\chapter{$B%$%G%"%k$NJ,2r(B}
$B%$%G%"%k(B $I \subset R=K[X]$ $B$KBP$7(B, $I=I_1\cap I_2$ $B$H=q$1$k;~(B, 
$V(I) = V(I_1) \cup V(I_2)$ $B$,@.$jN)$D(B. $B$9$J$o$A(B, $B%$%G%"%k$NJ,2r(B
$B$ONmE@$NJ,2r$rM?$($k(B. $B$h$j>\$7$/8@$($P(B, $BNmE@$NJ,2r$O(B radical $B$N(B
$BJ,2r$KBP1~$9$k(B. $BNmE@$r2DG=$J8B$jJ,2r$9$k$3$H$O(B, $BBe?tE*=89g$r(B
$B4{LsJ,2r$9$k$3$H$KBP1~$9$k(B. $BJ}Dx<0$K$D$$$F8@$($P(B, $B0lHL$K2r$N(B
$BJ,2r$K$h$j(B, $B$h$j07$$$d$9$$2r$NI=8=$rM?$($k$3$H$,$G$-$k(B. 

\section{$BAG%$%G%"%k(B, $B=`AG%$%G%"%k(B, $B=`AG%$%G%"%kJ,2r(B}

$B0J2<(B, $R=K[X]$ $B$r8GDj$7$F9M$($k(B. 

\begin{df}
$B%$%G%"%k(B $I \subset R$ $B$,AG%$%G%"%k(B (prime ideal)$B$H$O(B, 
\begin{center}
$ab \in I$ $B$+$D(B $a\notin I$ $B$J$i$P(B $b \in I$
\end{center}
$B$J$k$3$H(B. $B$3$l$O(B $R/I$ $B$,@00h$H$J$k$3$H$HF1CM(B. 
\end{df}

\begin{df}
$B%$%G%"%k(B $I \subset R$ $B$,=`AG%$%G%"%k(B (primary ideal)$B$H$O(B, 
\begin{center}
$ab \in I$ $B$+$D(B $a\notin I$ $B$J$i$P(B $b \in \sqrt{I}$
\end{center}
$B$J$k$3$H(B. 
\end{df}

\begin{df}
$B%$%G%"%k(B $I \subset R$ $B$,(B $I=\sqrt{I}$ $B$rK~$?$9$H$-(B $I$ $B$O(B radical $B%$%G%"%k(B
$B$G$"$k$H$$$&(B. $\sqrt{I}$ $B$O(B radical $B%$%G%"%k$G$"$k(B. 
\end{df}

\begin{lm}
$\sqrt{I} = \cap_{I\subset P:prime}P$
\end{lm}
\proof $I \subset P$ $B$J$i$P(B $\sqrt{I} \subset \sqrt{P}=P$ $B$h$j(B
$\subset$ $B$O(B OK. $B1&JU$,:8JU$r??$K4^$`$H$9$l$P(B, $B$"$k(B
$f \in \cap_{I\subset P:prime}P \setminus \sqrt{I}$ $B$,B8:_$9$k(B. $B$3$N$H$-(B, 
$S=\{f,f^2,\cdots,\}$ $B$H$*$1$P(B, 
$S \cap \sqrt{I} = \emptyset$. 
$F = \{J:$ $B%$%G%"%k(B $\mid \sqrt{I} \subset J$ $B$+$D(B $S \cap J = \emptyset \}$ $B$H(B
$B$*$/$H(B, $F \neq \emptyset$ $B$G(B, $BJq4^4X78$K4X$7$F5"G<E*(B. $B$h$C$F6KBg85(B $J_0$
$B$,B8:_$9$k(B. 

\noi
\underline{$B<gD%(B\,} $J_0$ $B$OAG(B. 

