

%% 
%% This is a LaTeX document generated by automatic conversion of a
%% Mathematica notebook using Mathematica 5.0.
%% 
%% This document uses special macros that are defined in a LaTeX 
%% package file named notebook2e.sty.  Appropriate commands for 
%% loading the package have been included in this document.
%% 
%% The LaTeX package files can be found in a directory with path:
%% 
%% D:\Programme\Wolfram Research\Mathematica\5.0\SystemFiles\IncludeFiles\TeX\
%% 
%% The LaTeX package used in this document may require that your 
%% LaTeX implementation support fonts other than the standard 
%% Computer Modern fonts that are used by LaTeX.  For more
%% information see the Additional Information under TeXSave
%% in the Help Browser, or at
%% 
%% http://documents.wolfram.com/v5/RefGuide/TeXSave.html
%% 
%% Implementation-specific instructions for installing TeX-related 
%% files can be found at
%% 
%% http://support.wolfram.com/mathematica/interface/export/virtualfonts.html
%% 
%% 

\documentclass{article}
\usepackage{notebook2e, latexsym}


\begin{document}

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\MathBegin{MathArray}{l}
\{\Mvariable{u3}-\Mvariable{x2},\Mvariable{u1}\multsp \Mvariable{u3}-\Mvariable{x1}\multsp \Mvariable{u3}+\Mvariable{u2}\multsp \Mvariable{x2},\Mvariable{u1}\multsp
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\hspace{1.em} \Mvariable{x1}\multsp \Mvariable{x4}-\Mvariable{x2}\multsp \Mvariable{x3},1-(-{{\Mvariable{x1}}^2}+2\multsp \Mvariable{x3}\multsp \Mvariable{x1}-{{\Mvariable{x2}}^2}+2\multsp
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\MathBegin{MathArray}{l}
\{\Mvariable{u3}-\Mvariable{x2},\Mvariable{u1}\multsp \Mvariable{u3}-\Mvariable{x1}\multsp \Mvariable{u3}+\Mvariable{u2}\multsp \Mvariable{x2},\Mvariable{u1}\multsp
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\hspace{1.em} \Mvariable{x1}\multsp \Mvariable{x4}-\Mvariable{x2}\multsp \Mvariable{x3},1-({{\Mvariable{u1}}^2}-2\multsp \Mvariable{x3}\multsp \Mvariable{u1}-{{\Mvariable{u2}}^2}-{{\Mvariable{u3}}^2}+2\multsp
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\Mfunction{GroebnerBasis}[\Mvariable{zw1},\Mvariable{vars},  \\
\noalign{\vspace{0.625ex}}
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\MathBegin{MathArray}{l}
\Mfunction{GroebnerBasis}[\Mvariable{zw2},\Mvariable{vars},  \\
\noalign{\vspace{0.625ex}}
\hspace{2.em} \Mvariable{CoefficientDomain}\rightarrow \Mvariable{RationalFunctions}]//\Mfunction{Timing}\\
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\MathBegin{MathArray}{l}
\Mvariable{h1}=\Mvariable{x2}-\Mvariable{u3};  \\
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\Mvariable{h4}=\Mvariable{x4}(\Mvariable{u2}-\Mvariable{u1})-(\Mvariable{x3}-\Mvariable{u1})\Mvariable{u3};  \\
\noalign{\vspace{0.625ex}}  \\
g=\Mvariable{x1}\RawWedge 2-2\Mvariable{x1}\multsp \Mvariable{x3}-2\Mvariable{x4}\multsp \Mvariable{x2}+\Mvariable{x2}\RawWedge 2;  \\
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\MathBegin{MathArray}{l}
\Mfunction{GroebnerBasis}[\{\Mvariable{h1},\Mvariable{h2},\Mvariable{h3},\Mvariable{h4},1-y\multsp g\},  \\
