This is a multi-part message in MIME format. ---------------0105020205818 Content-Type: text/plain; name="01-164.keywords" Content-Transfer-Encoding: 7bit Content-Disposition: attachment; filename="01-164.keywords" Singular Lagrangian manifolds, integrable Hamiltonian systems, bifurcations, Bohr-Sommerfeld rules, WKB, semi-classics, normal forms, versal deformations. ---------------0105020205818 Content-Type: application/postscript; name="lsing.ps" Content-Transfer-Encoding: 7bit Content-Disposition: inline; filename="lsing.ps" %!PS-Adobe-2.0 %%Creator: dvips(k) 5.86d Copyright 1999 Radical Eye Software %%Title: lsing.dvi %%Pages: 26 %%PageOrder: Ascend %%BoundingBox: 0 0 596 842 %%DocumentPaperSizes: a4 %%EndComments %DVIPSWebPage: (www.radicaleye.com) %DVIPSCommandLine: dvips lsing.dvi -o lsing.ps %DVIPSParameters: dpi=600, compressed %DVIPSSource: TeX output 2001.05.02:0833 %%BeginProcSet: texc.pro %! /TeXDict 300 dict def TeXDict begin/N{def}def/B{bind def}N/S{exch}N/X{S N}B/A{dup}B/TR{translate}N/isls false N/vsize 11 72 mul N/hsize 8.5 72 mul N/landplus90{false}def/@rigin{isls{[0 landplus90{1 -1}{-1 1}ifelse 0 0 0]concat}if 72 Resolution div 72 VResolution div neg scale isls{ landplus90{VResolution 72 div vsize mul 0 exch}{Resolution -72 div hsize mul 0}ifelse TR}if Resolution VResolution vsize -72 div 1 add mul TR[ matrix currentmatrix{A A round sub abs 0.00001 lt{round}if}forall round exch round exch]setmatrix}N/@landscape{/isls true N}B/@manualfeed{ statusdict/manualfeed true put}B/@copies{/#copies X}B/FMat[1 0 0 -1 0 0] N/FBB[0 0 0 0]N/nn 0 N/IEn 0 N/ctr 0 N/df-tail{/nn 8 dict N nn begin /FontType 3 N/FontMatrix fntrx N/FontBBox FBB N string/base X array /BitMaps X/BuildChar{CharBuilder}N/Encoding IEn N end A{/foo setfont}2 array copy cvx N load 0 nn put/ctr 0 N[}B/sf 0 N/df{/sf 1 N/fntrx FMat N df-tail}B/dfs{div/sf X/fntrx[sf 0 0 sf neg 0 0]N df-tail}B/E{pop nn A definefont setfont}B/Cw{Cd A length 5 sub get}B/Ch{Cd A length 4 sub get }B/Cx{128 Cd A length 3 sub get sub}B/Cy{Cd A length 2 sub get 127 sub} B/Cdx{Cd A length 1 sub get}B/Ci{Cd A type/stringtype ne{ctr get/ctr ctr 1 add N}if}B/id 0 N/rw 0 N/rc 0 N/gp 0 N/cp 0 N/G 0 N/CharBuilder{save 3 1 roll S A/base get 2 index get S/BitMaps get S get/Cd X pop/ctr 0 N Cdx 0 Cx Cy Ch sub Cx Cw add Cy setcachedevice Cw Ch true[1 0 0 -1 -.1 Cx sub Cy .1 sub]/id Ci N/rw Cw 7 add 8 idiv string N/rc 0 N/gp 0 N/cp 0 N{ rc 0 ne{rc 1 sub/rc X rw}{G}ifelse}imagemask restore}B/G{{id gp get/gp gp 1 add N A 18 mod S 18 idiv pl S get exec}loop}B/adv{cp add/cp X}B /chg{rw cp id gp 4 index getinterval putinterval A gp add/gp X adv}B/nd{ /cp 0 N rw exit}B/lsh{rw cp 2 copy get A 0 eq{pop 1}{A 255 eq{pop 254}{ A A add 255 and S 1 and or}ifelse}ifelse put 1 adv}B/rsh{rw cp 2 copy get A 0 eq{pop 128}{A 255 eq{pop 127}{A 2 idiv S 128 and or}ifelse} ifelse put 1 adv}B/clr{rw cp 2 index string putinterval adv}B/set{rw cp fillstr 0 4 index getinterval putinterval adv}B/fillstr 18 string 0 1 17 {2 copy 255 put pop}for N/pl[{adv 1 chg}{adv 1 chg nd}{1 add chg}{1 add chg nd}{adv lsh}{adv lsh nd}{adv rsh}{adv rsh nd}{1 add adv}{/rc X nd}{ 1 add set}{1 add clr}{adv 2 chg}{adv 2 chg nd}{pop nd}]A{bind pop} forall N/D{/cc X A type/stringtype ne{]}if nn/base get cc ctr put nn /BitMaps get S ctr S sf 1 ne{A A length 1 sub A 2 index S get sf div put }if put/ctr ctr 1 add N}B/I{cc 1 add D}B/bop{userdict/bop-hook known{ bop-hook}if/SI save N @rigin 0 0 moveto/V matrix currentmatrix A 1 get A mul exch 0 get A mul add .99 lt{/QV}{/RV}ifelse load def pop pop}N/eop{ SI restore userdict/eop-hook known{eop-hook}if showpage}N/@start{ userdict/start-hook known{start-hook}if pop/VResolution X/Resolution X 1000 div/DVImag X/IEn 256 array N 2 string 0 1 255{IEn S A 360 add 36 4 index cvrs cvn put}for pop 65781.76 div/vsize X 65781.76 div/hsize X}N /p{show}N/RMat[1 0 0 -1 0 0]N/BDot 260 string N/Rx 0 N/Ry 0 N/V{}B/RV/v{ /Ry X/Rx X V}B statusdict begin/product where{pop false[(Display)(NeXT) (LaserWriter 16/600)]{A length product length le{A length product exch 0 exch getinterval eq{pop true exit}if}{pop}ifelse}forall}{false}ifelse end{{gsave TR -.1 .1 TR 1 1 scale Rx Ry false RMat{BDot}imagemask grestore}}{{gsave TR -.1 .1 TR Rx Ry scale 1 1 false RMat{BDot} imagemask grestore}}ifelse B/QV{gsave newpath transform round exch round exch itransform moveto Rx 0 rlineto 0 Ry neg rlineto Rx neg 0 rlineto fill grestore}B/a{moveto}B/delta 0 N/tail{A/delta X 0 rmoveto}B/M{S p delta add tail}B/b{S p tail}B/c{-4 M}B/d{-3 M}B/e{-2 M}B/f{-1 M}B/g{0 M} B/h{1 M}B/i{2 M}B/j{3 M}B/k{4 M}B/w{0 rmoveto}B/l{p -4 w}B/m{p -3 w}B/n{ p -2 w}B/o{p -1 w}B/q{p 1 w}B/r{p 2 w}B/s{p 3 w}B/t{p 4 w}B/x{0 S rmoveto}B/y{3 2 roll p a}B/bos{/SS save N}B/eos{SS restore}B end %%EndProcSet %%BeginProcSet: special.pro %! 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Ft(the)d(bifurcation)h(of)f(p)r (erio)r(dic)g(orbits)g(of)h(a)f(Hamiltonian)g(system)g(where)g(the)h(P) n(oincar)n(\023)-39 b(e)538 1840 y(map)27 b(of)h(a)f(p)r(erio)r(dic)g (orbit)g(admits)h(an)f(eigen)n(v)-5 b(alue)27 b(whic)n(h)h(is)f(a)g (cubic)h(ro)r(ot)f(of)g(1)p Black 455 1999 a Fr(\017)p Black 41 w Ft(the)d(adiabatic)f(limit)i(or)e(the)i(Born-Opp)r(enheimer) d(appro)n(ximation)h(with)h(crossings)e(of)538 2099 y(more)k(than)i(2)f (eigen)n(v)-5 b(alues.)455 2251 y(This)22 b(w)n(a)n(y)-7 b(,)23 b(w)n(e)g(prop)r(ose)f(a)g(general)g(setting)g(inspired)h(b)n(y) g(\\Thom's)f(catastrophe)f(theory")330 2351 y(\(see)39 b([3]\))g(and)g(presen)n(t)f(a)h(sk)n(etc)n(h)n(y)f(study)h(of)g(the)g Fp(sadd)t(le-no)l(de)k(bifur)l(c)l(ation)80 b Ft(\(the)40 b(cusp\))330 2450 y Fq(\030)370 2420 y Fo(2)426 2450 y Ft(+)18 b Fq(x)556 2420 y Fo(3)617 2450 y Ft(=)k(0.)455 2550 y(A)27 b(more)g(algebraic)f(\(co-homological)f(approac)n(h\))h(is) h(presen)n(ted)g(in)h([37)o(].)455 2649 y(The)i(sub)5 b(ject)30 b(is)h(really)e(the)i(study)f(of)g(the)h Fp(singularities)j (of)f(L)l(agr)l(angian)g(manifolds)p Ft(,)g(of)330 2749 y(their)h Fp(deformations)h Ft(\(or)e Fp(bifur)l(c)l(ations)p Ft(\))i(and)f(of)g(the)g(asso)r(ciated)e Fp(semi-classic)l(al)37 b(A)n(nsatz's)p Ft(.)330 2849 y(Building)21 b(up)h(classical)e(and)h (semi-classical)f(normal)g(forms)h(leads)g(to)g(study)g(mo)r(del)h (problems)330 2948 y(dep)r(ending)h(on)f(a)h(\014nite)g(n)n(um)n(b)r (er)g(of)f(parameters)f(among)h(whose)g(the)h(simplest)g(w)n(ere)f (already)330 3048 y(describ)r(ed)32 b(in)g(the)h(litterature:)45 b(cubic)33 b(oscillators)d(\(see)i([9)o(],)i([8)o(],)g([19)o(]\),)g (quartic)d(oscillators)330 3148 y(\(see)e([30)o(],)g(and)g(p)r (olynomial)f(p)r(oten)n(tials)g(\(see)h([39)o(]\)\).)41 b(A)29 b(remarquable)e(fact)i(is)g(that)g(w)n(e)f(can)330 3247 y(use)34 b(the)h(same)f(metho)r(ds)h(for)f(the)h(classical)e(and)h (the)h(semi-classical)e(bifurcations)g(and)i(in)330 3347 y(particular)29 b(the)i(co)r(dimension)f(of)g(the)h(singularities)e (are)g(the)i(same.)45 b(Of)30 b(course,)g(the)h(study)330 3446 y(of)f(the)h(classical)e(Hamiltonian)h(dynamic)g(in)h(a)e(2D)i (phase)f(space)f(is)h(trivial,)h(but)g(this)f(is)g(no)330 3546 y(more)d(the)h(case)f(for)g(the)h(semi-classical)d(case)i(whic)n (h)h(w)n(e)f(reduce)g(to)g(sp)r(ecial)h(functions.)455 3646 y(The)38 b(reader)f(should)i(tak)n(e)f(care)f(of)i(the)g(fact)g (that)g(caustic)f(singularities)g(is)g(a)g(di\013er-)330 3745 y(en)n(t)30 b(problem)g(for)f(whic)n(h)h(Lagrangian)e(manifolds)h (usually)h(are)f(smo)r(oth.)44 b(W)-7 b(e)31 b(strongly)d(use)330 3845 y(canonical)e(transformations)g(whic)n(h)i(eliminate)f(the)h (problem)f(of)h(caustics.)455 3945 y(The)19 b(main)h(idea)f(is)g(to)h (forget)e(the)i(equations)f(of)h(the)g(manifolds)f(and)g(to)h(fo)r(cus) f(on)h(the)g Fp(ide)l(al)330 4044 y(of)36 b(functions)d Ft(whic)n(h)g(v)-5 b(anish)33 b(on)g(it.)54 b(The)33 b(same)g(idea)f(turned)i(out)f(to)g(b)r(e)g(v)n(ery)f(imp)r(ortan)n(t) 330 4144 y(in)41 b(algebraic)d(geometry)-7 b(.)75 b(On)40 b(the)h(quan)n(tum)f(side,)k(w)n(e)c(do)g(the)h(same)f(c)n(hange)f(of)i (p)r(oin)n(t)330 4243 y(of)36 b(view:)52 b(w)n(e)36 b(consider)e Fp(left)k(ide)l(als)g(of)g(pseudo-di\013er)l(ential)i(op)l(er)l(ators.) 62 b Ft(W)-7 b(e)36 b(can)f(do)h(that)330 4343 y(b)r(ecause)h(an)n(y)f (solution)g(of)1259 4322 y(^)1240 4343 y Fq(P)12 b(u)39 b Ft(=)f(0)f(satis\014es)f(also)2085 4322 y(^)2065 4343 y Fq(B)2151 4322 y Ft(^)2132 4343 y Fq(P)12 b(u)38 b Ft(=)h(0)d(for)h(an)n(y)f(op)r(erator)3132 4322 y(^)3112 4343 y Fq(B)t Ft(.)66 b(It)330 4443 y(app)r(ears)28 b(that)h(usual)f (singularities,)g(at)h(least)f(for)h(1)f(degree)g(of)h(freedom,)f(do)h (admit)g(normal)330 4542 y(forms)f(and)f(their)i(deformations)e(ha)n(v) n(e)g(a)g(univ)n(ersal)g(mo)r(del)h(dep)r(ending)h(of)f(a)g(\014nite)g (n)n(um)n(b)r(er)330 4642 y(of)34 b(parameters.)55 b(The)35 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y Ft(W)-7 b(e)31 b(giv)n(e)f(the)h(general)e(de\014nitions)i(for)f(an)n (y)g(dimension)h(and)g(w)n(e)f(restrict)g(after)g(section)330 5670 y(2)d(to)h(the)g(case)e(of)i(a)f(2)g(dimensional)g(phase)g(space.) p Black 1809 5919 a(2)p Black eop %%Page: 3 3 3 2 bop Black Black 455 390 a Ft(The)29 b(main)g(non)h(trivial)f (result)g(is)g(theorem)g(6)g(whic)n(h)g(is)h(an)f(holomorphic)f(v)n (ersal)g(defor-)330 490 y(mation)f(result)h(for)f(all)g (quasi-homogeneous)e(isolated)i(singularities)f(of)i(curv)n(es.)455 589 y(The)c(semi-classical)e(results)h(follo)n(w)g(then)i(from)f(the)g (tec)n(hniques)g(already)e(dev)n(elopp)r(ed)i(in)330 689 y([13)o(].)330 939 y Fs(Con)l(ten)l(ts)330 1088 y Fu(1)65 b(Singular)26 b(Lagrangian)e(manifolds)1766 b(4)436 1167 y Fy(1.1)72 b(De\014nitions)21 b(.)35 b(.)g(.)g(.)g(.)g(.)g(.)g(.) g(.)f(.)h(.)g(.)g(.)g(.)g(.)g(.)g(.)f(.)h(.)g(.)g(.)g(.)g(.)g(.)g(.)g (.)f(.)h(.)g(.)g(.)g(.)g(.)g(.)g(.)g(.)f(.)h(.)g(.)g(.)g(.)g(.)p Black 128 w(4)p Black 436 1246 a(1.2)72 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Fq(dx)2506 1053 y Fl(i)2534 1041 y Ft(;)14 b(0\).)40 b Fr(E)c Ft(will)29 b(denote)g(the)330 1140 y(algebra)d(of)h(germs)g(of)g(real)g(v)-5 b(alued)28 b(analytic)f(functions)h(\(or)f(smo)r(oth)g(functions\).)p Black 330 1323 a Fj(De\014nition)k(1)p Black Black 139 w Fp(1.)p Black 43 w(A)e(\(germ)g(of)9 b(\))30 b Ft(singular)c (Lagrangian)e(manifold)29 b Fq(L)g Fp(in)h Fq(Z)2890 1293 y Fo(2)p Fl(d)2990 1323 y Fp(is)g(a)g(germ)538 1423 y(of)g(r)l(e)l(al)f(analytic)i(variety)f(\(ie)g(c)l(omplex)g(variety)g (invariant)h(by)e(c)l(omplex)h(c)l(onjugation\))538 1522 y(of)k(dimension)i Fq(d)e Fp(which)h(is)g(L)l(agr)l(angian)g(ne)l(ar)f (al)t(l)h(smo)l(oth)f(p)l(oints.)52 b(We)34 b(wil)t(l)h(denote)538 1622 y(by)30 b Fr(L)g Fp(or)h Fr(L)898 1634 y Fl(L)977 1622 y Fp(the)g(ide)l(al)g(of)g Fr(E)37 b Fp(of)31 b(functions)f (vanishing)h(on)f Fq(L)p Fp(.)39 b(If)30 b Fq(F)2698 1634 y Fl(j)2734 1622 y Fq(;)44 b(j)28 b Ft(=)23 b(1)p Fq(;)14 b Fr(\001)g(\001)g(\001)27 b Fq(;)14 b(n)30 b Fp(is)538 1722 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Fo(3)1097 1702 y Fp(is)34 b(a)g(surfac)l(e)g(and)g Fq(V)52 b Fp(a)34 b(ve)l(ctor)g(\014eld)g(on)g Fq(X)40 b Fp(whose)34 b(inte)l(gr)l(al)g(curves)330 1802 y(ar)l(e)j(ge)l(o)l (desics,)i(the)e(set)f(of)h(a\016ne)f(lines)h(gener)l(ate)l(d)g(by)f (the)h(ve)l(ctors)f Fq(V)19 b Ft(\()p Fq(m)p Ft(\))p Fq(;)51 b(m)34 b Fr(2)h Fq(X)43 b Fp(is)37 b(a)330 1901 y(\(singular\))31 b(L)l(agr)l(angian)h(manifold)g(in)f(the)g(symple)l (ctic)h(manifold)g(of)g(a\016ne)f(lines)g(in)g Fi(R)3161 1871 y Fo(3)3204 1901 y Fp(.)41 b(It)330 2001 y(c)l(an)30 b(b)l(e)g(shown)g(that)g Fq(S)k Fp(is)c(not)g(a)g(c)l(omplete)g (interse)l(ction)2160 1971 y Fo(2)2198 2001 y Fp(.)330 2233 y Fm(1.3)112 b(Reduction)330 2386 y Ft(If)24 b Fq(Z)k Fr(\032)23 b Fq(X)30 b Ft(is)23 b(a)g(co-isotropic)e(manifold)j(of)f(a) g(symplectic)g(manifold)h Fq(X)29 b Ft(and)24 b Fq(Z)2822 2356 y Fl(o)2882 2386 y Ft(the)f(isotropic)330 2486 y(foliation)k(of)g