{VERSION 6 0 "SGI MIPS UNIX" "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 "2D Output" 2 20 "" 0 1 0 0 255 1 0 0 0 0 0 0 0 0 0 1 } {PSTYLE "Normal" -1 0 1 {CSTYLE "" -1 -1 "" 0 1 0 0 0 0 0 0 0 0 0 0 0 0 0 0 }0 0 0 -1 -1 -1 0 0 0 0 0 0 -1 0 }{PSTYLE "Heading 1" 0 3 1 {CSTYLE "" -1 -1 "" 1 18 0 0 0 0 0 1 0 0 0 0 0 0 0 0 }1 0 0 0 8 4 0 0 0 0 0 0 -1 0 }{PSTYLE "Maple Output" 0 11 1 {CSTYLE "" -1 -1 "" 0 1 0 0 0 0 0 0 0 0 0 0 0 0 0 0 }3 3 0 -1 -1 -1 0 0 0 0 0 0 -1 0 }{PSTYLE " " 11 12 1 {CSTYLE "" -1 -1 "" 0 1 0 0 0 0 0 0 0 0 0 0 0 0 0 0 }1 0 0 -1 -1 -1 0 0 0 0 0 0 -1 0 }} {SECT 0 {EXCHG {PARA 3 "" 0 "" {TEXT -1 28 "Ideal intersection example s." }}}{EXCHG {PARA 0 "" 0 "" {TEXT -1 42 "The example in the text on \+ page 186 where " }{XPPEDIT 18 0 "I = `<,>`(x^2*y);" "6#/%\"IG-%$<,>G6# *&%\"xG\"\"#%\"yG\"\"\"" }{TEXT -1 5 " and " }{XPPEDIT 18 0 "J = `<,>` (x*y^2);" "6#/%\"JG-%$<,>G6#*&%\"xG\"\"\"*$%\"yG\"\"#F*" }{TEXT -1 8 " then " }{XPPEDIT 18 0 "`intersect`(I,J);" "6#-%*intersectG6$%\"IG% \"JG" }{TEXT -1 3 " = " }{XPPEDIT 18 0 "`<,>`(x^2*y^2);" "6#-%$<,>G6#* &%\"xG\"\"#%\"yGF(" }{TEXT -1 2 " ." }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 23 "f := x^2*y;\ng := x*y^2;" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#>%\"fG*&)%\"xG\"\"#\"\"\"%\"yGF)" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#>%\"gG*&%\"xG\"\"\")%\"yG\"\"#F'" }}}{EXCHG {PARA 0 "" 0 "" {TEXT -1 30 "The algorithm is to eliminate " }{XPPEDIT 18 0 "t;" "6#%\"tG" }{TEXT -1 17 " from the ideal " }{XPPEDIT 18 0 "`<,>`(f*t,( 1-t)*g);" "6#-%$<,>G6$*&%\"fG\"\"\"%\"tGF(*&,&F(F(F)!\"\"F(%\"gGF(" } {TEXT -1 3 " . " }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 54 "G := Gro ebner[Basis]( [f*t, (1-t)*g], plex( t,x,y ) );" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#>%\"GG7%*&)%\"xG\"\"#\"\"\")%\"yGF)F*,&*&F(F*F+F*!\"\"* (F(F*F+F*%\"tGF*F**(F'F*F,F*F1F*" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 16 "remove(has,G,t);" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#7 #*&)%\"xG\"\"#\"\"\")%\"yGF'F(" }}}{EXCHG {PARA 0 "" 0 "" {TEXT -1 27 "My second example consider " }{XPPEDIT 18 0 "I = `<,>`(x,y);" "6#/%\" IG-%$<,>G6$%\"xG%\"yG" }{TEXT -1 5 " and " }{XPPEDIT 18 0 "J = `<,>`(x ,z);" "6#/%\"JG-%$<,>G6$%\"xG%\"zG" }{TEXT -1 18 ". Again, clearly " }{XPPEDIT 18 0 "`intersect`(I,J);" "6#-%*intersectG6$%\"IG%\"JG" } {TEXT -1 3 " = " }{XPPEDIT 18 0 "`<,>`(x,x*z, y*x, y*z);" "6#-%$<,>G6& %\"xG*&F&\"\"\"%\"zGF(*&%\"yGF(F&F(*&F+F(F)F(" }{TEXT -1 3 " = " } {XPPEDIT 18 0 "`<,>`(x,y*z);" "6#-%$<,>G6$%\"xG*&%\"yG\"\"\"%\"zGF)" } {TEXT -1 0 "" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 85 "G := Groebn er[Basis]( [x*t, y*t, (1-t)*x, (1-t)*z], plex(t,x,y,z) );\nremove(has, G,t);" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#>%\"GG7&*&%\"yG\"\"\"%\"zGF(% \"xG,&F)!