Luận văn Kiểu Tô-Pô của mầm đường cong giải tích bất khả qui tại điểm kỳ dị cô lập

41 Chuong4 " ,,? " ,,? ~ " KIEV TO-PO CUA MAM DUONG CONG ,,' ? BATKHAQVI Cho mall duOngcong (X,O) E cc2t';1idi~mki d! co l~p0, xac d!nhbai hamgifti tfch1 E CC{x,y} batkhftqui, 1 cob~cm > atheoy, l(a,y) = ymva m=multo(f).Khaitri~nPuiseuxcua1chomchu6inghi~mcual(x,y)=a: Yk(X) =y((kx1/m),k = 1,..,m. ( lacanb~cmnguyenthuycuadcmV!. 4.1 (m)-no xuyen XetmQtphiltuX vaoD,D = {xE CCilxlS;1}.Thi t';1im6idi~mXoE D\ {a},co m nghi~m{Yk(XO)}k=l,..,mcua phucmgtrinh l(xo, y) =a. Phil tu X\ {a}vaoD\ {a}la philkhongre nhanh(do 0 la di~mki d! co l~p cuaX), dodo,m6iduOngx(t)c D\{a},t E [a,1]vax(a)=x(1),duQ'Cnangthanh m duOngdi khacnhau: {(x(t),Y1(t)),...,(x(t),Ym(t))}eX. Mifltkhac,vi~cnghiencUuki~uto-pocuaciflp(B, B n X) duQ'cduav~nghien cUuki~uto-pocuaciflp(51X D2,Kd, v6i K1 = (51X D2)n X = {(x,y) E (51X 42 D2)lf(x,y) = O}.Do d6,tac6th~thallisoh6abienthvct chotQad(>phucx vay tren(51x D2)nhusail: Cho0 <5 ::;1. Xetx(t)=5e27rit,t E [0,1]v6'ix(O)=eo=x(l). V~y,x(t)cD. Va Yk(t)=y((kX1/m)=y(5e27ri(t+k/m)),t E [0,1],k =1,..,m. Khi d6, (51x D2)nX =K1 ={ (x(t),Yk(t))E (51x D2), t E [0,1],k=1,..m} M(>trt!diflcclingvaiquailh~dongnhatt<:tihaimifltdau-cuoicuatrt!thldong phoiv6'im(>txuyendiflc.Di~unaychopheptabi~udienK1 trentrt!([0,1]x D2)C ]R3, clingIa m duemgdi khacnhau: {(t'Y1(t)),...,(t,Ym(t))}C ([0,1]X D2). Vi dl;l4.1.1. ChoX ={(x,y) E ((;21f (x,y) = y2- x3= 0} Khai tri~nPuiseuxcuaf chohainghi~m: { Y1(X)=x3/2 Y2(X) =_X3/2 Thamsoh6abienthvct E [0,1],tadugc: { Y1(t) = 5e37rit Y2(t)= -5e37rit , 0<5::;1 Bi~udienK1trongtrt!([0,1]x D2)c ]R3Iahaiduemgtachrm nhau: { (t, 5e37rit),(t, -5e37rit)} D!nh nghia4.1.1. MQtnO'khdvi Ia ciflp(51x D2,K), gomm(>txuyendiflcvam(>tduemgcong dongphoivai 51vanamtrenxuyen. 43 Xet phepchieuPI : (SI XD2)~ SI. NeupIIK : K ~ Slla mQtphilb~cm >0 (m laSOdi~mlIen m6i thO'p-l(X), x E SI), thl na kha vi (SI X D2,K) duqc gQi la mQt(m)-na,clingcanduqcgQila mQtnaqUailm vangtheokinh tuyencuaxuyen. Vi dl;l4.1.2. (SI XD2,Kr) lamQt(m)-na,vaimlab~ccuadathucbatkhaquif theoy. D!nh nghia4.1.2. MQthimkhdvi m-nh{mhlamQthQm duOngcongkhavi {Yk}k=I,..,mtrongtrl;l (I x D2),J = [0,1]c JR: Yk : I ~ I x]R2 co Yk(I) c I x D2,k = 1,..,m sao cho: 1. Vt E I, himgiaovai latctttngang{t}XD2cuatrl;lI x D2tc;timdi~mkhacnhau. D~tphepchieuP'I : I X D2 ~ I. Khi do, { <I 0 Yk(O) = 0 ho(jc 1 PI 0 Yk(l) = 1ho(jc0 k = 1,..,m, Ta qui uO'cP'I 0 Yk(O) = 0 va P~0 Yk(l) =1, k =1,..,m. 2. m di~mtc;tihai dati trl;l: {Yl(O),...,Ym(O)}C {a}X D2 va {Yl(l), ...,Ym(1)}C {I} X D2 co cling tQadQph~ngtrongD2. 3. D~tala phepchuy~nvi mdi~m{Yl(O),...,Ym(O)}thanhmdi~m{Yl(l), ...,Ym(l)} trongD2 thl: dYa(l)(l)- dYk(O) I = 1,..,m;CJ(I)=k.