{VERSION 6 0 "IBM INTEL NT" "6.0" } {USTYLETAB {CSTYLE "Maple Input" -1 0 "Courier" 0 1 255 0 0 1 0 1 0 0 1 0 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 "" 0 1 0 0 0 0 0 1 0 0 0 0 0 0 0 0 }{CSTYLE "" -1 257 "" 1 12 0 0 0 0 0 0 0 0 0 0 0 0 0 0 }{CSTYLE "" -1 258 "" 1 12 0 0 0 0 0 0 0 0 0 0 0 0 0 0 }{CSTYLE "" -1 259 "" 1 12 0 0 0 0 0 0 0 0 0 0 0 0 0 0 }{PSTYLE "Normal" -1 0 1 {CSTYLE "" -1 -1 "Times" 1 14 0 0 1 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 "Text \+ Output" -1 2 1 {CSTYLE "" -1 -1 "Courier" 1 10 0 0 255 1 2 2 2 2 2 1 3 1 3 1 }1 1 0 0 0 0 1 0 1 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 1 0 1 0 2 2 0 1 }{PSTYLE "Normal" -1 256 1 {CSTYLE "" -1 -1 "Times" 1 14 0 0 1 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 }} {SECT 0 {EXCHG {PARA 256 "" 0 "" {TEXT 256 116 "IRO-2008-Groebner-Basi s-Assignment.mws - Buchberger's algorithm for computing Groebner basis .\nT. Pajdla, 24 Nov 2008" }}}{SECT 0 {PARA 3 "" 0 "" {TEXT 257 20 " P ackages & settings" }{TEXT -1 0 "" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 41 "restart;\nwith(ListTools):\nwith(Groebner):" }}}} {SECT 0 {PARA 3 "" 0 "" {TEXT 258 15 "Useful commands" }{TEXT -1 0 "" }}{EXCHG {PARA 0 "" 0 "" {TEXT -1 5 "While" }}}{EXCHG {PARA 0 "> " 0 " " {MPLTEXT 1 0 32 "while (...) and (...) do\nend do;" }}}{EXCHG {PARA 0 "" 0 "" {TEXT -1 3 "If " }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 25 "if ... then ... end if; " }}}{EXCHG {PARA 0 "" 0 "" {TEXT -1 9 "C onstants" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 12 "true;\nfalse;" }{TEXT -1 0 "" }}}{EXCHG {PARA 0 "" 0 "" {TEXT -1 14 "Print to debug" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 11 "print(...);" }}}{EXCHG {PARA 0 "" 0 "" {TEXT -1 42 "The least common multiple of two monomial s" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 19 "lcm(x*y^3,x*y^2*z);" } }}{EXCHG {PARA 0 "" 0 "" {TEXT -1 24 "Measuring time intervals" }}} {EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 32 "time(): 3^1000000:; time() - st;" }{TEXT -1 0 "" }}}}{SECT 0 {PARA 3 "" 0 "" {TEXT -1 11 " Algorit hms" }}{EXCHG {PARA 0 "" 0 "" {TEXT -1 30 "Cartesian product of two li sts" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 78 "LxL:=proc(X,Y)\n \+ Flatten(map(x->(map(y->Flatten([x,y]),Y)),X),1);\nend proc:" }}} {EXCHG {PARA 0 "" 0 "" {TEXT -1 19 "Division algortithm" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 64 "PolyDiv := proc(f,F,mo)\n# here co mes your code\n[r,a]:\nend proc:" }}}{EXCHG {PARA 0 "" 0 "" {TEXT -1 34 "Implement S-Polynomial computation" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 72 "SPol:=proc(f1,f2,mo)\n # here comes your code \n \+ expand();\nend proc:" }}}{EXCHG {PARA 0 "" 0 "" {TEXT -1 36 "Verify \+ on the following polynomials:" }}{PARA 0 "" 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.F'%\"zGF'F'*(\"#=F'F* F'%\"yGF.F'**\"#vF'F*F'F5F'F2F'F'" }{TEXT -1 0 "" }}{PARA 0 "" 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(*$F)\"\" #F(F**(F1F(%\"xGF(F-F1F(**\"\"%F(F3F(F-F(F)F(F(*(\"#%)F(F3F(F)F1F*" } {TEXT -1 38 "\nmo := plex(x,y,z);\ns:=SPol(f1,f2,mo);" }}{PARA 2 "" 0 "" {XPPEDIT 18 0 "s := 88/21*y^2+94/21*x*y^2+2/7*y^2*x^2+6/7*y^4*x+25/ 7*x*z*y^3+22*x*z^2-67/2*x*z^2*y+36*x*z^3-2*x^2*z^2*y+42*x^2*z^3;" "6#> %\"sG,6*(\"#))\"\"\"\"#@!