{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 59 "Examples of factoring radi cal ideals into prime components." }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 9 "restart; " }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 52 "interface(imaginaryunit=_i):\nwith(PolynomialIdeals):" }}}{EXCHG {PARA 0 "" 0 "" {TEXT -1 49 "This is the example from section 3.1 on p age 112." }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 37 "I := ;" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#>%\"IG-%$<,> G6%,*%\"xG\"\"\"*$)%\"yG\"\"#F*F*%\"zGF*F*!\"\",*F)F*F-F**$)F/F.F*F*F* F0,**$)F)F.F*F*F-F*F/F*F*F0" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 47 "IsRadical(I), IsPrime(I), IsZeroDimensional(I);" }}{PARA 11 "" 1 " " {XPPMATH 20 "6%%&falseGF#%%trueG" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 36 "G := Groebner[Basis](I,plex(x,y,z));" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#>%\"GG7&,**$)%\"zG\"\"'\"\"\"F+*&\"\"%F+)F)F-F+!\" \"*$)F)\"\"#F+F/*&F-F+)F)\"\"$F+F+,(*(F2F+F1F+%\"yGF+F+*$F.F+F+F0F/,** $)F8F2F+F+F0F/F8F/F)F+,*%\"xGF+F8F+F0F+F+F/" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 18 "f := factor(G[1]);" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#>%\"fG*()%\"zG\"\"#\"\"\",(*$F&F)F)*&F(F)F'F)F)F)!\"\"F)),&F'F)F )F-F(F)" }}}{EXCHG {PARA 0 "" 0 "" {TEXT -1 50 "One way to try to comp ute the radical is to throw " }{XPPEDIT 18 0 "f[red];" "6#&%\"fG6#%$re dG" }{TEXT -1 48 " the square-free part of this polynomial into I" }} }{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 29 "I := ;" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#>%\"IG-%$<,>G6&*(%\"zG\"\"\",(*$ )F)\"\"#F*F**&F.F*F)F*F*F*!\"\"F*,&F)F*F*F0F*,*%\"xGF**$)%\"yGF.F*F*F) F*F*F0,*F3F*F6F*F,F*F*F0,**$)F3F.F*F*F6F*F)F*F*F0" }}}{EXCHG {PARA 0 " > " 0 "" {MPLTEXT 1 0 47 "IsRadical(I), IsPrime(I), IsZeroDimensional( I);" }}{PARA 11 "" 1 "" {XPPMATH 20 "6%%%trueG%&falseGF#" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 36 "G := Groebner[Basis](I,plex(x,y,z)) ;" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#>%\"GG7&,**$)%\"zG\"\"%\"\"\"F+*$ )F)\"\"$F+F+*&F.F+)F)\"\"#F+!\"\"F)F+,(*(F1F+F)F+%\"yGF+F+F,F+F)F2,**$ )F5F1F+F+*$F0F+F2F5F2F)F+,*%\"xGF+F5F+F9F+F+F2" }}}{EXCHG {PARA 0 "> \+ " 0 "" {MPLTEXT 1 0 18 "f := factor(G[1]);" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#>%\"fG*(%\"zG\"\"\",(*$)F&\"\"#F'F'*&F+F'F&F'F'F'!