{VERSION 6 0 "IBM INTEL NT" "6.0" } {USTYLETAB {CSTYLE "Maple Input" -1 0 "Helvetica" 1 10 153 0 153 1 2 2 2 2 1 2 0 0 0 1 }{CSTYLE "2D Math" -1 2 "Times" 0 1 0 0 0 0 0 0 2 0 0 0 0 0 0 1 }{CSTYLE "2D Comment" 2 18 "" 0 1 0 0 0 0 0 0 0 0 0 0 0 0 0 1 }{CSTYLE "" -1 256 "" 1 14 0 0 0 0 0 0 0 0 0 0 0 0 0 0 }{CSTYLE " " -1 257 "" 1 8 0 0 0 0 0 0 0 0 0 0 0 0 0 0 }{CSTYLE "" -1 258 "" 1 14 0 0 0 0 0 0 0 0 0 0 0 0 0 0 }{CSTYLE "" -1 259 "" 1 14 0 0 0 0 0 0 0 0 0 0 0 0 0 0 }{CSTYLE "" -1 260 "" 1 14 0 0 0 0 0 0 0 0 0 0 0 0 0 0 }{PSTYLE "Normal" -1 0 1 {CSTYLE "" -1 -1 "Times" 1 12 0 0 0 1 2 1 2 2 2 2 1 1 1 1 }1 1 0 0 0 0 3 0 3 0 2 2 0 1 }{PSTYLE "Heading 1" -1 3 1 {CSTYLE "" -1 -1 "Times" 1 18 0 0 0 1 2 1 2 2 2 2 1 1 1 1 }1 1 0 0 8 4 2 0 2 0 2 2 0 1 }{PSTYLE "R3 Font 0" -1 256 1 {CSTYLE "" -1 -1 " Helvetica" 1 9 128 0 128 1 2 2 2 2 2 2 1 1 1 1 }1 1 0 0 0 0 1 0 1 0 2 2 0 1 }{PSTYLE "R3 Font 2" -1 257 1 {CSTYLE "" -1 -1 "Courier" 1 8 0 128 128 1 1 2 2 2 2 2 1 1 1 1 }1 1 0 0 0 0 1 0 1 0 2 2 0 1 }{PSTYLE "S eitenumbruch" -1 258 1 {CSTYLE "" -1 -1 "Times" 1 10 0 0 0 1 2 1 2 2 2 2 1 1 1 1 }1 1 0 0 0 0 1 0 1 0 1 2 0 1 }{PSTYLE "Normal" -1 259 1 {CSTYLE "" -1 -1 "Times" 1 14 0 0 0 1 2 1 2 2 2 2 1 1 1 1 }1 1 0 0 0 0 1 0 1 0 2 2 0 1 }{PSTYLE "Heading 1" -1 260 1 {CSTYLE "" -1 -1 "Time s" 1 18 0 0 0 1 2 1 2 2 2 2 1 1 1 1 }3 1 0 0 8 4 1 0 1 0 2 2 0 1 } {PSTYLE "Normal" -1 261 1 {CSTYLE "" -1 -1 "Times" 1 10 0 0 0 1 2 1 2 2 2 2 1 1 1 1 }3 1 0 0 0 0 1 0 1 0 2 2 0 1 }{PSTYLE "R3 Font 0" -1 262 1 {CSTYLE "" -1 -1 "Courier" 1 9 128 0 128 1 2 2 2 2 2 2 1 1 1 1 } 1 1 0 0 0 0 1 0 1 0 2 2 0 1 }{PSTYLE "R3 Font 2" -1 263 1 {CSTYLE "" -1 -1 "Courier" 1 8 0 0 0 1 2 1 2 2 2 2 1 1 1 1 }1 1 0 0 0 0 1 0 1 0 2 2 0 1 }} {SECT 0 {EXCHG {PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT 258 18 "Beispiel 8 in \247 13" }}{PARA 0 "" 0 "" {TEXT -1 7 "zu den " }{XPPEDIT 18 0 "mu;" "6#%#muG" }{TEXT -1 75 "-Aufl\366sungen nach Fabr ice Rouillier, bzw. zur Anwendung von Satz 7 in \247 13." }}{PARA 0 " " 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 26 "Bezeichnungen wie \+ in \247 13." }{TEXT 257 0 "" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 0 "" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 56 "restart:with(Groebne r):\nwith(linalg):\nwith(Ore_algebra):" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 0 "" }}}{SECT 1 {PARA 3 "" 0 "" {TEXT 260 58 " Die folgen den Prozeduren muessen zuerst aktiviert werden:" }}{EXCHG {PARA 0 "> \+ " 0 "" {MPLTEXT 1 0 953 "Kbasis := proc(PList,VList,torder)\n local B, Bs,C,G,i,v,t,l,m,pB,r;\n if is_finite(PList,VList) then\n \+ G := gbasis(PList,torder);\n B := [1];\n for v in VLis t do\n m := degree(univpoly(v,G,VList),v);\n C := B;\n for t in C do\n for l to m-1 do\n t := t*v;\n if normalf(t,G,to rder) = t then\n B := [op(B),t]\n \+ fi\n od\n od\n od;\n \+ Bs:=B;\n r:=nops(B);\n pB := 0;\n for i to \+ r do pB := pB+B[i] end do;\n i := 0; Bs := []; while pB <> 0 \+ do\n l := Groebner:-leadmon(pB,torder);\n i := i +1;\n Bs := [l[2], op(Bs)];\n pB := pB-l[2]\n \+ end do;\n B:=Bs;\n RETURN(B)\n else\n E RROR(`Das Ideal ist nicht null-dimensional, und es existiert keine end liche Basis.`)\n fi\n end:" }}{PARA 0 "> " 0 "" {MPLTEXT 1 0 1471 "holMatrix := proc(f,KbasisList,GBList,VList,torder)\n loc al KK,GG,r,ff,fff,XX,i,k,l,Ku,pKu;\n global B,Mf;\n \+ Ku := KbasisList;\n r := nops(Ku);\n \+ XX := VList;\n GG := GBList;\n # \n # zuerst wird \"Ku\" sortiert gem\344\337 \"torder\" : \n #\n pKu := 0;\n for i t o r do pKu := pKu+Ku[i] od;\n i := 0;\n \+ KK := [];\n while pKu <> 0 do l := leadmon(pKu,tor der); i := i+1; KK := [l[2],op(KK)]; pKu := pKu-l[2] od;\n \+ B:=KK;\n # print(`Die nach der vorgegebenen Monomord nung sortierte K-Basis ist : B = `,B);\n #\n \+ # jetzt wird die Matrix \"Mf\" bestimmt :\n #\n \+ ff := f;\n Mf := matrix(r,r,0);\n \+ for i to r do\n fff := ff*KK[i];\n \+ R||i := normalf(fff,GG,torder);\n wh ile R||i <> 0 do\n l := leadmon(R||i,torder); \n for k to r do if KK[k] = l[2] then Mf[k,i] := l[1]; k := r fi od;\n R||i := R||i-l[1]*l[ 2]\n od\n od;\n #p rint(`Die Matrix` ,` M`[f],` der Multiplikation mit f` = f,` in `,'K '*XX*`/ I`,\n # ` bezueglich dieser K-Basis ist` );\n evalm(Mf);\n end:" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 0 "" }}}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 0 "" }}}{SECT 1 {PARA 3 "" 0 "" {TEXT 259 50 " (a) Mit dem Radikalide al aus Beispiel 2 in \247 12" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 47 "A:=poly_algebra(x,y):T:=termorder(A,tdeg(x,y)):" }}}{EXCHG {PARA 0 "" 0 "" {TEXT -1 60 "Sei das Ideal I erzeugt von den folgende n Polynomen: " }}{PARA 0 "> " 0 "" {MPLTEXT 1 0 108 "f[1]:=x*y^2 +x^2-x*y-2*y^2-3*x+2*y+2:f[2]:=x*y-x-2*y+2:f[3]:=-2*x*y+y^2+4*x-y-2:F: =[f[1],f[2],f[3]]:X:=[x,y]:" }}{PARA 0 "> " 0 "" {MPLTEXT 1 0 105 "mat rix(3,3,['f[1]' , `=`, f[1], 'f[2]', `=`, f[2],'f[3]', `=`, f[3]]), ' F'=['f[1]','f[2]','f[3]'],'X'=X;" }}}{EXCHG {PARA 0 "" 0 "" {TEXT -1 28 "Die Gr\366bnerbasis GB von I:" }}{PARA 0 "> " 0 "" {MPLTEXT 1 0 27 "GB:=gbasis(F,T): 'GB'=GB;" }}}{EXCHG {PARA 0 "" 0 "" {TEXT -1 73 "Die nach der Monomordnung T=tdeg(x,y) sortierte Standardmonombasis ist : " }}{PARA 0 "> " 0 "" {MPLTEXT 1 0 63 "B:=Kbasis(GB,X,T):B:=sor t(B,(t1,t2)->testorder(t1,t2,T)):'B'=B;" }}}{EXCHG {PARA 0 "" 0 "" {TEXT -1 85 "Wahl einer Linearform h mit zuf\344lligen ganzzahligen Ko effizienten im Bereich [-a,a] :" }}{PARA 0 "> " 0 "" {MPLTEXT 1 0 7 "a :=100:" }}{PARA 0 "> " 0 "" {MPLTEXT 1 0 18 "c:=rand((-a)...a):" }} {PARA 0 "> " 0 "" {MPLTEXT 1 0 15 "h:=c()*x+c()*y;" }}}{EXCHG {PARA 0 "" 0 "" {TEXT -1 76 "Die Matrixdarstellung Mh der Multiplikation mit h bez\374glich der Basis B ist:" }}{PARA 0 "> " 0 "" {MPLTEXT 1 0 26 "M h:=holMatrix(h,B,GB,X,T);" }}}{EXCHG {PARA 0 "" 0 "" {TEXT -1 41 "Das \+ charakteristische Polynom von Mh ist:" }}{PARA 0 "> " 0 "" {MPLTEXT 1 0 19 "fh:=charpoly(Mh,t);" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 19 "gcd(fh,diff(fh,t));" }}}{EXCHG {PARA 0 "" 0 "" {TEXT -1 19 "Berech nung von g1h:" }}{PARA 0 "> " 0 "" {MPLTEXT 1 0 59 "Mhs1:=evalm(holMat rix(h,B,GB,X,T)+s*holMatrix(1,B,GB,X,T));" }}{PARA 0 "> " 0 "" {MPLTEXT 1 0 35 "subs(s=0,diff(charpoly(Mhs1,t),s));" }}{PARA 0 "> " 0 "" {MPLTEXT 1 0 60 "g1h:=-subs(s=0,diff(charpoly(Mhs1,t),s))/gcd(fh, diff(fh,t));" }}}{EXCHG {PARA 0 "" 0 "" {TEXT -1 24 "ggT(fh,g1h) muss \+ 1 sein:" }}{PARA 0 "> " 0 "" {MPLTEXT 1 0 12 "gcd(fh,g1h);" }}}{EXCHG {PARA 0 "" 0 "" {TEXT -1 37 "Die Berechnung der Polynome gxh, gyh:" }} {PARA 0 "> " 0 "" {MPLTEXT 1 0 59 "Mhsx:=evalm(holMatrix(h,B,GB,X,T)+s *holMatrix(x,B,GB,X,T));" }}{PARA 0 "> " 0 "" {MPLTEXT 1 0 60 "gxh:=-s ubs(s=0,diff(charpoly(Mhsx,t),s))/gcd(fh,diff(fh,t));" }}{PARA 0 "> " 0 "" {MPLTEXT 1 0 59 "Mhsy:=evalm(holMatrix(h,B,GB,X,T)+s*holMatrix(y, B,GB,X,T));" }}{PARA 0 "> " 0 "" {MPLTEXT 1 0 60 "gyh:=-subs(s=0,diff( charpoly(Mhsy,t),s))/gcd(fh,diff(fh,t));" }}}{EXCHG {PARA 0 "" 0 "" {TEXT -1 44 "Die Nullstellen von fh symbolisch berechnet:" }}{PARA 0 " > " 0 "" {MPLTEXT 1 0 