{VERSION 6 0 "IBM INTEL LINUX" "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 } {CSTYLE "" -1 256 "" 0 1 0 0 0 0 1 0 0 0 0 0 0 0 0 0 }{CSTYLE "" -1 257 "" 0 1 0 0 0 0 1 0 0 0 0 0 0 0 0 0 }{CSTYLE "" -1 258 "" 0 1 0 0 0 0 1 0 0 0 0 0 0 0 0 0 }{CSTYLE "" -1 259 "" 0 1 0 0 0 0 1 0 0 0 0 0 0 0 0 0 }{CSTYLE "" -1 260 "" 0 1 0 0 0 0 1 0 0 0 0 0 0 0 0 0 } {CSTYLE "" -1 261 "" 0 1 0 0 0 0 1 0 0 0 0 0 0 0 0 0 }{PSTYLE "Normal " -1 0 1 {CSTYLE "" -1 -1 "" 0 1 0 0 0 0 0 0 0 0 0 0 0 0 0 1 }0 0 0 -1 -1 -1 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 1 }3 3 0 -1 -1 -1 0 0 0 0 0 0 -1 0 }} {SECT 0 {EXCHG {PARA 0 "" 0 "" {TEXT -1 39 "Examples of computing Idea l Quotients " }{TEXT 261 5 "I : J" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 57 "interface(imaginaryunit=_i): # so we can use I for id eals" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 23 "with(PolynomialIdea ls):" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 33 "I, J, K := , , ;" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#>6%%\"IG%\"JG%\"KG6% -%$<,>G6$*&%\"xG\"\"\"%\"zGF.*&%\"yGF.F/F.-F*6$F-F1-F*6#F/" }}}{EXCHG {PARA 0 "" 0 "" {TEXT -1 28 "Check Proposition 9 (i): " }{TEXT 259 1 "I" }{TEXT -1 3 " : " }{XPPEDIT 18 0 "`<,>`(1);" "6#-%$<,>G6#\"\"\" " }{TEXT -1 3 " = " }{TEXT 260 1 "I" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 16 "Quotient(I,<1>);" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#- %$<,>G6$*&%\"xG\"\"\"%\"zGF(*&%\"yGF(F)F(" }}}{EXCHG {PARA 0 "" 0 "" {TEXT -1 30 "Check Proposition 9 (iii): If " }{XPPEDIT 18 0 "`subset`( J,I);" "6#-%'subsetG6$%\"JG%\"IG" }{TEXT -1 6 " then " }{TEXT 256 3 "I :J" }{TEXT -1 3 " = " }{XPPEDIT 18 0 "`<,>`(1);" "6#-%$<,>G6#\"\"\"" } {TEXT -1 0 "" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 18 "Quotient(I, );" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#-%$<,>G6#\"\"\"" }}}{EXCHG {PARA 0 "" 0 "" {TEXT -1 24 "To compute I : J where " }{XPPEDIT 18 0 "J = `<,>`(x)+`<,>`(y);" "6#/%\"JG,&-%$<,>G6#%\"xG\"\"\"-F'6#%\"yGF*" }{TEXT -1 39 " use Proposition 10 (3): I : J = I :" }{XPPEDIT 18 0 "`<,>`(x);" "6#-%$<,>G6#%\"xG" }{XPPEDIT 18 0 "`intersect`(``,``);" "6 #-%*intersectG6$%!GF&" }{TEXT -1 5 " I : " }{XPPEDIT 18 0 "`<,>`(y);" "6#-%$<,>G6#%\"yG" }{TEXT -1 36 " and compute these using Theorem 11. " }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 33 "Quotient(I,), Quotie nt(I,);" }}{PARA 11 "" 1 "" {XPPMATH 20 "6$-%$<,>G6#%\"zGF#" }}} {EXCHG {PARA 0 "" 0 "" {TEXT -1 9 "Thus I : " }{XPPEDIT 18 0 "`<,>`(x) ;" "6#-%$<,>G6#%\"xG" }{TEXT -1 3 " = " }{XPPEDIT 18 0 "`<,>`(z);" "6# -%$<,>G6#%\"zG" }{TEXT -1 6 " and " }{XPPEDIT 18 0 "I;" "6#%\"IG" } {TEXT -1 3 " : " }{XPPEDIT 18 0 "`<,>`(y);" "6#-%$<,>G6#%\"yG" }{TEXT -1 3 " = " }{XPPEDIT 18 0 "`<,>`(z);" "6#-%$<,>G6#%\"zG" }{TEXT -1 36 " and their intersection is clearly " }{XPPEDIT 18 0 "`<,>`(z);" "6#- %$<,>G6#%\"zG" }{TEXT -1 22 " . Check using Maple" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 14 "Quotient(I,J);" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#-%$<,>G6#%\"zG" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 11 "J := [x,y];" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#>%\"JG7$%\"xG%\"y G" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 33 "K := [y^2-x*z,x^3-y*z, x^2*y-z^2];" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#>%\"KG7%,&*$)%\"yG\"\"# \"\"\"F+*&%\"xGF+%\"zGF+!\"\",&*$)F-\"\"$F+F+*&F)F+F.F+F/,&*&)F-F*F+F) F+F+*$)F.F*F+F/" }}}{EXCHG {PARA 0 "" 0 "" {TEXT -1 10 "Construct " } {TEXT 257 1 "I" }{TEXT -1 3 " = " }{XPPEDIT 18 0 "`intersect`(J,K);" " 6#-%*intersectG6$%\"JG%\"KG" }{TEXT -1 40 " using the algorithm from s ection 4.3: " }{XPPEDIT 18 0 "`intersect`(J,K);" "6#-%*intersectG6$% \"JG%\"KG" }{TEXT -1 3 " = " }{XPPEDIT 18 0 "t*J;" "6#*&%\"tG\"\"\"%\" JGF%" }{TEXT -1 3 " + " }{XPPEDIT 18 0 "(1-t)*K;" "6#*&,&\"\"\"F%%\"tG !\"\"F%%\"KGF%" }{TEXT -1 1 " " }{XPPEDIT 18 0 "`intersect`(``,``);" " 6#-%*intersectG6$%!GF&" }{TEXT -1 1 " " }{XPPEDIT 18 0 "k;" "6#%\"kG" }{TEXT -1 1 "[" }{XPPEDIT 18 0 "x,y,z;" "6%%\"xG%\"yG%\"zG" }{TEXT -1 1 "]" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 69 "I := Groebner[Basis ]([seq(t*f,f=J),seq((1-t)*g,g=K)], plex(t,x,y,z));" }}{PARA 11 "" 1 " " {XPPMATH 20 "6#>%\"IG7*,&*$)%\"yG\"\"'\"\"\"F+*&)%\"zG\"\"%F+F)F+!\" \",&*$)F)\"\"#F+F0*&%\"xGF+F.F+F+,&*&)F)F/F+F6F+F+*&)F.\"\"$F+F)F+F0,& *&)F6F4F+F3F+F+*&F)F+)F.F4F+F0,&*$)F6F " 0 " " {MPLTEXT 1 0 21 "I := remove(has,I,t);" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#>%\"IG7',&*$)%\"yG\"\"'\"\"\"F+*&)%\"zG\"\"%F+F)F+!\"\",&*$)F)\" \"#F+F0*&%\"xGF+F.F+F+,&*&)F)F/F+F6F+F+*&)F.\"\"$F+F)F+F0,&*&)F6F4F+F3 F+F+*&F)F+)F.F4F+F0,&*$)F6F " 0 "" {MPLTEXT 1 0 25 "I, J, K := , , ;" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#>6%%\"IG%\"JG%\"KG6%-%$<,>G6',&*$)%\"yG\"\"#\"\"\"!