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" }{TEXT 248 0 "" }{TEXT 248 23 "\nT. Pajdla, 15 Nov 2010" }{TEXT 248 0 "" }}}{SECT 0 {PARA 235 "" 0 "" {TEXT 249 20 " Packages & settings" }{TEXT 249 0 "" }}{EXCHG {PARA 236 "> " 0 "" {MPLTEXT 1 250 8 "restart ;" }{MPLTEXT 1 250 0 "" }{MPLTEXT 1 250 17 "\nwith(ListTools):" } {MPLTEXT 1 250 0 "" }{MPLTEXT 1 250 16 "\nwith(Groebner):" }{MPLTEXT 1 250 0 "" }}{PARA 237 "" 1 "" {TEXT 251 57 "Warning, the assigned nam e Group now has a global binding" }{TEXT 251 0 "" }}}}{SECT 0 {PARA 238 "" 0 "" {TEXT 252 15 "Useful commands" }{TEXT 252 0 "" }}{EXCHG {PARA 239 "" 0 "" {TEXT 253 5 "While" }{TEXT 253 0 "" }}}{EXCHG {PARA 236 "> " 0 "" {MPLTEXT 1 250 25 "#while (...) and (...) do" }{MPLTEXT 1 250 0 "" }{MPLTEXT 1 250 9 "\n#end do;" }{MPLTEXT 1 250 0 "" }}} {EXCHG {PARA 239 "" 0 "" {TEXT 253 3 "If " }{TEXT 253 0 "" }}}{EXCHG {PARA 236 "> " 0 "" {MPLTEXT 1 250 26 "#if ... then ... end if; " } {MPLTEXT 1 250 0 "" }}}{EXCHG {PARA 239 "" 0 "" {TEXT 253 9 "Constants " }{TEXT 253 0 "" }}}{EXCHG {PARA 236 "> " 0 "" {MPLTEXT 1 250 6 "#tru e;" }{MPLTEXT 1 250 0 "" }{MPLTEXT 1 250 8 "\n#false;" }{MPLTEXT 1 250 0 "" }}}{EXCHG {PARA 239 "" 0 "" {TEXT 253 14 "Print to debug" } {TEXT 253 0 "" }}}{EXCHG {PARA 236 "> " 0 "" {MPLTEXT 1 250 12 "#print (...);" }{MPLTEXT 1 250 0 "" }}}{EXCHG {PARA 239 "" 0 "" {TEXT 253 42 "The least common multiple of two monomials" }{TEXT 253 0 "" }}} {EXCHG {PARA 236 "> " 0 "" {MPLTEXT 1 250 20 "#lcm(x*y^3,x*y^2*z);" } {MPLTEXT 1 250 0 "" }}}{EXCHG {PARA 239 "" 0 "" {TEXT 253 24 "Measurin g time intervals" }{TEXT 253 0 "" }}}{EXCHG {PARA 236 "> " 0 "" {MPLTEXT 1 250 33 "#time(): 3^1000000:; time() - st;" }{MPLTEXT 1 250 0 "" }}}}{SECT 0 {PARA 240 "" 0 "" {TEXT 254 11 " Algorithms" }{TEXT 254 0 "" }}{EXCHG {PARA 239 "" 0 "" {TEXT 253 30 "Cartesian product of two lists" }{TEXT 253 0 "" }}}{EXCHG {PARA 236 "> " 0 "" {MPLTEXT 1 250 14 "LxL:=proc(X,Y)" }{MPLTEXT 1 250 0 "" }{MPLTEXT 1 250 54 "\n \+ Flatten(map(x->(map(y->Flatten([x,y]),Y)),X),1);" }{MPLTEXT 1 250 0 "" }{MPLTEXT 1 250 10 "\nend proc:" }{MPLTEXT 1 250 0 "" }}}{EXCHG {PARA 239 "" 0 "" {TEXT 253 19 "Division algortithm" }{TEXT 253 