\begin{chapter}{Polynome f{\"u}r den Benchmarktest}\label{Polynome f{\"u}r den Benchmarktest}
Nachstehend sind die Polynome f{\"u}r den Benchmarktest aufgelistet. Die Menge der \texttt{Voraussetzungen} stammt aus dem
erweiterten Satz von Morley (Satz \ref{Erweiterter Satz von Morley}), die \gb\ dieser Menge ist hier nicht angegeben.
F{\"u}r die beiden Mengen \texttt{Beweis1} und \texttt{Beweis2}, die hier nicht angegeben sind, da sie die
\texttt{Voraussetzungen} nur um ein Polynom erweitern, muss die \gb\ $\{1\}$ sein; dann ist der erweiterte Satz von
Morley bewiesen.

\dispTFinmath{ \Mvariable{Voraussetzungen}=\Mfunction{Map}[\Mfunction{Numerator}[\Mfunction{Together}[\#]]\multsp
\&,\multsp \Mvariable{Gleichungen}] }

\dispTFoutmath{ \MathBegin{MathArray}[p]{l} \big\{3\multsp {{\Mvariable{u2}}^5}-6\multsp \Mvariable{u1}\multsp
{{\Mvariable{u2}}^4}-3\multsp \Mvariable{x5}\multsp {{\Mvariable{u2}}^4}+3\multsp
{{\Mvariable{u1}}^2}\multsp {{\Mvariable{u2}}^3}-10\multsp {{\Mvariable{u3}}^2}\multsp {{\Mvariable{u2}}^3}+  \\
\noalign{\vspace{0.726563ex}} \hspace{2.em} 6\multsp \Mvariable{u1}\multsp \Mvariable{x5}\multsp
{{\Mvariable{u2}}^3}+18\multsp \Mvariable{u1}\multsp {{\Mvariable{u3}}^2}\multsp {{\Mvariable{u2}}^2}-3\multsp
{{\Mvariable{u1}}^2}\multsp \Mvariable{x5}\multsp {{\Mvariable{u2}}^2}-6\multsp {{\Mvariable{u3}}^2}\multsp
\Mvariable{x5}\multsp
{{\Mvariable{u2}}^2}+  \\
\noalign{\vspace{0.601563ex}} \hspace{2.em} 3\multsp {{\Mvariable{u3}}^4}\multsp \Mvariable{u2}-9\multsp
{{\Mvariable{u1}}^2}\multsp {{\Mvariable{u3}}^2}\multsp \Mvariable{u2}+6\multsp \Mvariable{u1}\multsp
{{\Mvariable{u3}}^2}\multsp \Mvariable{x5}\multsp \Mvariable{u2}-3\multsp {{\Mvariable{u3}}^4}\multsp
\Mvariable{x5}+{{\Mvariable{u1}}^2}\multsp
{{\Mvariable{u3}}^2}\multsp \Mvariable{x5},  \\
\noalign{\vspace{0.601563ex}} \hspace{1.em} {{\Mvariable{u3}}^5}-3\multsp \Mvariable{x6}\multsp
{{\Mvariable{u3}}^4}-3\multsp {{\Mvariable{u1}}^2}\multsp {{\Mvariable{u3}}^3}-6\multsp {{\Mvariable{u2}}^2}\multsp
{{\Mvariable{u3}}^3}+6\multsp \Mvariable{u1}\multsp \Mvariable{u2}\multsp
{{\Mvariable{u3}}^3}+{{\Mvariable{u1}}^2}\multsp
\Mvariable{x6}\multsp {{\Mvariable{u3}}^2}-  \\
\noalign{\vspace{0.601563ex}} \hspace{2.em} 6\multsp {{\Mvariable{u2}}^2}\multsp \Mvariable{x6}\multsp
{{\Mvariable{u3}}^2}+6\multsp \Mvariable{u1}\multsp \Mvariable{u2}\multsp \Mvariable{x6}\multsp
{{\Mvariable{u3}}^2}+9\multsp {{\Mvariable{u2}}^4}\multsp \Mvariable{u3}-18\multsp \Mvariable{u1}\multsp
{{\Mvariable{u2}}^3}\multsp
\Mvariable{u3}+  \\
\noalign{\vspace{0.601563ex}} \hspace{2.em} 9\multsp {{\Mvariable{u1}}^2}\multsp {{\Mvariable{u2}}^2}\multsp
\Mvariable{u3}-3\multsp {{\Mvariable{u2}}^4}\multsp \Mvariable{x6}+6\multsp \Mvariable{u1}\multsp
{{\Mvariable{u2}}^3}\multsp \Mvariable{x6}-3\multsp {{\Mvariable{u1}}^2}\multsp {{\Mvariable{u2}}^2}\multsp
\Mvariable{x6},
 \\
\noalign{\vspace{0.6875ex}} \hspace{1.em} -{\sqrt{3}}\multsp {{\Mvariable{x5}}^2}\multsp
{{\Mvariable{u2}}^2}-{\sqrt{3}}\multsp {{\Mvariable{x6}}^2}\multsp {{\Mvariable{u2}}^2}+{\sqrt{3}}\multsp
\Mvariable{x3}\multsp \Mvariable{x5}\multsp {{\Mvariable{u2}}^2}-\Mvariable{x4}\multsp \Mvariable{x5}\multsp
{{\Mvariable{u2}}^2}+\Mvariable{x3}\multsp
\Mvariable{x6}\multsp {{\Mvariable{u2}}^2}+  \\
\noalign{\vspace{0.6875ex}} \hspace{2.em} {\sqrt{3}}\multsp \Mvariable{x4}\multsp \Mvariable{x6}\multsp
{{\Mvariable{u2}}^2}+{\sqrt{3}}\multsp \Mvariable{u1}\multsp {{\Mvariable{x5}}^2}\multsp
\Mvariable{u2}+{\sqrt{3}}\multsp \Mvariable{u1}\multsp {{\Mvariable{x6}}^2}\multsp \Mvariable{u2}-{\sqrt{3}}\multsp
\Mvariable{u1}\multsp \Mvariable{x3}\multsp
\Mvariable{x5}\multsp \Mvariable{u2}+  \\
