{VERSION 5 0 "IBM INTEL NT" "5.0" } {USTYLETAB {CSTYLE "Maple Input" -1 0 "Helvetica" 1 10 0 0 128 1 2 2 2 0 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 }{PSTYLE "Normal" -1 0 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 2 2 0 1 }{PSTYLE "R3 Font 0" -1 256 1 {CSTYLE "" -1 -1 "Helvetica" 1 9 128 0 128 1 2 2 2 0 0 0 0 0 0 1 }0 0 0 -1 -1 -1 0 0 0 0 0 0 -1 0 }{PSTYLE "R3 Font 2" -1 257 1 {CSTYLE "" -1 -1 "Courier" 1 8 0 128 128 1 1 2 2 0 0 0 0 0 0 1 }0 0 0 -1 -1 -1 0 0 0 0 0 0 -1 0 }{PSTYLE "Seitenumbruch" -1 258 1 {CSTYLE " " -1 -1 "Times" 1 10 0 0 0 0 2 1 2 0 0 2 0 0 0 1 }0 0 0 -1 -1 -1 0 0 0 0 1 0 -1 0 }{PSTYLE "R3 Font 0" -1 259 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" -1 260 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 "Seitenumb ruch" -1 261 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" -1 262 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" -1 263 1 {CSTYLE "" -1 -1 "Times" 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 "Norma l" -1 264 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 1 0 1 0 2 2 0 1 }{PSTYLE "Normal" -1 265 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 "" 0 266 1 {CSTYLE "" -1 -1 "" 0 1 0 0 0 0 0 0 0 0 0 0 0 0 0 0 }3 0 0 -1 -1 -1 0 0 0 0 0 0 -1 0 }{PSTYLE "" 0 267 1 {CSTYLE "" -1 -1 "" 0 1 0 0 0 0 0 0 0 0 0 0 0 0 0 0 }3 0 0 -1 -1 -1 0 0 0 0 0 0 -1 0 }} {SECT 0 {EXCHG {PARA 0 "" 0 "" {TEXT -1 24 "Algebra 2 WiSe 2004/2005" }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 29 "Bearbei tung der Aufgabe (38) " }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 3 "von" }}{PARA 0 "" 0 "" {TEXT -1 59 "Ina Burghaus, Ann e Lehmann, Imke Reimer und Insa Winzenborg" }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 15 "14. Januar 2004" }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}}{EXCHG {PARA 0 " " 0 "" {TEXT -1 45 "Benutzt werden folgender Divisionsalgorithmus" }} {PARA 0 "> " 0 "" {MPLTEXT 1 0 1051 "DIVALGO:=proc(f,F,mo,va)\nlocal p ,r,s,ltp,ltf,h,l,k,qq,var;\nvar:=seq(va[i],i=1..nops(va));\nh:=sort(f, va,mo);\np:=h;\nr:=0;\ns:=nops(F);\nltp:=leadterm(p,mo(var))*leadcoeff (p,mo(var));\nfor k from 1 to s do\n b||k:=op(k,F);\n ff||k:=op(k, F);\n ff||k:=sort(expand(ff||k),va,mo);\n a||k:=0;\n ltf||k:=lea dterm(ff||k,mo(var))*leadcoeff(ff||k,mo(var));\nend do;\nwhile p<>0 do \n k:=1;\n r2;\n while k <= s do \n if divide(ltp,(ltf||k), 'q')=true then\n qq:=q;\n qq:=sort(expand(qq),va,mo); \n a||k:=a||k+qq;\n a||k:=sort(expand(a||k),va,mo);\n \+ p:=p-qq*(ff||k);\n p:=sort(expand(p),va,mo);\n \+ ltp:=(leadterm(p,mo(var))*leadcoeff(p,mo(var)));\n k:=s+1;\n \+ else\n if k=s then\n r:=r+ltp;\n r: =sort(expand(r),va,mo);\n p:=p-ltp;\n p:=sort(ex pand(p),va,mo);\n ltp:=leadterm(p,mo(var))*leadcoeff(p,mo(v ar));\n k:=s+1;\n else k:=k+1;\n end if;\n \+ end if;\n end do;\nend do;\nh;\nF;\nmo;\n[p,seq(a||k,k=1..s),r] ;\nend proc:\n\n" }}}{EXCHG {PARA 0 "" 0 "" {TEXT -1 64 "und folgende \+ Funktionszuweisungen f\374r die einfachere Bedienung:" }}{PARA 0 "> \+ " 0 "" {MPLTEXT 1 0 40 "sor:= g -> sort(expand(g),[x,y,z],tdeg);" }}} {EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 59 "lt:= g -> leadterm(g,tdeg(x, y,z))*leadcoeff(g,tdeg(x,y,z));" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 69 "S:= (g,h) -> (lcm(lt(g),lt(h))/lt(g))*g - (lcm(lt(g),lt(h))/lt (h))*h;" }}}{EXCHG {PARA 0 "" 0 "" {TEXT -1 37 "Die Ausgabe von DIVALG O hat die Form " }}{PARA 267 "" 0 "" {TEXT -1 3 "[p," }{XPPEDIT 18 0 " a[1];" "6#&%\"aG6#\"\"\"" }{TEXT -1 7 ", ... ," }{XPPEDIT 18 0 "a[s]; " "6#&%\"aG6#%\"sG" }{TEXT -1 6 ", r]. " }}{PARA 0 "" 0 "" {TEXT -1 36 "Dabei sind r der Divisionsrest und " }{XPPEDIT 18 0 "a[1];" "6#&% \"aG6#\"\"\"" }{TEXT -1 7 ", ... ," }{XPPEDIT 18 0 "a[s];" "6#&%\"aG6# %\"sG" }{TEXT -1 45 " die Koeffizienten in der Linearkombination " }} {PARA 266 "" 0 "" {XPPEDIT 18 0 "f = a[1]*f[1]+` ... `+a[s]*f[s]+r;" " 6#/%\"fG,**&&%\"aG6#\"\"\"F*&F$6#F*F*F*%&~...