4 ) : p = (alx+a2Y) 3 = u 3 by changing coordinates, l u = alx + a2Y 3 x v ffi independent. 2 Case (3): p = u v where u, v are independent linear forms in x, y. 2 ~ x y Case (2): p = dld2d3, product of 3 linear forms, d i = aix + biY. We have kI = Ia2 ~23 ~ 0 because the root of d2 ~ the root of d3. Let a3 u + v = kld I = u'~ (*). We claim this is a nonsingular coordinate change. u - v k2d 2 v' U~V ~ U ! V t has a change of basis matrix with determinant = -2. xpy ~ U w, V t has a change of basis matrix with determinant = klk2 ~ = klk2k 3 ~ 0 Adding (*), 2u ffi kld I + k2d 2 = (a2b3-a3b2)(alX+blY) + (a3hl-alb3)(a2x+b2Y) = x(ala2b3-ala3b2+a2a3bl-ala2b3 ) + y(...) = a3x(a2bl-alb2 ) + b3Y(a2bl-alb 2) ffi -k3(a3x+b3Y) = -k3d 3. 3 2 So u - uv 2u(u2-v 2) = -klk2k3dld2d3 ~ p. Thus p ~ x 3 - xy 2. Case (i): This is the same as Case (2) except that a2 = el' b2 ffibl and a3 , b 3 arerld kl :: b3b21qla3 b3112  292 alb I - albl = ft, t E ~. Change coordinates, iu + v = kld I 1 (~)" k3= a2al ~i = iu - v k2d 2 We claim this is a real change. Adding, 2iu = k3d 3 = itd 3 and td 3 is real. Subtracting, 2v = kld I - k2d 2 = kld I + kldl, which is real. So both u and v are real. It is a non-singular change because i I = -2i # 0. The i -i product of (,) is 2u(-u2-v 2) = klk2t p ~ p. So p ~ 2(u3+uv2), absorbing into the u-coordlnate. 2 (u3+uv 2) ~ 2(u3+3uv 2) absorbing 3 89 into v. = u '3 + v '3 with u' = u + v ~ X 3 + y 3. V I U - V By calculation x 3 + y3 and x3 - xy 2 are both 3-determinate and both cod(x3+y 3) and cod(x3-xy 2) equal 3. Thus the orbits corresponding to these are of codimension 3 in 17 by Theorem 3.7. 2 Lemma 4.15. If n = q + p, q E Qn-2' p = x y + higher terms, then either (i) ~ ~ q + (x2y+y 4) and cod ~ = 4 (the parabolic umbilic) or (2) n ~ q + (x2y+y 5) and cod ~ = 5. 7 or (3) n belongs to E6" Proof. If k ~ 4, then if p = x2y + yk, cod p = k = det p. Lemma 4.16. If k >_ 4 and jk-lp = x2y then p ~ x 2y + y k , or p p, and jkp, = x2y. Lemma 4.16. clearly gives Lemma 4.15. Proof of Lemma 4.16. jkp = x2y + a polynomial of degree k = x 2 y + ax k + 2xyP + by k, where P is a homogeneous polynomial of degree k - 2 ~ 2. (x+p)2(y+ax k-2) = (x2+2xP)(y+ax k-2) = x2y + 2xyP + ax k in Ik. Put u = (x+P) and v = y + axk-2; v k = yk in I k. So jkp = u2v + bv k. There are two cases, b # 0: jkp ~ u2v + v k absorbing Ibl I/k into v, and absorbing i/Ibl I/2k into u. b = 0: jkp = u2v ~ x2y.  293 3 Lemma 4.17. If n = q + p, p E Qn- 2 and p = x + higher terms in x, y, then either (I) ~ ~ q + x 3 ~y4 and cod n = 5 or (2) ~ ~ Z7 6" Proof. 9 shows that x 3 ~ y4 = p, is 4-determinate and cod p' = 5. 