Fq(Z)6 b Ft(,)26 b Fq(X)930 2498 y Fl(R)1008 2486 y Ft(=)c Fq(Z)q(=)-5 b(Z)1253 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3054 y Fl(?)1768 3084 y Fi(R)39 b Ft(=)32 b Fr(f)p Ft(\()p Fq(t;)14 b(\034)9 b Ft(\))p Fr(g)34 b Ft(is)f(obtained)g(from)g(the)h(classical)330 3183 y(\015o)n(w)d Fq(')561 3195 y Fl(t)623 3183 y Ft(of)h(an)f(Hamiltonian)h Fq(H)39 b Ft(in)32 b(the)g(follo)n(wing)f(w)n(a)n(y:)45 b Fq(L)29 b Fr(\032)h Fq(T)2486 3153 y Fl(?)2523 3183 y Ft(\()p Fi(R)e Fr(\002)21 b Fq(M)30 b Fr(\002)21 b Fq(M)9 b Ft(\))30 b(=)g Fq(X)38 b Ft(is)330 3283 y(de\014ned)28 b(b)n(y)896 3383 y Fq(L)22 b Ft(=)h Fr(f)p Ft(\()p Fq(t;)14 b(\034)9 b Ft(;)14 b Fq(x;)g(\030)t Ft(;)g Fq(y)s(;)g(\021)s Ft(\))p Fr(j)p Fq(\034)33 b Ft(=)23 b Fq(H)7 b Ft(\()p Fq(x;)14 b(\030)t Ft(\))p Fq(;)g Ft(\()p Fq(x;)g(\030)t Ft(\))25 b(=)d Fq(')2439 3395 y Fl(t)2469 3383 y Ft(\()p Fq(y)s(;)14 b Fr(\000)p Fq(\021)s Ft(\))p Fr(g)330 3532 y Ft(and)27 b(w)n(e)h(reduce)f(using)g(the)h(co-normal)e(bundle)i(to)f (the)h(diagonal:)1229 3715 y Fq(Z)h Ft(=)22 b Fq(N)1478 3681 y Fl(?)1516 3715 y Fr(f)p Ft(\()p Fq(t;)14 b(x;)g(x)p Ft(\))p Fr(j)p 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g(generating)f(functions)330 4661 y(\(H\177)-42 b(ormander's)35 b Fp(phase)40 b(functions)p Ft(\):)55 b(if)37 b Fq(')i Ft(:)f Fq(X)31 b Fr(\002)24 b Fi(R)2024 4631 y Fl(N)2131 4661 y Fr(!)38 b Fi(R)43 b Ft(\(the)37 b(phase)f(function\),)k Fq(Z)j Ft(=)330 4761 y Fr(f)p Ft(\()p Fq(x;)14 b(\022)r Ft(;)g Fq(\030)t(;)g Ft(0\))p Fr(g)29 b(\032)g Fq(T)943 4731 y Fl(?)980 4761 y Ft(\()p Fq(X)e Fr(\002)21 b Fi(R)1248 4731 y Fl(N)1317 4761 y Ft(\),)32 b Fq(Z)1461 4773 y Fl(R)1545 4761 y Ft(=)c Fq(T)1699 4731 y Fl(?)1736 4761 y Fq(X)38 b Ft(and)31 b Fq(L)g Ft(is)g(the)h(graph)e(of)h Fq(d')p Ft(.)49 b(Semi-classical)330 4861 y(ob)5 b(jects)24 b(\(WKB-Maslo)n(v)e(Ansatz\))j(are)e(then)i(giv)n(en)f(b)n(y)g(the)h (follo)n(wing)e(oscillatory)g(in)n(tegrals:)1258 5086 y Fq(u)1306 5098 y Fl(h)1348 5086 y Ft(\()p Fq(x)p Ft(\))h(=)1571 4973 y Fk(Z)1617 5161 y Fh(R)1664 5145 y Fg(N)1730 5086 y Fq(e)1769 5051 y Fl(i')p Fo(\()p Fl(x;\022)r Fo(\))p Fl(=h)2056 5086 y Fq(a)p Ft(\()p Fq(x;)14 b(\022)r Ft(\))p 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y Fl(?)1274 841 y Fi(R)1328 811 y Fo(2)1408 841 y Fp(is)i(the)g(gr)l(aph)h(of)g Fq(d')g Fp(with)f Fq(')p Ft(\()p Fq(x;)14 b(y)s Ft(\))37 b(=)f Fq(y)s Ft(\()p Fq(x)2816 811 y Fo(2)2877 841 y Fr(\000)23 b Fq(y)3009 811 y Fo(2)3046 841 y Fq(=)p Ft(3\))36 b Fp(and)330 941 y Fq(Z)29 b Ft(=)22 b Fr(f)p Fq(\021)k Ft(=)d(0)p Fr(g)p Fp(,)29 b(we)h(get)g Fq(L)1152 953 y Fl(R)1229 941 y Ft(=)22 b Fr(f)p Ft(\()p Fq(x;)14 b Ft(2)p Fq(xy)s Ft(\))30 b Fr(j)g Fq(y)c Ft(=)c Fr(\006)p Fq(x)p Fr(g)h Ft(=)f Fr(f)p Fq(\030)2222 910 y Fo(2)2278 941 y Fr(\000)c Ft(4)p Fq(x)2450 910 y Fo(4)2510 941 y Ft(=)23 b(0)p Fr(g)p Fp(.)p Black 330 1116 a Fj(Example)30 b(1.6)p Black 41 w Fp(We)g(have)h(the)e(fol)t(lowing)k(\(se)l(e)c(also) i([35)q(]\))g(:)p Black 330 1276 a Ff(Pr)-5 b(op)g(osition)34 b(1)p Black 42 w Fp(The)42 b(germ)g(at)f Ft(0)g Fp(of)h(the)g(normal)g (bund)t(le)f(of)i(the)e(cusp)h(\(example)g(1.3\))330 1376 y(c)l(annot)29 b(b)l(e)h(obtaine)l(d)h(by)f(r)l(e)l(duction)g(of)h (a)f(germ)g(of)g 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Fq(T)2259 4141 y Fl(?)2296 4171 y Fi(R)34 b Fp(given)28 b(by)g Fq(F)2757 4183 y Fo(0)2818 4171 y Ft(=)22 b(0)p Fp(,)28 b(the)g Ft(space)330 4271 y(of)c (in\014nitesimal)g(deformations)i Fp(\(as)g(a)h(L)l(agr)l(angian)h (manifold\))g(of)g Fr(L)23 b Ft(=)p Fq(<)f(F)2771 4283 y Fo(0)2832 4271 y Fq(>)k Fp(is)h(the)f(sp)l(ac)l(e)330 4370 y(of)31 b(al)t(l)f(germs)g(of)h(functions)f Fr(E)7 b Fp(.)455 4531 y Ft(A)31 b(general)e(deformation)h(of)g(\()p Fq(F)1486 4543 y Fo(0)1524 4531 y Fq(;)14 b(!)1613 4543 y Fo(0)1650 4531 y Ft(\))31 b(is)g(giv)n(en)e(b)n(y)i(\()p Fq(F)2223 4543 y Fl(t)2253 4531 y Fq(;)14 b(!)2342 4543 y Fl(t)2370 4531 y Ft(\).)47 b(Using)30 b(Darb)r(oux,)h(w)n(e)f(can)330 4630 y(reduce)24 b(to)g(deformations)f(\()p Fq(F)1266 4642 y Fo(0)1316 4630 y Ft(+)12 b Fq(tK)k Ft(+)c Fq(O)r Ft(\()p Fq(t)1714 4600 y Fo(2)1751 4630 y Ft(\))p Fq(;)i(!)1872 4642 y Fo(0)1909 4630 y Ft(\).)36 b Fq(K)30 b Ft(is)24 b(an)g(arbitrary)e(germ)i(of)g(real)f(v)-5 b(alued)330 4730 y(function.)p Black 330 4906 a Fj(De\014nition)31 b(4)p Black 41 w Fp(A)g(deformation)h Fr(L)1481 4918 y Fl(t)1536 4906 y Ft(=)p Fq(<)24 b(F)1743 4918 y Fl(t)1797 4906 y Fq(>)31 b Fp(is)g Ft(trivial)f Fp(if)i(ther)l(e)e(exists)h(a)g (smo)l(oth)g(family)330 5005 y Fq(\037)382 5017 y Fl(t)436 5005 y Fp(of)c(c)l(anonic)l(al)f(tr)l(ansformations)g(and)g(a)f(smo)l (oth)h(family)h(of)f(functions)f Fq(E)2746 5017 y Fl(t)2799 5005 y Fr(2)e(E)7 b Fp(,)27 b(such)f(that:)1537 5181 y Fq(F)1590 5193 y Fl(t)1638 5181 y Fr(\016)18 b Fq(\037)1750 5193 y Fl(t)1802 5181 y Ft(=)k Fq(E)1950 5193 y Fl(t)1980 5181 y Fq(F)2033 5193 y Fo(0)2100 5181 y Fq(:)330 5356 y Fp(This)j(implies)g(that)e(ther)l(e)h(exists)f(germs)h(of)g (functions)g Fq(X)29 b Fp(and)24 b Fq(Y)43 b Fp(such)23 b(that)h(the)f(in\014nitesimal)330 5456 y(deformation)32 b Fq(K)c Ft(=)993 5422 y Fl(dF)1070 5430 y Fg(t)p 993 5437 105 4 v 1015 5484 a Fl(dt)1107 5489 y Fn(j)p Fl(t)p Fo(=0)1270 5456 y Fp(satis\014es:)1440 5650 y Fq(K)h Ft(=)22 b Fr(f)p Fq(X)r(;)14 b(F)1830 5662 y Fo(0)1867 5650 y Fr(g)k Ft(+)g Fq(Y)h(F)2130 5662 y Fo(0)2197 5650 y Fq(:)p Black 1809 5919 a Ft(6)p Black eop %%Page: 7 7 7 6 bop Black Black 330 390 a Ft(W)-7 b(e)28 b(can)f(no)n(w)g(giv)n(e)g (the)h(de\014nition)g(of)f(a)h(\014nite)g(co)r(dimensional)e(singular)h (germ)g(of)g(curv)n(e:)p Black 330 552 a Fj(De\014nition)k(5)p Black 41 w Fp(We)f(wil)t(l)h(say)f(that)g Fr(L)23 b Ft(=)p Fq(<)g(F)1775 564 y Fo(0)1835 552 y Fq(>)30 b Fp(is)g Ft(of)d(\014nite)h(co)r(dimension)g Fq(\026)h Fp(if)1163 729 y Ft(dim)1301 662 y Fk(\000)1339 729 y Fr(E)7 b Fq(=)14 b Ft(\()p Fr(fE)7 b Fq(;)14 b(F)1661 