{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 } {CSTYLE "" -1 256 "helvetica" 0 14 0 0 0 0 0 0 0 0 0 0 0 0 0 0 } {CSTYLE "" -1 257 "helvetica" 0 1 0 0 0 0 0 0 0 0 0 0 0 0 0 0 } {CSTYLE "" -1 258 "helvetica" 0 1 0 0 0 0 0 0 0 0 0 0 0 0 0 0 } {CSTYLE "" -1 259 "helvetica" 0 1 0 0 0 0 0 0 0 0 0 0 0 0 0 0 } {CSTYLE "" -1 260 "helvetica" 0 1 0 0 0 0 0 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 0 }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 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 0 "" 0 "" {TEXT -1 37 "Using Maple's Groebner Bas is Package." }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 15 "with(Groebne r);" }}{PARA 12 "" 1 "" {XPPMATH 20 "6#7:%&BasisG%%FGLMG%1HilbertDimen sionG%2HilbertPolynomialG%.HilbertSeriesG%,InterReduceG%)IsProperG%2Is ZeroDimensionalG%3LeadingCoefficientG%0LeadingMonomialG%,LeadingTermG% .MonomialOrderG%5MultiplicationMatrixG%+NormalFormG%*NormalSetG%'Reduc eG%.RememberBasisG%,SPolynomialG%&SolveG%*TestOrderG%0ToricIdealBasisG %5UnivariatePolynomialG%%WalkG%*fglm_algoG" }}}{EXCHG {PARA 0 "" 0 "" {TEXT -1 41 "The commands that we will mainly use are " }}{PARA 0 "" 0 "" {TEXT 256 5 "Basis" }{TEXT -1 33 " - for computing a Groebner bas is" }}{PARA 0 "" 0 "" {TEXT 257 10 "NormalForm" }{TEXT -1 76 " - for c omputing the remainder of a polynomial divided by a (Groebner) basis" }}}{EXCHG {PARA 0 "" 0 "" {TEXT -1 54 "Lets execute Buchberger's algor ithm on the ideal I = <" }{XPPEDIT 18 0 "f[1];" "6#&%\"fG6#\"\"\"" } {TEXT -1 2 ", " }{XPPEDIT 18 0 "f[2];" "6#&%\"fG6#\"\"#" }{TEXT -1 8 " > below." }}{PARA 0 "" 0 "" {TEXT -1 14 "We'll use the " }{TEXT 258 11 "SPolynomial" }{TEXT -1 5 " and " }{TEXT 259 15 "LeadingMonomial" } {TEXT -1 9 " commands" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 29 "f1 := x*y-y^2; f2 := x^3-z^2;" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#>%#f1G, &*&%\"xG\"\"\"%\"yGF(F(*$)F)\"\"#F(!\"\"" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#>%#f2G,&*$)%\"xG\"\"$\"\"\"F**$)%\"zG\"\"#F*!\"\"" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 65 "LeadingMonomial(f1,plex(x,y,z));\nL eadingMonomial(f2,plex(x,y,z));" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#*&% \"xG\"\"\"%\"yGF%" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#*$)%\"xG\"\"$\"\" \"" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 37 "f3 := SPolynomial(f1, f2,plex(x,y,z));" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#>%#f3G,&*&)%\"xG\" \"#\"\"\")%\"yGF)F*!\"\"*&F,F*)%\"zGF)F*F*" }}}{EXCHG {PARA 0 "" 0 "" {TEXT -1 4 "The " }{TEXT 260 10 "NormalForm" }{TEXT -1 49 " computes t he remainder of S(f1,f2) divided by G." }}}{EXCHG }{EXCHG {PARA 0 "> \+ " 0 "" {MPLTEXT 1 0 14 "G0 := [f1,f2];" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#>%#G0G7$,&*&%\"xG\"\"\"%\"yGF)F)*$)F*\"\"#F)!