-- ,dt dt Vi dl;l4.1.3. (m)-na(SI x D2,Kr) bi~udienlIen trl;lI x D2la m duOng,mam6idi~mlIen doco tQade:>Ianluqt: {(t,Yl(t)),..., (t,Ym(t))},t E [0,1] lamQthimkhavi m-nhanhlOpCoo. 44 Vi do 4.1.4. Xet: x = {(x,y) E C21f (x,y) = y2 - X3 = a} Khaitri~nPuiseuxcuaf chohainghi~m: { Y1(X) = X3/2 Y2(X) = _X3/2 Thamsoh6abienthlJct E [0,1]saochoduOngcongx(t)niimtrenbiencua D, nghiala: x(t) = e27rit, t E [0,1],x(O)=eo=x(l) Va, { Y1(t) = e37rit Y2(t) = -e37rit Khi d6, (2)-nO'niimtren51x 51,biencuaxuyend~c51x D2.NO'K1 qwlnd~u quanhtrenxuyen51x 51. Bim lienketv6i nO'K1 clingqUaild~utrenbienI x 51 cuatrl;lI x D2. Bi~udienhinhhQctrong}R3cua(2)-nO'(51x D2,Kd vacuabim2-nhanhU'ng v6i (2)-nO'(51x D2,Kd nhusau: (2)-nO'trenxuyen Bim trentn;l(2nh:inh) NO'K1 qUail2-vongtheokinh tuyencuaxuyen.Va chuyriing,thuhypphep chieup~: 51x 51-t ({to}X 51)rv51,toE I, trenK1 :P~IKI: K1 -t {to}X 51,sexac d!nhm9tphil b~c3. Khi d6, (2)-nO'(51x D2,Kd duqcgQila m9tnO'xuyenki~u (2,3),nO'qUail3-vongtheoVI tuyencuaxuyen. 0 Ta di denm9td!nhnghiat6ngquatsauday. 45 D!nhnghia4.1.3. Chohaisonguyenmvan, (m,n)= 1va1<m<n. X6thamgi:iitich:1(x,y)=ym- xn. (m)-na (51x D2,K1) U'ngvmham1 cod,;mgnhutrenduqcgQiIam('>tn(Jxuyln kilu (m,n). K1 Ia m9tna kh:ivi, qUaild6uquanhbien(51x 51)m vangtheokinh tuyen " vanvangtheovituyencuaxuyen.M9tnaxuyenkieu(m,n) lienketv6im9tbim m-nhanh,m6inhanhlam9tduO'ngxoanocquail~vangquanhbien([0,1]X51)cua trl;l([0,1]X D2). 4.2 Kieu to-pocuemamduongcongbatkhaqui D!nhIf 4.2.1.([5],p.94) Kilu to-pocuama'mduo'ngcongphdngheftkhdquiX tqidiemki dj co l{lp0 holmtoandu(/cxacdjnhbai caec(ipPuiseuxcua1 xacdjnhX. Chungminh. Gi:iSlrX duqcxacdinhbaihamgi:iitich1E C{x,y}batkh:iqui,1cob~c m > atheoy, 1(0,y) = ymvam = multo(J).Khai trienPuiseuxcua1 lay((x),( la canb~cmnguyenthuycuadanvi,va(m1,nd, ...,(mg,ng)lacaec~pPuiseuxcua1. TacokhaitrienPuiseuxchu~ntacU'ngValcaec~pPuiseuxcua1 la: 2:l... ---2.:2- ~ y(x) = Xml +xmlm2 +... +xml..m9 Quidi;lOcaediem(x,y(x))U'ngValkhaitrienchu~ntacnayduqcki hi~ula Kig),matasechU'ngminhrang,dayIam9tnaxuyen- gQila m9tn(Jxuylnl(ipva naxuyennaydongphoiVal(m)-naxuyen(51x D2,K1)cuaX. Va didenketlu~n. 1.Mo t:inaxuyenlap(51x D2,Kf): D~t(51x D2,Ki), 1:::;j :::;g,la(m1..mj)-naU'ngvmkhaitrienPuiseux: 2:l... ~ y(X)=Xml +...+xml..mj 46 trongdo,(ml,nd, ...,(mg,ng)Ia j ciftpPuiseuxdtiutiencuaf. 1.Bieudi~nhinhhQccua(ml..mj)-no: - V6i j = 1,hiennhien(51x D2,Kf) la m9tnoxuyenkieu(ml,nd. (ml)- no (51x D2,K{) qUaild~utrenbiencua xuyendiftcTo, Ia Ian c~nd~ngong cua (51x {a})c (51X D2),qUailml vangtheokinhtuyenvanl vangtheovi tuyentren biencuaTo. - Gia su v6i 1 ::; k ::;g, da:bietbieudi~nhinh hQccua (ml..mj)-no(51x D2,K{),Vj ::;k. Ta sematano (51x D2,K~+I). Kf hi~u(k+l(va(k)Iacanb~c(ml..mk+d(va(ml..mk))nguyenthuycuadan vi,vax=Ixle27rit,t E [0,1]. tffigvaim6it E [0,1]la(ml..mk+l)giatriyk+l((k+lX),(k+lnh~nltlnIuqttatca giatritrongt~p(ml..mk+dgiatricuacandanvD,tucIam6iIatciltngang{t}x D2 cuaxuyendiftc(51x D2)giaov6i K~+1t~i(ml..mk+ddiem.Xet cacdiemnay: k+l(1 ) I(ml"mk+d~!.:l. (ml"mk+l)~ ~ (ml"mk+d nk+1 ~y "k+lX = "k+l Xml +... + (k+l ml"mk xml"mk +(k+l ml"mk+l xml"mk+l l(nlm2 mk+d !.:l. (n m ) ~ n nk+l"k 1 " xml +...