\noalign{\vspace{0.6875ex}} \hspace{2.em} \Mvariable{u1}\multsp \Mvariable{x4}\multsp \Mvariable{x5}\multsp
\Mvariable{u2}-\Mvariable{u1}\multsp \Mvariable{x3}\multsp \Mvariable{x6}\multsp \Mvariable{u2}-{\sqrt{3}}\multsp
\Mvariable{u1}\multsp \Mvariable{x4}\multsp \Mvariable{x6}\multsp \Mvariable{u2}-{\sqrt{3}}\multsp
{{\Mvariable{u3}}^2}\multsp
{{\Mvariable{x5}}^2}-\Mvariable{u1}\multsp \Mvariable{u3}\multsp {{\Mvariable{x5}}^2}-  \\
\noalign{\vspace{0.6875ex}} \hspace{2.em} {\sqrt{3}}\multsp {{\Mvariable{u3}}^2}\multsp
{{\Mvariable{x6}}^2}-\Mvariable{u1}\multsp \Mvariable{u3}\multsp {{\Mvariable{x6}}^2}+{\sqrt{3}}\multsp
{{\Mvariable{u3}}^2}\multsp \Mvariable{x3}\multsp \Mvariable{x5}+\Mvariable{u1}\multsp \Mvariable{u3}\multsp
\Mvariable{x3}\multsp \Mvariable{x5}-{{\Mvariable{u3}}^2}\multsp
\Mvariable{x4}\multsp \Mvariable{x5}+  \\
\noalign{\vspace{0.6875ex}} \hspace{2.em} {\sqrt{3}}\multsp \Mvariable{u1}\multsp \Mvariable{u3}\multsp
\Mvariable{x4}\multsp \Mvariable{x5}+{{\Mvariable{u3}}^2}\multsp \Mvariable{x3}\multsp \Mvariable{x6}-{\sqrt{3}}\multsp
\Mvariable{u1}\multsp \Mvariable{u3}\multsp \Mvariable{x3}\multsp \Mvariable{x6}+{\sqrt{3}}\multsp
{{\Mvariable{u3}}^2}\multsp
\Mvariable{x4}\multsp \Mvariable{x6}+\Mvariable{u1}\multsp \Mvariable{u3}\multsp \Mvariable{x4}\multsp \Mvariable{x6},  \\
\noalign{\vspace{0.726563ex}} \hspace{1.em} -\Mvariable{x4}\multsp {{\Mvariable{u2}}^2}+2\multsp \Mvariable{u3}\multsp
\Mvariable{x3}\multsp \Mvariable{u2}+{{\Mvariable{u3}}^2}\multsp \Mvariable{x4},\Mvariable{x2}\multsp
{{\Mvariable{x5}}^3}-2\multsp \Mvariable{x4}\multsp {{\Mvariable{x5}}^3}-2\multsp \Mvariable{x2}\multsp
\Mvariable{x3}\multsp
{{\Mvariable{x5}}^2}+2\multsp \Mvariable{x1}\multsp \Mvariable{x4}\multsp {{\Mvariable{x5}}^2}+  \\
\noalign{\vspace{0.601563ex}} \hspace{2.em} 2\multsp \Mvariable{x3}\multsp \Mvariable{x4}\multsp
{{\Mvariable{x5}}^2}-\Mvariable{x1}\multsp \Mvariable{x6}\multsp {{\Mvariable{x5}}^2}+2\multsp \Mvariable{x3}\multsp
\Mvariable{x6}\multsp {{\Mvariable{x5}}^2}+\Mvariable{x2}\multsp {{\Mvariable{x3}}^2}\multsp
\Mvariable{x5}-\Mvariable{x2}\multsp
{{\Mvariable{x4}}^2}\multsp \Mvariable{x5}+\Mvariable{x2}\multsp {{\Mvariable{x6}}^2}\multsp \Mvariable{x5}-  \\
\noalign{\vspace{0.601563ex}} \hspace{2.em} 2\multsp \Mvariable{x4}\multsp {{\Mvariable{x6}}^2}\multsp
\Mvariable{x5}-2\multsp \Mvariable{x1}\multsp \Mvariable{x3}\multsp \Mvariable{x4}\multsp \Mvariable{x5}-2\multsp
{{\Mvariable{x3}}^2}\multsp \Mvariable{x6}\multsp \Mvariable{x5}+2\multsp {{\Mvariable{x4}}^2}\multsp
\Mvariable{x6}\multsp
\Mvariable{x5}-\Mvariable{x1}\multsp {{\Mvariable{x6}}^3}+2\multsp \Mvariable{x3}\multsp {{\Mvariable{x6}}^3}-  \\
\noalign{\vspace{0.601563ex}} \hspace{2.em} 2\multsp \Mvariable{x2}\multsp \Mvariable{x3}\multsp
{{\Mvariable{x6}}^2}+2\multsp \Mvariable{x1}\multsp \Mvariable{x4}\multsp {{\Mvariable{x6}}^2}-2\multsp
\Mvariable{x3}\multsp \Mvariable{x4}\multsp {{\Mvariable{x6}}^2}+\Mvariable{x1}\multsp {{\Mvariable{x3}}^2}\multsp
\Mvariable{x6}-\Mvariable{x1}\multsp
{{\Mvariable{x4}}^2}\multsp \Mvariable{x6}+2\multsp \Mvariable{x2}\multsp \Mvariable{x3}\multsp \Mvariable{x4}\multsp \Mvariable{x6},  \\
\noalign{\vspace{0.601563ex}} \hspace{1.em} -2\multsp \Mvariable{u3}\multsp {{\Mvariable{u1}}^2}+\Mvariable{x2}\multsp
{{\Mvariable{u1}}^2}+2\multsp \Mvariable{u2}\multsp \Mvariable{u3}\multsp
\Mvariable{u1}+2\multsp \Mvariable{u3}\multsp \Mvariable{x1}\multsp \Mvariable{u1}-  \\
\noalign{\vspace{0.726563ex}} \hspace{2.em} 2\multsp \Mvariable{u2}\multsp \Mvariable{x2}\multsp
\Mvariable{u1}-2\multsp \Mvariable{u2}\multsp \Mvariable{u3}\multsp \Mvariable{x1}+{{\Mvariable{u2}}^2}\multsp
\Mvariable{x2}-{{\Mvariable{u3}}^2}\multsp \Mvariable{x2}\big\}\\
\MathEnd{MathArray} }