\"\",&*$)F(\"\"$F)F )*$)%\"zGF-F)F." }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 36 "f3 := No rmalForm(f3,G0,plex(x,y,z));" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#>%#f3G ,&*&%\"yG\"\"\")%\"zG\"\"#F(F(*$)F'\"\"%F(!\"\"" }}}{EXCHG {PARA 0 "> \+ " 0 "" {MPLTEXT 1 0 17 "G1 := [f1,f2,f3];" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#>%#G1G7%,&*&%\"xG\"\"\"%\"yGF)F)*$)F*\"\"#F)!\"\",&*$)F (\"\"$F)F)*$)%\"zGF-F)F.,&*&F*F)F4F)F)*$)F*\"\"%F)F." }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 36 "map(LeadingMonomial,G1,plex(x,y,z));" }} {PARA 11 "" 1 "" {XPPMATH 20 "6#7%*&%\"xG\"\"\"%\"yGF&*$)F%\"\"$F&*$)F '\"\"%F&" }}}{EXCHG {PARA 0 "" 0 "" {TEXT -1 77 "Now S(f2,f3) reduces \+ to 0 by Proposition 4 of 2.9 so we only need to consider" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 37 "f4 := SPolynomial(f1,f3,plex(x,y,z) );" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#>%#f4G,&*$)%\"yG\"\"&\"\"\"F**(% \"xGF*F(F*)%\"zG\"\"#F*!\"\"" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 36 "f4 := NormalForm(f4,G1,plex(x,y,z));" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#>%#f4G\"\"!" }}}{EXCHG {PARA 0 "" 0 "" {TEXT -1 99 "So \+ G1 is a Groebner basis for I = . It happens to be also reduced . Let's check with Maple." }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 32 "G := Basis([f1,f2],plex(x,y,z));" }}{PARA 11 "" 1 "" {XPPMATH 20 " 6#>%\"GG7%,&*$)%\"yG\"\"%\"\"\"F+*&F)F+)%\"zG\"\"#F+!\"\",&*&%\"xGF+F) F+F+*$)F)F/F+F0,&*$)F3\"\"$F+F+*$F-F+F0" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 3 "G1;" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#7%,&*&%\"xG\"\" \"%\"yGF'F'*$)F(\"\"#F'!\"\",&*$)F&\"\"$F'F'*$)%\"zGF+F'F,,&*&F(F'F2F' F'*$)F(\"\"%F'F," }}}{EXCHG {PARA 0 "" 0 "" {TEXT -1 51 "This polynomi al is in the ideal I. Let's test this" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 23 "f := expand(x*f1+y*f2);" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#>%\"fG,**&)%\"xG\"\"#\"\"\"%\"yGF*F**&)F+F)F*F(F*!\"\"*&)F(\"\"$ F*F+F*F**&F+F*)%\"zGF)F*F." }}}{EXCHG {PARA 0 "" 0 "" {TEXT -1 0 "" }} }{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 28 "NormalForm(f,G,plex(x,y,z)) ;" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#\"\"!" }}}{EXCHG {PARA 0 "> " 0 " " {MPLTEXT 1 0 32 "NormalForm(f+x+1,G,plex(x,y,z));" }}{PARA 11 "" 1 " " {XPPMATH 20 "6#,&%\"xG\"\"\"F%F%" }}}{EXCHG {PARA 0 "" 0 "" {TEXT -1 98 "Let's try to color the graph C3, a cycle on three vertices with k=2 colors (it's not 2-colorable)." }}}{EXCHG {PARA 0 "" 0 "" {TEXT -1 4 " " }{GLPLOT2D 153 151 151 {PLOTDATA 2 "6'-%'CURVESG6'7$7$!\" \"\"\"!7$F)\"\"\"7$F'7$F+F)7$F*F--%*THICKNESSG6#\"\"$-%&COLORG6&%$RGBG F)F)F+-%*AXESSTYLEG6#%%NONEG-%%TEXTG6%7$F)$\"#6F(Q\"16\"-%%FONTG6$%*HE LVETICAG\"#=-F<6%7$$!