0 "" } }}{EXCHG {PARA 236 "> " 0 "" {MPLTEXT 1 250 23 "PolyDiv := proc(f,F,mo )" }{MPLTEXT 1 250 0 "" }{MPLTEXT 1 250 60 "\n local a,s,p,divisionoc cured ,i,lt_fi, LT_fi,lt_p, LT_p,r;" }{MPLTEXT 1 250 0 "" }{MPLTEXT 1 250 10 "\n p := f;" }{MPLTEXT 1 250 0 "" }{MPLTEXT 1 250 16 "\n s := nops(F);" }{MPLTEXT 1 250 0 "" }{MPLTEXT 1 250 24 "\n a := [seq(0,i= 1..s)];" }{MPLTEXT 1 250 0 "" }{MPLTEXT 1 250 10 "\n r := 0;" } {MPLTEXT 1 250 0 "" }{MPLTEXT 1 250 18 "\n while (p<>0) do" } {MPLTEXT 1 250 0 "" }{MPLTEXT 1 250 12 "\n i := 1;" }{MPLTEXT 1 250 0 "" }{MPLTEXT 1 250 30 "\n divisionoccured := false;" } {MPLTEXT 1 250 0 "" }{MPLTEXT 1 250 1 "\n" }{MPLTEXT 1 250 53 "\n w hile (i <= s) and (divisionoccured = false) do" }{MPLTEXT 1 250 0 "" }{MPLTEXT 1 250 56 "\n LT_fi := leadmon(F[i],mo)[1]*leadmon(F[i], mo)[2];" }{MPLTEXT 1 250 0 "" }{MPLTEXT 1 250 49 "\n LT_p := lead mon(p,mo)[1]*leadmon(p,mo)[2];" }{MPLTEXT 1 250 0 "" }{MPLTEXT 1 250 128 "\n if divide(LT_p,LT_fi) and (LT_p<>0) then #ZDE byl o nutne pridat podminku nuloveho leading termu deleneho polynomu" } {MPLTEXT 1 250 0 "" }{MPLTEXT 1 250 115 "\n a[i] := expand(a[i] + LT_p/LT_fi); #funkce se jinak zacyklila kdyz byly po lynomy delitelne." }{MPLTEXT 1 250 0 "" }{MPLTEXT 1 250 46 "\n \+ p := expand(p - (LT_p/LT_fi) * F[i]);" }{MPLTEXT 1 250 0 "" }{MPLTEXT 1 250 32 "\n divisionoccured = true;" }{MPLTEXT 1 250 0 "" } {MPLTEXT 1 250 12 "\n else " }{MPLTEXT 1 250 0 "" }{MPLTEXT 1 250 20 "\n i := i + 1;" }{MPLTEXT 1 250 0 "" }{MPLTEXT 1 250 14 "\n end if;" }{MPLTEXT 1 250 0 "" }{MPLTEXT 1 250 13 "\n en d do; " }{MPLTEXT 1 250 0 "" }{MPLTEXT 1 250 1 "\n" }{MPLTEXT 1 250 38 "\n if (divisionoccured = false) then" }{MPLTEXT 1 250 0 "" } {MPLTEXT 1 250 49 "\n LT_p := leadmon(p,mo)[1]*leadmon(p,mo)[2];" }{MPLTEXT 1 250 0 "" }{MPLTEXT 1 250 29 "\n r := expand(r + LT_p );" }{MPLTEXT 1 250 0 "" }{MPLTEXT 1 250 29 "\n p := expand(p - L T_p);" }{MPLTEXT 1 250 0 "" }{MPLTEXT 1 250 12 "\n end if;" } {MPLTEXT 1 250 0 "" }{MPLTEXT 1 250 1 "\n" }{MPLTEXT 1 250 8 "\nend do ;" }{MPLTEXT 1 250 0 "" }{MPLTEXT 1 250 80 "\n[r,a]; # return the list containting the remainder and the list of coefficients" }{MPLTEXT 1 250 0 "" }{MPLTEXT 1 250 11 "\nend proc: " }{MPLTEXT 1 250 0 "" }}} {EXCHG {PARA 241 "> " 0 "" {MPLTEXT 1 255 23 "PolyDiv(x,[x],plex(x));" }{MPLTEXT 1 255 0 "" }}{PARA 242 "" 1 "" {XPPMATH 20 "6#7$\"\"!7#\"\" \"" }{TEXT 20 0 "" }}}{EXCHG {PARA 243 "" 1 "" {TEXT 256 0 "" }}} {EXCHG {PARA 239 "" 0 "" {TEXT 253 34 "Implement S-Polynomial computat ion" }{TEXT 253 0 "" }}}{EXCHG {PARA 236 "> " 0 "" {MPLTEXT 1 250 18 " SPol:=proc(f,g,mo)" }{MPLTEXT 1 250 0 "" }{MPLTEXT 1 250 10 "\n local \+ S;" }{MPLTEXT 1 250 0 "" }}{PARA 236 "> " 0 "" {MPLTEXT 1 250 28 " S:= f*lcm(leadmon(f,mo)[2]," }{MPLTEXT 1 250 16 "leadmon(g,mo)[2]" } {MPLTEXT 1 250 5 ") / (" }{MPLTEXT 1 250 7 "leadmon" }{MPLTEXT 1 250 10 "(f,mo)[1]*" }{MPLTEXT 1 250 7 "leadmon" }{MPLTEXT 1 250 13 "(f,mo) [2]) - " }{MPLTEXT 1 250 0 "" }}{PARA 236 "> " 0 "" {MPLTEXT 1 250 11 " g*lcm(" }{MPLTEXT 1 250 16 "leadmon(f,mo)[2]" }{MPLTEXT 1 250 1 "," }{MPLTEXT 1 250 16 "leadmon(g,mo)[2]" }{MPLTEXT 1 250 5 ") / (" } {MPLTEXT 1 250 7 "leadmon" }{MPLTEXT 1 250 10 "(g,mo)[1]*" }{MPLTEXT 1 250 7 "leadmon" }{MPLTEXT 1 250 11 "(g,mo)[2]);" }{MPLTEXT 1 250 0 " " }}{PARA 244 "> " 0 "" {MPLTEXT 1 257 10 "expand(S);" }{MPLTEXT 1 258 0 "" }{MPLTEXT 1 258 10 "\nend proc:" }{MPLTEXT 1 258 0 "" }}} {EXCHG {PARA 239 "" 0 "" {TEXT 253 36 "Verify on the following polynom ials:" }{TEXT 253 0 "" }}{PARA 245 "" 0 "" {XPPEDIT 18 0 "f1 := 88+94* x+6*x^2+21*x^2*z+18*x*y^2+75*x*y*z;" "6#>%#f1G,.\"#))\"\"\"*&\"#%*F'% \"xGF'F'*&\"\"'F'*$F*\"\"#F'F'*(\"#@F'F-F'%\"zGF'F'*(\"#=F'F*F'%\"yGF. F'**\"#vF'F*F'F4F'F1F'F'" }{TEXT 259 0 "" }}{PARA 246 "" 0 "" {XPPEDIT 18 0 "f2 := -44*z+67*y*z-72*z^2+2*x*y^2+4*x*y*z-84*x*z^2;" "6 #>%#f2G,.*&\"#W\"\"\"%\"zGF(!\"\"*(\"#nF(%\"yGF(F)F(F(*&\"#sF(*$F)\"\" #F(F**(F1F(%\"xGF(F-F1F(**\"\"%F(F3F(F-F(F)F(F(*(\"#%)F(F3F(F)F1F*" } {TEXT 260 0 "" }{TEXT 260 19 "\nmo := plex(x,y,z);" }{TEXT 260 0 "" } {TEXT 260 19 "\ns:=SPol(f1,f2,mo);" }{TEXT 260 0 "" }}{PARA 247 "" 0 " " {XPPEDIT 18 0 "s := 88/21*y^2+94/21*x*y^2+2/7*y^2*x^2+6/7*y^4*x+25/7 *x*z*y^3+22*x*z^2-67/2*x*z^2*y+36*x*z^3-2*x^2*z^2*y+42*x^2*z^3;" "6#>% \"sG,6*(\"#))\"\"\"\"#@!