\"\"F ',&F&F'F'F-F'" }}}{EXCHG {PARA 0 "" 0 "" {TEXT -1 50 "Now we are going to split I into three components " }{XPPEDIT 18 0 "P[1],P[2],P[3];" " 6%&%\"PG6#\"\"\"&F$6#\"\"#&F$6#\"\"$" }{TEXT -1 39 " corresponding to \+ the three factors of " }{XPPEDIT 18 0 "f;" "6#%\"fG" }{TEXT -1 24 " a nd then we will have " }{XPPEDIT 18 0 "I;" "6#%\"IG" }{TEXT -1 3 " = \+ " }{XPPEDIT 18 0 "`intersect`(P[1],P[2]);" "6#-%*intersectG6$&%\"PG6# \"\"\"&F'6#\"\"#" }{TEXT -1 1 " " }{XPPEDIT 18 0 "`intersect`(``,P[3]) ;" "6#-%*intersectG6$%!G&%\"PG6#\"\"$" }{TEXT -1 2 " ." }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 105 "P1 := Quotient(I,);\nP 2 := Quotient(I,);\nP3 := Quotient(I,);" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#>%#P1G-%$<,>G6%%\"zG*&,&%\"xG\" \"\"F,!\"\"F,F+F,,(%\"yGF,F+F,F,F-" }}{PARA 11 "" 1 "" {XPPMATH 20 "6# >%#P2G-%$<,>G6%%\"xG%\"yG,&%\"zG\"\"\"F,!\"\"" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#>%#P3G-%$<,>G6%,(*$)%\"zG\"\"#\"\"\"F-*&F,F-F+F-F-F-!\" \",&%\"xGF-F+F/,&F+F/%\"yGF-" }}}{EXCHG {PARA 0 "" 0 "" {TEXT -1 68 "W e could have done this by putting the three irreducible factors of " } {XPPEDIT 18 0 "f;" "6#%\"fG" }{TEXT -1 31 " into the basis for I this way" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 117 "Groebner[Basis](,plex(x,y,z));\nGroebner[Basis](,plex(x,y,z));\nGroebner[Bas is](,plex(x,y,z));" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#7%% \"zG,&*$)%\"yG\"\"#\"\"\"F*F(!\"\",(F(F*%\"xGF*F*F+" }}{PARA 11 "" 1 " " {XPPMATH 20 "6#7%,&%\"zG\"\"\"F&!\"\"%\"yG%\"xG" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#7%,(*$)%\"zG\"\"#\"\"\"F)*&F(F)F'F)F)F)!\"\",&F'F+%\"yG F),&%\"xGF)F'F+" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 38 "IsPrime( P1), IsPrime(P2), IsPrime(P3);" }}{PARA 11 "" 1 "" {XPPMATH 20 "6%%&fa lseG%%trueGF$" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 45 "G := facto r(Groebner[Basis](P1,plex(x,y,z)));" }}{PARA 11 "" 1 "" {XPPMATH 20 "6 #>%\"GG7%%\"zG*&%\"yG\"\"\",&F(F)F)!\"\"F),(F(F)%\"xGF)F)F+" }}} {EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 47 "P4, P5 := Quotient(P1,), \+ Quotient(P1,);" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#>6$%#P4G%#P5G6$ -%$<,>G6%%\"xG%\"zG,&%\"yG\"\"\"F/!\"\"-F)6%F.F,,&F+F/F/F0" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 25 "IsPrime(P4), IsPrime(P5);" }}{PARA 11 "" 1 "" {XPPMATH 20 "6$%%trueGF#" }}}{EXCHG {PARA 0 "" 0 "" {TEXT -1 40 "We are done: the prime decomposition of " }{XPPEDIT 18 0 "sqrt( I);" "6#-%%sqrtG6#%\"IG" }{TEXT -1 34 " is the following four compone nts" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 15 "P2, P3, P4, P5;" }} {PARA 11 "" 1 "" {XPPMATH 20 "6&-%$<,>G6%%\"xG%\"yG,&%\"zG\"\"\"F*!\" \"-F$6%,(*$)F)\"\"#F*F**&F1F*F)F*F*F*F+,&F&F*F)F+,&F)F+F'F*-F$6%F&F),& F'F*F*F+-F$6%F'F),&F&F*F*F+" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 28 "J := Intersect(P2,P3,P4,P5);" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#>% \"JG-%$<,>G6',*%\"xG\"\"\"*$)%\"yG\"\"#F*F*%\"zGF*F*!