\noi
\proof $ab\in J_0$ $B$+$D(B $a, b\notin J_0$ $B$H$9$k(B. $J_0$ $B$N6KBg@-$h$j(B, 
$S \cap (J_0+Id(a)) \neq \emptyset$ $B$+$D(B $S \cap (J_0+Id(b)) \neq \emptyset$.
$B$9$J$o$A(B, $B$"$k<+A3?t(B $s,t > 0$, $c,d \in J_0$, $e,f \in R$ $B$,B8:_$7$F(B, 
$f^s=c+ae$, $f^t=d+bf$ $B$H=q$1$k(B. $B$9$k$H(B $f^{s+t}=abef+cd+aed+cbf \in J_0$
$B$H$J$jL7=b(B. $B$h$C$F(B $J_0$ $B$OAG(B. \qed

\noi
$B$3$N<gD%$K$h$j(B, $f \notin J_0$ $B$+$D(B $I \subset J_0$ $B$J$kAG%$%G%"%k(B $J_0$ $B$,(B
$BB8:_$9$k(B. $B$3$l$OL7=b(B. \qed

\begin{lm}
$B%$%G%"%k(B $I \subset R$ $B$,=`AG$J$i$P(B, $\sqrt{I}$ $B$OAG%$%G%"%k(B.
\end{lm}

\begin{df}
$B%$%G%"%k(B $I \subset R$ $B$,=`AG$G(B $\sqrt{I}=P$ $B$N$H$-(B, $I$ $B$O(B $P$-$B=`AG$H$$$&(B. 
$B$^$?(B $P$ $B$rIUB0AG%$%G%"%k(B (associated prime ideal)$B$H8F$V(B. 
\end{df}

\begin{df}
$B%$%G%"%k(B $I \subset R$ $B$,(B
\begin{center}
$I = I_1 \cap I_2 \Rightarrow I = I_1$ $B$^$?$O(B $I = I_2$
\end{center}
$B$rK~$?$9$H$-(B, $I$ $B$O4{Ls$H$$$&(B. 
\end{df}

\begin{lm}
$B4{Ls%$%G%"%k$O=`AG(B. 
\end{lm}
\proof $I$ $B$,4{Ls$H$7(B, $fg \in I$ $B$+$D(B $f\notin I$ $B$H$9$k(B. 
$I:g^\infty = I:g^s$ $B$J$k(B $s$ $B$r$H$k$H(B, 
$$I = (I+Id(g^s)) \cap (I+Id(f)).$$
$B$3$l$O(B, $h = a+bg^s = c+df$ ( $a,c \in I$ ) $B$J$k(B $h$ $B$r$H$k$H(B, 
$$ bg^{s+1} = (c-a)g+dfg \in I \Rightarrow b \in I:g^{s+1}=I:g^s \Rightarrow
bg^s \in I \Rightarrow h \in I$$
$B$+$i$o$+$k(B. $I \neq I+Id(f)$ $B$@$+$i(B $I=I+Id(g^s)$. $B$9$J$o$A(B $g \in \sqrt{I}$.
\qed

\begin{th}
$BG$0U$N%$%G%"%k(B $I \subset R$ $B$OM-8B8D$N=`AG%$%G%"%k$N8r$o$j$H$7$F=q$1$k(B. 
\end{th}
\proof $I$ $B$,M-8B8D$N4{Ls%$%G%"%k$NM-8B8D$N8r$o$j$G=q$1$k$3$H$r(B
$B$$$($P$h$$(B. $I$ $B$,4{Ls$G$J$1$l$P(B, $I = I_1 \cap I_2$ $B$H(B, $I$ $B$r??$K(B
$B4^$`%$%G%"%k$N8r$o$j$G=q$1$k(B. $I_1$ $B$,4{Ls$G$J$1$l$P(B, $I_1$ $B$OF1MM$N(B
$B8r$o$j$G=q$1$k(B. $B$b$7(B, $B$3$NA`:n$,M-8B2s$G=*$i$J$1$l$P(B, $B%$%G%"%k$N??$N(B
$BL58BA}BgNs$,B8:_$9$k$3$H$K$J$j(B, $R$ $B$N(B Noether $B@-$KH?$9$k(B. \qed