\"\"%\"yG\"\"#F(**\"#%*F(F)F*%\"xGF(F+F,F(**F, F(\"\"(F*F+F,F/F,F(**\"\"'F(F1F*F+\"\"%F/F(F(*,\"#DF(F1F*F/F(%\"zGF(F+ \"\"$F(*(\"#AF(F/F(F7F,F(*,\"#nF(F,F*F/F(F7F,F+F(F**(\"#OF(F/F(F7F8F(* *F,F(*$F/F,F(F7F,F+F(F**(\"#UF(F@F(F7F8F(" }{TEXT 259 0 "" }}}{EXCHG {PARA 236 "> " 0 "" {MPLTEXT 1 250 47 "f1 := 88+94*x+6*x^2+21*x^2*z+18 *x*y^2+75*x*y*z;" }{MPLTEXT 1 250 0 "" }{MPLTEXT 1 250 52 "\nf2 := -44 *z+67*y*z-72*z^2+2*x*y^2+4*x*y*z-84*x*z^2;" }{MPLTEXT 1 250 0 "" } {MPLTEXT 1 250 19 "\nmo := plex(x,y,z);" }{MPLTEXT 1 250 0 "" } {MPLTEXT 1 250 19 "\ns:=SPol(f1,f2,mo);" }{MPLTEXT 1 250 0 "" } {MPLTEXT 1 250 16 "\n#leadmon(f1,mo)" }{MPLTEXT 1 250 0 "" }{MPLTEXT 1 250 1 "\n" }}{PARA 242 "" 1 "" {XPPMATH 20 "6#>I#f1G6\",.\"#))\"\"\" I\"xGF%\"#%**$F)\"\"#\"\"'*&F)F,I\"zGF%F(\"#@*&F)F(I\"yGF%F,\"#=*(F)F( F2F(F/F(\"#v" }{TEXT 20 0 "" }}{PARA 242 "" 1 "" {XPPMATH 20 "6#>I#f2G 6\",.I\"zGF%!#W*&I\"yGF%\"\"\"F'F+\"#n*$F'\"\"#!#s*&I\"xGF%F+F*F.F.*(F 1F+F*F+F'F+\"\"%*&F1F+F'F.!#%)" }{TEXT 20 0 "" }}{PARA 242 "" 1 "" {XPPMATH 20 "6#>I#moG6\"-I%plexGF%6%I\"xGF%I\"yGF%I\"zGF%" }{TEXT 20 0 "" }}{PARA 242 "" 1 "" {XPPMATH 20 "6#>I\"sG6\",6*$I\"yGF%\"\"##\"#) )\"#@*&I\"xGF%\"\"\"F(F)#\"#%*F,*&F(F)F.F)#F)\"\"(*&F(\"\"%F.F/#\"\"'F 4*(F(\"\"$F.F/I\"zGF%F/#\"#DF4*&F.F/F;F)\"#A*(F;F)F.F/F(F/#!#nF)*&F;F: F.F/\"#O*(F;F)F.F)F(F/!\"#*&F;F:F.F)\"#U" }{TEXT 20 0 "" }}}{EXCHG {PARA 239 "" 0 "" {TEXT 253 22 "Buchberger's algorithm" }{TEXT 253 0 " " }}}{EXCHG {PARA 236 "> " 0 "" {MPLTEXT 1 250 20 "BuchGB := proc(F,mo )" }{MPLTEXT 1 250 0 "" }}{PARA 236 "> " 0 "" {MPLTEXT 1 250 35 "local G,G1,pridano,poc,i,j,s,sPoly;" }{MPLTEXT 1 250 0 "" }}{PARA 236 "> " 0 "" {MPLTEXT 1 250 5 "G:=F;" }{MPLTEXT 1 250 0 "" }}{PARA 244 "> " 0 "" {MPLTEXT 1 257 6 "G1:=0;" }{MPLTEXT 1 258 0 "" }{MPLTEXT 1 258 14 " \npridano := 1;" }{MPLTEXT 1 257 0 "" }{MPLTEXT 1 257 19 "\nwhile prid ano>0 do" }{MPLTEXT 1 258 0 "" }}{PARA 236 "> " 0 "" {MPLTEXT 1 250 9 " G1 := G;" }{MPLTEXT 1 250 0 "" }}{PARA 236 "> " 0 "" {MPLTEXT 1 250 14 " pridano := 0;" }{MPLTEXT 1 250 0 "" }{MPLTEXT 1 250 18 "\n poc := nops(G1);" }{MPLTEXT 1 250 0 "" }{MPLTEXT 1 250 24 "\n for i from 1 t o poc do" }{MPLTEXT 1 250 0 "" }}{PARA 236 "> " 0 "" {MPLTEXT 1 250 26 " for j from i+1 to poc do" }{MPLTEXT 1 250 0 "" }}{PARA 