\"\",*F)F*F-F**$) F/F.F*F*F*F0,**$)F)F.F*F*F-F*F/F*F*F0,&*&F/F*F-F*F**&F)F*F/F*F0,&*&F)F *F-F*F*F9F0" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 31 "Groebner[Bas is](J,plex(x,y,z));" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#7&,**$)%\"zG\" \"%\"\"\"F)*$)F'\"\"$F)F)*&F,F))F'\"\"#F)!\"\"F'F),(*(F/F)F'F)%\"yGF)F )F*F)F'F0,**$)F3F/F)F)*$F.F)F0F3F0F'F),*%\"xGF)F3F)F7F)F)F0" }}} {EXCHG {PARA 0 "" 0 "" {TEXT -1 173 "The above example worked because \+ when we computed the Groebner basis for I we found a polynomial that f actored. This does not always happen. Consider the following example " }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 25 "I := ;" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#>%\"IG-%$<,>G6%,&*$)%\"zG\"\"#\" \"\"F-F-F-,&*$)%\"yGF,F-F-F-F-,&*$)%\"xGF,F-F-F-F-" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 36 "G := Groebner[Basis](I,plex(x,y,z));" }} {PARA 11 "" 1 "" {XPPMATH 20 "6#>%\"GG7%,&*$)%\"zG\"\"#\"\"\"F+F+F+,&* $)%\"yGF*F+F+F+F+,&*$)%\"xGF*F+F+F+F+" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 10 "factor(G);" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#7%,&*$) %\"zG\"\"#\"\"\"F)F)F),&*$)%\"yGF(F)F)F)F),&*$)%\"xGF(F)F)F)F)" }}} {EXCHG {PARA 0 "" 0 "" {TEXT -1 76 "Notice that G is a Groebner basis \+ for I in every monomial ordering since LT(" }{XPPEDIT 18 0 "x^2+1;" "6 #,&*$%\"xG\"\"#\"\"\"F'F'" }{TEXT -1 4 ") = " }{XPPEDIT 18 0 "x^2;" "6 #*$%\"xG\"\"#" }{TEXT -1 88 " in every monomial ordering. But it is \+ not prime over the field of rational numbers Q." }}}{EXCHG {PARA 0 "> \+ " 0 "" {MPLTEXT 1 0 25 "IsPrime(I), IsRadical(I);" }}{PARA 11 "" 1 "" {XPPMATH 20 "6$%&falseG%%trueG" }}}{EXCHG {PARA 0 "" 0 "" {TEXT -1 86 "We can solve this problem by inspection. The second generator minus \+ the first factor." }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 21 "f := ( x^2+1)-(y^2+1);" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#>%\"fG,&*$)%\"xG\" \"#\"\"\"F**$)%\"yGF)F*!\"\"" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 10 "factor(f);" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#*&,&%\"xG\"\"\"%\" yG!\"\"F&,&F%F&F'F&F&" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 47 "P1 , P2 := Quotient(I,), Quotient(I,);" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#>6$%#P1G%#P2G6$-%$<,>G6%,&*$)%\"zG\"\"#\"\"\"F0F0F0,&*$ )%\"yGF/F0F0F0F0,&%\"xGF0F4F0-F)6%,&F6!\"\"F4F0F+,&*$)F6F/F0F0F0F0" }} }{EXCHG {PARA 0 "" 0 "" {TEXT -1 57 "Again, the second generator minus the third factors into " }{XPPEDIT 18 0 "(y-z)*(y+z);" "6#*&,&%\"yG\" \"\"%\"zG!