\"\" *&%\"xGF1%\"zGF1F1,&*$)F4\"\"$F1F1*&F/F1F5F1F2,&*&)F4F0F1F.F1F1*&F/F1) F5F0F1F2,&*&)F/\"\"%F1F4F1F1*&)F5F9F1F/F1F2,&*$)F/\"\"'F1F1*&)F5FCF1F/ F1F2-F*6$F/F4-F*6%,&F-F1F3F2F6,&*&F=F1F/F1F1*$F?F1F2" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 20 "I := Intersect(J,K);" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#>%\"IG-%$<,>G6$,&*$)%\"yG\"\"#\"\"\"F-*&%\"xGF-%\"zGF-! \"\",&*$)F/\"\"$F-F-*&F+F-F0F-F1" }}}{EXCHG {PARA 0 "" 0 "" {TEXT -1 98 "The basis we computed is different from the one Intersect( ... ) c omputed. The difference is just" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 34 "Groebner[Basis]( I, tdeg(x,y,z) );" }}{PARA 11 "" 1 " " {XPPMATH 20 "6#7$,&*$)%\"yG\"\"#\"\"\"F)*&%\"xGF)%\"zGF)!\"\",&*$)F+ \"\"$F)F)*&F'F)F,F)F-" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 17 "Qu otient( I, K );" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#-%$<,>G6$%\"yG%\"xG " }}}{EXCHG {PARA 0 "" 0 "" {TEXT -1 67 "To compute I : K using Propos ition 9 (3) and Theorem 11, this time " }{XPPEDIT 18 0 "K = `<,>`(g1)+ `<,>`(g2)+`<,>`(g3);" "6#/%\"KG,(-%$<,>G6#%#g1G\"\"\"-F'6#%#g2GF*-F'6# %#g3GF*" }{TEXT -1 2 " ." }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 19 "g := Generators(K);" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#>%\"gG<%,&*$)% \"yG\"\"#\"\"\"F+*&%\"xGF+%\"zGF+!\"\",&*$)F-\"\"$F+F+*&F)F+F.F+F/,&*& )F-F*F+F)F+F+*$)F.F*F+F/" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 49 "T1, T2, T3 := seq( Intersect(I,), i=1..3 );" }}{PARA 11 "" 1 " " {XPPMATH 20 "6#>6%%#T1G%#T2G%#T3G6%-%$<,>G6#,&*$)%\"yG\"\"#\"\"\"F1* &%\"xGF1%\"zGF1!\"\"-F*6#,&*$)F3\"\"$F1F1*&F/F1F4F1F5-F*6$,&*&F:F1F/F1 F1*&F3F1)F4F0F1F5,&*&)F3F0F1F.F1F1*&F/F1FBF1F5" }}}{EXCHG {PARA 0 "" 0 "" {TEXT -1 39 "From this we see that a basis for I : <" }{XPPEDIT 18 0 "g[1];" "6#&%\"gG6#\"\"\"" }{TEXT -1 29 "> is <1> and similarly \+ I : <" }{XPPEDIT 18 0 "g[2];" "6#&%\"gG6#\"\"#" }{TEXT -1 5 "> = " } {XPPEDIT 18 0 "`<,>`(1);" "6#-%$<,>G6#\"\"\"" }{TEXT -1 3 ". " }}} {EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 43 "< seq( normal(f/g[3]), f=Gen erators(T3) )>;" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#-%$<,>G6$%\"yG%\"xG " }}}{EXCHG {PARA 0 "" 0 "" {TEXT -1 5 "Thus " }{TEXT 258 5 "I : K" } {TEXT -1 3 " = " }{XPPEDIT 18 0 "`intersect`(`<,>`(1),`<,>`(x,y));" "6 #-%*intersectG6$-%$<,>G6#\"\"\"-F'6$%\"xG%\"yG" }{TEXT -1 4 " = " } {XPPEDIT 18 0 "`<,>`(x,y);" "6#-%$<,>G6$%\"xG%\"yG" }{TEXT -1 2 " ." } }}}{MARK "8 0 2" 21 }{VIEWOPTS 1 1 0 1 1 1803 1 1 1 1 }{PAGENUMBERS 0 1 2 33 1 1 }