Luận văn Phương pháp xác định phương trình dạng ẩn cho đường cong và mặt tham số hữu tỷ

Phl;t ll;tcA " " " "A? ? ,./' THU~TTOANTINHB~CANCUAD~NGTHAMSO Chungtabierdingyoi m6id~;lllgthams6hoacuam~thuuty se t6n t(;limOts6nguyenn ~1 saochomOtdiSmtrenm~tu'dnglingyoi n giatri thams6.s6 n du'QcgQi1fts6tu'dnglingcuad(;lngthams6boa.Khi n = 1~ ta noi d(;lngthams6 hoa1ftchinhqui,ngu'Qcl(;li1ftkhongchinhqui.Theo Dinh 19Bezout [14],b~ccua da thlic f b~ngs6 giaodiSmcua m~td(;lis6 f =0 . yft hai m~tph~ng.Nhu'nggiao cua hai m~tph~ng1ftmOtdu'ongth~ng,y~y sudvnghaidu'ongth~ngkhacnhauchungtaseHnhdu'Qcb~ccuadathlicf. Thm}ttoanA.I. Tinhb~cin cuam~thams6huuty(1.3)[4]. Nh4p: Xudt: Rude1: Rude2: Rude3: Rude4: Cac dathlic a,b,c,dE k[s,t]. I Ia b~cin cuam~thams6huuty (1.3). TImbiSudi~nin cuamOtdu'ongth~ngtoy9 ca:tm~tham s6huuty alx+(31Y+ "YlZ+81=a2x+(32Y+"f2Z +82=o. Cacgiatri s tu'dnglingyoi cacgiaodiSmcuadu'ongth~ng trongRude1 yoi m~tthams6 1ftnghi~mcua k~tthuc h(s) = Res(ala + (31b+ "flC+ 81d,a2a+ (32b+ "f2C+ 82d,t). Neum~thams6cochliacacdiSmcosothl h(s)cocac nghi~mngo(;lilai. DS lo(;libo chung,chungta HmbiSu di~nin cuamOtdu'ongth~ngtoy9khacyoi du'ongth~ng trongRude1 Ia alx +!31Y+ilZ +81= a2x+!32Y+i2Z +82 =o. Tu'dngtvxacdinhdathlicco clings6nghi~mngo(;liai yoi Budc5: 41 da thuc h(s)trongBudc 2 Ia fi(s)= Res(ala+,BIb+"YIC+ 81d,a2a + ,B2b+ "Y2C+ 82d,i). Xua'tl:= deg(h(s))- deg(gcd(h(s),fi(s))). Ph1;t l1;tcB CACTHU!TTOANCOBANvtMATR!NDATHDc Trangph§nnaychungtoi se trinhbaytomtiltmQts6khainit?mva cac thu~troanHnhroancd banvS matr~nda thuclam cd sd chovit?cxa~dinh modunsyzygy[14,19]. Ma tr~nF ca'p mx l voi cac ph§n td'thuQck[xl'...,xIJ duQcgQi Ia ma tr~nda thuc, ky hit?u FE k1nXI[Xl"",Xn].Ma tr~n F co h~ngIa T', ky hi~u rankF = T',ne'utan t~imQtdinhthucconca'pT'khac khongva ta'tca cac dinh thucconca'pT'+ 1 dSub~ngkhong.Ne'u rank(F) = min(m,l) tanoi F 1ah~ng d§yduoB~ccuaF, ky hit?udeg(F),1agiatri IOnnha'tcuab~ccuacacph§ntd' cuaF. Cho ME k1nX1n[Xl"",XIJ,khi do dinh thuc cua M 1amQtda thuc thuQc k[Xl"",Xn]' Ta noi M khong suy bie'nne'u det(M) +=O. Ne'u det(M) Ia mQt h~ngkhackhongthuQck tanoi M Ia ddnmodun. Djnh nghiaB.l. Cha FE k1nXI[Xl'...,xnJ,m < l. Khi do F dll(1CgQi[a: (1) Nghi~mnguyen((5trai (ZLP) ntu khongt6n tCfimQtn-bQ (zf,...,z~)E kn [a nghi~mchungcilatatcdcacdinhthaccancapm cila F. (2) Dinhthacnguyent6trai (MLP) ntu tatcdcacdinhthaccancap m cila F nguyento'cungnhau. (3) Thaas6nguyent6trai (FLP) ntu F =~F;,trangdo~danmodun. Cackhaini~mnghi~mnguyent6phdi(ZRP),dinhthacnguyent6phdi(MRP)va thaas6 nguyentlfphdi (FRP) dU:(lcdinhnghiatllangfT!. 