\"\"F&,&F%F&F'F&F&" }{TEXT -1 51 " so we repeat this decompo sition on each component." }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 123 "P11, P12, P21, P22 := Quotient(P1,), Quotient(P1,),\n \+ Quotient(P2,), Quotient(P2,);" }}{PARA 12 "" 1 "" {XPPMATH 20 "6#>6&%$P11G%$P12G%$P21G%$P22G6&-%$<,>G6%,&%\"x G!\"\"%\"yGF/,&F0F/%\"zGF/,&*$)F0\"\"#\"\"\"F/F7F/-F+6%,&F4F7F7F7,&F.F 7F0F7,&F2F7F0F/-F+6%,&F0F7F2F7,&F2F7F.F7,&*$)F2F6F7F7F7F7-F+6%,&F.F/F2 F7,&*$)F.F6F7F7F7F7,&F2F/F0F7" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 37 "map( IsPrime, [P11, P12, P21, P22] );" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#7&%%trueGF$F$F$" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 32 "J := Intersect(P11,P12,P21,P22);" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#>%\"JG-%$<,>G6%,&*$)%\"zG\"\"#\"\"\"F-F-F-,&*$)%\"yGF,F-F-F-F-,& *$)%\"xGF,F-F-F-F-" }}}{EXCHG {PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 99 "Now how would we \+ factor this ideal in general (if we cannot \"spot\" a polynomial in I \+ that factors)." }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 2 "I;" }} {PARA 11 "" 1 "" {XPPMATH 20 "6#-%$<,>G6%,&*$)%\"zG\"\"#\"\"\"F+F+F+,& *$)%\"yGF*F+F+F+F+,&*$)%\"xGF*F+F+F+F+" }}}{EXCHG {PARA 0 "" 0 "" {TEXT -1 60 "One way is to use field extensions - which can be expensi ve." }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 23 "alias(i=RootOf(z^2+1 )):" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 45 "G[1] = factor(G[1],i );\nG[2] = factor(G[2],i);" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#/,&*$)% \"zG\"\"#\"\"\"F)F)F)*&,&%\"iGF)F'F)F),&F'F)F,!\"\"F)" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#/,&*$)%\"yG\"\"#\"\"\"F)F)F)*&,&F'F)%\"iGF)F),&F'F )F,!\"\"F)" }}}{EXCHG {PARA 0 "" 0 "" {TEXT -1 26 "From here we can id entify " }{XPPEDIT 18 0 "i;" "6#%\"iG" }{TEXT -1 7 " with " } {XPPEDIT 18 0 "x;" "6#%\"xG" }{TEXT -1 35 " so we have found the poly nomials " }{XPPEDIT 18 0 "(x+z)*(-x+z);" "6#*&,&%\"xG\"\"\"%\"zGF&F&,& F%!\"\"F'F&F&" }{TEXT -1 6 " and " }{XPPEDIT 18 0 "(x+y)*(y-x);" "6#* &,&%\"xG\"\"\"%\"yGF&F&,&F'F&F%!\"\"F&" }{TEXT -1 29 " in I. We proce ed as before." }}}{EXCHG {PARA 0 "" 0 "" {TEXT -1 147 "Another approac h is to make a linear subsitution to try to \"put the ideal in general position\" so that when we compute the lex Groebner basis with " } {XPPEDIT 18 0 "x;" "6#%\"xG" }{TEXT -1 3 " > " }{XPPEDIT 18 0 "y;" "6# %\"yG" }{TEXT -1 3 " > " }{XPPEDIT 18 0 "z;" "6#%\"zG" }{TEXT -1 20 " \+ we get the form [ " }{XPPEDIT 18 0 "f(z);" "6#-%\"fG6#%\"zG" }{TEXT -1 3 " , " }{XPPEDIT 18 0 "x-g(z);" "6#,&%\"xG\"\"\"-%\"gG6#%\"zG!\"\" " }{TEXT -1 3 " , " }{XPPEDIT 18 0 "y-h(z);" "6#,&%\"yG\"\"\"-%\"hG6#% \"zG!