#6F(F)Q\"3FBFC-F<6%7$F?F)Q\"2FBFC" 1 2 0 1 10 0 2 9 1 1 2 1.000000 339.000000 320.000000 0 0 "Curve 1" "Curve 2" "Curv e 3" "Curve 4" }}}}{EXCHG }{EXCHG }{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 36 "k := 2;\nS := [x1^k-1,x2^2-1,x3^2-1];" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#>%\"kG\"\"#" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#>%\"SG7% ,&*$)%#x1G\"\"#\"\"\"F+F+!\"\",&*$)%#x2GF*F+F+F+F,,&*$)%#x3GF*F+F+F+F, " }}}{EXCHG {PARA 0 "" 0 "" {TEXT -1 28 "The colors are the roots of \+ " }{XPPEDIT 18 0 "x^2-1;" "6#,&*$)%\"xG\"\"#\"\"\"F(F(!\"\"" }{TEXT -1 226 " which are 1 and -1. These equations say each vertex can be c olored with either color. But vertex x1 may not be the same color as \+ vertex x2. So we should add the equation x1 + x2 = 0. Same for the o ther two edges. We have" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 35 "S := [op(S), x1+x2, x1+x3, x2+x3 ];" }}{PARA 11 "" 1 "" {XPPMATH 20 " 6#>%\"SG7(,&*$)%#x1G\"\"#\"\"\"F+F+!\"\",&*$)%#x2GF*F+F+F+F,,&*$)%#x3G F*F+F+F+F,,&F)F+F0F+,&F)F+F4F+,&F0F+F4F+" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 25 "Basis(S,grlex(x1,x2,x3));" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#7#\"\"\"" }}}{EXCHG {PARA 0 "" 0 "" {TEXT -1 56 "This m eans G is not 2-colorable. But it is 3-colorable." }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 141 "k := 3;\nS := [x1^k-1,x2^k-1,x3^k-1,\n \+ normal((x1^k-x2^k)/(x1-x2)),\n normal((x1^k-x3^k)/(x1-x3)),\n \+ normal((x2^k-x3^k)/(x2-x3))];" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#> %\"kG\"\"$" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#>%\"SG7(,&*$)%#x1G\"\"$ \"\"\"F+F+!\"\",&*$)%#x2GF*F+F+F+F,,&*$)%#x3GF*F+F+F+F,,(*$)F)\"\"#F+F +*&F0F+F)F+F+*$)F0F8F+F+,(F6F+*&F)F+F4F+F+*$)F4F8F+F+,(F>F+*&F0F+F4F+F +F:F+" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 25 "Basis(S,grlex(x1,x 2,x3));" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#7%,(%#x1G\"\"\"%#x3GF&%#x2G F&,(*$)F'\"\"#F&F&*&F(F&F'F&F&*$)F(F,F&F&,&*$)F'\"\"$F&F&F&!\"\"" }}} {EXCHG {PARA 0 "" 0 "" {TEXT -1 108 "This means x3 can be any color, x 2 must be different from x3, and x1, x2, x3 must have all different co lors." }}}{EXCHG {PARA 0 "" 0 "" {TEXT -1 10 "Example 3." }}{PARA 0 " " 0 "" {TEXT -1 66 "Lets work with one of the ideals we saw in class, \+ namely \n I = < " }{XPPEDIT 18 0 "x^2+y+z-1x+y^2+z-1x+y+z^2-1;" "6%,* *$)%\"xG\"\"#\"\"\"F(%\"yGF(%\"zGF(F(!\"\",*F&F(*$)F)F'F(F(F*F(F(F+,*F &F(F)F(*$)F*F'F(F(F(F+" }{TEXT -1 2 " >" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 37 "F := [x^2+y+z-1,x+y^2+z-1,x+y+z^2-1];" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#>%\"FG7%,**$)%\"xG\"\"#\"\"\"F+%\"yGF+%\"zGF+F+!