248 "> " 0 "" {MPLTEXT 1 261 12 " sPoly := " }{MPLTEXT 1 261 4 "SPol" } {MPLTEXT 1 261 19 "(G1[i], G1[j], mo);" }{MPLTEXT 1 261 0 "" } {MPLTEXT 1 261 35 "\n s := PolyDiv(sPoly, G1, mo)[1];" }{MPLTEXT 1 261 0 "" }{MPLTEXT 1 261 4 "\n " }{MPLTEXT 1 261 12 "if s<>0 then" } {MPLTEXT 1 261 0 "" }}{PARA 236 "> " 0 "" {MPLTEXT 1 250 20 " G := \+ [op(G), s];" }{MPLTEXT 1 250 0 "" }{MPLTEXT 1 250 28 "\n pridano := pridano + 1;" }{MPLTEXT 1 250 0 "" }{MPLTEXT 1 250 6 "\n fi" } {MPLTEXT 1 250 0 "" }}{PARA 236 "> " 0 "" {MPLTEXT 1 250 9 " end do;" }{MPLTEXT 1 250 0 "" }}{PARA 236 "> " 0 "" {MPLTEXT 1 250 8 " end do; " }{MPLTEXT 1 250 0 "" }{MPLTEXT 1 250 8 "\nend do;" }{MPLTEXT 1 250 0 "" }}{PARA 236 "> " 0 "" {MPLTEXT 1 250 2 "G;" }{MPLTEXT 1 250 0 "" }}{PARA 236 "> " 0 "" {MPLTEXT 1 250 9 "end proc:" }{MPLTEXT 1 250 0 " " }}}}{SECT 0 {PARA 249 "" 0 "" {TEXT 262 1 " " }{TEXT 263 21 "Groebne r basis - Test" }{TEXT 263 0 "" }}{EXCHG {PARA 246 "" 0 "" {TEXT 260 32 "Test the Buchberger's algorithm " }{XPPEDIT 18 0 "f1 := x^2*y-1;" "6#>%#f1G,&*&%\"xG\"\"#%\"yG\"\"\"F*F*!\"\"" }{TEXT 260 0 "" }{TEXT 260 1 "\n" }}{PARA 245 "" 0 "" {XPPEDIT 18 0 "f2 := x*y-1;" "6#>%#f2G, &*&%\"xG\"\"\"%\"yGF(F(F(!\"\"" }{TEXT 259 0 "" }}{PARA 239 "" 0 "" {TEXT 253 26 "BuchGB([f1,f2],plex(x,y));" }{TEXT 253 0 "" }}{PARA 245 "" 0 "" {XPPEDIT 18 0 "[x^2*y-1, x*y-1, -1+x, -x+1, -1+y, -1+y, 1-y, 1 -y];" "6#7*,&*&%\"xG\"\"#%\"yG\"\"\"F)F)!\"\",&*&F&F)F(F)F)F)F*,&F)F*F &F),&F&F*F)F),&F)F*F(F)F/,&F)F)F(F*F0" }{TEXT 259 0 "" }}{PARA 239 "" 0 "" {TEXT 253 0 "" }}{PARA 239 "" 0 "" {TEXT 253 0 "" }}}{EXCHG {PARA 236 "> " 0 "" {MPLTEXT 1 250 14 "f1 := x^2*y-1;" }{MPLTEXT 1 250 0 "" }{MPLTEXT 1 250 13 "\nf2 := x*y-1;" }{MPLTEXT 1 250 0 "" } {MPLTEXT 1 250 27 "\nBuchGB([f1,f2],plex(x,y));" }{MPLTEXT 1 250 0 "" }{MPLTEXT 1 250 1 "\n" }}{PAGEBK }{PARA 242 "" 1 "" {XPPMATH 20 "6#>I# f1G6\",&*&I\"xGF%\"\"#I\"yGF%\"\"\"F+!\"\"F+" }{TEXT 20 0 "" }}{PARA 242 "" 1 "" {XPPMATH 20 "6#>I#f2G6\",&*&I\"xGF%\"\"\"I\"yGF%F)F)!\"\"F )" }{TEXT 20 0 "" }}{PARA 242 "" 1 "" {XPPMATH 20 "6#7&,&*&I\"xG6\"\" \"#I\"yGF'\"\"\"F*!