15 "L:=[solve(fh)];" }}}{EXCHG {PARA 0 "" 0 "" {TEXT -1 22 "Die L\366sungen sind nun:" }}{PARA 0 "> " 0 "" {MPLTEXT 1 0 73 "seq([subs(t=L[k],gxh),subs(t=L[k],gyh)]*1/subs(t=L[k],g1h),k=1 ..nops(L));" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 0 "" }}}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 0 "" }}}{SECT 1 {PARA 3 "" 0 "" {TEXT -1 1 " " }{TEXT 256 45 "(b) Mit dem Ideal aus Beispiel 7 (b) in \247 \+ 10" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 52 "A:=poly_algebra(x,y,z) :\nT:=termorder(A,tdeg(x,y,z)):" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 63 "f[1]:=x^2+y^2+z^2-2*x:f[2]:=x^3-y*z-x:f[3]:=x-y+2*z:X:=[x,y,z] :" }}{PARA 0 "> " 0 "" {MPLTEXT 1 0 132 "F:=[f[1],f[2],f[3]]:X:=[x,y,z ]:matrix(3,3,['f[1]', `=`, f[1],'f[2]', `=`,f[2],'f[3]', `=`,f[3]]); \+ 'F'=['f[1]','f[2]','f[3]'],'X'=X;" }}}{EXCHG {PARA 0 "" 0 "" {TEXT -1 28 "Die Gr\366bnerbasis GB von I:" }}{PARA 0 "> " 0 "" {MPLTEXT 1 0 27 "GB:=gbasis(F,T): 'GB'=GB;" }}}{EXCHG {PARA 0 "" 0 "" {TEXT -1 75 "Die nach der Monomordnung T=tdeg(x,y,z) sortierte Standardmonombas is ist : " }}{PARA 0 "> " 0 "" {MPLTEXT 1 0 63 "B:=Kbasis(GB,X,T):B:=s ort(B,(t1,t2)->testorder(t1,t2,T)):'B'=B;" }}}{EXCHG {PARA 0 "" 0 "" {TEXT -1 85 "Wahl einer Linearform h mit zuf\344lligen ganzzahligen Ko effizienten im Bereich [-a,a] :" }}{PARA 0 "> " 0 "" {MPLTEXT 1 0 7 "a :=100:" }}{PARA 0 "> " 0 "" {MPLTEXT 1 0 18 "c:=rand((-a)...a):" }} {PARA 0 "> " 0 "" {MPLTEXT 1 0 21 "h:=c()*x+c()*y+c()*z;" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 26 "Mh:=holMatrix(h,B,GB,X,T):" }} {PARA 0 "> " 0 "" {MPLTEXT 1 0 19 "fh:=charpoly(Mh,t);" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 19 "gcd(fh,diff(fh,t));" }}}{EXCHG {PARA 0 "" 0 "" {TEXT -1 19 "Berechnung von g1h:" }}{PARA 0 "> " 0 "" {MPLTEXT 1 0 59 "Mhs1:=evalm(holMatrix(h,B,GB,X,T)+s*holMatrix(1,B,GB, X,T)):" }}{PARA 0 "> " 0 "" {MPLTEXT 1 0 72 "g1h:=simplify((-subs(s=0, diff(charpoly(Mhs1,t),s))/gcd(fh,diff(fh,t))));" }}}{EXCHG {PARA 0 "> \+ " 0 "" {MPLTEXT 1 0 12 "gcd(fh,g1h);" }}}{EXCHG {PARA 0 "" 0 "" {TEXT -1 42 "Die Berechnung der Polynome gxh, gyh, gzh:" }}{PARA 0 "> " 0 " " {MPLTEXT 1 0 59 "Mhsx:=evalm(holMatrix(h,B,GB,X,T)+s*holMatrix(x,B,G B,X,T)):" }}{PARA 0 "> " 0 "" {MPLTEXT 1 0 72 "gxh:=simplify((-subs(s= 0,diff(charpoly(Mhsx,t),s)))/gcd(fh,diff(fh,t)));" }}{PARA 