{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 "" 18 256 "Helvetica" 0 1 0 0 0 0 0 1 0 0 0 0 0 0 0 0 } {PSTYLE "Normal" -1 0 1 {CSTYLE "" -1 -1 "Times" 1 12 0 0 0 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 "Warning" -1 7 1 {CSTYLE "" -1 -1 "Courier" 1 10 0 0 255 1 2 2 2 2 2 1 1 1 3 1 }1 1 0 0 0 0 1 0 1 0 2 2 0 1 }{PSTYLE "Maple Output" -1 11 1 {CSTYLE "" -1 -1 "Times" 1 12 0 0 0 1 2 2 2 2 2 2 1 1 1 1 }3 3 0 0 0 0 1 0 1 0 2 2 0 1 }{PSTYLE "Maple Output" -1 12 1 {CSTYLE "" -1 -1 "Times" 1 12 0 0 0 1 2 2 2 2 2 2 1 1 1 1 }1 3 0 0 0 0 1 0 1 0 2 2 0 1 }} {SECT 0 {EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 53 "interface(imaginaryu nit=_i):\nwith(PolynomialIdeals):\n" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 85 "h1 := x2-u3; \nh2 := (x1-u1)*u3-u2*u3;\nh3 := x1*x4-x 2*x3;\nh4 := u3*(u1-x3)-x4*(u1-u2);" }}{PARA 11 "" 1 "" {XPPMATH 20 "6 #>%#h1G,&%#x2G\"\"\"%#u3G!\"\"" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#>%#h 2G,&*&,&%#x1G\"\"\"%#u1G!\"\"F)%#u3GF)F)*&%#u2GF)F,F)F+" }}{PARA 11 " " 1 "" {XPPMATH 20 "6#>%#h3G,&*&%#x1G\"\"\"%#x4GF(F(*&%#x2GF(%#x3GF(! \"\"" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#>%#h4G,&*&%#u3G\"\"\",&%#u1GF( %#x3G!\"\"F(F(*&%#x4GF(,&F*F(%#u2GF,F(F," }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 86 "g1 := x3^2+x4^2 - ((x1-x3)^2+(x2-x4)^2);\ng2 := (u1-x 3)^2+x4^2 - ((x3-u2)^2+(x4-u3)^2);" }}{PARA 11 "" 1 "" {XPPMATH 20 "6# >%#g1G,**$)%#x3G\"\"#\"\"\"F**$)%#x4GF)F*F**$),&%#x1GF*F(!\"\"F)F*F2*$ ),&%#x2GF*F-F2F)F*F2" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#>%#g2G,**$),&% #u1G\"\"\"%#x3G!\"\"\"\"#F*F**$)%#x4GF-F*F**$),&F+F*%#u2GF,F-F*F,*$),& F0F*%#u3GF,F-F*F," }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 19 "I := < h1,h2,h3,h4>;" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#>%\"IG-%$<,>G6&,**&%# u3G\"\"\"%#u1GF+F+*&F*F+%#x3GF+!\"\"*&%#x4GF+F,F+F/*&F1F+%#u2GF+F+,&%# x2GF+F*F/,(*&F*F+%#x1GF+F+F)F/*&F3F+F*F+F/,&*&F8F+F1F+F+*&F5F+F.F+F/" }}}{EXCHG {PARA 0 "" 0 "" {TEXT -1 75 "Note, by default, all uknowns a re treated as variables, not parameters, so " }{XPPEDIT 18 0 "`subset` (I,Q);" "6#-%'subsetG6$%\"IG%\"QG" }{TEXT -1 1 "[" }{XPPEDIT 18 0 "x1, x2,x3,x4,u1,u2,u3;" "6)%#x1G%#x2G%#x3G%#x4G%#u1G%#u2G%#u3G" }{TEXT -1 2 "]." }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 24 "IdealInfo[Variable s](I);" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#<)%#x2G%#u3G%#x1G%#u1G%#u2G% #x4G%#x3G" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 25 "IdealInfo[Para meters](I);" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#<\"" }}}{EXCHG {PARA 0 "" 0 "" {TEXT -1 39 "The test if g1 and g2 are in I is false" }}} {EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 22 "IdealMembership(g1,I);" }} {PARA 11 "" 1 "" {XPPMATH 20 "6#%&falseG" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 22 "IdealMembership(g2,I);" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#%&falseG" }}}{EXCHG {PARA 0 "" 0 "" {TEXT -1 24 "The test if the y are in " }{XPPEDIT 18 0 "sqrt(I);" "6#-%%sqrtG6#%\"IG" }{TEXT -1 12 " also fails" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 26 "R := ;" }}{PARA 12 "" 1 "" {XPPMATH 20 "6#>%\"RG-%$<,>G6',&* &%#u3G\"\"\",&%#u1GF+%#x3G!\"\"F+F+*&%#x4GF+,&F-F+%#u2GF/F+F/,&*&%#x1G F+F1F+F+*&%#x2GF+F.F+F/,&F8F+F*F/,&*&,&F6F+F-F/F+F*F+F+*&F3F+F*F+F/,&F +F+*&,**$)F.