\"\"F*,&*&F&F*F)F*F*F+F*,&F+F*F&F*,&F+F*F)F*" } {TEXT 20 0 "" }}}{EXCHG {PARA 241 "> " 0 "" {MPLTEXT 1 255 0 "" }}} {EXCHG {PARA 239 "" 0 "" {TEXT 253 99 "Polynomials in the Groebner bas is repeat. Why? Can you speed up the algorithm? Explain how and why." }{TEXT 253 0 "" }}}{EXCHG {PARA 236 "> " 0 "" {MPLTEXT 1 250 0 "" }}}} {SECT 0 {PARA 240 "" 0 "" {TEXT 254 24 " Groebner basis - Task 1" } {TEXT 254 0 "" }}{EXCHG {PARA 239 "" 0 "" {TEXT 253 96 "Compute the Gr oebner basis of the following system w.r.t. all lexicographical monomi al orderings" }{TEXT 253 0 "" }{TEXT 253 28 "\nand measure time it tak es. " }{TEXT 253 0 "" }}}{EXCHG {PARA 236 "> " 0 "" {MPLTEXT 1 250 12 "g1 := x*y-1:" }{MPLTEXT 1 250 0 "" }{MPLTEXT 1 250 13 "\ng2 := x+z-1: " }{MPLTEXT 1 250 0 "" }{MPLTEXT 1 250 13 "\ng3 := x*z+1:" }{MPLTEXT 1 250 0 "" }{MPLTEXT 1 250 32 "\nmo:=combinat[permute]([x,y,z]):" } {MPLTEXT 1 250 0 "" }{MPLTEXT 1 250 17 "\nfor i from 1 to " }{MPLTEXT 1 250 15 "Factorial(3) do" }{MPLTEXT 1 250 0 "" }{MPLTEXT 1 250 28 "\n print(plex(op(mo[IRON])));" }{MPLTEXT 1 250 0 "" }{MPLTEXT 1 250 14 " \n st:= time();" }{MPLTEXT 1 250 0 "" }{MPLTEXT 1 250 26 "\n print(Buc hGB([g1,g2,g3]," }{MPLTEXT 1 250 18 "plex(op(mo[IRON]))" }{MPLTEXT 1 250 3 "));" }{MPLTEXT 1 250 0 "" }{MPLTEXT 1 250 9 "\n cas := " } {MPLTEXT 1 250 12 "time() - st;" }{MPLTEXT 1 250 1 " " }{MPLTEXT 1 250 0 "" }{MPLTEXT 1 250 15 "\n print('cas');" }{MPLTEXT 1 250 0 "" } {MPLTEXT 1 250 13 "\n print(cas);" }{MPLTEXT 1 250 0 "" }{MPLTEXT 1 250 8 "\nend do:" }{MPLTEXT 1 250 0 "" }}{PARA 250 "> " 0 "" {MPLTEXT 1 0 0 "" }}{PARA 242 "" 1 "" {XPPMATH 20 "6#-I%plexG6\"6%I\"zGF%I\"yGF %I\"xGF%" }{TEXT 20 0 "" }}{PARA 242 "" 1 "" {XPPMATH 20 "6#7(,&*&I\"x G6\"\"\"\"I\"yGF'F(F(!\"\"F(,(F&F(I\"zGF'F(F*F(,&*&F,F(F&F(F(F(F(,&*&F &\"\"#F)F(F*F&F(,(F)F*F&F(F*F(,(*$F&F1F(F&F*F*F(" }{TEXT 20 0 "" }} {PARA 242 "" 1 "" {XPPMATH 20 "6#I$casG6\"" }{TEXT 20 0 "" }}{PARA 242 "" 1 "" {XPPMATH 20 "6#$\"#%*!\"$" }{TEXT 20 0 "" }}{PARA 242 "" 1 "" {XPPMATH 20 "6#-I%plexG6\"6%I\"zGF%I\"yGF%I\"xGF%" }{TEXT 20 0 "" }}{PARA 242 "" 1 "" {XPPMATH 20 "6#7(,&*&I\"xG6\"\"\"\"I\"yGF'F(F(!\" \"F(,(F&F(I\"zGF'F(F*F(,&*&F,F(F&F(F(F(F(,&*&F&\"\"#F)F(F*F&F(,(F)F*F& F(F*F(,(*$F&F1F(F&F*F*F(" }{TEXT 20 0 "" }}{PARA 242 "" 1 "" {XPPMATH 20 "6#I$casG6\"" }{TEXT 20 0 "" }}{PARA 242 "" 1 "" {XPPMATH 20 "6#$\" #$*!