~GF**&&F(6#%\"sGF*&F$6#F1 F*F*%\"rGF*" }{TEXT -1 2 " ." }}}{EXCHG {PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 16 "Nun zur Aufgabe:" }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "> " 0 "" {MPLTEXT 1 0 15 "with(Groebner):" }}}{EXCHG {PARA 0 "" 0 "" {TEXT -1 3 "Mit" }}{PARA 0 "> " 0 "" {MPLTEXT 1 0 74 "f1:=x^2-y; f2:=x*(x-1)*(x -2); f2:=sor(f2); f3:=y*(y-1)*(y-4); f3:=sor(f3);" }}}{EXCHG {PARA 0 " " 0 "" {TEXT -1 25 "gilt: I = < f1, f2, f3 >." }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 37 "Vorgehen nach Buchberger- Algorithmus:" }}{PARA 0 "> " 0 "" {MPLTEXT 1 0 17 "t:=sor(S(f1,f2));" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 35 "DIVALGO(t,[f1,f2,f3],tdeg ,[x,y,z]);" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 17 "f4:=-x*y-2*x+ 3*y;" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 17 "t:=sor(S(f1,f3));" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 38 "DIVALGO(t,[f1,f2,f3,f4],t deg,[x,y,z]);" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 17 "t:=sor(S(f 2,f3));" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 38 "DIVALGO(t,[f1,f2 ,f3,f4],tdeg,[x,y,z]);" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 23 "f 5:=30*y^2+180*x-210*y;" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 17 "t :=sor(S(f1,f4));" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 41 "DIVALGO (t,[f1,f2,f3,f4,f5],tdeg,[x,y,z]);" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 17 "t:=sor(S(f1,f5));" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 41 "DIVALGO(t,[f1,f2,f3,f4,f5],tdeg,[x,y,z]);" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 17 "t:=sor(S(f2,f4));" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 41 "DIVALGO(t,[f1,f2,f3,f4,f5],tdeg,[x,y,z]); " }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 17 "t:=sor(S(f2,f5));" }}} {EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 41 "DIVALGO(t,[f1,f2,f3,f4,f5],t deg,[x,y,z]);" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 17 "t:=sor(S(f 3,f4));" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 41 "DIVALGO(t,[f1,f2 ,f3,f4,f5],tdeg,[x,y,z]);" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 17 "t:=sor(S(f3,f5));" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 41 "DI VALGO(t,[f1,f2,f3,f4,f5],tdeg,[x,y,z]);" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 17 "t:=sor(S(f4,f5));" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 41 "DIVALGO(t,[f1,f2,f3,f4,f5],tdeg,[x,y,z]);" }}}{EXCHG {PARA 0 "" 0 "" {TEXT -1 75 "Damit bilden die folgenden Polynome eine \+ (nicht-reduzierte!) Gr\366bner-Basis:" }}{PARA 0 "> " 0 "" {MPLTEXT 1 0 69 "'f1'=sor(f1); 'f2'=sor(f2); 'f3'=sor(f3); 'f4'=sor(f4); 'f5'=sor (f5);" }}}{EXCHG {PARA 0 "" 0 "" {TEXT -1 50 "Ist das folgende Polynom eine Einheit im Restring?" }{MPLTEXT 1 0 0 "" }}{PARA 0 "> " 0 "" {MPLTEXT 1 0 14 "f:=sor(1+x-y);" }}}{EXCHG {PARA 0 "" 0 "" {TEXT -1 87 "Also, gibt es ein g im Polynomring so, dass f * g = 1 mod I bzw. f * g - 1 in I liegt? " }}{PARA 0 "" 0 "" {TEXT -1 123 "Dies ist genau \+ dann der Fall, falls 1 in < I, f > liegt. Also wird nun die reduzierte Gr\366bner-Basis von < I, f > bestimmt. " }}{PARA 0 "" 0 "" {TEXT -1 70 "Wenn diese die Menge \{1\} ist, ist f invertierbar, anderenfalls n icht.\n" }}{PARA 0 "" 0 "" {TEXT -1 83 "Zur Bestimmung einer Gr\366bne r-Basis k\366nnen die vorherigen Rechnungen benutzt werden:" }}} {EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 16 "t:=sor(S(f1,f));" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 43 "DIVALGO(t,[f1,f2,f3,f4,f5,f],tdeg,[ x,y,z]);" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 9 "f6:=-y+3;" }}} {EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 16 "t:=sor(S(f2,f));" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 46 "DIVALGO(t,[f1,f2,f3,f4,f5,f,f6],tde g,[x,y,z]);" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 6 "f7:=1;" }}} {EXCHG {PARA 0 "" 0 "" {TEXT -1 223 "Da hier 1 der Basis hinzugef\374g t wird, kann man aufh\366ren, weil damit alle Polynome erzeugt werden \+ k\366nnen und somit die reduzierte Gr\366bner-Basis nur \{1\} sein kan n. Damit ist g invertierbar und es gibt eine Darstellung der Art" }} {PARA 0 "" 0 "" {TEXT -1 2 " " }}{PARA 265 "" 0 "" {TEXT -1 59 "1 = h (1) * f(1)+ ... + h(5) * f(5) + g * f = g * f mod I." }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 0 "" }}}}{MARK "0 0 0" 24 }{VIEWOPTS 1 1 0 1 1 1803 1 1 1 1 }{PAGENUMBERS 0 1 2 33 1 1 }