\begin{df}
$B%$%G%"%k(B $I$ $B$N=`AGJ,2r$H$O(B, $I=\cap_{i=1}^r Q_i$ ($Q_i$:$B=`AG(B) $B$J$kI=(B
$B<($N$3$H(B. $B3F(B $Q_i$ $B$r=`AG@.J,$H8F$V(B. $B=`AGJ,2r$,(B minimal $B$H$O(B, $BA4$F$N(B 
$\sqrt{Q_i}$ $B$,Aj0[$J$j(B, $\cap_{j\neq i} Q_j {\not \subset} Q_i$ $B$J$k(B
$B$3$H(B.
\end{df}

\begin{lm}
$B=`AG%$%G%"%k(B $I$, $J$ $B$K$D$$$F(B, $\sqrt{I} = \sqrt{J}$ $B$J$i$P(B $I\cap J$ $B$b=`AG(B.
\end{lm}
\proof $ab \in I\cap J$ $B$+$D(B $a\notin I\cap J$ $B$H$9$k(B. $a\notin I$
$B$N$H$-(B, $b \in \sqrt{I}$, $a\notin J$ $B$N$H$-(B $b\in \sqrt{J}$ $B$,@.$jN)$D(B. 
$\sqrt{I\cap J} = \sqrt{I} \cap \sqrt{J}$ $B$h$j(B, $b \in \sqrt{I\cap J}$. \qed

\begin{th}
$BG$0U$N%$%G%"%k(B $I \subset R$ $B$O(B minimal $B$J=`AGJ,2r$r;}$D(B. 
\end{th}
\proof $BJdBj$h$j(B, $BIUB0AG%$%G%"%k$,0lCW$9$k=`AG@.J,$O8r$o$j$r$H$k$3$H$K(B
$B$h$j0l$D$K$^$H$a$k$3$H$,$G$-$k(B. $B$=$N8e(B, $\cap_{j\neq i} Q_j \subset Q_i$
$B$J$k(B $Q_i$ $B$r<h$j=|$$$F$b;D$j$O(B $I$ $B$N=`AGJ,2r$H$J$k$+$i(B, $B$3$l$r(B
$B7+$jJV$7$F(B minimal $B$J=`AGJ,2r$rF@$k(B. \qed

\noi
$B$5$i$K(B, $B<!$,@.$jN)$D(B. 
\begin{th}
minimal $B$J=`AGJ,2r$ND9$5$O0l0UE*$G(B, $BIUB0AG%$%G%"%k$O=89g$H$7$F0lCW$9$k(B. 
\end{th}

\begin{df}
$B%$%G%"%k(B $I$ $B$N=`AG@.J,(B $Q$ $B$NIUB0AG%$%G%"%k$,6K>.$N;~8IN)(B (isolated) $B$H(B
$B$$$&(B. $B$=$&$G$J$$$H$-KdKW(B (embedded) $B$H$$$&(B. 
\end{df}

\begin{re}
$B8IN)(B, $BKdKW$H$$$&L>>N$O(B, $B$=$N(B variety $B$NMM;R$K$h$k(B. $B=`AG@.J,(B $Q$ $B$NIUB0(B
$BAG%$%G%"%k$r(B $P$ $B$H$9$k(B. $Q$ $B$,8IN)$N$H$-(B, $P$ $B$OB>$NIUB0AG%$%G%"%k$r(B
$B4^$^$J$$$+$i(B, variety $B$G$_$l$P(B, $V(P)$ $B$OB>$N@.J,$,Dj5A$9$k(B variety $B$K(B
$B4^$^$l$J$$(B. $B0lJ}(B, $Q$ $B$,KdKW$J$i$P(B, $P$ $B$O$"$kIUB0AG%$%G%"%k(B $P'$ $B$r4^(B
$B$`$+$i(B, variety $B$G$_$l$P(B $V(P) \subset V(P')$ $B$9$J$o$AKdKW$7$F$$$k(B. 
\end{re}

\section{$B=`AGJ,2r$N35N,(B}

$B=`AGJ,2r$N$?$a$N<g$J<jCJ$O(B, {\bf $B%$%G%"%k>&(B},
{\bf extension}, {\bf contraction} $B$*$h$SB?9`<0$N0x?tJ,2r$G$"$k(B. 