3.4P = x 3 + aOx4 + alx3y + a2x2y2 + a3xy3 + a4Y 4 - a4 # 0: Put v = y + 4a 4. a3x Then j4p = x 3 + 3x2p + a4v4 ' where P is a homogeneous polynomial of degree 2 in x, v. In 14 j4p = (x+p)3 + a4v4 u 3 ~ v 4 ' putting u = x + P and absorbing llla41 88 into v. a 4 = 0: As above we find that 3.4 P ~ x 3 + xy 3, which is 4-determlnate as stated in Chapter 2. (This is Siersma's germ) In any case a short calculation gives cod n = cod(x3+xy 3) = 6, so q Z7 6" Lemma 4.14 and a straightforward calculation produce the following facts, The symbolic umbilic (S) is a twisted cubic curve of dimension 1 in R 3. The parabolic umbilic (P) is a quartic surface with a cusp edge along S. The elliptic umbilic (E) is inside the cusp. The hyperbolic umbilic (H) is outside the cusp. (4.18) CHAPTER 5. THE PREPARATION THEOREM. This chapter is self-contained and is devoted to proving a major result~ the Preparation Theorem, which we need for Chapter 6. The words "near 0" will always be understood to mean "in some neighborhood of 0." Theorem 5.1. (Division Theorem) Let D be a C~ function defined near 0, from ~ x I~n to IR, such that D(t,0) = d(t)t k where d(0) # 0 and d is  294 C~ near 0 in R. Then given any C~E: IR l~n -+ I~. defined near 0, 3 C '~' functions q and r such that: (i) E = qD + r near 0 in IR x IRn, k-I where (2) r(t,x) = Z ri(x)ti for (t,x) ~ IR i=O near 0. Notation. Let Pk: IR x IRk +lR be the polynomial Pk(t,%) = t k + k-i ~iti" i=0 Theorem 5.2. (Polynomial Division Theorem) Let E(t,x) be a E-valued C~ function defined near 0 in l~ x IRn. Then 3 C-valued C~ functions q(t,x,l) and r(t,x,~) defined near 0 in I{ x IRn x IRk satisfying: (l) E(t,x) = q(t,x,l)Pk(t,%) + r(t,x,A), and k-I i (2) r(t,x,%) = Z r.(x,%)t , i=0 I where each ri is a C= function defined near 0 in IRn x l~k. Moreover if E is l~-valued, then q and r may be chosen l~-valued. Note that if E is R-valued we merely equate real parts of (i) in Theorem 5.2 to give the last part. Proof of Theorem 5.1 using Theorem 5.2. Given D, E we can apply Theorem 5.2 to find qD' rD' qE' rE such that D = qDPk + r D and E = qEPk + rE; let now k-i D rD(t,x,~) = E ri(x,~)ti (*). i=0 Now tkd(t) = D(t,0) = q D ~ , O ) p k ~ , 0 ) + rD(=,0 ) (I = 0% = qD(t,O)t k + k~l r~(O)ti. i=O Comparing coefficients of powers of t, r~(0) = 0 and qD(0) # 0 (d(0) # 0). ~si(0) Write si(~) = r~(0,~). We claim that ~lj ~ 0. k k-I i k-I tkd(t) = D(t,0) = qD(t,0,%)(t + Z l.t ) + Z si(l)ti. Differentiati~ i=O i i=O ~qD. ^. k tj k-I ~s i i with respect to ~'3 and setting ~ = 0, 0 = ~ - ~ t , u ) t + qD(t'0) + i=0E ~-~(0)t. ~s. ~s J ~s i J Thus ~ (0) = 0 if i < j and ~(0) = -qD(O). So (3--~.(0)) is a lower J ~ ~'~-~I " J triangular matrix, and as qD(O) # (O) # 0  295 By the implicit function theorem, 3 C| functions 8i(x) (0 ~ i ~ k - i) such that (a) rj(x,e) ~ 0, and (b) 6(0) = 0 (recall r (0) = 0). Let q(t,x) = qD(t,x,8) and P(t,x) = Pk(t,0). Then D(t,x) = q(t,x)P(t,x) (as rD(t,x,8) ~ 0 by (a).) As q(0) = qD(0) # 0, P(t,x) = D(t~x) near 0 in Rn. q(t,x) By (*), E(t,x) = qE(t,x,0)Pk(t,O ) + rE(t,x,O ) = q(t,x)D(t,x) + r(t,x), qEtt,x, 0)'" k-i E i where q(t,x) q(t,x) and r(t,x) = rg(t,x,e ) = i=0Z ri(x,6)t. Finally E let ri(x) = ri(x,8 ). Suppose f: ~ -~ E, f = u + iv and u, v: E + ~. If z = x + iy, then Du Du Dx + D__u. ~y l[~u iDu] ~z Dx ~z Dy Dz- = 2 ~ x + Dy. A similar result for v gives us that -Df ~z i (~u - = ~[ ~v) ~fx - ~ + i(~u Dv) 3y + ~x ] (5.3) Lemma 5.4. Let f: ~ + K he C as a function ~2 +~2. Let y be a simple closed curve in C whose interior is U. Then for w E U, I dz^dz f(w) = ~ I 4 z f(z) -w dz + 2--~iffU ~f(z) 3~ z-w (If f is holomorphic this reduces to the Cauchy Integral Formula since f is holomorphic = D_~f 0.) DE Proof. Let w ~ U and choose Z < min{lw-z]:zEy}. Let U = U - (disc radius E about w), and Ye = DUe" Recall Green's Theorem for ~2. If M, N: U § are C| on ~e, E then fy(Mdx+Ndy) = ff (aN_ ~ ) d x A dy. U ~x E Green's Theorem and (5.3) for f = u + iv give f f dz = f (u+iv) Cdx+idy) = 2i Ifu ~~f d x A dy. ye ye E 2i dxAdy = -dz Adz, so f f d~=-ff ~-~-fd~^d~ (*) y u ~ E  296 Apply (*) to f(z) , noting that - -i is holomorphic on U. Z--W Z--W _~ ~f(z) dzAdz = [ f(z) dz = ~ f(z___~)dz - ~ f(-~z, (:) u ~z z-w -~ z-w Y Z --W CE Z --W where Cg is the circle, radius e, centre w. 2~ With polar coordinates at w, f f(z) dz = f f(w+eeiS)idB. As Z--W c 0 * f(z) E + O, R.H.S. of (,) + ~ dz - 2nif(w), and L.H.S. of (**) + _ff ~f(z) dzhdz y z-w U ~z z-w (The limit exists because ~ is bounded on U, and !. is integrable over U.) 3~ z-w Proof of Theorem 5.2. Let E(z,x,l) be a C~ function defined near 0 in IRn x ~k such that E(t,x,%) = E(t,x) V t I~, i.e. E is an extension of E. Then E(w,x,l)= ~I f ECZ)z_wdz + 2~--iI-'U ~ z [ ~E(z) [ dzAdZz_~,by Lemma 5.4. Let Y k-i Pk(Z, ~) Pk(W, ~) k-1 Pk(Z,~) - Pk(W,~) = (z-w) E pi(z,l)wi, i.e. - - + E p.(z,l)w i. i=O z-w z-w i=O I In the expression for E(w,x,%) multiply top and bottom inside the integrals Pk(Z, ~) by Pk(Z,A) and expand giving E = qPk + r on E IRn Ek where Z--W q(w,x,l) = I r E(z,x~k) dz i ~E(z,x,l).l.dzAdz U ~z Pk(Z,X) (z-w) and ri(x,~ ) = ~ i ~ ~(z~x,~) jypk(z,l ) 9pi(z,X).dz + ~ i ffg~z~E(z'x'~)" " Pi (z'~) (z'x------~" Pk dz Adz, so long as these integrals are well defined and yield C~ functions. The first integral in the definition of both q and r is well- defined and C as long as the zeros of Pk(Z,l) do not occur on the curve y for ~ near 0 in ~k. Such a y is easily chosem. But U may contain zeros of Pk" So we need E such that ~E vanishes on zeros of Pk and for real z to ensure q, r well-defined. As the integrands are bounded we need C~ E such that B~E vanishes to infinite order on zeros of Pk and for real z to ensure q and r C ~. Lemma I. (Nirenberg Extension Lemma) Let E(t,x) be a C~ E-valued function defined near 0 in ~ ~n. Then 3 a C~ E-valued function E(z,x,%) defined near 0 in ~ ~ n Kk such that,  297 (I) ~(t,x,~) = E(t,x) V t ~. (2) -- vanishes to infinite order on {Im z = 0}. (3) -- vanishes to infinite order on {Pk(Z,~) = 0}. DE Lemma 2. (E. Borel's Theorem) Let f0' fl' ''" be a sequence of C~ functions on a given neighborhood N of 0 in Rn. Then 3 a C~ function F(t,x) on %i F a neighborhood of 0 in R ~n such that ~ (0,x) = fi(x) V i. C~ Proof. Let p: R > l~ be such that p(t) = ~I Itl - i t i Let F(t,x) = l ~ 0(uit)fi(x) , where {~i} is a rapidly increasing i=0 sequence of real numbers tending to ~, so that F is C~ near 0. (Lemma 2 may be used to show that for any power series about 0 in Rn 3 a C" real-valued function with its Taylor series at 0 the given power series.) Lemma 3. Let V, W be complementary subspaces or R n (= V+W). Let g, h be C= functions near 0 in Rn, such that for all multl-indlees a, T I ~l=Ig(x) 3!