{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 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 "Heading 2" 3 4 1 {CSTYLE "" -1 -1 "" 1 14 0 0 0 0 0 0 0 0 0 0 0 0 0 0 }0 0 0 -1 8 2 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 4 "" 0 "" {TEXT -1 79 "Computing minimal polynomi als using Groebner bases and the elimination theorem." }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 25 "alpha := sqrt(2)+sqrt(3);" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#>%&alphaG,&*$\"\"##\"\"\"F'F)*$\"\"$F(F)" }} }{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 25 "F := [z-s-t,s^2-2,t^2-3];" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#>%\"FG7%,(%\"zG\"\"\"%\"sG!\"\"%\"tG F*,&*$)F)\"\"#F(F(F/F*,&*$)F+F/F(F(\"\"$F*" }}}{EXCHG }{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 37 "G := Groebner[gbasis](F,plex(s,t,z));" }} {PARA 11 "" 1 "" {XPPMATH 20 "6#>%\"GG7%,(\"\"\"F'*&\"#5F')%\"zG\"\"#F '!\"\"*$)F+\"\"%F'F',(*&\"#6F'F+F'F-*&F,F'%\"tGF'F'*$)F+\"\"$F'F',(*&F ,F'%\"sGF'F'F6F-*&\"\"*F'F+F'F'" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 11 "m := G[1]; " }}{PARA 11 "" 1 "" {XPPMATH 20 "6#>%\"mG,(\"\"\"F &*&\"#5F&)%\"zG\"\"#F&!\"\"*$)F*\"\"%F&F&" }}}{EXCHG {PARA 0 "> " 0 " " {MPLTEXT 1 0 10 "factor(m);" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#,(\" \"\"F$*&\"#5F$)%\"zG\"\"#F$!\"\"*$)F(\"\"%F$F$" }}}{EXCHG {PARA 0 "> \+ " 0 "" {MPLTEXT 1 0 16 "eval(m,z=alpha);" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#,(\"\"\"F$*&\"#5F$),&*$\"\"##F$F*F$*$\"\"$F+F$F*F$!\"\"*$)F(\"\" %F$F$" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 24 "expand(eval(m,z=al pha));" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#\"\"!" }}}{EXCHG {PARA 0 "> \+ " 0 "" {MPLTEXT 1 0 35 "alpha := 2^(1/3)+sqrt(1+2^(1/3))-1;" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#>%&alphaG,(*$)\"\"##\"\"\"\"\"$F*F**$,&F*F*F &F*#F*F(F*F*!\"\"" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 31 "F := [ z-s-t+1,s^3-2,t^2-(1+s)];" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#>%\"FG7%, *%\"zG\"\"\"%\"sG!\"\"%\"tGF*F(F(,&*$)F)\"\"$F(F(\"\"#F*,(*$)F+F0F(F(F (F*F)F*" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 43 "G := Groebner[gb asis](F,lexdeg([s,t],[z]));" }}{PARA 12 "" 1 "" {XPPMATH 20 "6#>%\"GG7 %,0*&\"#I\"\"\")%\"zG\"\"#F)!\"\"*&\"#sF)F+F)F-\"#]F-*&\"\"%F))F+\"\"$ F)F)*&\"#7F))F+F2F)F)*&\"\"'F))F+\"\"&F)F)*$)F+F9F)F),0\"$H&F)*&\"$F#F )%\"tGF)F)*&\"$n#F)F+F)F)*&\"$9\"F)F3F)F-*&\"#RF)F*F)F)*&\"#pF)F7F)F-* &\"#5F)F:F)F-,0*&FAF)%\"sGF)F)*&FLF)F:F)F)*&FJF)F7F)F)*&FFF)F3F)F)*&FH F)F*F)F-*&\"$%\\F)F+F)F-\"$c(F-" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 18 "m := factor(G[1]);" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#>%\"mG,0 *&\"#I\"\"\")%\"zG\"\"#F(!\"\"*&\"#sF(F*F(F,\"#]F,*&\"\"%F()F*\"\"$F(F (*&\"#7F()F*F1F(F(*&\"\"'F()F*\"\"&F(F(*$)F*F8F(F(" }}}{EXCHG }{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 24 "expand(eval(m,z=alpha));" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#\"\"!" }}}{EXCHG }{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 27 "alpha := I+sqrt(2)+sqrt(3);" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#>%&alphaG,(^#\"\"\"F'*$\"\"##F'F)F'*$\"\"$F*F'" }}} {EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 33 "F := [z-s-t-i,i^2+1,s^2-2,t^ 2-3];" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#>%\"FG7&,*%\"zG\"\"\"%\"sG!\" \"%\"tGF*%\"iGF*,&*$)F,\"\"#F(F(F(F(,&*$)F)F0F(F(F0F*,&*$)F+F0F(F(\"\" $F*" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 45 "G := Groebner[gbasis ](F,lexdeg([s,t,i],[z]));" }}{PARA 12 "" 1 "" {XPPMATH 20 "6#>%\"GG7&, ,\"$W\"\"\"\"*&\"$#>F()%\"zG\"\"#F(F(*&\"#))F()F,\"\"%F(F(*&\"#;F()F, \"\"'F(!\"\"*$)F,\"\")F(F(,,*&\"%_6F(%\"iGF(F(*&\"\"&F()F,\"\"(F(F6*& \"#uF()F,F?F(F(*&\"$!QF()F,\"\"$F(F6*&\"%;9F(F,F(F6,,*&\"$%QF(%\"tGF(F (*&FAF(F@F(F(*&\"$=\"F(FDF(F6*&\"$C(F(FGF(F(*&\"$/&F(F,F(F(,,*&F'F(%\" sGF(F(*&F-F(F@F(F6*&\"#NF(FDF(F(*&\"$C#F(FGF(F6*&\"$c\"F(F,F(F6" }}} {EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 18 "m := factor(G[1]);" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#>%\"mG,,\"$W\"\"\"\"*&\"$#>F')%\"zG\"\"#F'F' *&\"#))F')F+\"\"%F'F'*&\"#;F')F+\"\"'F'!\"\"*$)F+\"\")F'F'" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 0 "" }}}}{MARK "20 0 0" 0 }{VIEWOPTS 1 1 0 1 1 1803 1 1 1 1 }{PAGENUMBERS 0 1 2 33 1 1 }