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\hspace{1.em} \{\Mvariable{u1},\Mvariable{u2},\Mvariable{u3},\Mvariable{x1},\Mvariable{x2},\Mvariable{x3},\Mvariable{x4},y\},  \\
\noalign{\vspace{0.625ex}}
\hspace{1.em} \Mvariable{MonomialOrder}\rightarrow \Mvariable{Lexicographic}]\\
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\MathBegin{MathArray}{l}
\{-2\multsp \Mvariable{x2}\multsp y\multsp {{\Mvariable{x4}}^3}+{{\Mvariable{x2}}^2}\multsp y\multsp {{\Mvariable{x4}}^2}-{{\Mvariable{x4}}^2}-2\multsp
\Mvariable{x2}\multsp {{\Mvariable{x3}}^2}\multsp y\multsp \Mvariable{x4}+{{\Mvariable{x2}}^2}\multsp {{\Mvariable{x3}}^2}\multsp y,  \\
\noalign{\vspace{0.601563ex}}
\hspace{1.em} \Mvariable{x2}\multsp \Mvariable{x3}-\Mvariable{x1}\multsp \Mvariable{x4},\Mvariable{x4}\multsp y\multsp {{\Mvariable{x2}}^2}-2\multsp
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y\multsp \Mvariable{x2}-\Mvariable{x4},  \\
\noalign{\vspace{0.601563ex}}
\hspace{1.em} -y\multsp {{\Mvariable{x1}}^2}+2\multsp \Mvariable{x3}\multsp y\multsp \Mvariable{x1}-{{\Mvariable{x2}}^2}\multsp y+2\multsp \Mvariable{x2}\multsp
\Mvariable{x4}\multsp y+1,  \\
\noalign{\vspace{0.625ex}}
\hspace{1.em} \Mvariable{x2}-\Mvariable{u3},\Mvariable{x2}\multsp \Mvariable{x3}-\Mvariable{u2}\multsp \Mvariable{x4},\Mvariable{x1}\multsp \Mvariable{x2}-\Mvariable{u2}\multsp
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\MathBegin{MathArray}{l}
\Mvariable{dis}[  \\
\noalign{\vspace{0.625ex}}
\hspace{1.em} f=\Mfunction{GroebnerBasis}[\{\Mvariable{h1},\Mvariable{h2},\Mvariable{h3},\Mvariable{h4}\},  \\
\noalign{\vspace{0.625ex}}
\hspace{3.em} \{\Mvariable{x1},\Mvariable{x2},\Mvariable{x3},\Mvariable{x4},\Mvariable{u1},\Mvariable{u2},\Mvariable{u3}\},  \\
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\\
	\Mvariable{x2}-\Mvariable{u3} \\
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\Mvariable{u3}
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\MathBegin{MathArray}{l}
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\noalign{\vspace{0.625ex}}
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\MathBegin{MathArray}{l}
\{\Mvariable{v1}=\Mfunction{GroebnerBasis}[f,\{\Mvariable{x1},\Mvariable{x2},\Mvariable{x3},\Mvariable{x4},\Mvariable{u1},\Mvariable{u2},\Mvariable{u3}\}]\}//
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\noalign{\vspace{0.625ex}}
\hspace{2.em} \Mvariable{Transpose}//\Mfunction{Factor}\\
\MathEnd{MathArray}
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\Bigg(\NoBreak \MathBegin{MathArray}[c]{c}
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\MathBegin{MathArray}{l}
\Muserfunction{v2}[[2]]=-\Mvariable{x4};  \\
\noalign{\vspace{0.625ex}}  \\
\Muserfunction{dis}[\Mvariable{v21}=\Mfunction{GroebnerBasis}[\Mvariable{v2},\{\Mvariable{x1},\Mvariable{x2},\Mvariable{x3},\Mvariable{x4},\Mvariable{u1},\Mvariable{u2},\Mvariable{u3}\}]]
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\Bigg(\NoBreak \MathBegin{MathArray}[c]{c}
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}