\dispTFinmath{ \MathBegin{MathArray}{l}
(*\Mvariable{conclusion}*)  \\
\noalign{\vspace{0.625ex}}\Mvariable{con1}=\Muserfunction{sqrdist}[D,E]-\Muserfunction{sqrdist}[E,F]\\
\MathEnd{MathArray} }

\dispTFoutmath{ {{\Mvariable{u2}}^2}-2\multsp \Mvariable{x1}\multsp
\Mvariable{u2}+{{\Mvariable{u3}}^2}-{{\Mvariable{x3}}^2}-{{\Mvariable{x4}}^2}-2\multsp \Mvariable{u3}\multsp
\Mvariable{x2}+2\multsp \Mvariable{x1}\multsp \Mvariable{x3}+2\multsp \Mvariable{x2}\multsp \Mvariable{x4} }

\dispTFinmath{ \Mvariable{con2}=\Muserfunction{sqrdist}[D,E]-\Muserfunction{sqrdist}[D,F] }

\dispTFoutmath{ {{\Mvariable{x1}}^2}-2\multsp \Mvariable{u2}\multsp
\Mvariable{x1}+{{\Mvariable{x2}}^2}-{{\Mvariable{x3}}^2}-{{\Mvariable{x4}}^2}-2\multsp \Mvariable{u3}\multsp
\Mvariable{x2}+2\multsp \Mvariable{u2}\multsp \Mvariable{x3}+2\multsp \Mvariable{u3}\multsp \Mvariable{x4} }

\dispTFinmath{ \Mvariable{Beweis1}=\Mfunction{Union}[\Mvariable{Voraussetzungen},\{1-y\multsp \Mvariable{con1}\}]; }

\dispTFinmath{ \Mvariable{Beweis2}=\Mfunction{Union}[\Mvariable{Voraussetzungen},\{1-y\multsp \Mvariable{con2}\}]; }

Die Demoversionen der Computeralgebrasysteme finden sich f{\"u}r \ma\ unter \cite{mathematica}, Maple unter \cite{maple},
MuPad unter \cite{mupad} und Reduce unter \cite{reduce}. Zu Magma\cite{magma} habe ich leider keine Demoversion im Netz
finden k{\"o}nnen.
\end{chapter}