\"\"#F+F+*$)F1FCF+F+*$),&F6F+F.F/FCF+F/*$),&F8F+F1F/FCF+F/ F+%\"yGF+F/" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 21 "IdealMembers hip(1,R);" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#%&falseG" }}}{EXCHG {PARA 0 "" 0 "" {TEXT -1 39 "The problem is that u1 and u3 can be 0." }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 51 "G := Groebner[Basis](I,td eg(x1,x2,x3,x4,u1,u2,u3));" }}{PARA 12 "" 1 "" {XPPMATH 20 "6#>%\"GG7( ,&%#x2G\"\"\"%#u3G!\"\",(*&F)F(%#x1GF(F(*&F)F(%#u1GF(F**&%#u2GF(F)F(F* ,**&%#x4GF(F/F(F(*&F4F(F1F(F**&F)F(%#x3GF(F(F.F*,&*&F-F(F4F(F(F6F*,(** \"\"#F(F4F(F1F(F)F(F(*(FF(F(,(**F " 0 "" {MPLTEXT 1 0 10 "factor(G);" }}{PARA 12 "" 1 "" {XPPMATH 20 "6#7(,&%#x 2G\"\"\"%#u3G!\"\"*&F'F&,(%#x1GF&%#u1GF(%#u2GF(F&,**&%#x4GF&F,F&F&*&F0 F&F-F&F(*&F'F&%#x3GF&F&*&F'F&F,F&F(,&*&F+F&F0F&F&F2F(*&F'F&,(*(\"\"#F& F0F&F-F&F&*(F:F&F'F&F3F&F(F4F&F&,$*(F'F&F,F&,(*&F:F&F3F&F(F,F&F-F&F&F( " }}}{EXCHG {PARA 0 "" 0 "" {TEXT -1 127 "Note, the test for whether I is radical does not terminate. I don't know why. I could not comput e the prime decomposition of " }{XPPEDIT 18 0 "sqrt(I);" "6#-%%sqrtG6# %\"IG" }{TEXT -1 8 " either." }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 11 "Radical(I);" }}{PARA 7 "" 1 "" {TEXT -1 33 "Warning, computation interrupted\n" }}}{EXCHG {PARA 0 "" 0 "" {TEXT -1 19 "Let's specify t hat " }{XPPEDIT 18 0 "u1 <> 0;" "6#0%#u1G\"\"!" }{TEXT -1 5 " and " } {XPPEDIT 18 0 "u3 <> 0;" "6#0%#u3G\"\"!" }{TEXT -1 2 " ." }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 19 "J := ;" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#>%\"JG-%$<,>G6',&*&%#u3G\"\"\",&%#u1GF+%#x3G!\"\"F +F+*&%#x4GF+,&F-F+%#u2GF/F+F/,&*&%#x1GF+F1F+F+*&%#x2GF+F.F+F/,&F8F+F*F /,&*&,&F6F+F-F/F+F*F+F+*&F3F+F*F+F/,&F+F+*(F-F+F*F+%\"tGF+F/" }}} {EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 24 "IdealInfo[Variables](J);" }} {PARA 11 "" 1 "" {XPPMATH 20 "6#<)%#x2G%#u3G%#x1G%#u1G%#u2G%#x4G%#x3G " }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 25 "IdealInfo[Parameters](J );" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#<#%\"tG" }}}{EXCHG {PARA 0 "" 0 "" {TEXT -1 73 "Notice that t is not a variable, it is a parameter -- \+ annoying. Okay, so " }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 29 "J := ;" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#>%\"JG-%$ <,>G6',&*&%#u3G\"\"\",&%#u1GF+%#x3G!\"\"F+F+*&%#x4GF+,&F-F+%#u2GF/F+F/ ,&*&%#x1GF+F1F+F+*&%#x2GF+F.F+F/,&F8F+F*F/,&*&,&F6F+F-F/F+F*F+F+*&F3F+ F*F+F/,&F+F+*(F-F+F*F+%\"tGF+F/" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 24 "IdealInfo[Variables](J);" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#<* %#x2G%#u3G%#x1G%#u1G%#u2G%#x4G%#x3G%\"tG" }}}{EXCHG {PARA 0 "" 0 "" {TEXT -1 46 "Okay, so we want to eliminate t. We compute " } {XPPEDIT 18 0 "`intersect`(I,Q);" "6#-%*intersectG6$%\"IG%\"QG" } {TEXT -1 1 "[" }{XPPEDIT 18 0 "x1,x2,x3,x4,u1,u2,u3;" "6)%#x1G%#x2G%#x 3G%#x4G%#u1G%#u2G%#u3G" }{TEXT -1 2 "]." }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 48 "J := EliminationIdeal(J,\{x1,x2,x3,x4,u1,u2,u3\});" } }{PARA 11 "" 1 "" {XPPMATH 20 "6#>%\"JG-%$<,>G6&,&%#x2G\"\"\"%#u3G!\" \",(%#u2GF*%#u1GF*%#x1GF,,&*&\"\"#F*%#x4GF*F*F+F,,&*&F3F*%#x3GF*F*F0F, " }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 13 "IsRadical(J);" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#%%trueG" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 22 "PrimeDecomposition(J);" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#-%$<,>G6&,&%#x2G\"\"\"%#u3G!