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\MathBegin{MathArray}{l}
\Muserfunction{v2}[[2]]=\Mvariable{u1}-\Mvariable{u2};  \\
\noalign{\vspace{0.625ex}}  \\
\Muserfunction{dis}[\Mvariable{v22}=\Mfunction{GroebnerBasis}[\Mvariable{v2},\{\Mvariable{x1},\Mvariable{x2},\Mvariable{x3},\Mvariable{x4},\Mvariable{u1},\Mvariable{u2},\Mvariable{u3}\}]]
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\Bigg(\NoBreak \MathBegin{MathArray}[c]{c}
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\MathBegin{MathArray}{l}
\Muserfunction{v22}[[4]]=\Mvariable{x1};  \\
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\MathEnd{MathArray}
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\Mfunction{General}::\Mvariable{spell1}:\multsp \Mvariable{Possible}\multsp \Mvariable{spelling}\multsp \Mvariable{error}:\multsp \Mvariable{new}\multsp
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\noalign{\vspace{0.5ex}}
\hspace{2.em} \Mvariable{v221}\backslash "\multsp \Mvariable{is}\multsp \Mvariable{similar}\multsp \Mvariable{to}\multsp \Mvariable{existing}\multsp
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\MathEnd{MathArray}\)}

\dispSFoutmath{
\Bigg(\NoBreak \MathBegin{MathArray}[c]{c}
	\Mvariable{u3} \\
	\Mvariable{u1}-\Mvariable{u2} \\
	\Mvariable{x2} \\
	\Mvariable{x1} 
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}

\dispSFinmath{
\MathBegin{MathArray}{l}
\Muserfunction{v22}[[4]]=\Mvariable{x4};  \\
\noalign{\vspace{0.625ex}}  \\
\Muserfunction{dis}[\Mvariable{v222}=\Mfunction{GroebnerBasis}[\Mvariable{v22},\{\Mvariable{x1},\Mvariable{x2},\Mvariable{x3},\Mvariable{x4},\Mvariable{u1},\Mvariable{u2},\Mvariable{u3}\}]]
\MathEnd{MathArray}
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\dispSFoutmath{
\Bigg(\NoBreak \MathBegin{MathArray}[c]{c}
	\Mvariable{u3} \\
	\Mvariable{u1}-\Mvariable{u2} \\
	\Mvariable{x4} \\
	\Mvariable{x2} 
  \MathEnd{MathArray}\NoBreak \Bigg)
}

nun V1

\dispSFinmath{
\MathBegin{MathArray}{l}
\Muserfunction{v1}[[1]]=\Mvariable{u1};  \\
\noalign{\vspace{0.625ex}}  \\
\Muserfunction{dis}[\Mvariable{v11}=\Mfunction{GroebnerBasis}[\Mvariable{v1},\{\Mvariable{x1},\Mvariable{x2},\Mvariable{x3},\Mvariable{x4},\Mvariable{u1},\Mvariable{u2},\Mvariable{u3}\}]]
\MathEnd{MathArray}
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\Bigg(\NoBreak \MathBegin{MathArray}[c]{c}
	\Mvariable{u1} \\
	\Mvariable{u3}\multsp \Mvariable{x3}-\Mvariable{u2}\multsp \Mvariable{x4} \\
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	\Mvariable{x1}-\Mvariable{u2} 
  \MathEnd{MathArray}\NoBreak \Bigg)
}

\dispSFinmath{
\MathBegin{MathArray}{l}
\Muserfunction{v1}[[1]]=\Mvariable{u3}-2\Mvariable{x4};  \\
\noalign{\vspace{0.625ex}}  \\
\Muserfunction{dis}[\Mvariable{v12}=\Mfunction{GroebnerBasis}[\Mvariable{v1},\{\Mvariable{x1},\Mvariable{x2},\Mvariable{x3},\Mvariable{x4},\Mvariable{u1},\Mvariable{u2},\Mvariable{u3}\}]]
\MathEnd{MathArray}
}

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\Bigg(\NoBreak \MathBegin{MathArray}[c]{c}
	\Mvariable{u3}-2\multsp \Mvariable{x4} \\
	\Mvariable{u3}\multsp (\Mvariable{u1}+\Mvariable{u2}-2\multsp \Mvariable{x3}) \\
	\Mvariable{x2}-\Mvariable{u3} \\
	-\Mvariable{u1}-\Mvariable{u2}+\Mvariable{x1} 
  \MathEnd{MathArray}\NoBreak \Bigg)
}

\dispSFinmath{
\MathBegin{MathArray}{l}
\Muserfunction{v12}[[2]]=\Mvariable{u3};  \\
\noalign{\vspace{0.625ex}}  \\
\Muserfunction{dis}[\Mvariable{v121}=\Mfunction{GroebnerBasis}[\Mvariable{v12},\{\Mvariable{x1},\Mvariable{x2},\Mvariable{x3},\Mvariable{x4},\Mvariable{u1},\Mvariable{u2},\Mvariable{u3}\}]]
\MathEnd{MathArray}
}

\Message{\(\MathBegin{MathArray}{l}
\Mfunction{General}::\Mvariable{spell}:\multsp \Mvariable{Possible}\multsp \Mvariable{spelling}\multsp \Mvariable{error}:\multsp \Mvariable{new}\multsp
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\noalign{\vspace{0.5ex}}
\hspace{2.em} \Mvariable{v121}\backslash "\multsp \Mvariable{is}\multsp \Mvariable{similar}\multsp \Mvariable{to}\multsp \Mvariable{existing}\multsp
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\MathEnd{MathArray}\)}

\dispSFoutmath{
\Bigg(\NoBreak \MathBegin{MathArray}[c]{c}
	\Mvariable{u3} \\
	\Mvariable{x4} \\
	\Mvariable{x2} \\
	-\Mvariable{u1}-\Mvariable{u2}+\Mvariable{x1} 
  \MathEnd{MathArray}\NoBreak \Bigg)
}

