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{SECT 0 {EXCHG {PARA 0 "" 0 "" {TEXT -1 87 "Second examples of the Syl
vester resultant: applying proposition 3 to solve a system\n  " }
{XPPEDIT 18 0 "\{f(x,y,z) = 0, g(x,y,z) = 0, h(x,y,z) = 0\};" "6#<%/-%
\"fG6%%\"xG%\"yG%\"zG\"\"!/-%\"gG6%F(F)F*F+/-%\"hG6%F(F)F*F+" }{TEXT 
-1 3 " .\n" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 15 "with(Groebner
):" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 48 "f,g,h := x^2+y^2+z^2-
1, x*y+x*z+y*z-1, x*y*z-1;\n" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#>6%%\"
fG%\"gG%\"hG6%,**$)%\"xG\"\"#\"\"\"F.*$)%\"yGF-F.F.*$)%\"zGF-F.F.F.!\"
\",**&F,F.F1F.F.*&F,F.F4F.F.*&F1F.F4F.F.F.F5,&*(F,F.F1F.F4F.F.F.F5" }}
}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 60 "G := gbasis([f,g,h],plex(x,
y,z)):\nmap(sort,G,[x,y,z],plex);\n" }}{PARA 12 "" 1 "" {XPPMATH 20 "6
#7%,.*$)%\"zG\"\"'\"\"\"F)*$)F'\"\"%F)!\"\"*&\"\"#F))F'\"\"$F)F-*$)F'F
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8F)F0F)F)*(F1F)F8F)F3F)F)*&F8F)F'F)F)F8F)*$F:F)F-*$F0F)F)*&F/F)F3F)F)F
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0 "" 0 "" {TEXT -1 0 "" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 57 "S
 := resultant( resultant(f,g,x), resultant(f,h,x), y );\n" }}{PARA 11 
"" 1 "" {XPPMATH 20 "6#>%\"SG,<*$)%\"zG\"#C\"\"\"F**&\"\"%F*)F(\"#AF*!
\"\"*&\"#5F*)F(\"#?F*F**&F3F*)F(\"#=F*F/*&\"\"(F*)F(\"#;F*F**&\"\")F*)
F(\"#9F*F**&\"#GF*)F(\"#7F*F/*&\"#cF*)F(F1F*F**&\"$.\"F*)F(F<F*F**&F@F
*)F(\"\"'F*F**&F>F*)F(F,F*F/*&F,F*)F(\"\"#F*F/F*F*" }}}{EXCHG {PARA 0 
"> " 0 "" {MPLTEXT 1 0 10 "factor(S);" }}{PARA 11 "" 1 "" {XPPMATH 20 
"6#*&),.*$)%\"zG\"\"'\"\"\"F**$)F(\"\"%F*!\"\"*&\"\"#F*)F(\"\"$F*F**$)
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*F0F*" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 75 "Rfg := resultant(f
,g,x);\nRfh := resultant(f,h,x);\nRgh := resultant(g,h,x);\n" }}{PARA 
11 "" 1 "" {XPPMATH 20 "6#>%$RfgG,4*$)%\"yG\"\"%\"\"\"F**(\"\"#F*)F(\"
\"$F*%\"zGF*F**(F.F*)F(F,F*)F/F,F*F**$F1F*!\"\"*(F,F*F(F*)F/F.F*F**(F)
F*F(F*F/F*F4*$)F/F)F*F**$F2F*F4F*F*" }}{PARA 11 "" 1 "" {XPPMATH 20 "6
#>%$RfhG,**&)%\"yG\"\"%\"\"\")%\"zG\"\"#F*F**&)F(F-F*)F,F)F*F**&F/F*F+
F*!\"\"F*F*" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#>%$RghG,**&)%\"yG\"\"#
\"\"\")%\"zGF)F*!\"\"*&F(F*F,F*F*F(F-F,F-" }}}{EXCHG }{EXCHG }{EXCHG }
{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 52 "map( normalf, [Rfg,Rfh,Rgh],
 G[1..2], plex(x,y,z) );" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#7%\"\"!F$F
$" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 117 "Rfhgh := factor(resul
tant(Rfh,Rgh,y));\nRfgfh := factor(resultant(Rfg,Rfh,y));\nRfggh := fa
ctor(resultant(Rfg,Rgh,y));\n" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#>%&Rf
hghG*&)%\"zG\"\"#\"\"\"),.*$)F'\"\"'F)F)*$)F'\"\"%F)!\"\"*&F(F))F'\"\"
$F)F2*$F&F)F)*&F(F)F'F)F2F)F)F(F)" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#>
%&RfgfhG*&),.*$)%\"zG\"\"'\"\"\"F,*$)F*\"\"%F,!\"\"*&\"\"#F,)F*\"\"$F,
F,*$)F*F2F,F,*&F2F,F*F,F,F,F,F2F,),.F(F,F-F0*&F2F,F3F,F0F5F,*&F2F,F*F,
F0F,F,F2F," }}{PARA 11 "" 1 "" {XPPMATH 20 "6#>%&RfgghG*&),&*$)%\"zG\"
\"#\"\"\"F,F,F,F+F,),.*$)F*\"\"'F,F,*$)F*\"\"%F,!\"\"*&F+F,)F*\"\"$F,F
5F(F,*&F+F,F*F,F5F,F,F+F," }}}{EXCHG }{EXCHG }{EXCHG }{EXCHG }{EXCHG }
{EXCHG }{EXCHG }{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 33 "g := gcd(gcd
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