\"\"" }{TEXT -1 84 " ] from where the problem is easily solved - \+ we just need to look at the factors of " }{XPPEDIT 18 0 "f(z);" "6#-% \"fG6#%\"zG" }{TEXT -1 43 " since the other polynomials are LINEAR in \+ " }{XPPEDIT 18 0 "x;" "6#%\"xG" }{TEXT -1 5 " and " }{XPPEDIT 18 0 "y; " "6#%\"yG" }{TEXT -1 159 " . This idea only works if the ideal is ze ro dimensional, i.e. the variety of the ideal has finitely many soluti ons. In our example the variety has 8 points." }}}{EXCHG {PARA 0 "> \+ " 0 "" {MPLTEXT 1 0 44 "J := < subs( z=z+5*x-3*y, Generators(I) ) >;" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#>%\"JG-%$<,>G6%,&*$),(%\"zG\"\"\"*& \"\"&F-%\"xGF-F-*&\"\"$F-%\"yGF-!\"\"\"\"#F-F-F-F-,&*$)F3F5F-F-F-F-,&* $)F0F5F-F-F-F-" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 36 "G := Groe bner[Basis](J,plex(x,y,z));" }}{PARA 12 "" 1 "" {XPPMATH 20 "6#>%\"GG7 %,,*$)%\"zG\"\")\"\"\"F+*&\"&g3%F+)F)\"\"#F+F+\"&@d$F+*&\"%y_F+)F)\"\" %F+F+*&\"$S\"F+)F)\"\"'F+F+,,*&\"(;7;\"F+%\"yGF+F+*&\"#>F+)F)\"\"(F+F+ *&\"'HN7F+)F)\"\"$F+F+*&\"(:>G\"F+F)F+F+*&\"%\\GF+)F)\"\"&F+F+,,*&\"(! 31eF+%\"xGF+F+*&\"&*G;F+FIF+F+*&\"'lijF+FCF+F+*&\"(rhU'F+F)F+F+*&\"$: \"F+F?F+F+" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 18 "f := factor(G [1]);" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#>%\"fG**,&*$)%\"zG\"\"#\"\"\" F+\"\"*F+F+,&F'F+\"#\\F+F+,&F'F+\"#\")F+F+,&F'F+F+F+F+" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 28 "S1 := Simplify();" }} {PARA 11 "" 1 "" {XPPMATH 20 "6#>%#S1G-%$<,>G6%,&%\"zG\"\"\"*&\"\"$F*% \"xGF*F*,&F-!\"\"%\"yGF*,&*$)F-\"\"#F*F*F*F*" }}}{EXCHG {PARA 0 "" 0 " " {TEXT -1 76 "Undoing the linear change of variables we get the first prime component of I" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 53 "P1 := Simplify( );" }}{PARA 11 "" 1 " " {XPPMATH 20 "6#>%#P1G-%$<,>G6%,&%\"xG!\"\"%\"yG\"\"\",&%\"zGF,F)F,,& *$)F)\"\"#F,F,F,F," }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 93 "for k from 2 to 4 do\n P||k := Simplify( ))> );\nod;" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#>%#P2G-%$<,>G6 %,&%\"xG!\"\"%\"zG\"\"\",&*$)F)\"\"#F,F,F,F,,&F)F,%\"yGF," }}{PARA 11 "" 1 "" {XPPMATH 20 "6#>%#P3G-%$<,>G6%,&%\"zG\"\"\"%\"xGF*,&*$)F+\"\"# F*F*F*F*,&F+F*%\"yGF*" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#>%#P4G-%$<,>G 6%,&%\"xG!\"\"%\"yG\"\"\",&F)F*%\"zGF,,&*$)F)\"\"#F,F,F,F," }}}{EXCHG {PARA 0 "" 0 "" {TEXT -1 75 "These components are all prime over the f ield of rational numbers. A check" }}}{EXCHG }{EXCHG }{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 35 "Simplify( Intersect(P1,P2,P3,P4) );" }} {PARA 11 "" 1 "" {XPPMATH 20 "6#-%$<,>G6%,&*$)%\"zG\"\"#\"\"\"F+F+F+,& *$)%\"yGF*F+F+F+F+,&*$)%\"xGF*F+F+F+F+" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 0 "" }}}}{MARK "39 1 0" 0 }{VIEWOPTS 1 1 0 1 1 1803 1 1 1 1 }{PAGENUMBERS 0 1 2 33 1 1 }