43 Vdi n = 1,2thlMLP FLP va MRP FRP. Vdi n 2:: 3 thl MLP $. FLP vaMRP $. FLP. Vdi n 2::1 thl MLP ~ FLPvaMRP~ FRP [18]. B.I Phantichnhantii'matr~ndathuc ~ Thu~ttminB.l. Phantichnhantii'tniichomatr~ndathuchaibie'nh~ngd~ydu [12,16]. Nhljp: Ma tr~nFE kmXI[XpX2Jco h~ngd~ydu, m <l. Xudt: Hai ma tr~n L E kmxm[xpx2J,R E kmXI[XpX2JsaDcho F = LR va detL = 9 vdi 9 E k[XIJIa dungcuatidechungldnnha'tcua BucJe1: caedinhthucca'pm cuaF. TIm dathuc9E k[XIJIa dungcuatidechungldnnha'tcuacac dinhthucca'pm cuaF. Phantich9 thanhtichcuacaedathuc ba'tkha guytrangk[XIJ. L = 1m;R = F. Bude2: Ne'u da:xet he'tcae thanhph~nba'tkha guy thl de'nRude 6. NgtiQcI~i,vdi thanhph~nba'tkha guy p E k[XIJ i =1;j =1;R =R (modp). Rude3: Trang matr~nR,ne'ut6nt~imQthangio,vdi i <io<m, ma ta'tca caeph~ntii'deub~ngkh6ngthl Do=diag(l,...,p,...,l) Ia mQttidctraicuaF L = LDo;R = D;;lR. QuayI~iBude2 vdi thanhph~nba'tkhaguyke'tie'p. BUdC4: BUdC5: 44 Trangmatr~nR, tlmcQtjo d~utien,vdi j <jo < l, co it nha't mQtph~ntakhackh6ngtrongcachangtu i de'nm ) =)0' Trang cQt j cua matr~nR, tuhangi de'nm Hmph~ntaco ~ b~ctheox2nhonha't,gQila Rj~'D~tD1la matr~ncodu<;1ctu 1mbiingcachhoanvi haidongi va ~ - - -1 R =D1R; L =LD1;R =D1R. Giii saCQtj cod~ng(...,ai,...,am)T.Hc$s6chud~ocuacacda thuc ai' ..., amE k(X1)[X2]la cac da thuc bi, ..., bmE k[xd. Do biva p nguyent6clIngnhaunentheothu~t toanchiaEuclide tant~icacdathucg, h E k[X1]saocho gbi=1- hp. D~t a; =gai (modp)E k[XpX2]'Khi dotant~icacc~pdathuc qk' ..., rkE k[X1,X2]saocho * ak= qkak+ rk' vdi k = i +1,...,m va degtrk<degta; ho~c rk =O. D~t 1 1 D3= 1 -xqi+1 1 -xqm 1 RUdC6: 45 - - -1 R = D3R;L = LD3 ;R = D3R. Khi docQtj cod(;lng(...,ai,1i+1,...,r;n)T. R=R (modp). Ne'ucac dathuc1i+1=...=r;n= 0 thl i = i +1;j = j +1; va quayl(;liRUdC3.NguQcl(;li,quayl(;liRUdC5. Xua'tL, R. Ne'um> l, taclingco phanrichnhantii'phaicho P, nghlala tlmduQc hai ma tr~nL E k1nXl[Xp2]va R E k1Xl[X1'X2]saocho P =LR. Apd\lllgmatr~n chuyS'nvi taco thS'xaydvngthu~troanphanrichnhantii'phainhusau. Thm}ttminB.2.Phanrichnhantii'phaichomatr~ndathuchaibie'nh(;lngdfiy duo Nhqp: Xudt: RUdC1: RUdC2: RUdC3: Ma tr~nP E k1nXl[Xp2]co h(;lngdfiydu, m >l. Hai ma tr~nL E k1nXl[X1'2]'R E k1Xl[X1'X2]saocho P =LR va detR = g voi gE k[X1]la dungcua uoc chungIOnnha'tcua cacdinhthucca'pl cuaP. Di;it p' = pT E k1X1n[X1'X2]'dungThu~troanB.1 tlm hai ma tr~nL' E k1Xl[X1'X2]va R' E k1X1n[xpX2]saocho p' =L'R'. Taco(pT)T=pIT=(L'R')ThayP =R'TLff=LR. Di;it L =R'T;R=Lff. Xua'tL,R. B.2 Tim d~ngHermite 46 Cho ma tr;%nF E k1nXI[Xl'X2]'d,.lllgHermite cua F la ma tr;%nH = UF vdi hij = 0 nSu i > j va degx2hii> degx2hij nSu i < j, trongdo U E k1nX1n[xl'X2]va ddnmodun.Sa dl;lllgky thu;%tkha Gaussde timd~ngHermitechotru'onghQp mQtbiSn[12],tacothu;%ttoansau. Thu~t toan B.3. Tim d~ngHermitecho ma tr;%nda thti'chai biSn h~ngd~ydu [12,14,19]. Nhqp: Ma tr;%nF E k1nXI[Xl'2]co h~ngd~yduo Xudt: Ma tr;%nHermite HE k1nXI[Xl'X2]cua F. Bu(]c1: Xem F E k1nX\Xl)[X2]'sa d\lngky thu;%tkha Gaussta timdu'Qc matr;%nHermite II E k1nXI(Xl)[X2]cua F va matr;%n0 E k1nX1n( Xl)[X2]saocho II = OF. Bu(]c2: GQi hi vdi i = 1,...,m la bQichungnhonha'tcuacaem§:uthti'c hangthti'i. D~tD =diag(~,...,h1n)' - - H=DH;U=DU. Bu(]c3: Xua'tH. B.3 Rut trichtidechungIOnnha't A,B la haimatr;%ndathti'c ungsO'hang(cQt).A,B du'QcgQila nguyen to'cungnhautrai (phai) nSu t6n t~ihai ma tr;%nA,B va matr;%nda thti'cddn modunC saochoA =CA,B=CB (A=AC,B =BC). 47 Ma tr;%nda thucvuong D duQcgQila uoc chungtnii (phiii) IOnnha'tcua A va B ne'utant~ihaimatr;%ndathuc1,13nguyento'cungnhautrai (phiii) saocho A =DA,B=DB (A=AD,B=BD). Thu~ttminB.4.TImuocchungphiiiIonnha'tcuahaimatr;%ndathuc[12,16]. ~ Nhqp: Hai ma tr;%nda thuc A E kmXI[Xl'X2]va B E knXI[XpX2] sao cho Xu{{t: (AT,BT)T E k(m+n)xl[Xl,X2]la h~ngd~yduo Ma tr;%ndathuc DE kIXI[Xl,X2]la uoc chungIon nha'tphiii cua A vaB. RUdC1: Dung Thu;%ttoanB.2 phantfchnhantuphiii matr;%n [~]=[~]R. RUdC2: Dung Thu;%ttoanB.3 tlmd~ngHermitecuamatr;%n(iF, IF)T u[~J=[~J. RUdC3: Dung Thu;%ttoanB.l phantichnhantu trai matr;%n Rv=LR. RUdc4: Xua'tD =RR. B.4 Dc)phuct~pHnhtoaD BQ phuct~pcuaThu;%ttoanBA phl;1thuQcvao dQphuct~pcuaThu;%t toanB.l d RUdC1va 3 va dQphuct~pcuaThu;%ttoanB.3d RUdC2.CacThu;%t toanB.l va B.3 d€u duQcxay d1.,1'ngd1.,1'atrencachtie'pc;%nc6 dienla phuong phapkhii'Gauss.Banh gia dQphuct~pcua Thu;%toanB.l tuykhongdon giiin 48 nhrtngchUngtaco th€ thiy cdbanno se g6rndi)ph",ctaPd€ Hnh(7] dinhthuc ca'pm d Budc 1 va dQphuc