+1 k k+l Xml"mk + 1 k+lXml"mk+l+ "k+1 "k+1 [( /mk+l ) nlm2"mk !.:l. ( mk ) nk ~ ] n nk+l "k+l Xml +...+ (k+t Xml"mk +(k~t1Xml"mk+l. T~p(ml..mk+dSOphuc(k+l,Ia canb~c(ml..mk+l)cuadanvi, makhi m6i phtintutrongdo duQ'cliiy thuamk+1thi t~pnaytIlingv6i t~p(ml..mk)sophuc(k, Ia canb~c(ml..mk)cuadanvi. Do do,taco thenhomt~p(ml..mk+dsophucthanh (ml..mk)nhom,m6inhomco mk+lsophuc,kf hi~u(k+1,k,saDcho: Imk+l "k+l,k= (k Khi do, k+l(1 ) [ /nlm2"mk!.:l. n ~ ] n :.'.k:t y "k+lkX = "k Xml +...+1 kxml"mk +1 k+lxml"mk+l, "k "k+l,k k ( 1 ) nk+l - y ((kX)+ (k+l,kXml"mk+l [ 1 ] nk+l - yk((kX) + ((kxml"~k+l) mk+l , (mk+1,nk+d= 1. tffigvaim6igiatri (k,cot~pg6mmk+lsophuc(k+l,kmam6iphtintutrong 1 1 dothi:(;;;~t= (k.Tucla t~p{((kXml"mk+l)mk+l}comk+lsophuc,Iacanb~cmk+l 47 1 cua(m1..mk)SOphuc((kXml..mk). Dien gi:hhlnhhQc,trongm6ihitcat{t}X D2cuaxuyend~c(51X D2) , co (m1..mk+1)di~mIagiaocuaduemgcongkhavi K~+lv6'i({t}XD2)nftmtren(m1..mk) nk+l duemgtroll,m6iduemgtrOllbankinh Ixlml"mk+l vatamt(;tim6idi~mtrong(m1..mk) di~mgiaocuaKf v6'i({t}x D2),phanbod~utrenm6iduemgtrollIamk+1di~m. (2)B6 d~4.2.1.1: LuanchQnduqcIxl dunhosaDchotrenm6iIatcat({t}XD2)cua"xuyend~c (51x D2), (m1..mk)duemgtrollmataa trenIa khanggiaonhau. Chungminh. TasechUngminhdug,khoangcachgiUahaidi~mbatkl cuaKf n ({t}XD2), clingIa tamcuacacduemgtroll,Ianhon2 Hinbankinhm6iduemgtroll,tucIa Ian nk+l hon21xlml,mk+l v6'iIxl dunho. D~t, dk(lxl)= inf,lyk((kx)- yk((~x)l, (d(k l:;'k:;'g nk+l TadinchUngminh:dk(lxl)>21xlml..mk+l, v6'iIxl dunho. Di~unaytuongduongv6'i: ;\ k (1 I) ~ dk(lxl) 2LJ. X nk+l >, Ixl ml..mk+l vai Ixl dTlnho. Qui u6'c,2l0(lxl)= dO(lxl)=O. Ta tinh,6,k(lxl): 2lk(lxl)= 1nk+l inf Ixlml..mk+1 (k7'=(~ nlm2..mk 2:L nk Imk nk-l ~ (k xml +... + (k - Xml..mk-l + (;kXml..mk ;-'nlm2"mk ~ /nk-lmk nk-l ,~ -"k X 1 - ... - "k Xml"mk-l - (knkXml..mk (~- nk+l ) I nlm2"mk-nk , inf, IXI ml..mk ml..mk+l X ml"mk ((;lm2"mk - (knlm2"mk) +...+ (d(k nk-lmk-nk , , +X ml"mk ((;k-lmk - (knk-lmk) + ((;k - (knk) 48 nkmk+1-nk+1 nk-1mk-nk ( n1m2..mk-1-nk-1 [ ]in!,lxl m1..mk+1 X m1..mk X m1..mk-1 ((;;k)n1m2..mk-1 - ((~mk)n1m2..mk-1 +(d(k +. + [((;"'J"'-' - ((~m'J"'_'J)+((;' - (~n,) nkmk+1-nk+1 in! Ixl m1,.mk+1l~k-1(lxl) +((;k - (~nk) (d(~ - TruOnghqpk = 1: ~l(lxl)= in! Ix! nl;:?;':2n2 1 (n1 - ( 'n1 I 1 1 (1#(1 Tfnhchatcuacacc~pPuiseux:n1m2<n2=}n1m2- n2 <0 n] m2-n2 1 =} Ixl m1m2 = n2-n]m2 -+ 00 khi Ixl -+0 Ixl m1m2 I Do (1 1=(1 I =} (1 - (1 1=0 =}~l(lxl) -+00 khi Ixl -+ o. V~y,v6iIxldonhothl~l(!xl)>2. - Giasuv6im9is6nguyenl ::;k - 1thl~1(lxl)>2v6iIxldunho,tachUng minhdi~unayclingdungv6il =k. ~k(lxl)= nkmk+1-nk+1 I in! Ixl m1..mk+1l~k-1(lxl) +((;k- (tk) (k,i(~ TrangtruOnghqpxaunhat,thl ((;k - (~nk)=-2. Do do, ~k-1(lxl)> 2 =}~k-1(lxl)- 2 > O.Va: nkmk+1-nk+1 Ixl m1..mk+1 1 Ixl-+O nk+1-nkmk+1 ~ 00. IXI m1..mk+1 =}~k(lxl) ;::: Ixlnk:~~,;.