741 y Fo(0)1698 729 y Fr(g)k Ft(+)g Fr(E)7 b Fq(:F)1968 741 y Fo(0)2006 729 y Ft(\))2052 662 y Fk(\001)2113 729 y Ft(=)23 b Fq(\026)g(<)g Fr(1)29 b Fq(;)727 b Ft(\(1\))330 907 y Fp(wher)l(e)30 b Fr(f)p Fq(:)p Fr(j)p Fq(:)p Fr(g)f Fp(is)h(the)g(Poisson)h(br)l (acket,)455 1006 y(A)e(b)l(asis)i Fq(K)817 1018 y Fl(\013)888 1006 y Fr(2)24 b Fq(D)1036 1018 y Fn(L)1085 1006 y Fq(;)44 b(\013)24 b Ft(=)g(1)p Fq(;)14 b Fr(\001)g(\001)g(\001)27 b Fq(;)14 b(\026)p Fp(,)30 b(of)h(a)g(supplementary)f(sp)l(ac)l(e)h(of) g Fr(fE)7 b Fq(;)14 b(F)2882 1018 y Fo(0)2919 1006 y Fr(g)19 b Ft(+)f Fr(E)7 b Fq(:F)3190 1018 y Fo(0)3258 1006 y Fp(in)330 1106 y Fq(D)399 1118 y Fn(L)479 1106 y Fp(wil)t(l)31 b(b)l(e)e(c)l(al)t(le)l(d)j(a)e Ft(\(uni\)v)n(ersal)d (deformation)f(of)k Fr(L)p Fp(.)455 1205 y(Mor)l(e)38 b(pr)l(e)l(cisely,)j(we)d(ask)g(that)f(e)l(quation)h(\(1\))f(is)h(true) f(with)h Fr(E)7 b Ft(\()p Fq(U)2601 1217 y Fl(j)2636 1205 y Ft(\))38 b Fp(for)g(a)g(b)l(asis)g Fq(U)3190 1217 y Fl(j)3262 1205 y Fp(of)330 1305 y(neighb)l(ourho)l(o)l(ds)32 b(of)e Fq(O)j Fp(\(with)d(the)g(same)g(functions)g Fq(K)2091 1317 y Fl(\013)2138 1305 y Fp(\).)p Black 330 1482 a Fj(Question)g(3)p Black 42 w Fp(Give)35 b(a)f(natur)l(al)g(extension)g (of)g(the)h(de\014nition)f(1.1)i(to)e(the)g(c)l(ase)g(of)h(systems)330 1582 y(of)c(op)l(er)l(ators,)g(i.e.)39 b(matrix)30 b(value)l(d)h(germs) e(of)i(functions)f(\(se)l(e)g([7]\).)330 1856 y Fs(3)135 b(Examples)p Black 431 2038 a Ft(1.)p Black 42 w Fj(The)41 b(smo)s(oth)e(case:)53 b Ft(the)36 b(di\013eren)n(tials)f Fq(dF)2042 2050 y Fl(j)2113 2038 y Ft(are)g(linearly)g(indep)r(enden)n (t)i(in)f(some)538 2137 y(neigh)n(b)r(ourho)r(o)r(d)48 b(of)24 b(the)h(origine.)35 b(Then)25 b Fr(L)f Ft(is)h(a)f(germ)g(of)g (smo)r(oth)h(Lagrangian)c(mani-)538 2237 y(fold.)36 b(This)26 b Fr(L)g Ft(is)g(of)g(co)r(dimension)g(0.)35 b(Moreo)n(v)n(er)24 b(Darb)r(oux)h(theorem)g(implies)i(that)f(up)538 2336 y(to)h(canonical)f(transformation)g Fr(L)e Ft(=)p Fq(<)e(\030)1838 2348 y Fo(1)1875 2336 y Fq(;)14 b Fr(\001)g(\001)g(\001)28 b Fq(;)14 b(\030)2110 2348 y Fl(d)2172 2336 y Fq(>)p Ft(.)p Black 431 2500 a(2.)p Black 42 w Fj(The)32 b(Morse)g Ft(\()p Fq(d)25 b Ft(=)e(1\))33 b Fj(case:)k Ft(let)29 b Fq(F)1728 2512 y Fl(")1788 2500 y Ft(=)24 b Fq(F)1930 2512 y Fo(0)1986 2500 y Ft(+)19 b Fq(O)r Ft(\()p Fq(")p Ft(\))29 b(where)e Fq(F)2560 2512 y Fo(0)2626 2500 y Ft(is)h(a)g(non)g(degenerate)538 2600 y(quadratic)34 b(form)h(on)g Fq(T)1305 2570 y Fl(?)1342 2600 y Fi(R)p Ft(.)67 b(By)35 b(the)h Fp(lemme)h(de)h(Morse)g(iso)l(chor)l(e)f Ft(\(see)e([14)o(]\),)j(there)538 2700 y(exists)26 b Fq(\037)818 2712 y Fl(")880 2700 y Ft(a)h(germ)f(of)g(canonical)g (transformations)f(smo)r(othly)h(dep)r(ending)i(of)e Fq(")h Ft(and)f(a)538 2799 y(smo)r(oth)h(function)h(\010)1216 2811 y Fl(")1279 2799 y Ft(suc)n(h)f(that)1619 2977 y Fq(F)1672 2989 y Fl(")1726 2977 y Fr(\016)18 b Fq(\037)1838 2989 y Fl(")1897 2977 y Ft(=)k(\010)2044 2989 y Fl(")2098 2977 y Fr(\016)c Fq(F)2211 2989 y Fo(0)538 3154 y Ft(and)27 b(\010)759 3124 y Fn(0)759 3174 y Fo(0)796 3154 y Ft(\(0\))c Fr(6)p Ft(=)g(0.)36 b(Hence)28 b(\010)1421 3166 y Fl(")1484 3154 y Ft(admits)g(a)f(non)h(degenerate)e(zero)g Fq(t)p Ft(\()p Fq(")p Ft(\))i(and)g(w)n(e)f(ha)n(v)n(e)1224 3331 y Fq(F)1277 3343 y Fl(")1331 3331 y Fr(\016)18 b Fq(\037)1443 3343 y Fl(")1479 3331 y Ft(\()p Fq(x;)c(\030)t Ft(\))24 b(=)f Fq(E)1840 3343 y Fl(")1875 3331 y Ft(\()p Fq(x;)14 b(\030)t Ft(\)\()p Fq(F)2148 3343 y Fo(0)2187 3331 y Ft(\()p Fq(x;)g(\030)t Ft(\))20 b Fr(\000)e Fq(t)p Ft(\()p Fq(")p Ft(\)\))538 3508 y(from)27 b(whic)n(h)g(it)h(is)g(clear) e(that)i Fq(<)23 b(F)1657 3520 y Fo(0)1713 3508 y Fr(\000)18 b Fq(t)23 b(>)k Ft(is)h(a)f(v)n(ersal)f(deformation)h(of)g Fq(<)c(F)3029 3520 y Fo(0)3090 3508 y Fq(>)p Ft(.)p Black 431 3672 a(3.)p Black 42 w Fj(The)33 b(Eliasson)e(case)f Ft(\([21)o(])f(or)f(the)h(non)g(degenerate)e(case)h(of)h([33)o(],)g(d)n (\023)-39 b(e\014nition)29 b(2.1.\).)538 3772 y(It)e(is)h(an)f (extension)g(of)h(the)g(previous)e(case)h(to)h(sev)n(eral)d(quadratic)i (forms.)36 b(Let)1785 3949 y Fq(q)1822 3961 y Fo(1)1859 3949 y Fq(;)14 b Fr(\001)g(\001)g(\001)g Fq(q)2044 3961 y Fl(d)538 4127 y Ft(b)r(e)22 b Fq(d)g Ft(indep)r(enden)n(t)g(comm)n (uting)g(quadratic)e(forms)h(on)h Fq(T)2359 4097 y Fl(?)2396 4127 y Fi(R)2450 4097 y Fl(d)2517 4127 y Ft(where)f(\()p Fq(q)2820 4139 y Fo(1)2857 4127 y Fq(;)14 b Fr(\001)g(\001)g(\001)28 b Fq(;)14 b(q)3093 4139 y Fl(d)3131 4127 y Ft(\))23 b(is)e(of)538 4226 y(t)n(yp)r(e)i(\()p Fq(m)825 4238 y Fl(e)861 4226 y Fq(;)14 b(m)971 4238 y Fl(h)1014 4226 y Fq(;)g(m)1124 4238 y Fl(f)1167 4226 y Ft(\))24 b(and)f Fq(d)g Ft(=)g Fq(m)1607 4238 y Fl(e)1653 4226 y Ft(+)10 b Fq(m)1801 4238 y Fl(h)1854 4226 y Ft(+)g(2)p Fq(m)2044 4238 y Fl(f)2110 4226 y Ft(where)23 b Fq(m)2419 4238 y Fl(e)2478 4226 y Ft(is)h(the)g(n)n(um)n(b)r(er)f(of)h(elliptic)538 4326 y(forms,)j Fq(m)863 4338 y Fl(h)933 4326 y Ft(the)h(n)n(um)n(b)r(er)g (of)f(h)n(yp)r(erb)r(olic)g(one's)g(and)h Fq(m)2322 4338 y Fl(f)2392 4326 y Ft(the)g(n)n(um)n(b)r(er)g(of)f(fo)r(cus-fo)r(cus) 538 4426 y(one's.)42 b(W)-7 b(e)30 b(ha)n(v)n(e)e Fq(\026)f Ft(=)e Fq(d)p Ft(.)44 b(This)29 b(v)-5 b(alue)30 b(is)f(minimal)h(for)f (rank)g(0)g(singular)f(p)r(oin)n(t)i(of)f(an)538 4525 y(in)n(tegrable)d(system.)p Black 431 4689 a(4.)p Black 42 w Fj(Cusp)h Ft(\()p Fq(A)872 4701 y Fo(2)910 4689 y Ft(\))h(:)37 b Fq(F)1083 4701 y Fo(0)1143 4689 y Ft(=)23 b Fq(\030)1271 4659 y Fo(2)1327 4689 y Ft(+)18 b Fq(x)1457 4659 y Fo(3)1522 4689 y Ft(\()p Fq(d)24 b Ft(=)e(1\))28 b(and)f Fq(\026)c Ft(=)g(2:)1613 4866 y Fq(K)1684 4878 y Fo(1)1744 4866 y Ft(=)g(1)p Fq(;)41 b(K)2009 4878 y Fo(2)2069 4866 y Ft(=)22 b Fq(x)28 b(:)538 5044 y Ft(W)-7 b(e)32 b(will)h(see)f(that)h(up)g(to)g(canonical)e(transformation)g(an) n(y)h Fq(F)44 b Ft(whic)n(h)33 b(admits)f(a)g(non)538 5143 y(degenerate)26 b(cusp)h(is)h(equiv)-5 b(alen)n(t)27 