\" \",*F)F+*$)F,F*F+F+F-F+F+F.,*F)F+F,F+*$)F-F*F+F+F+F." }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 27 "G := Basis(F,grlex(x,y,z));" }}{PARA 11 " " 1 "" {XPPMATH 20 "6#>%\"GG7%,*%\"xG\"\"\"%\"yGF(*$)%\"zG\"\"#F(F(F(! \"\",*F'F(*$)F)F-F(F(F,F(F(F.,**$)F'F-F(F(F)F(F,F(F(F." }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 26 "H := Basis(F,plex(x,y,z));" }} {PARA 11 "" 1 "" {XPPMATH 20 "6#>%\"HG7&,**$)%\"zG\"\"'\"\"\"F+*&\"\"% F+)F)\"\"$F+F+*&F-F+)F)F-F+!\"\"*$)F)\"\"#F+F2,(*(F5F+%\"yGF+F4F+F+*$F 1F+F+F3F2,**$)F8F5F+F+F3F2F)F+F8F2,*%\"xGF+F8F+F3F+F+F2" }}}{EXCHG {PARA 0 "" 0 "" {TEXT -1 79 "We can see that the first polynomial has \+ repeated roots. Let's eliminate them." }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 13 "factor(H[1]);" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#*()% \"zG\"\"#\"\"\",(*$F$F'F'*&F&F'F%F'F'F'!\"\"F'),&F%F'F'F+F&F'" }}} {EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 42 "H := [z*(z-1)*(z^2+2*z-1),H[ 2],H[3],H[4]];" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#>%\"HG7&*(%\"zG\"\" \",&F'F(F(!\"\"F(,(*$)F'\"\"#F(F(*&F.F(F'F(F(F(F*F(,(*(F.F(%\"yGF(F-F( F(*$)F'\"\"%F(F(F,F*,**$)F2F.F(F(F,F*F'F(F2F*,*%\"xGF(F2F(F,F(F(F*" }} }{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 26 "H := Basis(H,plex(x,y,z)); " }}{PARA 11 "" 1 "" {XPPMATH 20 "6#>%\"HG7&,**$)%\"zG\"\"%\"\"\"F+*$) F)\"\"$F+F+*&F.F+)F)\"\"#F+!\"\"F)F+,(*(F1F+F)F+%\"yGF+F+F,F+F)F2,**$) F5F1F+F+*$F0F+F2F)F+F5F2,*%\"xGF+F5F+F9F+F+F2" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 17 "Solve(H,[x,y,z]);" }}{PARA 12 "" 1 "" {XPPMATH 20 "6#<&7%7%%\"yG%\"xG,&%\"zG\"\"\"F*!\"\"-%%plexG6%F)F'F&<\"7%7%F',&F &F*F*F+F)-F-6%F)F&F'F/7%7%,&F'F*F*F+F&F)F3F/7%7%,(*$)F)\"\"#F*F**&F=F* F)F*F*F*F+,&F&F*F)F+,&F'F*F)F+-F-6%F'F&F)<$F)F(" }}}{EXCHG {PARA 0 "" 0 "" {TEXT -1 74 "This has split up the solutions. We can solve them \+ explicitly using solve" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 46 "_ EnvExplicit := true;\nV := \{solve(H,\{x,y,z\})\};" }}{PARA 11 "" 1 " " {XPPMATH 20 "6#>%-_EnvExplicitG%%trueG" }}{PARA 12 "" 1 "" {XPPMATH 20 "6#>%\"VG<'<%/%\"xG,&\"\"\"!\"\"*$\"\"##F*F-F+/%\"zGF)/%\"yGF)<%/F0 \"\"!/F(F*/F2F5<%/F(,&F*F+F,F*/F0F:/F2F:<%F4/F(F5/F2F*<%F>F7/F0F*" }}} {EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 8 "nops(V);" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#\"\"&" }}}{EXCHG {PARA 0 "" 0 "" {TEXT -1 48 "There are 5 distinct solutions in the variety V." }}}{EXCHG }{EXCHG {PARA 0 "" 0 "" {TEXT -1 66 "The PolynomialIdeals package has additional operatio ns for ideals." }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 23 "with(Poly nomialIdeals);" }}{PARA 12 "" 1 "" {XPPMATH 20 "6#7B%$<,>G%$AddG%)Cont ractG%1EliminationIdealG%+GeneratorsG%1HilbertDimensionG%1IdealContain mentG%*IdealInfoG%0IdealMembershipG%*IntersectG%*IsMaximalG%*IsPrimary