~:~+l(l k-1(lxl)l- 2) -+00khiIxl-+ O. V~y,~k(lxl)>2v6iIxldunho. . 49 (3lKf+1IamQt(mk+1)-naxuyen: Ta ch<;mIxldunhod~coduQ'cacdi6unhuBe;d64.2.1.1. XetIanc~nd~ngongTk cuaduemgcongkhavi Kf magiaocuaKf vaim6i hitcat{t}x D2cuaxuyendiftc(Sl x D2)I~pthanh(m1..mk)cfia(duQ'cxacdinhnhu Be;d64.2.1.1).Vi duemgcongKf dongphoivmSl, nenTkdongphoivmmQtxuyen dac. Duemgcongkhavi Kf+1duQ'cvenQitieptrenbiencuaTknenxac~dinhduQ'c mQtvi phoigili'abiencuaIanc~nongTk vmxuyenKf x Sl, clingvi phoigili'aTk vaKf x D2.Codinhvi phoinay,tadongnhatTk vaiKf x D2. TheochUngminhaBe;d64.2.1.1,m6iIatcatngangxuyenKf x Sl IamQt duemgtroll chuamk+1di~mcuaKf+1. Nhu v~y,(Kf X D2,Kf+1) Ia mQtmk+1-na qUaild6utrenbien(Kf x Sl) cua(Kf x D2). (4)ChUngminh(mk+d-na(Kf x D2,Kf+1)clingki~uvmnaxuyenki~u(mk+1,nk+1): 1 Difttf = Ixl dunho,xetduemgtrollSE'tam0 bankinhf' = fml..mktrongmiftt phingphuc<C.Choanhx~h: h: SE'-'rKf C (Sl x D2). t H(t'ml..mk,yk(t')) Tabietdmg,h IadongphoivaduQ'CmarQngthanhdongphoiH nhusail: H : SE'x D2 -'rKf X D2 (t',y') H(t'ml..mk,yk(t') + y') Xet mQtna xuyenki~u(mk+1,nk+1)trenbiencuaxuyendiftc(SE'x D2), na Ungvmkhaitri~nPuiseux: Y(~k+1t')=(:nk+l ~<" 1xmk+l , ~k+1 fa din bcJcmk+1nguyenthuycuadO'nvi. anhcuam6idi~m(t',Y(~k+1t'))quaH Ia: , ,~ H(t', Y(~k+1t'))= (tml..mk,yk(t')+~;~i1tmk+l) T~inhli'ngdi~mkhactren(SE'x D2),thamsox theothamsothlfct'baiquail 50 he: x = t'ml..mk Ch9nt'HimQtr!trong(m1..mk)tr!canb~c(m1..mk)cuax,ki hi~ugiatr!do Ia: , L-t = (kxml..mk Khi do, H(t',Y(~k+1t'))=(x, =(x, =(x, L- nk+1 yk((kX) + [~k+1(kmk+lrk+1Xml"mk+l) k n nk+l y ((kX)+ [(k+1,k]k+1Xml"mk+l) yk+1((k+1X)). DayIat9adQcuadi~mcuaduangcongkhftvi Kf+1tren(Kf x D2).Nhuv~y dongphoiH giii'ahaixuyen(5E,x D2)va (Kf x D2),clingIa dongphoigiii'anO' K k.:! ( ) A (5 D2) ' K k+1 A (Kk D2) Ch ' ? ( )xuyen leU mk+1, nk+1 tren E' x va 1 tren 1 x . ungto, mk+1-nO' (Kf x D2,Kf+1)clingki~uv6'inO'xuyenki~u(mk+1,nk+d. (5)V6'i Ixl dunho,taxaydvngduqcnO'(Kf-1 x D2,Kf). Nhuv~ytadfidvngduqcnO'xuyenKt ki~u(m1,n1)trenbiencuaxuyend~c To- IaIanc~ndC;lngongcua{a}x D2trong(51x D2).Saildo,xetIanC~llldC;lngong T1cuaKt. Lanc~nT1viphoiv6'imQtxuyend~c(Kt xD2),trenbiencuaxuyend~c (Kt x D2),taxaydvngnO'(Kt x D2,Kn ki~u(m2,n2)'Theoeachnhuv~y,tadvng duqcnO'(Kf-1 x D2,Kf) ki~u(mg,ng),lingv6'ic~pPuiseuxthu9cuaf. (m)-nO'tim duqcIamQtnO'xuyenki~u(mg,ng),duqcg9iHimQtno'xuye'nl(ip(gIan). II. Chlingminhcaem-nO'(51x D2,Kd va(51x D2,Kf) Iaclingki~u: D~t(51x D2,Kij)) va (51x D2,KFJ), 1 ::;j ::;g, Iacae(m1..mj)-nO'IanIuqt lingv6'ikhaitri~nPuiseux: . ..':.L ~ y(J)(x) = alOxml+... +ajOXml"mjva, . ..':.L ~ y[J](x)= la10lxml+ ... + lajolxml,.mj Trongdo,a10,..,ajOIa caeh~sotrongcaesohC;lngtuanglingtrongkhaitri~n Puiseuxcuaf. 