t""pcua cac bu'ockhii'Gausscon I""i.Nhu'v~ydQ phuc t""pcua Thu~troanB.I cling Ia mQtham da thuc theo m,Z.DQ phuc t~p , cuaThu~troanB.3Ia O(mZ2d2)[20],trongdo d Ia b~ccuamatr~ndathuc.V~y dQphuct""pcuaThu~troanBA Ia mQthamdathuc. Phl;t ll;tc C A "" ? THU~T TOAN XAC DJNH JL-COSO Trangph~nnay chungWi trlnhbay thu~ttoantim f.L-cdsd cua duongcQng ph~ngva m~tthallis6 hull ty duqcsad1;lllgtrongThu~ttoan3.3va 3.4trong Chuang3. c.t Thn~ttoaDOmp,-cdsO'cuadu'ongcongthams(fhunty phdng Cd sd de xay dl;1'ngthu~ttoanxac dinh f.L-cdsd cua duongcong thalli s6 hulltyph~ngla cacke'tquasau. DinhIy C.t. Chop,q E Syz(a,b,c)la haidudngthdngdi d(}ngsinhdudngGong thamsf;'(1.2)bfjc n. Gid sii deg(p)<deg(q), khi do p va q tCJothanhf.L-casO cuadudngGong(1.2)niu vachiniu m(}trongcacddu ki~nsauthoa (1) M(}tdudngthdngdi d(}ngL sinhdudngGongthams{f(1.2)co thi duf/cbiiu diln boi (2.3)wii deg(~p):::;deg(L)va deg(h2q)<deg(L). (2) M(}tdudngthdngdi d(}ngL sinhdudngGongthams{f(1.2)co thi duf/cbilu diln boi (2.3)va haivectdLV(p) va LV(q) d(}clfjp tuyin tinhtren JR. (3) M(}tdudngthdngdi d(}ngL sinhdudngGongthams{f(1.2)co thi duf/cbilu diln boi (2.3)va deg(p)+deg(q)=n. Chungminh:xem[2,tr.374]. M~nhd~C.2. (1) ModunsyzygySyz(a,b,c) cuadudngGongthams{fhilu typhdng (1.2)duf/c 50 sinh biJi ba duilngthlingdi dQngVI=(-b,a,0), V2= (-c,0,a) va V3= (D,c, -b). (2) rank(vI,v2,V3)=2. (3) rank(LV(vI)'LV(V2)'LV(vJ) =2. Chung minh: xem[2,tr. 376- 377]. Thu~t toaDC.l. Tim p,-edsa euadttongeongthams6huuty phing [2]. Nh(jp: Cae dathlie a,b,cE k[t]. Xudt: Hai da thlie p,q E k[x,y,t] 13.p,-edsa euadttongeongthams6 huutyphing. Rude1: Df;it VI = (-b,a,D),V2= (-c,D,a),V3= (D,c,-b); mI= LV(VI)' m2=LV(v2)'m3=LV(v3)' Rude2: Df;it ni = deg(vi)'vdi i = 1,2,3.Khongma'ttinhtangquat,ta siip xe'p1~icae Vi theothli t1!giamd~neuacae ni. Rude3: Tim caes6 th1!eaI' a2'a3 (co it nha't2 s6 khaethong) saoeho aImI +a2m2+ a3m3=D. Rude 4: Ne'u al -:;r:. Dthi V - a V +a tnl-n2v +a tnl-n3v .1 - 1 1 2 2 3 3' ~ =LV(VI); nI =deg( VI)' Ne'u al = D khi do a2,a3-:;r:.0 thi V - r" V + tn2-n3V '2 - "'2 2 a3 3' 51 m2 =LV(V2); n2 =deg(V2). Buac5: Ne'umQttrongcac Vi=0, gia SITla VI= 0, thld~tp = V2va q = V3Ia hai da thuc t<:;lOthanhf-J,-Cdsa cua du'ongcongt~ham s6huutyph£ng.Ngu'qcl<:;li,quayv6Buac2. C.2 Thn~t toaDtimp,-cdsdcuam~t thamsf{hunty Cd sa quantrQngdegiupHmf-J,-cdsacuam~tthams6hUllty