\dispSFinmath{
\MathBegin{MathArray}{l}
\Muserfunction{v12}[[2]]=\Mvariable{u1}+\Mvariable{u2}-2\Mvariable{x3};  \\
\noalign{\vspace{0.625ex}}  \\
\Muserfunction{dis}[\Mvariable{v122}=\Mfunction{GroebnerBasis}[\Mvariable{v12},\{\Mvariable{x1},\Mvariable{x2},\Mvariable{x3},\Mvariable{x4},\Mvariable{u1},\Mvariable{u2},\Mvariable{u3}\}]]
\MathEnd{MathArray}
}

\Message{\(\MathBegin{MathArray}{l}
\Mfunction{General}::\Mvariable{spell1}:\multsp \Mvariable{Possible}\multsp \Mvariable{spelling}\multsp \Mvariable{error}:\multsp \Mvariable{new}\multsp
\Mvariable{symbol}\multsp \Mvariable{name}\multsp \backslash "  \\
\noalign{\vspace{0.5ex}}
\hspace{2.em} \Mvariable{v122}\backslash "\multsp \Mvariable{is}\multsp \Mvariable{similar}\multsp \Mvariable{to}\multsp \Mvariable{existing}\multsp
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\MathEnd{MathArray}\)}

\dispSFoutmath{
\Bigg(\NoBreak \MathBegin{MathArray}[c]{c}
	\Mvariable{u3}-2\multsp \Mvariable{x4} \\
	\Mvariable{u1}+\Mvariable{u2}-2\multsp \Mvariable{x3} \\
	\Mvariable{x2}-\Mvariable{u3} \\
	-\Mvariable{u1}-\Mvariable{u2}+\Mvariable{x1} 
  \MathEnd{MathArray}\NoBreak \Bigg)
}

\dispSFinmath{
\{\Mvariable{v11},\Mvariable{v121},\Mvariable{v122},\Mvariable{v21},\Mvariable{v221},\Mvariable{v222}\}
}

\dispSFoutmath{
\MathBegin{MathArray}{l}
\{\{\Mvariable{u1},\Mvariable{u3}\multsp \Mvariable{x3}-\Mvariable{u2}\multsp \Mvariable{x4},\Mvariable{x2}-\Mvariable{u3},\Mvariable{x1}-\Mvariable{u2}\},\{\Mvariable{u3},\Mvariable{x4},\Mvariable{x2},-\Mvariable{u1}-\Mvariable{u2}+\Mvariable{x1}\},
 \\
\noalign{\vspace{0.625ex}}
\hspace{1.em} \{\Mvariable{u3}-2\multsp \Mvariable{x4},\Mvariable{u1}+\Mvariable{u2}-2\multsp \Mvariable{x3},\Mvariable{x2}-\Mvariable{u3},-\Mvariable{u1}-\Mvariable{u2}+\Mvariable{x1}\},
 \\
\noalign{\vspace{0.625ex}}
\hspace{1.em} \{\Mvariable{u3},\Mvariable{x4},\Mvariable{x2}\},\{\Mvariable{u3},\Mvariable{u1}-\Mvariable{u2},\Mvariable{x2},\Mvariable{x1}\},\{\Mvariable{u3},\Mvariable{u1}-\Mvariable{u2},\Mvariable{x4},\Mvariable{x2}\}\}\\
\MathEnd{MathArray}
}

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\Mvariable{v122}
}

\dispSFoutmath{
\{\Mvariable{u3}-2\multsp \Mvariable{x4},\Mvariable{u1}+\Mvariable{u2}-2\multsp \Mvariable{x3},\Mvariable{x2}-\Mvariable{u3},-\Mvariable{u1}-\Mvariable{u2}+\Mvariable{x1}\}
}

\dispSFinmath{
\Mvariable{g1}=-\Mvariable{con}
}

\dispSFoutmath{
-\Mvariable{con}
}

\dispSFinmath{
\Mfunction{GroebnerBasis}[\{\Mvariable{v122},1-y\multsp \Mvariable{g1}\},\{\Mvariable{u1},\Mvariable{u2},\Mvariable{u3},\Mvariable{x1},\Mvariable{x2},\Mvariable{x3},\Mvariable{x4}\}]
}

\dispSFoutmath{
\{\Mvariable{con}\multsp y+1,2\multsp \Mvariable{x4}-\Mvariable{x2},2\multsp \Mvariable{x3}-\Mvariable{x1},\Mvariable{u3}-2\multsp \Mvariable{x4},\Mvariable{u1}+\Mvariable{u2}-2\multsp
\Mvariable{x3}\}
}


\end{document}