51 PhanI, tadiixaydlJIlgdugcnaxuyenl~p(51X D2,Kf), 1 ~ k ~ g,naKf qUailtrenbiencuaIanc~nd~ngongTk-1cuaKk-1mat~im6ilatci\tngangTk-1, tadugcm9tilia trollbankfnhIxl ml~~mk. V6i cachlamhoantoantuangtl;r,taclingdlJIlgdugcna (51x D2,KIk]),1 ~ k ~ g, bangcachxetIanc~nd~ngongT[k-1]cuaKIk-1])saochom6ilat ci\tngang ~ cuaT[k-1]tadugcm9tiliatrollbankfnhlakollxlml..mk.ChUngminhm9tketquatuang tvB6d~4.2.1.1,vaiIxl dunho,tanh~ndugcm9tna(51xD2,KIk])cling~i~uvaina xuyenki~u(mk,nk)'Quatrlnhl~pchodenc~pPuiseuxthug,(m)-na(51x D2,KIg]) lam9tnaxuyenki~u(mk,nk)' (1)ChUngminhcac(m)-na(51x D2,K~g])va(51x D2,K~g))laclingki~u: CachxaydlJIlgcho(51xD2,KIg])Iam9tnaxuyenl~p,hlnhanhlam9tduemg congqUaild~ulientiepnhau. Cach~sophucaiOva laiOl,1 ~i ~g,satkhacnhaum9tg6cquayquanh0, clingvaitfnhchatqUaild~ulientiepnhaucuana (51x D2,KIg]),thl(51x D2,Kig)) chfnhlana (51x D2,KIg])quaym9tg6cquanhO. Va nhuv~y,(51x D2,KIg])va (51x D2,Kig))Iaclingki~uto-po. (2)ChUngminhcac(m)-na(51x D2,Kf) va(51x D2,K~g])laclingki~u: D~chUngminh(51x D2,Kf) va (51x D2,KIg])clingki~u,tachUngminhton t~idongphoih : 51XD2 -+51x D2vaanhx~thuh~phlKf :Kf -+KIg]clinglam9t dongphoi.Dongthaih camsinhm9tphepdongnhattu-51x {O}den51x {O}vatu- 51x 51den51x 51. MQilatci\t({t}x D2)cua(m)-na(51x D2,Kf) vacua(m)-na(51x D2,KIg]) ladongnhat.Dod6,taxaydlJIlgm9tvi phoih, h: (51 X D2) ~ ({t}X D2) -+ ({t}x D2) C (51 X D2) thoa: . hbien m di~mcua Kf n ({t}x D2) thanhm di~mcuaKIg]n ({t}x D2). . hcamsinhphepdongnhattu-({t} x {O})den({t} x {O})va tu-({t} X 51)den ({t}X 51). V6i vi phoih naythlh=hhoantoandugcxacdinh. ** D~c6vi phoih, tadlJIlgtruemgvec-taV C }R3lapCOOtrongl~nc~ncua 52 tr1;1({t}X D2) X [0,1]nhusau: Xet tr1;1C = ({t}X D2)X[0,1],vaquailh~dongnh~thitcat({t}xD2)x {a}c ({t} x D2) cua nO'(51 x D2,Kf) v6i Iat cat ({t} x D2) x {1}C ({t} x D2) cua nO' (51 x D2,KIg]). Ki hi~uA Iat9ad9theotr1;1c({t}x {a})x [0,1]C C, va{5ih=1,..,mIa m do~n th~ngnoi tudi~m(t,yg((~x(t)))dendi~m(t,y[g]((~x(t)))xacdinhbCri: y = yg((;x) +A[y[g]((;x) - yg((;x)J, a ::;A ::;1 ((gIacanb~cmnguyenthuycuadonvi) V6i Ixldunho,trongmotanO'(51x D2,Kf) va(51x D2,KIg]),mdi~mcua mbinO'trenIatcat({t}x D2)Iadoixlingnhauvalamchomdo~n{5ih=1,..,mtUng doim9trOinhau,khongglaDv6itr1;1c({t}x {a})x [0,1]vaciingkhongglaDvoibien ({t}X 51)X [0,1]cuatr1;1C. Nhuv~y,taciingxaydvngduqcmIanc~nd~ngong {Tdi=l,..,mcuado~n{5ih=1,..,mtuonglingsaDchotUngdoim9t,Ianc~nIi vaTj rOi nhau,khongglaDvoitr1;1c,ciingkhongglaDv6ibien({t}X51)X [0,1]cuatr1;1C. Bd d~4.2.1.2: Ton t~im9ttruangvec-tO'V lOpCOOtrongIanc~ncuatr1;1C trong]R3thoacac tinhch~tsau: . Thanhphancuavec-tO'V