la haike't quasau. Dinh Iy C.3.Chomiftthamsdhiiuty (1.3),khid6fuontan((;iibaphdngdi dqng p,q,r saGcho(2.4)thoa.Banniia,mqtcasa p,q,r tflyycaamodunSyz(a,b,c, d) Gangthoa(2.4). Chung minh: xem[1,tr.694]. Dinh Iy C.4. Cho p,q,r fa mQtf-J,-casa caamiftthamso'hiiu ty (1.3).Khi do p,q,r famQtcasacaamodunSyz(a,b,c,d),ngh'iafavaiba'tkyphdngdidQngP sinhmiftthamso'co bilu ddn duy nha't P =~p+~q+h3r vai ~,~, h3E IR.[s,t]. Ban niia degt(~p),degt(~q),degt(h3r)<degt(P)+degt(p) +degt(q)+degt(r)- n; degs(~p),degs(~q),degs(h3r):::; degs(P)+degs(p)+degs( q)+degs(r)- m ntumiftfasongbtJ-c(m,n) va deg(~p),deg(~q),deg(h3r)<deg (P) +deg(p)+deg(q)+deg(r)- n ntu miftfaphantamgidcbtJ-cn. Chung minh: xem[1,tr. 695]. 52 Do d6 de rim f-L-cosd cua mi,itthams6 huu ty (1.3)chungta chi cffnxac dinh co sd cua m6dunSyz(a,b,c,d). Chungtabie'tding vi~cxac dinh co sd cho mQt m6dun tl,l'do la kh6ng d~dang. Trong khi d6 ta c6 the sa d\lng ytudngcua thu~troanBuchbergerde xay dl,l'ngt~phqp cacphffnta sinhchom6dunsy-zygy [10]va sa d\lngke'tquacuaM~nhd€ 2.20de rimco sd chom6dunSyz(a,b,c,d). Nhungphuongphapnay ding chuahi~uqua. Sa d\lng cac ke'tqua nghien CUllv€ m6dunsyzygycua h~ cac phuong trinhtuye'ntinhthuffnnha'tvoi cach~s6 la da thucnhi€u bie'ndl,l'atrenly thuye'tcacmatr~ndathuc[18],chungtac6thexaydl,l'ngthu~troanxacdinhf-L- co sd cho mi,itthams6huuty [12].Honnuaphuongphapnayclingc6theduqc ap d\lngde rim f-L-cosd choduongcongthams6 huuty hi~uquahonThu~troan D.l. Cho matr~ndathucF = (A,...,ft)E k7nXI[Xl"",XrJKhi d6 (~,...,hl)T E kl[ Xl,...,X,JduqcgQiIa mQtsyzygycua F ne'u I Lhd:=o. i=l (D.1) T~phqpta'tca cacsyzygycua F duqcgQila m6dunsyzygycuaF, kyhi~ula Syz(F).T~phqpcac~,...,hsE kl[xl1'",xnJduqcgQiIa t~pcacphffntasinhcua Syz(F) ne'u Fhi =0 voi i =1,...,8 (D.2) va voi mQisyzygyt E Syz(F) t6nt~iduynha'tcac da thuc fl1""fs E k[xl1""x,J saDcho 53 t =h~+... Ishs' (D.3) Ne'utad~th =(hi,...,fmJT,hj =(~j,...,hlj)Tvat =(~,...,tl)Tvoii =1,...,[va j = 1,...,sthlcacbi€u thuc(C.2)va(C.3)trdthanh hI hi )(~I ... ~s =0 Imi ... ImLjlhn ~ .. F ... his Ii va ; _f;I ... ;sH~. - tl Ihn ... hisIlls Khi do H =(~,...,hs)E kIXS[xu...,x,JduQcgQila matr~nsinhcua Syz(F).Cau hoi d~tra la khinaothlmatr~nH co s6cQtnhonha'tva co tant~ihaykh6ng mQtmatr~nsinhco s6cQtnhonha'tvoi F ba'tky? Cacke'tquasause traloi chova'nd€ nay. Mc$nhd@C.S. Cho F =(-fib) E kmXI[Xl1'..,X,Jco h{mgfa m, wii 1> m va b E kmxm[xu...,x,J khongsuybitn.Diit T=[- m, khidomodunSyz(F) comatrlJ.n sinhdip [x T ntu va chi ntu tant{lim(}tmatrtJ.nMRP HE kIXr[XI,...,x,JSaDcho FH = O.Hefnnila H chinhfa matrtJ.nsinh. Chung minh: xem[18,tr. 80]. Voi n ::;2 talu6nHmduQcmQtmatr~nsinhco s6cQtnhonha'tchoF tuyy. 54 Mc%nhd~C.6. Cho F = (-Njj) E klnXI[Xl,X2]co hc;mgla m, velil> m va DE klnXIn[xl'X2]khong suy bien.Dijt r =l- m, khi do tan tc;limQtma tr4nsinh H E klXr[xl'X2]cho modun Syz(F). Chung minh: xem[18,tr. 81]. D~t F = (a,b,c,d)E eX4[s,t],theoM~nhde C.6 m6dunSyz(F) hay Syz(a, b,c,d) comatr~nsinhH E k4X3[S,t].TheoM~nhde2.20,suyracaccQtcuaH t(;lOthanhmQtcdsacuam6dunSyz(a,b,c,d).V~ytacothu~toansauxacdinh p,-cdsachom~thamsO'hiluty. Thu~ttoaDC.2.Timp,-cdsachom~thamsO'hiluty [12]. Nh4p: Cac dathuc a,b,c,dE k[s,t]. Xw1't: Badathucp,q,r E k[x,y,z,s,t] la p,-cdsacuam~thamsO'hilu Buelc1: ? ty. D~tN=(-a,-b,-c) va jj = (d).Xay dllngmatr~n p = jj-lN =(-ajd,-bjd,-cjd) =ND-1. BuelC2: Trongdo N =(-a,-b,-c) va D =diag(d,d,d). DungThu~toanB.4till u'occhungIOnnhtt C cua N va D. Khi do t6nt(;lihaimatr~nN,jj E k[s,t] saocho - - N =NC;D =DC; trongdoN va jj nguyento' cungnhauphai.D~t -T -T T H =(N ,D ) =(hr,~,h3); P =hrT(x,y,z,l);q=h;(x,y,z,l);r=hJ(x,y,z,l). 55 Blide3: Xua'tp,q,r. C.3 Dc}phuc t~p Hnh tmin DQ phuct(;lpcuaThu~troanC.! trongtrlionghQptrungbinh la O(n2)[2]. Trangkhi do vi~cdanhgiadQphuct(;lpcuaThu~troanC.2 la khongdongian va noph\lthuQcchinhvaodQphuct(;lpcuacacthu~troanv€ Hnhroanmatr~n dathuctrongPh\ll\lc B. C\l the,neuboquadQphuct(;lpcuavi~cxayd1;fngcac matr~nN va D t(;liBlide1thidQphuct(;lpcuaThu~troanC.2ph\lthuQc hinh vaodQphuct(;lpcuaThu~troanB.4 d Blide2.V~ytaco thexemdQphuct(;lp cuaThu~troanC.2xa'pXl dQphuct(;lpcuaThu~troanB.4va clingla mQtham dathuc. Phl;tll;tcD cA C PHUONG TRINH M4.T HUU TY TrangphgnnaychungWi lit%tke phu'dngtrinhcacm~thuuty du'Qcdung d€ daubgiacacthu~troangnhoatrongChu'dng3. Sl = -3s3t- 3,-4i +6st,9s- 6,t; s = it +2st5- st s3e- s2t3- 2it2 +S2- st - 2s s - t - 2 s2t- se - 2st,2' ", S3 = S2 +t, s2t+st4+st2- 3s+t5- 3t3, st+e- 3, s2t+2se- 3s+t3- 3t; S4 = -18s4t+27s3t2- s2t3- 8st4+ 2t5, 4s3t3+6s3t2+2s3t- 2s2t4- 3it3 - s2t2, -3s3t2- 3s3t- 8it3 - 8s2t2- 6s2t- 6i + 3st4+ 3st3+ 2st2+ 2st, 6s3t2+6s3t- 5s2t3- 5it2 +st4+st3; S5=S3 +5st2+2st+t3- 