0 "> " 0 " " {MPLTEXT 1 0 59 "Mhsy:=evalm(holMatrix(h,B,GB,X,T)+s*holMatrix(y,B,G B,X,T)):" }}{PARA 0 "> " 0 "" {MPLTEXT 1 0 72 "gyh:=simplify((-subs(s= 0,diff(charpoly(Mhsy,t),s)))/gcd(fh,diff(fh,t)));" }}{PARA 0 "> " 0 " " {MPLTEXT 1 0 59 "Mhsz:=evalm(holMatrix(h,B,GB,X,T)+s*holMatrix(z,B,G B,X,T)):" }}{PARA 0 "> " 0 "" {MPLTEXT 1 0 72 "gzh:=simplify((-subs(s= 0,diff(charpoly(Mhsz,t),s)))/gcd(fh,diff(fh,t)));" }}}{EXCHG {PARA 0 " " 0 "" {TEXT -1 44 "Die Nullstellen von fh symbolisch berechnet:" }} {PARA 0 "> " 0 "" {MPLTEXT 1 0 15 "L:=[solve(fh)]:" }}}{EXCHG {PARA 0 "" 0 "" {TEXT -1 148 "Die exakten (!) L\366sungen sind nun nach Satz 8 (a) wie folgt zu berechnen. Da die Ausdr\374cke i.A. recht aufw\344nd ig werden, ist die Anzeige unterdr\374ckt:" }}{PARA 0 "> " 0 "" {MPLTEXT 1 0 135 "L\366sungen:=[seq([subs(t=L[k],gxh)/subs(t=L[k],g1h) ,subs(t=L[k],gyh)/subs(t=L[k],g1h),subs(t=L[k],gzh)/subs(t=L[k],g1h)], \nk=1..nops(L))]:" }}{PARA 0 "> " 0 "" {MPLTEXT 1 0 12 "nL:=nops(%);" }}}{EXCHG {PARA 0 "" 0 "" {TEXT -1 6 "Probe:" }}{PARA 0 "> " 0 "" {MPLTEXT 1 0 66 "seq(simplify(subs(seq(X[i]=L\366sungen[k][i],i=1..3), f[1])),k=1..nL);" }}}{EXCHG {PARA 0 "" 0 "" {TEXT -1 37 "N\344herungsw erte f\374r die L\366sungen sind:" }}{PARA 0 "> " 0 "" {MPLTEXT 1 0 76 "Digits:=3;concat(vector([seq(k,k=1..nL)]),matrix(nL,3,map(evalf,L \366sungen)));" }}}{EXCHG {PARA 0 "" 0 "" {TEXT -1 93 "Da nur mit 3 St ellen gerechnet wurde, sind die Ergebnisse allerdings u.U. noch etwas \+ ungenau." }}{PARA 0 "" 0 "" {TEXT -1 57 "Rechnung mit 15 Stellen und A nzeige mit 5 Stellen ergibt:" }}{PARA 0 "> " 0 "" {MPLTEXT 1 0 11 "Dig its:=15;" }}{PARA 0 "> " 0 "" {MPLTEXT 1 0 70 "LM:=concat(vector([seq( k,k=1..nL)]),matrix(nL,3,map(evalf,L\366sungen))):" }}{PARA 0 "> " 0 " " {MPLTEXT 1 0 10 "Digits:=5;" }}{PARA 0 "> " 0 "" {MPLTEXT 1 0 10 "ev alm(LM);" }}}{EXCHG {PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 147 "Insbesondere sieht man am Ergebnis, dass bei diesem Beis piel drei reelle Nullstellen vorliegen und dass zwei Nullstellen, die \+ Vielfachheit 2 haben." }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 0 "" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 0 "" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 0 "" }}}} {MARK "2 0 0" 3 }{VIEWOPTS 1 1 0 1 1 1803 1 1 1 1 }{PAGENUMBERS 0 1 2 33 1 1 }