G%(IsPrimeG%)IsProperG%*IsRadicalG%2IsZeroDimensionalG%6MaximalIndepen dentSetG%)MultiplyG%*OperatorsG%0PolynomialIdealG%5PrimaryDecompositio nG%3PrimeDecompositionG%)QuotientG%(RadicalG%2RadicalMembershipG%)Satu rateG%)SimplifyG%5UnivariatePolynomialG%/VanishingIdealG%=ZeroDimensio nalDecompositionG%#inG%'subsetG" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 31 "interface(imaginaryunit = iii);" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#^#\"\"\"" }}}{EXCHG {PARA 0 "" 0 "" {TEXT -1 61 "The above is to let me use the capital letter I for an ideal." }}}{EXCHG {PARA 0 "> \+ " 0 "" {MPLTEXT 1 0 2 "F;" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#7%,**$)% \"xG\"\"#\"\"\"F)%\"yGF)%\"zGF)F)!\"\",*F'F)*$)F*F(F)F)F+F)F)F,,*F'F)F *F)*$)F+F(F)F)F)F," }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 9 "I := < F>;" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#>%\"IG-%$<,>G6%,**$)%\"xG\"\"# \"\"\"F-%\"yGF-%\"zGF-F-!\"\",*F+F-*$)F.F,F-F-F/F-F-F0,*F+F-F.F-*$)F/F ,F-F-F-F0" }}}{EXCHG }{EXCHG {PARA 0 "" 0 "" {TEXT -1 57 "The radical \+ operation gets rid of all repeated solutions." }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 16 "J := Radical(I);" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#>%\"JG-%$<,>G6&,**$)%\"xG\"\"#\"\"\"F-%\"yGF-%\"zGF-F-!\"\",*F+F -*$)F.F,F-F-F/F-F-F0,*F+F-F.F-*$)F/F,F-F-F-F0,**$)F+\"\"%F-F-*$)F+\"\" $F-F-*&F=F-F*F-F0F+F-" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 21 "Ba sis(J,plex(x,y,z));" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#7&,**$)%\"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)F0F'F)F3F0,*%\"xGF)F3F)F7F)F)F0" }}} {EXCHG {PARA 0 "" 0 "" {TEXT -1 126 "The PrimeDecompostion operation s plits the ideal J into prime components, each of which corresponds to \+ an irreducible variety." }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 29 " P := [PrimeDecomposition(J)];" }}{PARA 12 "" 1 "" {XPPMATH 20 "6#>%\"P G7&-%$<,>G6',**$)%\"xG\"\"#\"\"\"F.%\"yGF.%\"zGF.F.!\"\",*F,F.*$)F/F-F .F.F0F.F.F1,*F,F.F/F.*$)F0F-F.F.F.F1,(F*F.*&F-F.F,F.F.F.F1,**$)F,\"\"% F.F.*$)F,\"\"$F.F.*&F@F.F+F.F1F,F.-F'6(F,F)F2F5,&F0F.F.F1F:-F'6'F)F2F5 ,&F,F.F.F1F:-F'6(F,F0F)F2F5F:" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 21 "P := map(Simplify,P);" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#>%\"PG7 &-%$<,>G6%,&%\"zG\"\"\"%\"xG!\"\",(*$)F,\"\"#F+F+*&F1F+F,F+F+F+F-,&%\" yGF+F,F--F'6%F,F4,&F*F+F+F--F'6%F*F4,&F,F+F+F--F'6%F,F*,&F4F+F+F-" }}} {EXCHG {PARA 0 "" 0 "" {TEXT -1 112 "From which we can see 1, 1, 2, 1 \+ solutions. From which we can understand the output of the Solve comma nd above." }}}{EXCHG }{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 0 "" }}}} {MARK "65 0 0" 0 }{VIEWOPTS 1 1 0 3 2 1804 1 1 1 1 }{PAGENUMBERS 0 1 2 33 1 1 }