\"\"%\"yG\"\"#F(**\"#%*F(F)F*%\"xGF(F+F,F(**F ,F(\"\"(F*F+F,F/F,F(**\"\"'F(F1F*F+\"\"%F/F(F(*,\"#DF(F1F*F/F(%\"zGF(F +\"\"$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,F(F7F8F(" }{TEXT -1 0 "" }}} {EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 137 "f1 := 88+94*x+6*x^2+21*x^2* z+18*x*y^2+75*x*y*z;\nf2 := -44*z+67*y*z-72*z^2+2*x*y^2+4*x*y*z-84*x*z ^2;\nmo := plex(x,y,z);\ns:=SPol(f1,f2,mo);" }{TEXT -1 0 "" }}}{EXCHG {PARA 0 "" 0 "" {TEXT -1 22 "Buchberger's algorithm" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 56 "BuchGB := proc(F,mo)\n # here comes you r code\nend proc:" }}}}{SECT 0 {PARA 3 "" 0 "" {TEXT -1 0 "" }{TEXT 259 1 " " }{TEXT -1 21 "Groebner basis - Test" }}{EXCHG {PARA 0 "" 0 " " {TEXT -1 33 "Test the Buchberger's algorithm \n" }{XPPEDIT 18 0 "f1 \+ := x^2*y-1;" "6#>%#f1G,&*&%\"xG\"\"#%\"yG\"\"\"F*F*!\"\"" }}{PARA 0 " " 0 "" {XPPEDIT 18 0 "f2 := x*y-1;" "6#>%#f2G,&*&%\"xG\"\"\"%\"yGF(F(F (!\"\"" }}{PARA 0 "" 0 "" {TEXT -1 26 "BuchGB([f1,f2],plex(x,y));" }} {PARA 0 "" 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)F)F(F*,&F)F)F(F*" }{TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 54 "f1 := x^2*y-1;\nf2 := x*y-1;\nBuchGB([f1,f2],plex(x,y ));" }}}{EXCHG {PARA 0 "" 0 "" {TEXT -1 99 "Polynomials in the Groebne r basis repeat. Why? Can you speed up the algorithm? Explain how and w hy." }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 0 "" }}}}{SECT 0 {PARA 3 "" 0 "" {TEXT -1 24 " Groebner basis - Task 1" }}{EXCHG {PARA 0 "" 0 "" {TEXT -1 124 "Compute the Groebner basis of the following system \+ w.r.t. all lexicographical monomial orderings\nand measure time it tak es.\005" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 94 "g1 := x*y-1;\ng2 := x+z-1;\ng3 := x*z+1;\nst:= time(): BuchGB([g1,g2,g3],plex(...)); t ime() - st;" }{TEXT -1 0 "" }}}}{SECT 0 {PARA 3 "" 0 "" {TEXT -1 24 " \+ Groebner basis - Task 2" }}{PARA 0 "" 0 "" {TEXT -1 130 "Compute the G roebner basis of the following system w.r.t. the lexicographical order ing generated\nby your personal IRO number IRON." }}{EXCHG {PARA 0 "> \+ " 0 "" {MPLTEXT 1 0 182 "IRON:= # your IRO personal number;\nmo:=combi nat[permute]([x,y,z,w,u]):\nmo:=plex(op(mo[IRON]));\nf1:=x*y+1;\nf2:=y *z-2;\nf3:=z*w+3;\nf4:=w*u-4;\nf5:=u*x+5;\nGB:=BuchGB([f1,f2,f3,f4,f5] ,mo);" }}}{EXCHG {PARA 0 "" 0 "" {TEXT -1 75 "Show the GB is a Groebne r basis generated by polynomial f1, f2, f3, f4, f5." }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 0 "" }}}}}{MARK "0 0 0" 8 }{VIEWOPTS 1 1 0 3 4 1802 1 1 1 1 }{PAGENUMBERS 0 1 2 33 1 1 }