=Lh(x) V x ( V n W. Then 3 C~ F near 0 in Rn, such that 3x e = ~x ~ Ve, ~x ~ ~x a T T ~lU~h(x) x E W (A m u l t i - i n d e x a = (a I.... ,a n ) and lul = a I +... + a n so that ~]a[SCx) ~al+...+a ngcx).) ~x ~ aI an ~x I... Bx n Proof. Without loss of generality h ~ 0, for if FI is the required extension for (g-h) and 0, then F = FI + h is the required extension for g and h. Choose coordinates YI' "''' Yn so that V E Yl = "'" = YJ = 0  298 and W - YJ+I = "'" = Yk = 0. Let J 2 : I$:o ~ye co..... o,yj+1..... yn) ( l li lYi ) , where 0 is as in a = (a I..... aj ,0..... 0) Lemma 2 and {~i } increases to rapidly enough so that F is C= near 0, If y ~ W, each term of a18]F(Y) contains a factor a[Y!$ (0..... O,Yk+ 1..... y~. ay 8 aYY ~lSlF(y ) Since (0..... O,Yk+ I.... ,yn ) E V ~ W, this factor = 0 (h-0). So = 0. ~yB a ]'Y[ J y2) If y 6 V, note that aYY P(PI~Ii=EI i = ~1 y = 0, YI=...=yj=O (0 y # 0 and then ay8 lai: 0 ~y8 ! aya j -...=yj=O If b i # ai some i S j, then this term is 0. In fact the only nonzero te~m is aiSig(y ) ir ay 8 Lemma 4. Let f be a C~ K-valued function near 0 in ~n and let X be a vector field on Bn with coefficient. Then 3 Cm ~-valued F near 0 in ~ ~n so that (a) F(0,x) = f(x) V x E Rn. aF (b) ~ agrees to infinite order with XF at all (0,x) E ~ ~ n tk Proof. Try F(t,x) = etXf = Z xkf Differentiating termwlse at t = 0 k=0 ~.' " gives (b). Clearly (a) holds. To ensure that F is C~ use Lemma 2 to choose C" F such that F = k=0 ~ ~tk Txkf0(~kt) 9 Proof of Lemma i. We use induction on k. If k = 0, Pk(Z,l) , i, so we need ~ C~ E(z,x) such that E(t,x) = E(t,x) Vt ( R and aE(t,x) vanishes to ~z infinite order V t ~ R. Let z = s + it, 2 a__= _as a_ + i ~a. (Compare 5.3) ai Then Lemma 4 with X = -i ~ss gives such an E. Suppose Lemma i is proved for k - i. We show 3 C=F(z,x,l) and G(z,x,~) such that  299 (I)' F and G agree to infinite order on {Pk(Z,%) = 0} (2)' F is an extension of E. ~F (3)' -- vanishes to infinite order on {Im z = 0}. (4)' Let M = FI{Pk(Z,k). = 0}" 3 M vanishes to infinite order on ;Pk {~-~- (z,~) = 0}. ~G (5)' -- vanishes to infinite order on {Pk(Z,%) = 0}. Existence of F and G proves Lemma I. Let u = P(z,k) K Pk(Z,k) and ~' = (kl,...,kk_l). Consider (z,k0,k') ~ (z,u,k') on E x E Kk-l. This ~u is a valid coordinate change because ~--~0 ~ i. In the new coordinates, {Pk(Z,k) = 0} is given by u = 0. By Lemma 3 3 E agreeing to infinite order with G on u = 0 and to infinite order with F on Im z = 0. (u = 0 and Im z = 0 intersect transversally in R2k+2.) (2)', (3)' and (5)' now imply is the desired extension of E. Existence of F and G. Suppose we have that F exists. In (z,u,%')- coordinates, ~z becomes ~ z + ~z~P ~Bu' and ~z becomes ~---+~ 3z~-~P~---~" So in these coordinates we need G(z,x,u,%') such that (a) F = G to i