\"\"*&F)F(%\"tGF(F(*&F'F(F.F(" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#7$*&%\"yG\"\"\"%\"zGF&%\"xG" }}}{EXCHG {PARA 0 "" 0 "" {TEXT -1 96 "You can also use the PolynomialIdeals package in Maple 10 which do es this for you automatically." }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 23 "with(PolynomialIdeals):" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 31 "J := Intersect( , );" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#>%\"JG-%$<,>G6$%\"xG*&%\"yG\"\"\"%\"zGF+" }}}{EXCHG {PARA 0 "" 0 "" {TEXT -1 169 "The second example in the text on page 186 is also \+ simple because each ideal is a principal ideal except that the Groebne r basis is quite big so I have not printed it. " }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 61 "f := (x+y)^4*(x^2+y)^2*(x-5*y);\ng := (x+y)*( x^2+y)^3*(x+3*y);" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#>%\"fG*(),&%\"xG \"\"\"%\"yGF)\"\"%F)),&*$)F(\"\"#F)F)F*F)F0F),&F(F)*&\"\"&F)F*F)!\"\"F )" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#>%\"gG*(,&%\"xG\"\"\"%\"yGF(F(),& *$)F'\"\"#F(F(F)F(\"\"$F(,&F'F(*&F/F(F)F(F(F(" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 72 "G := Groebner[Basis]([f*t,(1-t)*g], plex(t,x,y) \+ ):\nE := remove(has,G,t);" }}{PARA 12 "" 1 "" {XPPMATH 20 "6#>%\"EG7#, Z*&)%\"xG\"\"'\"\"\")%\"yG\"\"$F+F+*(\"\"#F+)F)\"\"&F+)F-\"\"%F+F+*(\" # " 0 "" {MPLTEXT 1 0 18 "h := factor(E[1]);" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#>%\"hG**,&%\"xG\"\"\"*&\"\"$F(%\"yGF (F(F(,&F'F(*&\"\"&F(F+F(!\"\"F(),&*$)F'\"\"#F(F(F+F(F*F(),&F'F(F+F(\" \"%F(" }}}{EXCHG {PARA 0 "" 0 "" {TEXT -1 16 "This is the LCM(" } {XPPEDIT 18 0 "f,g;" "6$%\"fG%\"gG" }{TEXT -1 10 " ) so GCD(" } {XPPEDIT 18 0 "f,g;" "6$%\"fG%\"gG" }{TEXT -1 4 ") is" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 18 "simplify( f*g/h );" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#*&,&%\"xG\"\"\"%\"yGF&F&),&*$)F%\"\"#F&F&F'F&F,F& " }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 9 "gcd(f,g);" }}{PARA 11 " " 1 "" {XPPMATH 20 "6#*&,&%\"xG\"\"\"%\"yGF&F&),&*$)F%\"\"#F&F&F'F&F,F &" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 0 "" }}}}{MARK "18 0 0" 0 }{VIEWOPTS 1 1 0 1 1 1803 1 1 1 1 }{PAGENUMBERS 0 1 2 33 1 1 }