theohu6'ngtr1;1ccuaC Ia hang1. . T~im9idi~mx trongIanc~ncuatf1;1cvabien({t}x 51)x [0,1]cuatr1;1C,V(x) songsongvoitr1;1ccuatr1;1C. . T~im9idi~mx thu9CIanc~nIi cua5i,1::;i ::;m,V(x)songsongvoi5i. Chungminh. Trong]R3,taxaydvngphilmachotr1;1C gom:t~pmacua0 chuatr1;1c({t}x {a})x [0,1],bien ({t}x 51)X [0,1]vaphankhongglaDv6'icac5i,1::; i::; m; vam t~pmaTi cua5i,1::;i ::;m. R6rang,diaphuongtrongt~pmacua0 haytrongt~pmaTi,1::;i ::;m,cling tont~itwangvec-tO'thoacacdi~uki~ntren. 53 Truemgvec-taduqcdvngtucactruemgvec-tO'diaphuongnaynhb'm(>tphan ho<;tchdonvi lapCOOtheophuma. . ** V6i truemgvec-taV lap CoovUaKaydvng,tuongtv nhuchUngminh dinhIy diu trucnon,tahoantoanxacdinhduqcvi ph6iit tu ({t}X D2)X {O}vao ({t}XD2)x {I}thoacacyellCalldfld~tfa.Nghialaclingxacdinhduqcdongph6i h gilracacc~p(51x D2,Kf) va(51x D2,Kig]). (3)Nhuv~ytadflchUngminhduqc(51x D2,Kf) va(51x D2,Kig])la..clIngki~u. TheoketquachUngminha do<;tnILl thl (51x D2,Kig])va(51x D2,Kig))laclIng ki~u.Suyfa, (51x D2,Kf) va(51x D2,Kig))laclIngki~u. Ml;1cdichcuatalachUngminh(51x D2,Kf) va(51x D2,Kd clIngki~u,do v~y,tachiconcanchUngminh(51x D2,Kig))va(51x D2,K1) laclIngki~u. (42ChUngminh(51x D2,K~g))va(51x D2,K1) laclIngki~u: D~chUngminhduqcdi~unay,tacanKaydvngduqcm(>tdongph6ih : (51X D2) -+ (51x D2),clingdongph6igilraK1 vaKig). Dongthaih camsinhm(>tphep dongnhattrenbien(51x 51)vadongnhattrentUnglatcat({t}XD2)cua(51x D2). Sosanhkhaitri~nPuiseuxy(x)vay(g)(x),chungsaikhacnhautoidala (g+1) nhomcacsohnglambiend<;tngchutit nO'Ung v6'iy(x) sov6'inO'xuyenl~p(51x D2,Kig)) Ungv6'iy(g)(x). Dov~y,tasephanbi~thaitruemghqp: a.Neuy(x) y(g)(x).Talayh laphepdongnhat. ChUngminhketthuc. b.Neuy(x)=I-y(g)(x),d~tZ(x) = y(x) - y(g)(x)=I-o. Ta tienhanhchUngminhtuongtVnhuIL2 (nhusail). Trang]R4,ki hi~uT = (51XD2)X [0,1]va'\ la tQad(>theo[0,1].5lam(>tm~t trangT duqcxacdinhbOiphuongtrlnh: J(x, y,'\)= II (y- ['\Y((gx)+ (1- ,\)y(g)((gx)J) =O. (m=l T<;ti,\ = 0 : 54 J(x, y,0)= II (y- y(g)((gx))=O. (m=l T(;li),= 1: J(x,y,1)= II (y- y((gx)) =f (x,y) =O. (m=l Nhu v~y,thu hypcua 8 t(;li(81X D2) X {a}la dongnhatv6i K~g),va t(;li (81X D2)X {1}la dongnhatvmK1. Ch9ncact9ad(>cua]R4la: t (U'ngvmx = Ixle27rit),Re(y),Im(y)va),. Phuongtrlnhf(x, y,),) = 0 dugctachthanh: { Re~x,y,),)=0 Imf(x,y,),)=0 *Trenm~t8, chi ton t(;lim(>ts6 hii'uh(;lnduangcongn(>itieptren8 lam 8 kh6ngchinhqui.Cacduangcongnaydugcxacdtnhla giaocua8 v6ihii'uh(;lnsieu m~t({ti}X D2)X [0,1],1:::;i:::;N. D~codugcketquanay,tatinhh(;lngcuamatr~ A dumday,lamatr~ncon cuamatr~nJacobicuaSt(;lidi~mbatki thu(>c8. Tabietrang,rank(A) ::J2t(;linhUngdi~mthu(>c8 madet(A)=0(dayladi~m ki dt cua8). Ta setinhdtnhthucdet(A)t(;lidi~mthu(>c8 co y = ),Y((gx)+ (1- ),)y(g)((gx): ReM,(x,y,),) Re¥y(x,y,),) det(A)= ImM,(x,y,),) Im¥y(x,y,),) Tu day,nh~nxetrangdet(A)= 0{::}Im(Y((gx)- y(g)((gx))=ImZ((gx)=0 Nhuv~y,d~coketlu~nv~di~mthu(>c8 nay,tachicantimnhUng iatrtcua t d~ImZ((gx)=O. aRe!