b(to)h(the)g(standard)e(example)i Fq(\030)2577 5113 y Fo(2)2632 5143 y Ft(+)18 b Fq(x)2762 5113 y Fo(3)2800 5143 y Ft(.)p Black 431 5307 a(5.)p Black 42 w Fj(Quartic)37 b(oscillator)30 b Ft(\()p Fq(A)1394 5272 y Fo(+)1394 5329 y(3)1450 5307 y Ft(\))i Fq(F)1567 5319 y Fo(0)1635 5307 y Ft(=)d Fq(\030)1769 5277 y Fo(2)1828 5307 y Ft(+)21 b Fq(x)1961 5277 y Fo(4)2030 5307 y Ft(\()p Fq(d)31 b Ft(=)e(1\))j(and)f Fq(\026)f Ft(=)g(3:)44 b Fq(K)2856 5319 y Fo(1)2923 5307 y Ft(=)30 b(1)p Fq(;)45 b(K)3199 5319 y Fo(2)3265 5307 y Ft(=)538 5407 y Fq(x;)d(K)721 5419 y Fo(3)780 5407 y Ft(=)23 b Fq(x)915 5377 y Fo(2)953 5407 y Ft(.)p Black 431 5571 a(6.)p Black 42 w Fj(Quartic)35 b(an)m(ti-oscillator)30 b Ft(\()p Fq(A)1584 5535 y Fn(\000)1584 5593 y Fo(3)1640 5571 y Ft(\))h Fq(F)1756 5583 y Fo(0)1821 5571 y Ft(=)c Fq(\030)1953 5541 y Fo(2)2011 5571 y Fr(\000)19 b Fq(x)2142 5541 y Fo(4)2210 5571 y Ft(or)30 b Fq(F)2368 5583 y Fo(0)2433 5571 y Ft(=)d Fq(x)p Ft(\()p Fq(x)21 b Fr(\000)f Fq(\030)2797 5541 y Fo(2)2834 5571 y Ft(\))31 b(\()p Fq(d)d Ft(=)f(1\))j(and)538 5670 y Fq(\026)23 b Ft(=)f(3.)p Black 1809 5919 a(7)p Black eop %%Page: 8 8 8 7 bop Black Black Black 431 390 a Ft(7.)p Black 42 w Fj(T)-8 b(riple)31 b(crossing)c Ft(\()p Fq(D)1280 355 y Fn(\000)1278 412 y Fo(4)1336 390 y Ft(\))h Fq(F)1449 402 y Fo(0)1510 390 y Ft(=)22 b Fq(x\030)t Ft(\()p Fq(x)e Fr(\000)e Fq(\030)t Ft(\))28 b(\()p Fq(d)c Ft(=)f(1\))k(and)g Fq(\026)d Ft(=)e(4:)1266 573 y Fq(K)1337 585 y Fo(1)1397 573 y Ft(=)h(1)p Fq(;)41 b(K)1662 585 y Fo(2)1722 573 y Ft(=)22 b Fq(x;)42 b(K)1992 585 y Fo(3)2052 573 y Ft(=)23 b Fq(\030)t(;)41 b(K)2315 585 y Fo(4)2375 573 y Ft(=)23 b Fq(x\030)32 b(:)p Black 431 789 a Ft(8.)p Black 42 w Fj(Hyp)s(erb)s(olic)e(um)m(bilic)c Ft(\()p Fq(D)1468 753 y Fo(+)1466 811 y(4)1523 789 y Ft(\))i Fq(F)1636 801 y Fo(0)1697 789 y Ft(=)23 b Fq(x)p Ft(\()p Fq(x)1911 759 y Fo(2)1967 789 y Ft(+)18 b Fq(\030)2090 759 y Fo(2)2128 789 y Ft(\))28 b(\()p Fq(d)23 b Ft(=)g(1\))k(and)h Fq(\026)23 b Ft(=)g(4.)p Black 330 971 a Fj(Question)30 b(4)p Black 42 w Fp(Describ)l(e)g(al)t(l)h(singular)f(L)l(agr)l(angian)h(manifolds) g(of)g(smal)t(l)g(c)l(o)l(dimension.)330 1246 y Fs(4)135 b(In)l(tegrable)46 b(systems)330 1444 y Fm(4.1)112 b(Singularities)35 b(of)j(in)m(tegrable)e(systems)p Black 330 1598 a Fj(De\014nition)31 b(6)p Black 41 w Fp(A)n(n)24 b Ft(in)n(tegrable)c(Hamiltonian)i(system) i Fp(is)h(given)g(by)f(a)h(map)g(\(the)g(momentum)330 1697 y(map\):)1304 1797 y 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Fq(=)p Ft(Jac)o(\()p Fq(F)12 b Ft(\))26 b(where)e(Jac)o(\()p Fq(F)12 b Ft(\))26 b(is)f(the)g(ideal)g(generated)f(b)n(y)330 589 y(the)35 b(t)n(w)n(o)e(partial)h(deriv)-5 b(ativ)n(es)33 b(of)h Fq(F)12 b Ft(.)57 b(The)35 b(germs)e(of)h(the)h(non)f(singular)f (curv)n(es)g Fr(f)p Fq(F)46 b Ft(=)34 b Fq(")p Fr(g)330 689 y Ft(\()p Fq(")23 b Fr(6)p Ft(=)g(0)h(and)h(small\))g(ha)n(v)n(e)f (the)i(homotop)n(y)e(t)n(yp)r(e)h(of)g(a)f(b)r(ouquet)i(of)f Fq(\026)g Ft(circles)f(\(the)i(\\v)-5 b(anishing)330 789 y(cycles"\).)330 888 y Fp(Pr)l(o)l(of.{)p Black Black 567 1054 a Ft(Let)30 b(us)f(denote)h(b)n(y)f(\012)1273 1024 y Fl(j)1337 1054 y Ft(the)h(germs)e(of)i(di\013eren)n(tial)f (forms)g(of)g(degree)g Fq(j)34 b Ft(near)29 b(0)538 1154 y(in)e Fi(C)688 1124 y Fo(2)732 1154 y Ft(.)37 b(F)-7 b(rom)27 b(the)h(results)f(of)g(Sebastiani)g(\(see)h([26)o(])g (p.416\),)f(w)n(e)g(kno)n(w)g(that)1577 1337 y(\012)1637 1302 y Fo(2)1675 1337 y Fq(=dF)1813 1349 y Fo(0)1868 1337 y Fr(^)19 b Fq(d)p Ft(\012)2045 1302 y Fo(0)538 1519 y Ft(is)27 b(a)g(free)h(mo)r(dule)f(of)h(rank)f Fq(\026)g Ft(o)n(v)n(er)f Fi(C)15 b Fr(f)p Fq(F)1833 1531 y Fo(0)1876 1519 y Fr(g)p Ft(.)37 b(W)-7 b(e)28 b(get)f(the)h(consequence)f(that:)1388 1702 y(\012)1448 1668 y Fo(2)1485 1702 y Fq(=)1541 1635 y Fk(\000)1578 1702 y Fq(dF)1674 1714 y Fo(0)1731 1702 y Fr(^)18 b Fq(d)p Ft(\012)1907 1668 y Fo(0)1963 1702 y Ft(+)g Fq(F)2099 1714 y Fo(0)2137 1702 y Ft(\012)2197 1668 y Fo(2)2234 1635 y Fk(\001)538 1885 y Ft(is)25 b(of)h(dimension)f Fq(\026)h Ft(o)n(v)n(er)d Fi(C)15 b Ft(.)43 b(The)25 b(result)h(follo)n(ws)e(from)h(the)h(natural)f(iden)n(ti\014ca-)538 1984 y(tions)e(of)h(the)h(2-forms)d(with)j(the)f(functions)g(and)g(of)g (the)g(w)n(edge)f(pro)r(duct)h Fq(d)-14 b(f)20 b Fr(^)11 b Fq(dg)538 2084 y Ft(with)28 b(the)g(P)n(oisson)d(brac)n(k)n(et.)3265 2250 y Fe(\003)455 2350 y Ft(A)37 b(simple)h(pro)r(of)f(of)g(theorem)g (1)h(in)f(the)h(quasi-homogeneous)d(case)h(will)i(b)r(e)g(giv)n(en)f (in)330 2449 y(section)27 b(7.)455 2549 y(Theorem)32 b(1)g(admits)h(a)g(v)n(ery)f(nice)h(geometrical)f(in)n(terpretation)g (whic)n(h)h(w)n(e)f(can)h(deriv)n(e)330 2648 y(from)27 b(the)h(pap)r(er)f([29)o(].)37 b(If)28 b Fq(\037)f Ft(is)h(a)f(germ)g (of)g(canonical)f(transformation)g(near)h(the)g(origin,)g(ac-)330 2748 y(tions)j(in)n(tegrals)f(o)n(v)n(er)f(small)i(cycles)g(are)f (preserv)n(ed.)43 b(Hence)31 b(an)n(y)e(\(uni\)v)n(ersal)h(deformation) 330 2848 y(should)g(b)r(e)h(able)f(to)h(repro)r(duce)e(the)i(v)-5 b(ariations)29 b(of)i(the)g(action)f(in)n(tegrals)f(o)n(v)n(er)g(the)i (v)-5 b(anish-)330 2947 y(ing)33 b(cycles.)53 b(This)33 b(is)g(strongly)f(consisten)n(t)h(with)h(the)f(fact)h(that)f Fq(\026)g Ft(is)g(also)g(the)g(n)n(um)n(b)r(er)g(of)330 3047 y(v)-5 b(anishing)26 b(cycles)f(as)g(sho)n(wn)h(in)g([27)o(].)36 b(This)26 b(is)g(exactly)g(the)g(w)n(a)n(y)f(things)h(w)n(ork)e(in)j (the)f(quasi-)330 3147 y(homogeneous)i(case)h(as)g(sho)n(wn)g(in)i 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(ector)f(subspaces)g(of)330 3844 y(the)k(same)f(co)r(dimension)g(alw)n (a)n(ys)f(admit)i(a)f(comm)n(un)g(supplemen)n(tary)g(subspace.)455 3944 y(F)-7 b(or)21 b(example)i(of)f(a)g(non)h(quasi-homogeneous)c (singularit)n(y)-7 b(,)22 b(w)n(e)h(can)f(tak)n(e)g(the)g(singularit)n (y)330 4043 y(called)k Fq(Z)621 4055 y Fo(11)718 4043 y Ft(\()p Fq(\026)d Ft(=)g(11\))j(in)h([3],)g(whic)n(h)f(is)h(giv)n(en) f(b)n(y)g