\"$" }{TEXT 20 0 "" }}{PARA 242 "" 1 "" {XPPMATH 20 "6#-I%plexG6\" 6%I\"zGF%I\"yGF%I\"xGF%" }{TEXT 20 0 "" }}{PARA 242 "" 1 "" {XPPMATH 20 "6#7(,&*&I\"xG6\"\"\"\"I\"yGF'F(F(!\"\"F(,(F&F(I\"zGF'F(F*F(,&*&F,F (F&F(F(F(F(,&*&F&\"\"#F)F(F*F&F(,(F)F*F&F(F*F(,(*$F&F1F(F&F*F*F(" } {TEXT 20 0 "" }}{PARA 242 "" 1 "" {XPPMATH 20 "6#I$casG6\"" }{TEXT 20 0 "" }}{PARA 242 "" 1 "" {XPPMATH 20 "6#$\"#z!\"$" }{TEXT 20 0 "" }} {PARA 242 "" 1 "" {XPPMATH 20 "6#-I%plexG6\"6%I\"zGF%I\"yGF%I\"xGF%" } {TEXT 20 0 "" }}{PARA 242 "" 1 "" {XPPMATH 20 "6#7(,&*&I\"xG6\"\"\"\"I \"yGF'F(F(!\"\"F(,(F&F(I\"zGF'F(F*F(,&*&F,F(F&F(F(F(F(,&*&F&\"\"#F)F(F *F&F(,(F)F*F&F(F*F(,(*$F&F1F(F&F*F*F(" }{TEXT 20 0 "" }}{PARA 242 "" 1 "" {XPPMATH 20 "6#I$casG6\"" }{TEXT 20 0 "" }}{PARA 242 "" 1 "" {XPPMATH 20 "6#$\"#%*!\"$" }{TEXT 20 0 "" }}{PARA 242 "" 1 "" {XPPMATH 20 "6#-I%plexG6\"6%I\"zGF%I\"yGF%I\"xGF%" }{TEXT 20 0 "" }} {PARA 242 "" 1 "" {XPPMATH 20 "6#7(,&*&I\"xG6\"\"\"\"I\"yGF'F(F(!\"\"F (,(F&F(I\"zGF'F(F*F(,&*&F,F(F&F(F(F(F(,&*&F&\"\"#F)F(F*F&F(,(F)F*F&F(F *F(,(*$F&F1F(F&F*F*F(" }{TEXT 20 0 "" }}{PARA 242 "" 1 "" {XPPMATH 20 "6#I$casG6\"" }{TEXT 20 0 "" }}{PARA 242 "" 1 "" {XPPMATH 20 "6#$\"#y! \"$" }{TEXT 20 0 "" }}{PARA 242 "" 1 "" {XPPMATH 20 "6#-I%plexG6\"6%I \"zGF%I\"yGF%I\"xGF%" }{TEXT 20 0 "" }}{PARA 242 "" 1 "" {XPPMATH 20 " 6#7(,&*&I\"xG6\"\"\"\"I\"yGF'F(F(!\"\"F(,(F&F(I\"zGF'F(F*F(,&*&F,F(F&F (F(F(F(,&*&F&\"\"#F)F(F*F&F(,(F)F*F&F(F*F(,(*$F&F1F(F&F*F*F(" }{TEXT 20 0 "" }}{PARA 242 "" 1 "" {XPPMATH 20 "6#I$casG6\"" }{TEXT 20 0 "" } }{PARA 242 "" 1 "" {XPPMATH 20 "6#$\"#%*!\"$" }{TEXT 20 0 "" }}}} {SECT 0 {PARA 240 "" 0 "" {TEXT 254 24 " Groebner basis - Task 2" } {TEXT 254 0 "" }}{PARA 239 "" 0 "" {TEXT 253 96 "Compute the Groebner \+ basis of the following system w.r.t. the lexicographical ordering gene rated" }{TEXT 253 0 "" }{TEXT 253 34 "\nby your personal IRO number IR ON." }{TEXT 253 0 "" }}{EXCHG {PARA 248 "> " 0 "" {MPLTEXT 1 261 37 "I RON:= 6; # your IRO personal number;" }{MPLTEXT 1 261 0 "" }{MPLTEXT 1 261 36 "\nmo:=combinat[permute]([x,y,z,w,u]):" }{MPLTEXT 1 261 0 "" }{MPLTEXT 1 261 24 "\nmo:=plex(op(mo[IRON]));" }{MPLTEXT 1 261 0 "" } {MPLTEXT 1 261 11 "\nf1:=x*y+1;" }{MPLTEXT 1 261 0 "" }{MPLTEXT 1 261 11 "\nf2:=y*z-2;" }{MPLTEXT 1 261 0 "" }{MPLTEXT 1 261 11 "\nf3:=z*w+3 ;" }{MPLTEXT 1 261 0 "" }{MPLTEXT 1 261 11 "\nf4:=w*u-4;" }{MPLTEXT 1 261 0 "" }{MPLTEXT 1 261 11 "\nf5:=u*x+5;" }{MPLTEXT 1 261 0 "" } {MPLTEXT 1 261 1 "\n" }{MPLTEXT 1 261 1 "\n" }{MPLTEXT 1 261 1 "\n" }} {PARA 242 "" 1 "" {XPPMATH 20 "6#>I%IRONG6\"\"\"'" }{TEXT 20 0 "" }} {PARA 242 "" 1 "" {XPPMATH 20 "6#>I#moG6\"-I%plexGF%6'I\"xGF%I\"yGF%I \"uGF%I\"wGF%I\"zGF%" }{TEXT 20 0 "" }}{PARA 242 "" 1 "" {XPPMATH 20 " 6#>I#f1G6\",&*&I\"xGF%\"\"\"I\"yGF%F)F)F)F)" }{TEXT 20 0 "" }}{PARA 242 "" 1 "" {XPPMATH 20 "6#>I#f2G6\",&*&I\"yGF%\"\"\"I\"zGF%F)F)!\"#F) " }{TEXT 20 0 "" }}{PARA 242 "" 1 "" {XPPMATH 20 "6#>I#f3G6\",&*&I\"zG F%\"\"\"I\"wGF%F)F)\"\"$F)" }{TEXT 20 0 "" }}{PARA 242 "" 1 "" {XPPMATH 20 "6#>I#f4G6\",&*&I\"wGF%\"\"\"I\"uGF%F)F)!\"%F)" }{TEXT 20 0 "" }}{PARA 242 "" 1 "" {XPPMATH 20 "6#>I#f5G6\",&*&I\"uGF%\"\"\"I\"x GF%F)F)\"\"&F)" }{TEXT 20 0 "" }}}{EXCHG {PARA 239 "" 0 "" {TEXT 253 75 "Show the GB is a Groebner basis generated by polynomial f1, f2, f3 , f4, f5." }{TEXT 253 0 "" }}}{EXCHG {PARA 236 "> " 0 "" {MPLTEXT 1 250 48 "GB:=BuchGB([f1,f2, f3, f4, f5],plex(x,y,z,w,u));" }{MPLTEXT 1 250 0 "" }}{PARA 242 "" 1 "" {XPPMATH 20 "6#>I#GBG6\"7W,&*&I\"xGF%\"\" \"I\"yGF%F*F*F*F*,&*&F+F*I\"zGF%F*F*!\"#F*,&*&F.F*I\"wGF%F*F*\"\"$F*,& *&F2F*I\"uGF%F*F*!\"%F*,&*&F6F*F)F*F*\"\"&F*,&F.F*F)\"\"#,&F6F*F+!\"&, &F2F/F+!\"$,&F-F>\"#5F*,&F6F3F.\"\"%,&F1F>!#:F*,&F)F7F2F>,&F*F*F5#!\" \"FDFH,&F2F/F6#F@F:,&FBF**$F6F<#F3FD,&*$F6F3#!\"*\"$g\"F6#F@FD,&F/F*FN #F@\"#?FVFK,&*&F2F*F6F<#F3\"#;F6FU,&FFF*F5#\"#:FD,&FZ#FS\"#KF6#\"\"*\" \"),&F6#F3F:FZFW,&F6Fao*&F2FF6#F@F<,&FZ#F3F_oF6Fio,&FZ#F*F:F6#F7F:,&Fco#F/F3F6F^p,&FZFUF6F3,&Fco #F>FDF6Fio,&F:F*FNF[p,&!#5F*FNFU,&FinF*F5#FFFD,&FJF*FN#F@\"#S,&F5FIFNF jp,&FQ#F^oFTF6FO,&FhnF*FN#F^oF\\o,&FZF`qF6#FSF_o,&F2FcpF6#F@F_o,&F6#FJ F:F2F`p,&F " 0 "" {MPLTEXT 1 0 0 "" }}}}{PARA 252 "" 0 "" {TEXT 217 0 "" }}{PARA 253 "" 0 "" {TEXT 265 0 "" }}{PARA 254 "" 0 "" {TEXT -1 0 "" }}}{MARK "0 0 0" 0 } {VIEWOPTS 1 1 0 75 100 1802 1 1 1 1 }{PAGENUMBERS 0 1 2 33 1 1 }