(x y),) aRe!(x y ),)a). " a ey" A=I I ' (x,y,A)E 8. aIml( ),) aIm!(x y A)a). x,y, aRey" 55 Xethaitruemghqp: [I]Z((l)x)cohfiuhans6'hflnf.? Vai giathietnay,ImZ((gx)duqcvietd~ng: <00 ImZ((gx) =L ajsin(bjt+Cj,k) j=l Trangdo,aj,bjt,Cj,kE IR;Cj,knh~nm giatr~khacnhauUngv6'i(gnh~nm gia tr~canb~cm cuadonv~. Khi do,ImZ((gx)= 0t~imQtsohfulh~ngiatr~t E [0,1],ki hi~ucaegiatr~ naylati,1::;i ::;N. V~ylatadaxacd~nhduqccaegiatr~cuat lamm~tS khongchinhqui. Clingchuydmg,ImZ((gx)=I0v6'iIxidunho. [Ii] Z((l)x)co v6hanso'hflnf.? D~ty((g))(x)gomcaeso h~ngdtlutien,lay denso h~ngagOXml~gmgcuakhai tri~nPuiseuxy(x). Sailday,tasechUngminhrang,cae(m)-na(Sl x D2,K1) va (Sl x D2,Ki(g))) Ungvai khaitri~nPuiseuxy(x) vay((g))(x) Iaclingki~u. TuongtvBdd64.2.1.1,v6'iIxl dunho,chUngminhduqcsvtont~imQtIanc~n d~ngongT' cuaKi(g))chuaK1 saochoT' vi phoivaimQtxuyend~c.D~t(S{,x D2) la xuyend~cnay,vaH la vi phoi: H : (S{,x D2)-+T'c (Sl x D2) saGcho: (t',y) H(t'm,y((g))(x)+y) R5 rang,H bien(S{,x {O})thanhKi(g)),dongthaibienduemgcongxacd~nh bai (t',y(t')- y((g))(t'))thanhK1. Ma rQngH thanhdongphoiH' :T -+T bienKi(g))thanhK1, vathuhyptren bien:H'I&T: aT -+aT la dongnhilt. Sailcling,marQngH' thanhdongphoiii : (Sl x D2) -+ (Sl x D2)vai phep dongnhilttren(Sl x D2)\T. Tont~idongphoiii, chUngtocae(m)-na(Sl x D2,Kd va(Sl x D2,Ki(g))) 56 Ia clingki~u. . Sauday,tatiept1;lchUngminhchotnranghqpZ((gx)covoh(;lnsoh(;lng(y(x) co voh(;lnsoh(;lng)bangcachtheml~nluqttUngsoh(;lngcuanhomthu(9+ 1)cua y(x) vaoy((g»(x).Nhu v~y,clingchUngminhchotruanghqpZ((gx)co hii'uh(;lnso h(;lngm(>tcacht6ngquat. Ta da tlm duqcN gia tti ti,1 ::; ti ::; N, tuc la tlm duqcN duangcong 5 n ({ti} x D2) x [0,1],1 ::;ti ::;N trongT lam 5 khongchinhqui. Lo<;yN duang congnayrakhoiT bangcachd~t: T = T\{5 n ({ti}x D2) X [0,1],1::;ti ::;N} Khi do,vai t E [0,1]\{t1,..,tN}thlm duangcong5 n ({t}x D2) x [0,1](tren tUnglat cat)la chinhqui. M~tkhac,tUngduangcongtrongm duangcong5 n ({t}x D2) x [0,1],t co dinh,chinhla m(>tdo(;lnthing trongtr1;l({t} X D2) X [0,1]duqcxacdinhberi: y =AY((glxle27rit)+ (1- A)y(g)((glxle27rit) Do v~y,t(;lim(>tdi~mbatkl thu(>c5 n T, m~tphingtiepxucvm5nTchua do(;lnthingcothanhph~nt9ad(>theoAkhacO. Trong]R4\{5n({ti}x D2)x [0,1],1::;ti::; N},t(;lim(>tIanc~ncuaT, tadvng duqctruangvec-tO'lapCOOthoa: . Thanhph~ncuaV theotr1;lct9ad(>A lahang1. . T(;liIanc~ncuabien51x 51X [0,1]cuaT, V songsongvai tr1;lct9ad(>A. . Vai m6igiatrit,V namtrong({t}x D2)X [0,1]. . T(;lim6idi~mthu(>c(5 n T), V tiepxucvai (5 n T). Theoketquavli'athuduqctren(5n T), taxaydvngduqctruangvec-tO'dia phuangthoacaeyellc~utren.Saudo,tuangtVchUngminhdinhly Calltruenon,ta dvngtruangvec-tO'toanC1;lCnham(>tphanho(;lchdanvi lapCoo. Vm