Fq(F)1989 4055 y Fl(a)2053 4043 y Ft(=)c Fq(x)2187 4013 y Fo(3)2225 4043 y Fq(\030)f Ft(+)16 b Fq(\030)2403 4013 y Fo(5)2457 4043 y Ft(+)g Fq(ax\030)2669 4013 y Fo(4)2707 4043 y Ft(.)36 b(Di\013eren)n(t)27 b(v)-5 b(alues)330 4143 y(of)28 b Fq(a)f Ft(giv)n(e)g(non-equiv)-5 b(alen)n(t)27 b(singularities)f(of)h(functions,)h(but)h(equiv)-5 b(alen)n(t)27 b(ideals.)455 4242 y(If)f Fq(F)589 4254 y Fo(0)650 4242 y Ft(=)c(0)k(is)g(a)g(germ)f(of)h(singular)f(curv)n(e,)h(w)n(e)f(can)h (asso)r(ciate)f(to)h(it)g(a)g(de)g(Rham)h(complex)330 4342 y(as)g(in)h([23)o(]:)1462 4442 y(0)22 b Fr(!)h(E)31 b(!)23 b Ft(\012)1873 4407 y Fo(1)1910 4442 y Fq(=K)28 b Fr(!)23 b Ft(0)330 4591 y(where)34 b(the)h(non)f(trivial)f(arro)n(w)g (is)h Fq(d)g Ft(and)h Fq(K)k Ft(is)c(the)f(set)h(of)f(1-form)f(whic)n (h)h(v)-5 b(anish)35 b(on)f(the)330 4691 y(tangen)n(t)27 b(v)n(ectors)f(to)i(the)g(smo)r(oth)f(stratum)g(of)h Fq(F)1914 4703 y Fo(0)1975 4691 y Ft(=)22 b(0.)37 b(Then)1171 4873 y Fq(H)1247 4839 y Fo(1)1240 4894 y Fl(de)27 b(Rham)1522 4873 y Ft(\()p Fq(<)c(F)1695 4885 y Fo(0)1755 4873 y Fq(>)p Ft(\))g(=)g(\012)2023 4839 y Fo(1)2060 4873 y Fq(=)p Ft(\()p Fq(K)h Ft(+)18 b Fq(d)p Fr(E)7 b Ft(\))28 b Fq(:)330 5056 y Ft(There)23 b(is)h(a)g(subspace)f(of)h(the)g(space)f (of)h(in\014nitesimal)h(deformations)e(whic)n(h)g(w)n(e)h(can)g(iden)n (tify)330 5156 y(with)32 b Fq(H)599 5126 y Fo(1)592 5179 y Fl(de)27 b(Rham)874 5156 y Ft(\()p Fq(<)h(F)1052 5168 y Fo(0)1119 5156 y Fq(>)p Ft(\).)47 b(If)32 b Fq(\013)d Fr(2)g Ft(\012)1599 5126 y Fo(1)1668 5156 y Ft(is)i(a)f(germ)h(of)g (1-form,)g(it)g(giv)n(es)f(a)h(deformation)f(of)330 5255 y(\()p Fq(F)415 5267 y Fo(0)453 5255 y Fq(;)14 b(!)542 5267 y Fo(0)579 5255 y Ft(\))31 b(de\014ned)h(b)n(y)e(\()p Fq(F)1135 5267 y Fo(0)1173 5255 y Fq(;)14 b(!)1262 5267 y Fo(0)1320 5255 y Ft(+)20 b Fq("d\013)p Ft(\).)48 b(It)32 b(is)f(equiv)-5 b(alen)n(t)31 b(to)g(\014x)g Fq(!)2500 5267 y Fo(0)2568 5255 y Ft(and)g(to)g(deform)g Fq(F)3174 5267 y Fo(0)3242 5255 y Ft(b)n(y)330 5355 y Fq(F)383 5367 y Fo(0)443 5355 y Ft(+)21 b Fq("dF)664 5367 y Fo(0)702 5355 y Ft(\()p Fq(X)803 5367 y Fl(\013)850 5355 y Ft(\))33 b(where)f Fq(X)1229 5367 y Fl(\013)1309 5355 y Ft(is)h(de\014ned)g(b)n (y)f Fq(\023)p Ft(\()p Fq(X)1939 5367 y Fl(\013)1987 5355 y Ft(\))p Fq(!)2071 5367 y Fo(0)2140 5355 y Ft(=)f Fq(\013)p Ft(.)53 b(It)33 b(is)g(easy)f(to)g(c)n(hec)n(k)g(that,)j(if) 330 5455 y Fq(\013)17 b Fr(\000)f Fq(dh)27 b Ft(v)-5 b(anishes)27 b(on)f Fq(C)1098 5467 y Fo(0)1136 5455 y Ft(,)h(the)g(deformation)f(is)g(trivial)g(and)h(con)n(v)n(ersely)e(if)i 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Ft(=)2226 3029 y Fq(@)g(\025)p 2226 3066 98 4 v 2226 3142 a(@)g(x)2352 3085 y Ft(+)2445 3029 y Fq(@)g(\027)p 2445 3066 95 4 v 2448 3142 a(@)g(\030)2577 3085 y(;)538 3293 y Ft(whic)n(h)30 b(admits)h(an)f(unique)h(solution)g Fq(Y)18 b Ft(:)43 b(w)n(e)31 b(solv)n(e)e(\014rst)i(inside)g(formal)f (series,)538 3393 y(then)23 b(inside)h(\015at)f(functions.)36 b(W)-7 b(e)23 b(can)g(tak)n(e)g(for)g(the)g Fq(U)2266 3405 y Fl(j)2301 3393 y Ft('s)g(a)g(basis)g(of)g(neigh)n(b)r(our-)538 3492 y(ho)r(o)r(ds)k(star-shap)r(ed)f(with)i(resp)r(ect)g(to)f (quasi-homogeneous)e(dilatations.)3265 3647 y Fe(\003)330 3919 y Fs(8)135 b(V)-11 b(ersal)44 b(deformations)h(for)e (quasi-homogeneous)i(sin-)532 4068 y(gularities)330 4267 y Fm(8.1)112 b(Holomorphic)35 b(case)330 4420 y Ft(W)-7 b(e)29 b(will)f(pro)n(v)n(e)f(the)i(v)n(ersal)e(deformation)g(theorem)h (for)g(all)g(quasi-homogeneous)e(singulari-)330 4520 y(ties.)p Black 330 4688 a Fj(Lemma)k(1)p Black 41 w Fp(L)l(et)i Fq(F)957 4700 y Fl(a)998 4688 y Ft(\()p Fq(x;)14 b(\030)t Ft(\))34 b(\()p Fq(a)29 b Fr(2)g Fi(C)1463 4658 y Fl(\026)1514 4688 y Ft(\))k Fp(b)l(e)g(a)h(versal)g(deformation)h(of) e(a)h(quasi-homo)l(gene)l(ous)330 4788 y(singularity)41 b(and)f Fq(\015)965 4800 y Fl(j)1039 4788 y Fp(a)g(lo)l(c)l(al)t(ly)i (c)l(onstant)d(b)l(asis)h(of)h(the)f(vanishing)h(homolo)l(gy.)70 b(Then)41 b(the)330 4887 y(Jac)l(obian)g(determinant)e Fq(J)8 b Ft(\()p Fq(a)p Ft(\))40 b Fp(of)g Fq(a)h Fr(!)f Ft(\()1711 4821 y Fk(R)1751 4917 y Fl(\015)1786 4925 y Fg(j)1817 4917 y Fo(\()p Fl(a)p Fo(\))1923 4887 y Fq(\030)t(dx)p Ft(\))g Fp(which)h(is)f(wel)t(l)h(de\014ne)l(d)e(outsise)h(the)330 5004 y(discriminant)g(set)e(\(the)g(set)g(of)h Fq(a)p Fp('s)g(for)g(which)h(the)e(curve)h Fq(F)2345 5016 y Fl(a)2424 5004 y Ft(=)f(0)g Fp(is)h(singular\))f(extends)330 5104 y(to)31 b Fi(C)485 5073 y Fl(\026)567 5104 y Fp(as)h(a)g Ft(non)d(v)-5 b(anishing)31 b Fp(holomorphic)k(function.)44 b(If)31 b(we)h(take)g(the)g(versal)g(deformation)330 5203 y(gener)l(ate)l(d)e(by)g(monomials,)i Fq(J)38 b Fp(is)30 b(c)l(onstant.)455 5372 y Ft(As)k(a)g(corollary)e(w)n(e)i(get) g(that)h(there)f(exists)g(a)g(canonical)f(measure)h(on)g(the)h(v)n (ersal)d(de-)330 5471 y(formation)27 b(\(b)r(ecause)h(the)g(v)-5 b(anishing)27 b(homology)f(has)i(a)f(canonical)g(Leb)r(esgue)g (measure\).)37 b(It)330 5571 y(w)n(ould)27 b(b)r(e)h(nice)g(to)f(ha)n (v)n(e)g(a)g(geometric)f(de\014nition)i(of)g(that)g(measure.)330 5670 y Fp(Pr)l(o)l(of.