\"\",(%#u2GF(%#u1GF(%#x1GF*,&*&\"\"# F(%#x4GF(F(F)F*,&*&F1F(%#x3GF(F(F.F*" }}}{EXCHG {PARA 0 "" 0 "" {TEXT -1 75 "Well, the ideal is now radical and prime (and linear) so it sho uld be easy " }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 22 "IdealMember ship(g1,J);" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#%%trueG" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 22 "IdealMembership(g2,J);" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#%%trueG" }}}{EXCHG {PARA 0 "" 0 "" {TEXT -1 20 "The correct test in " }{XPPEDIT 256 0 "Q(u1,u2,u3);" "6#-%\"QG6%%#u1G %#u2G%#u3G" }{TEXT -1 1 "[" }{XPPEDIT 18 0 "x1,x2,x3,x4;" "6&%#x1G%#x2 G%#x3G%#x4G" }{TEXT -1 30 "] where we don't need to say " }{XPPEDIT 18 0 "u1 <> 0;" "6#0%#u1G\"\"!" }{TEXT -1 5 " and " }{XPPEDIT 18 0 "u3 <> 0;" "6#0%#u3G\"\"!" }{TEXT -1 2 " ." }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 45 "K := ;" }} {PARA 11 "" 1 "" {XPPMATH 20 "6#>%\"KG-%$<,>G6&,&*&%#u3G\"\"\",&%#u1GF +%#x3G!\"\"F+F+*&%#x4GF+,&F-F+%#u2GF/F+F/,&*&%#x1GF+F1F+F+*&%#x2GF+F.F +F/,&F8F+F*F/,&*&,&F6F+F-F/F+F*F+F+*&F3F+F*F+F/" }}}{EXCHG {PARA 0 "> \+ " 0 "" {MPLTEXT 1 0 24 "IdealInfo[Variables](K);" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#<&%#x2G%#x1G%#x4G%#x3G" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 25 "IdealInfo[Parameters](K);" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#<%%#u3G%#u1G%#u2G" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 13 "IsRadical(K);" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#%%tr ueG" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 22 "IdealMembership(g1,K );" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#%%trueG" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 22 "IdealMembership(g2,K);" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#%%trueG" }}}{EXCHG {PARA 0 "" 0 "" {TEXT -1 15 "The tes t for " }{XPPEDIT 18 0 "`in`(g1,K);" "6#-%#inG6$%#g1G%\"KG" }{TEXT -1 8 " is if " }{XPPEDIT 18 0 "`in`(1,sqrt(I+`<,>`(1-g1*y)));" "6#-%# inG6$\"\"\"-%%sqrtG6#,&%\"IGF&-%$<,>G6#,&F&F&*&%#g1GF&%\"yGF&!\"\"F&" }{TEXT -1 1 " " }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 54 "K := ;" }}{PARA 12 "" 1 "" {XPPMATH 20 "6#>%\"KG-%$<,>G6',&*&%#u3G\"\"\",&%#u1GF+%#x3G!\"\"F+F+*& %#x4GF+,&F-F+%#u2GF/F+F/,&*&%#x1GF+F1F+F+*&%#x2GF+F.F+F/,&F8F+F*F/,&*& ,&F6F+F-F/F+F*F+F+*&F3F+F*F+F/,&F+F+*&,**$)F.\"\"#F+F+*$)F1FCF+F+*$),& F6F+F.F/FCF+F/*$),&F8F+F1F/FCF+F/F+%\"yGF+F/" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 21 "IdealMembership(1,K);" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#%%trueG" }}}{EXCHG {PARA 0 "" 0 "" {TEXT -1 9 "Well, if " }{XPPEDIT 18 0 "`in`(1,K);" "6#-%#inG6$\"\"\"%\"KG" }{TEXT -1 15 " \+ then how come " }{XPPEDIT 18 0 "K <> `<1>`;" "6#0%\"KG%$<1>G" }{TEXT -1 105 " . Because \"forming\" and ideal with < .... > does not cause a Groebner basis computation. But this does" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 12 "Simplify(K);" }}{PARA 11 "" 1 "" {XPPMATH 20 "6# -%$<,>G6#\"\"\"" }}}{EXCHG }{EXCHG }{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 0 "" }}}}{MARK "0 0 0" 53 }{VIEWOPTS 1 1 0 1 1 1803 1 1 1 1 } {PAGENUMBERS 0 1 2 33 1 1 }