1,2S2+3st+2t3- 2,5s3+2it +3se+5st+5t2- 5, S4 + t4- 1; S6 = 16s3+32it -120i - 56st+128s+24t- 24, 48it +8i +16se- 456st+384s+16e-16t2+392t- 392, 16s3+32it +192i +48se-1328st+1040s+64e- 48e+1232t-1248, 16s3- 240st+224s+16e+224t- 240; S7 = 16s3+32it -120i - 56st+128s+24t- 24, 48s2t+8i +16se- 456st+384s+16t3-16e +392t- 392, 16s3+32it +192i +48st2-1328st+1040s+64t3- 48e+1232t-1248, - 2i +8s- 6; S8 = 684288s3-1419264it- 608256i+684288st2+1495296st- 836352s- 684288t2- 76032t+760320,983808s2t+1043712i- 8711424st+ 57 7803648s+684288t3+1119744t2+7043328t- 8847360,4790016s3- 6983424s2t-1126656s2 -15667200st + 17556480s+ 1430784t2+ 20597760t- 21219840,6158592s3-19660032s2t-12780288i+ 74437632st- 62152704s- 4790016e- 13996800e- 44987584t+ 68774400; S9 = -84ie +7716st-14004s- 7632t+14004,- 84se+1056st-1692s- 972t+ 1692,- 252s2t- 84st2+ 3360st- 4284s- 3024t+4284,6i - 57s+ 6t+ 51; SlO= - 2209it3+ 7359ie - 4800it - 350i +2225st3- 7425se+ 4800st+ 400s,- 2209it3- 732s2t2+3291s2t+2225st3+750se- 3375st, 168ie-168it - i -168se+168st+7, 24it2- 24it - 24st2+24st+1; S11= -2209it3- 732ie+ 3291s2t+2225st3+750se- 3375st, 2209s2t3- 7359ie+4800it+350s2- 2225st3+7425se- 4800st- 400s, 168ie -168it +S2-168se +168st+7, 24s2e- 24s2t- 24st2+ 24st+ 1; S12= 2209it3- 7359it2+4800it+350i - 2225st3+7425st2- 4800st- 400s, 2209ie+732ie- 3291s2t- 2225st3- 750se+3375st, 168ie -168it - S2-168se +168st+ 7, 24it2 - 24s2t- 24st2+24st+1; S13=2209s2t3+732ie- 3291it - 2225se- 750st2+3375st, - 2209s2t3+7359ie- 4800s2t- 350s2+2225st3- 7425st2+4800st+400s, -2209it3 +7359s2e- 4800it- 350i +2225st3- 7425se+4800st+400s, 24s2t2- 24it - 24st2+24st+1; 58 814 = 73682e-11182t2- 600it - 25i - 7448t3+1448e+6008t+32t3-132e + 100,- 2072it3+171982e+378it +20648t3-17288t2- 3368t+32t3+ 36t2-168t,54ie - 3682t2-1882t+ i - 548e+368t2+188t- 68+6, 18it3-12ie - 682t-188e +128e+68t+1; 815 = - 207282t3+ 171982t2+ 37882t+ 20648t3- 17288e- 3368t+ 32t3+ 36t2-168t, -100 - 32e +25i - 73682t3- 6008t+ 132e-1448t2+ 7448t3+ 60082t+11182t2,5482t3- 1882t- 3682t2+ i - 548t3+368e + 188t- 68+6,1882t3-1282t2- 6s2t-188e +128e+68t+1; 816 = - 736it3 + 111it2+ 60082t+ 25i + 7448e-1448e - 6008t- 32t3+ 132t2-100, 207282t3-171982t2- 37882t- 20648t3+17288t2+ 3368t- 32t3- 36t2+ 168t,54ie - 36it2 -18it + i - 548t3+ 368t2+ 168t- 68+ 6, 18it3 -1282t2- 682t-188e +128t2+ 68t+ 1;