{)p Black 1788 5919 a Ft(13)p Black eop %%Page: 14 14 14 13 bop Black Black Black Black Black 637 523 a Fr(\017)p Black 41 w Ft(W)-7 b(e)28 b(\014rst)g(c)n(hec)n(k)e(that:)1483 672 y Fq(@)p 1438 709 140 4 v 1438 785 a(@)5 b(a)1531 797 y Fl(\013)1601 615 y Fk(Z)1648 804 y Fl(\015)t Fo(\()p Fl(a)p Fo(\))1792 728 y Fq(\030)t(dx)24 b Ft(=)2033 615 y Fk(Z)2080 804 y Fl(\015)t Fo(\()p Fl(a)p Fo(\))2224 728 y Fq(K)2295 740 y Fl(\013)2342 728 y Fq(dt)720 967 y Ft(where)j Fq(dt)g Ft(is)f(the)i(time)f(for)f(the)h(dynamics)g (induced)g(b)n(y)g(the)g(Hamiltonian)720 1067 y Fq(H)789 1079 y Fo(0)845 1067 y Ft(+)928 1005 y Fk(P)1029 1067 y Fq(a)1073 1079 y Fl(\013)1121 1067 y Fq(K)1192 1079 y Fl(\013)1266 1067 y Ft(on)g(the)h(surface)f Fq(H)1873 1079 y Fo(0)1929 1067 y Ft(+)2012 1005 y Fk(P)2113 1067 y Fq(a)2157 1079 y Fl(\013)2205 1067 y Fq(K)2276 1079 y Fl(\013)2345 1067 y Ft(=)c(0.)p Black 637 1200 a Fr(\017)p Black 41 w Ft(W)-7 b(e)28 b(then)g(pro)n(v)n(e)e(using)h(Picard-Lefsc)n (hetz)e(form)n(ula)i(that)g Fq(J)36 b Ft(is)27 b(univ)-5 b(alen)n(t:)720 1299 y(the)31 b(P)n(oincar)n(\023)-39 b(e)28 b(group)h(of)i(the)g(complemen)n(t)g(of)f(the)h(discriminan)n(t) g(is)f(gen-)720 1399 y(erated)d(b)n(y)h(small)g(lo)r(ops)f(around)g (the)h(stratum)g(corresp)r(onding)e(to)h(1)h(v)-5 b(an-)720 1499 y(ishing)30 b(cycle)g(sa)n(y)g Fq(\015)1361 1511 y Fo(1)1398 1499 y Ft(.)46 b(F)-7 b(ollo)n(wing)29 b(suc)n(h)h(a)g(lo)r (op)g(will)h(add)f(to)g(the)h(lines)g(of)720 1598 y(the)d(Jacobian)e (determinan)n(t)i(a)f(linear)g(com)n(bination)f(of)i(the)g(\014rst)f (one.)p Black 637 1731 a Fr(\017)p Black 41 w Fq(J)36 b Ft(is)28 b(b)r(ounded)g(near)f(the)h(co)r(dimension)g(1)f(stratum)h (of)g(the)g(discriminan)n(t.)720 1831 y(Hence)d Fq(J)32 b Ft(is)24 b(holomorphic)f(near)h(the)g(co)r(dimension)g(1)g(strata)f (and)h(b)n(y)g(Har-)720 1930 y(togs)36 b(ev)n(erywhere.)63 b Fq(J)45 b Ft(is)37 b(clearly)f(quasi-homogeneous.)62 b(Being)37 b(non)n(v)-5 b(a-)720 2030 y(nishing)34 b(b)n(y)g([5])g (p.95,)i Fq(J)42 b Ft(is)34 b(quasi-homogeneous)e(of)i(degree)f(0,)j (hence)e(a)720 2130 y(non-zero)26 b(constan)n(t.)3265 2296 y Fe(\003)455 2395 y Ft(Using)h(the)h(strategy)e(of)i(Pham)f(in)h ([29)o(],)f(w)n(e)h(can)f(pro)n(v)n(e)f(the)i(follo)n(wing:)p Black 330 2561 a Fj(Theorem)i(6)p Black 42 w Fp(L)l(et)d Fq(<)c(F)1107 2573 y Fo(0)1168 2561 y Fq(>)k Fp(a)i(quasi-homo)l(gene)l (ous)g(singularity)g(with)g Fq(F)2683 2573 y Fl(a)2747 2561 y Ft(=)22 b Fq(F)2887 2573 y Fo(0)2940 2561 y Ft(+)3020 2499 y Fk(P)3121 2561 y Fq(a)3165 2573 y Fl(\013)3212 2561 y Fq(K)3283 2573 y Fl(\013)330 2661 y Fp(\()p Fq(K)435 2673 y Fl(\013)512 2661 y Fp(monomials\))31 b(as)g(a)f(versal)h (deformation.)42 b(L)l(et)30 b Fq(<)23 b(F)2170 2673 y Fl(t)2223 2661 y Fq(>)30 b Fp(b)l(e)g(any)h(analytic)g(deformation) 330 2761 y(of)i Fq(<)26 b(F)574 2773 y Fo(0)638 2761 y Fq(>)p Fp(.)44 b(Ther)l(e)32 b(exists)g(an)f(analytic)i(family)h(of)e (germs)g(of)h(c)l(anonic)l(al)f(di\013e)l(omorphisms)330 2860 y Fq(\037)382 2872 y Fl(t)441 2860 y Fp(such)e(that)1416 2960 y Fq(<)23 b(F)1557 2972 y Fl(t)1605 2960 y Fr(\016)18 b Fq(\037)1717 2972 y Fl(t)1769 2960 y Fq(>)p Ft(=)p Fq(<)k(F)2039 2975 y Fl(a)p Fo(\()p Fl(t)p Fo(\))2179 2960 y Fq(>)330 3109 y Fp(wher)l(e)30 b(the)g(functions)g Fq(a)1107 3121 y Fl(j)1142 3109 y Ft(\()p Fq(t)p Ft(\))g Fp(ar)l(e)g(analytic.)330 3275 y(Pr)l(o)l(of.{)p Black Black 563 3441 a Ft(W)-7 b(e)26 b(will)g(giv)n(e)f(the)h(pro)r(of)f (for)g Fq(A)1571 3453 y Fo(2)1634 3441 y Ft(\(the)i(cusp\),)f(it)g(is)g (then)g(trivial)f(to)g(see)h(ho)n(w)f(to)538 3541 y(extend)i(the)h(pro) r(of)f(to)h(the)g(general)e(case.)662 3641 y(Using)34 b(Moser's)f(metho)r(d,)j(the)f(idea)f(is)g(to)g(\014t)g(the)h(action)e (in)n(tegrals.)55 b(The)538 3740 y(details)27 b(run)g(as)g(follo)n(ws:) p Black 637 3873 a Fr(\017)p Black 41 w Ft(W)-7 b(e)20 b(can)g(assume,)g(using)g(the)g(v)n(ersal)e(deformation)h(theorem)g (\(see)h([3)o(]\),)i(that)720 3973 y(w)n(e)j(start)g(with)i Fq(F)1278 3985 y Fl(a)1341 3973 y Ft(=)c Fq(F)1482 3985 y Fo(0)1534 3973 y Ft(+)14 b Fq(a)1657 3985 y Fo(1)1694 3973 y Fq(x)h Ft(+)f Fq(a)1879 3985 y Fo(2)1942 3973 y Ft(and)25 b Fq(!)2153 3985 y Fl(c)2210 3973 y Ft(=)d Fq(!)2349 3985 y Fo(0)2401 3973 y Ft(+)14 b Fq(O)r Ft(\()p Fq(c)p Ft(\))27 b(and)e(think)h(as)720 4072 y Fq(t)h Ft(=)f(\()p Fq(a;)14 b(c)p Ft(\).)44 b(W)-7 b(e)31 b(c)n(ho)r(ose)d Fq(\025)1575 4084 y Fl(c)1639 4072 y Ft(suc)n(h)i(that)g Fq(d\025)2102 4084 y Fl(c)2163 4072 y Ft(=)d Fq(!)2307 4084 y Fl(c)2360 4072 y Fr(\000)20 b Fq(!)2497 4084 y Fo(0)2564 4072 y Ft(an)29 b(assume)g(that)720 4172 y Fq(\025)768 4184 y Fl(c)826 4172 y Ft(=)22 b Fq(O)r Ft(\()p Fr(j)p Fq(c)p Fr(j)p Ft(\).)p Black 637 4305 a Fr(\017)p Black 41 w Ft(Let)28 b Fq(\016)f Ft(=)d Fr(f)p Ft(4)p Fq(a)1150 4275 y Fo(3)1150 4325 y(1)1205 4305 y Ft(+)18 b(27)p Fq(a)1416 4275 y Fo(2)1416 4325 y(2)1476 4305 y Ft(=)24 b(0)p Fr(g)j Ft(the)h(discriminan)n(t)g(set.)38 b(W)-7 b(e)29 b(w)n(an)n(t)e(to)h(de\014ne)720 4404 y(a)j(smo)r(oth)h (family)g(of)f(holomorphic)g(di\013eomorphisms)g Fq(a)e Fr(!)h Fq(')2788 4416 y Fl(c)2822 4404 y Ft(\()p Fq(a)p Ft(\))h(=)e Fq(a)3099 4374 y Fn(0)720 4504 y Ft(suc)n(h)e(that)h Fq(')1141 4516 y Fo(0)1202 4504 y Ft(=)23 b Fq(I)7 b(d)27 b Ft(and)h(for)f(all)g(cycles)g Fq(\015)2088 4516 y Fl(j)2151 4504 y Ft(of)g Fq(Z)2302 4516 y Fl(a)2365 4504 y Ft(=)c Fr(f)p Fq(F)2548 4516 y Fl(a)2611 4504 y Ft(=)g(0)p Fr(g)j Ft(w)n(e)i(ha)n(v)n(e)1305 4619 y Fk(Z)1351 4808 y Fl(\015)1386 4816 y Fg(j)1417 4808 y Fo(\()p Fl(a)1479 4791 y Fc(0)1502 4808 y Fo(\))1546 4732 y Fq(\030)t(dx)23 b Ft(=)1787 4619 y Fk(Z)1833 4808 y Fl(\015)1868 4816 y Fg(j)1899 4808 y Fo(\()p Fl(a)p Fo(\))2005 4732 y Fq(\030)t(dx)c Ft(+)2237 4619 y Fk(Z)2283 4808 y Fl(\015)2318 4816 y Fg(j)2349 4808 y Fo(\()p Fl(a)p Fo(\))2455 4732 y Fq(\025)2503 4744 y Fl(c)p Black 637 4990 a Fr(\017)p Black 41 w Ft(This)33 b(implicit)h(equation)e(can)g(b)r(e)h(uniquely)g(solv)n(ed)f(for)g Fq(c)h Ft(small)f(enough)720 5089 y(outside)19 b Fq(\016)i Ft(b)r(ecause)d(the)h(Jacobian)e(determinan)n(t)i(of)f Fq(a)23 b Fr(!)g Ft(\()2576 5022 y Fk(R)2616 5119 y Fl(\015)2651 5127 y Fg(j)2682 5119 y Fo(\()p Fl(a)p Fo(\))2788 5089 y Fq(\030)t(dx)p Ft(\))2950 5101 y Fl(j)s Fo(=1)p Fl(;)p Fo(2)720 5206 y Ft(is)28 b(a)f(nonzero)f(constan)n(t)h(\(see)g(lemma)h (1\).)p Black 637 5339 a Fr(\017)p Black 41 w Ft(Near)h(the)g(stratum)h (of)f(the)h(discriminan)n(t)e(where)h(the)h(v)-5 b(anishing)29 b(cycle)g(is)720 5438 y Fq(\015)763 5450 y Fo(1)800 5438 y Ft(,)37 b(the)e(in)n(tegrals)1353 5371 y Fk(R)1392 5468 y Fl(\015)1427 5476 y Fd(1)1498 5438 y Ft(and)1667 5371 y Fk(R)1706 5468 y Fl(\015)1741 5476 y Fd(2)1792 5438 y Fr(\006)1871 5371 y Fk(R)1909 5468 y