truangvec-tO'da dvng,ta hoantoanxac dinh duqcvi phoi h' : (51x D2)\{m.Ndie'm}-+ (51x D2)\{m.Ndie'm}.Merr(>ngh' thanhdongphoih : (51X 57 D2) -+ (51x D2),bientUngdi~mcuaKig)trenm6ilat(trongN lat)tOth<;1ncua (51x D2) thanhm<)tdi~mtucmgUng(trongmdi~m)cuaK1 trenlattucmgUngtrong (51x D2).V~y,h la dongphoithoacacyellCalldad~tra bandati. Ton t<;1idongphoih dakhepl<;1ichUngminh(51x D2,K1) va (51x D2,Kig)) clingki~utrongtatcacactruemghqpcuakhattri~nPuiseuxy(x). III. Ketlu~n: ChomamduemgcongphlmgX batkhaquit<;1idi~mki d!co l~pb, tahoan toanxacd!nhduqccacc~pPuiseuxcuaf xacd!nhX. Tucacc~pPuiseuxnay,xily dvngm<)tnaxuyenl~p(51x D2,Kf) nhumotaa I. M~tkhacki~uto-pocuamam(X,0) duqcxacd!nhbCrlki~uto-pocua(m)-na xuyen(51x D2,Kd. (m)-naxuyen(51x D2,K1)vanaxuyenl~p(51x D2,Kf) la clingki~uto-po,theochUngminhaII. Nhuv~y,ki~uto-pocuamamduemgcong(X,0) hoantoanduqcxacd!nhbat cacc~pPuiseux. 0 Nh~nxet4.2.1.([5],p.ll2) PhanI trongchUngminhD!nhly 4.2.1damotach~tchebangtoanh9Ccach xilydvngm<)tnaxuyenl~ptucacc~pPuiseux.Tudo,coth~chUngminhduqcrang, hatnaxuyenla clingki~uto-pokhivachikhi chungduqcdvngtuclingcacc~p Puiseux. Djnh ly 4.2.2.([5],p.ll1) ChohaimdmduO'ngcongphdngbatkhdquiX vaX' tgidiemO. 0 fadiem chinhquicuaX. Khi do,X vaX' facungkiiu to-pokhivachikhi0 dingfadiem chinhquicuaX'. Chungminh. Tabietrang,nalienketvOlX lanatamthuemg,X vaX' clingki~uto-po,nen (m')-nalienketVOlX' clingki~uto-povOlvOlnatamthuemg.Di~unaychixayra khim'= 1vadodo,0 phailadi~mchinhquicuaX'. Di~unguqcl<;1iladung,theo D!nhly 1.4.1. 0 Dfnh ly 4.2.3.([5],p.ll1) 58 Hai mcimduCingcongphdngbatkhdquiX vaX' tqzidiemki dj cott;2p0 ta cungkilu to-ponlu vachinlu chungcocungCClc(ipPuiseux. Chltngminh. Neuhaimam(X,O) va(X',0) clIngki~uto-po,thlnO'(51x D2,Kl) lienket v6iX vanO'(51x D2,K~)lienketv6iX' Ia clIngki~u,nhuvi;ly,cacnO'xuyenl~p tUO'llgtl'ngcuaX vaX' Ia (51XD2,Kf) va(51xD2,Kt) laclIngki~u.Di~unaychi KayrakhichungcoclIngcacc~pPuiseux.Di~unguqcl(;lihi~nnhienth€oDinh19 4.2.1. 0 Vi dl;!4.2.1. Chocacc~psotvnhien:(2;3),(2;7),thoacacdi~uki~ntrongNhi;lnxet2.2.1, khaitri~nPuiseuxchu~ntilctl'ngv6i cacc~p(2;3),(2;7) la: 3 7 3 7 y(x)=x"2 +x~ =x"2 +x"4. Mo tahlnhh9CcuanO'xuyenl~p(5x D2,Kf) nhusau: Trenxuyen(5 x D2,Kf), nO'Kt IanO'xuyenki~u(2,3)quaild~ulIenxuyen, qUail2vangtheokinhtuyenva3vangtheovi tuyen: NO'xuyenki~u(2,3) Bim 2-nhanhtu'dnglingtrentn;! T(;lilatciltngangxuyen: DlJIlgm<)tIanc~nd~ngongT1cuanamabankinhm6iIatcatngangT1 Ia m<)thlnhtrail bankinh Ixl m~;"2.Chli yding,v& Ixl dunh6thlT1 namhoanloan trongxuyen51x D2. Lanc~nd~ng6ngT1cuanaKj' Uin c~nd~ng6ngtu'onglinglIentr\! TrenIanc~nT1,tadlJIlgnaxuyenKl ki6u(2;7)qUail2 vangtheokinhtuyen va7 vangtheoVItuyencuaT1: 60 ~ ........ / ~ NO'xuyenKl2trenIanc~nongTI ,"""""""...,.. Bi~'tadng'iing't~6~' tn~ Lit dlt ngangxuyen51x D2(chuyvmIxldlinh6):