Fl(\015)1944 5476 y Fd(1)1994 5438 y Ft(log)2115 5371 y Fk(R)2155 5468 y Fl(\015)2190 5476 y Fd(1)2261 5438 y Ft(are)f(univ)-5 b(alen)n(t)35 b(and)g(holo-)720 5538 y(morphic,)26 b(thanks)g(to)g(the) g(Picard-Lefsc)n(hetz)e(form)n(ula)h(and)h(the)g(Jacobian)720 5638 y(determinan)n(t)i(is)f(the)h(same:)36 b(so)27 b(w)n(e)h(can)f (also)f(solv)n(e.)p Black 1788 5919 a(14)p Black eop %%Page: 15 15 15 14 bop Black Black Black 637 390 a Fr(\017)p Black 41 w Ft(No)n(w)34 b(w)n(e)g(ha)n(v)n(e)f(solv)n(ed)g(the)i(equation)f (outside)g(a)g(set)g(of)g(co)r(dimension)g(2)720 490 y(and)g(w)n(e)f(conclude)h(b)n(y)g(the)g(fact)g(that)g(holomorphic)f (functions)h(ha)n(v)n(e)f(no)720 589 y(singularities)27 b(of)g(co)r(dimension)g Fr(\025)c Ft(2)k(\(Hartog's)g(theorem\).)p Black 637 709 a Fr(\017)p Black 41 w Ft(P)n(erforming)17 b(the)i(reparametrization)d(of)j(the)g(v)n(ersal)e(deformation)h(w)n(e) g(need)720 809 y(to)23 b(sho)n(w)e(that)i(\()p Fq(<)g(F)1365 821 y Fl(a)1429 809 y Fq(>;)14 b Ft(\()p Fq(')1617 778 y Fn(\000)p Fo(1)1617 829 y Fl(c)1706 809 y Ft(\))1738 778 y Fl(?)1777 809 y Ft(\()p Fq(d\030)e Fr(^)c Fq(dx)p Ft(\)\))25 b(and)e(\()p Fq(<)g(F)2472 821 y Fl(a)2535 809 y Fq(>;)14 b(d\030)e Fr(^)c Fq(dx)g Ft(+)g Fq(d\025)3053 821 y Fl(c)3090 809 y Ft(\))720 908 y(are)33 b(equiv)-5 b(alen)n(t.)55 b(The)34 b(di\013erence)f(of)h(these)g(2)f(symplectic)h (forms)f(is)g Fq(d\014)3088 920 y Fl(c)720 1008 y Ft(where)25 b(the)h(in)n(tegral)e(of)h Fq(\014)1538 1020 y Fl(c)1598 1008 y Ft(o)n(v)n(er)e(all)i(v)-5 b(anishing)25 b(cycles)g(of)g(all)g Fq(Z)2749 1020 y Fl(a)2789 1008 y Ft('s)h(v)-5 b(anish.)p Black 637 1127 a Fr(\017)p Black 41 w Ft(It)26 b(remains)f(no)n(w)f(to) i(\014nd)g Fq(f)1591 1139 y Fl(a;c)1680 1127 y Ft(\()p Fq(x;)14 b(\030)t Ft(\))26 b(whose)f(di\013eren)n(tial)g(on)g Fq(Z)2720 1139 y Fl(a)2786 1127 y Ft(is)g Fq(\014)2914 1139 y Fl(c)2948 1127 y Ft(.)36 b(W)-7 b(e)720 1227 y(de\014ne)30 b Fq(f)1003 1239 y Fl(a;c)1118 1227 y Ft(=)c Fq(g)1249 1239 y Fl(a)1285 1247 y Fd(1)1317 1239 y Fl(;c)1370 1227 y Ft(.)43 b(The)29 b(restriction)g(of)g Fq(g)2144 1239 y Fl(a)2180 1247 y Fd(1)2212 1239 y Fl(;c)2295 1227 y Ft(to)h(all)f Fq(Z)2573 1239 y Fl(a)2609 1247 y Fd(1)2641 1239 y Fl(;b)2723 1227 y Ft(is)h(obtained)720 1326 y(b)n(y)i(in)n (tegration)f(from)g(a)h(p)r(oin)n(t)g Fq(m)1835 1338 y Fl(a)1871 1346 y Fd(1)1903 1338 y Fl(;b)1987 1326 y Fr(2)e Fq(Z)2129 1338 y Fl(a)2165 1346 y Fd(1)2198 1338 y Fl(;b)2272 1326 y Fr(\\)22 b(fk)p Fq(z)t Fr(k)28 b Ft(=)i(1)p Fr(g)h Ft(whic)n(h)h(can)720 1426 y(b)r(e)38 b(c)n(ho)r(osen)f(an)h(analytic)f(function)h(of)g(\()p Fq(a)2131 1438 y Fo(1)2168 1426 y Fq(;)14 b(b)p Ft(\))38 b(of)g(the)g(forms)f Fq(\014)2855 1438 y Fl(c)2889 1426 y Ft(.)67 b(The)720 1526 y(smo)r(othness)39 b(of)h Fq(f)49 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a([2])p Black 41 w(V.)22 b(Arnold,)29 b(First)23 b(Steps)f(in)h(Lo)r(cal)e(Symplectic)i (Algebra.)k Fp(A)n(mer.)e(Math.)i(So)l(c.)e(T)-6 b(r)l(ansl.,)501 2149 y Ft(\(2\)194:1-8,)25 b(1999.)p Black 372 2315 a([3])p Black 41 w(V.)33 b(Arnold,)g(A.)g(V)-7 b(arc)n(henk)n(o)31 b(et)i(S.)f(Goussein-Sad)n(\023)-39 b(e,)85 b(Singularit)n(\023)-39 b(es)31 b(des)h(applications)501 2415 y(di\013)n(\023)-39 b(eren)n(tiables)26 b(I.)37 b Fp(Mir)30 b(\(Mosc)l(ou\),)h(1986.)p Black 372 2581 a Ft([4])p Black 41 w(V.)d(Arnold,)36 b(Mathematical)27 b(Metho)r(ds)h(of)g(Classical)e(Mec)n(hanics.)36 b Fp(Springer,)31 b(1989.)p Black 372 2747 a Ft([5])p Black 41 w(V.)d(Arnold,)36 b(Dynamical)27 b(Systems)h(VI.)37 b Fp(Springer,)31 b(1993.)p Black 372 2913 a Ft([6])p Black 41 w(W.)25 b(Balser,)f(F)-7 b(ormal)24 b(P)n(o)n(w)n(er)e(Series) i(and)h(Linear)f(Systems)h(of)f(Meromorphic)g(Ordinary)501 3013 y(Di\013eren)n(tial)j(Equations.)g Fp(Springer,)k(2000.)p Black 372 3179 a Ft([7])p Black 41 w(P)-7 b(.)29 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a([11])p Black 41 w(Y.)e(Colin)g(de)g(V)-7 b(erdi)n(\022)-39 b(ere)21 b(et)h(B.)g(P)n(arisse,)27 b(Equilibre)22 b(instable)g(en)g(r) n(\023)-39 b(egime)20 b(semi-classique)h(:)501 4341 y(I{Concen)n (tration)k(microlo)r(cale.)36 b Fp(Commun.)30 b(PDE)p Ft(,)e(19:1535-1563,)22 b(1994.)p Black 330 4507 a([12])p Black 41 w(Y.)g(Colin)g(de)g(V)-7 b(erdi)n(\022)-39 b(ere)21 b(et)h(B.)g(P)n(arisse,)27 b(Equilibre)22 b(instable)g(en)g(r)n(\023) -39 b(egime)20 b(semi-classique)h(:)501 4607 y(I)r(I{Conditions)35 b(de)h(Bohr-Sommerfeld.)61 b Fp(A)n(nnales)38 b(de)g(l'IHP)g (\(Physique)h(th)n(\023)-40 b(eorique\))p Ft(,)501 4706 y(61:347-367,)23 b(1994.)p Black 330 4872 a([13])p Black 41 w(Y.)31 b(Colin)g(de)h(V)-7 b(erdi)n(\022)-39 b(ere)29 b(and)j(B.)f(P)n(arisse,)46 b(Singular)31 b(Bohr-Sommerfeld)e(rules.)48 b Fp(Com-)501 4972 y(mun.)29 b(Math.)i(Phys.)p Ft(,)e(205:459-500,)23 b(1999.)p Black 330 5138 a([14])p Black 41 w(Y.)i(Colin)g(de)g(V)-7 b(erdi)n(\022)-39 b(ere)24 b(et)h(J.)g(V)-7 b(ey)g(,)33 b(Le)25 b(lemme)h(de)f(Morse)f(iso)r(c)n(hore.)31 b Fp(T)-6 b(op)l(olo)l(gy)p Ft(,)28 b(18:283-)501 5238 y(293,)e(1979.)p Black 330 5404 a([15])p Black 41 w(Y.)32 b(Colin)g(de)g(V)-7 b(erdi)n(\022)-39 b(ere)30 b(and)i(San)f(V)r(~)-44 b(u)33 b(Ngo)1876 5421 y(.)1908 5404 y(c,)g(Singular)e(Bohr-Sommerfeld)g (Rules)h(for)501 5503 y(2d)24 b(In)n(tegrable)g(Systems.)g Fp(Pr)n(\023)-40 b(epublic)l(ation)29 b(no)e(508)i(de)e(l'Institut)g(F) -6 b(ourier)27 b(\(mai)h(2000\),)501 5603 y Ft(h)n (ttp://www-fourier.ujf-grenoble.fr/~)c(ycolv)n(er.)p Black 1788 5919 a(23)p Black eop %%Page: 24 24 24 23 bop Black Black Black 330 390 a Ft([16])p Black 41 w(E.)31 b(Delabaere,)h(H.)g(Dillinger)g(and)f(F.)h(Pham,)h(Exact)e (semi-classical)e(Expansions)i(for)501 490 y(one)21 b(Dimensional)h (Quan)n(tum)f(Oscillators,)g Fp(Journal)j(Math.)i(Phys.,)f Ft(38)c(\(12\):6126-6184)501 589 y(\(1997\).)p Black 330 756 a([17])p Black 41 w(E.)33 b(Delabaere,)h(H.)h(Dillinger)e(et)h 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