\begin{lm}
$B%$%G%"%k(B $I$, $f \in R \setminus I$ $B$KBP$7(B, $I:f^m = I:f^\infty$ $B$J$k(B
$m$ $B$r$H$l$P(B, 
$$I = I:f^\infty \cap (I+Id(f^m))$$
\end{lm}
\proof $h \in$ $B1&JU$H$9$k$H(B, $hf^m \in I$ $B$+$D(B $h=a+bf^m$ ($a \in I$)
$B$H$+$1$k(B. $B$h$C$F(B $bf^{2m} = hf^m-af^m \in I$ $B$9$J$o$A(B $b \in I:f^{2m}=I:f^m$
$B$3$l$+$i(B $h \in I$. \qed

\begin{df}(extension)\\
$Y \subset X$ $B$KBP$7(B, $I^e=K(Y)[X\setminus Y]I$ $B$HDj5A$9$k(B.
\end{df}

\begin{df}
$B%$%G%"%k(B $\dim(I)=d$ $B$H$9$k$H(B, $B<!85$NDj5A$K$h$j(B, 
$|Y|=d$ $B$J$k(B independent set $Y \subset X$ $B$,$H$l$k(B. 
$B$3$l$r(B maximally independent set $B$H$h$V(B. 
\end{df}

\begin{lm}
$Y$ $B$,(B maximally independent set $B$N$H$-(B
$I^e$ $B$O(B $K(Y)$ $B>e(B 0 $B<!85%$%G%"%k(B. 
\end{lm}
\proof $BDj5A$K$h$j(B, $BA4$F$N(B $x \in X\setminus Y$ $B$KBP$7(B, 
$I \cap K[\{x\} \cup Y] \neq 0$ $B$@$+$i(B, $K(Y)$ $B>e$G9M$($l$P(B
$x$ $B$N0lJQ?tB?9`<0$,(B $I^e$ $BCf$KB8:_$9$k$3$H$K$J$k(B. $B$h$C$F(B $I^e$ $B$O(B
0 $B<!85(B.\qed

\begin{lm}
$<$ $B$r(B $Y < (X\setminus Y)$ $B$J$k(B block order $B$H$9$k(B. $G$ $B$,(B $I$ $B$N(B
$<$ $B$K4X$9$k%0%l%V%J4pDl$J$i$P(B, $G$ $B$O(B, $I^e$ $B$N(B, $BF1$8(B order $B$K(B
$B4X$9$k%0%l%V%J4pDl(B. 
\end{lm}

\begin{df}(contraction)\\
$B%$%G%"%k(B $J \subset K(Y)[X\setminus Y]$ $B$KBP$7(B, $J\cap K[X]$ $B$r(B
$J$ $B$N(B contraction $B$H$h$S(B, $J^c$ $B$H=q$/(B. 
\end{df}

\begin{pr}
$I$ $B$r%$%G%"%k(B, $<$ $B$r(B $Y < (X\setminus Y)$ $B$J$k(B block order $B$H$7(B, $G$ 
$B$r(B $<$ $B$K4X$9$k(B $I$ $B$N%0%l%V%J4pDl$H$9$k(B. $B$3$N$H$-(B, 
$$f = \LCM\{HC(g) \mid g \in G\}$$
($B$?$@$7(B, $HC(g)$ $B$O(B $K(Y)[X\setminus Y]$ $B$N85$H$7$F$H$k(B) $B$H$9$l$P(B, 
$I^{ec} = I:f^\infty$
\end{pr}

$B$3$l$i$H(B, $B8e$G=R$Y$k(B 0 $B<!85%$%G%"%k$N=`AGJ,2r$rMQ$$$F(B, $B<!$N%"%k%4%j%:%`(B
$B$,F@$i$l$k(B. 

\begin{al} \cite{GTZ}
\label{gtz}
\begin{tabbing}
Input : $B%$%G%"%k(B $I \in R=K[X]$\\
Output : \= $I = \cap Q_i$ ($I$ $B$N(B minimal $B$J=`AGJ,2r(B)\\
\> $P_i = \sqrt{Q_i}$ ($Q_i$ $B$NIUB0AG%$%G%"%k(B)\\

$Y \leftarrow$ maximally independent set modulo $I$\\
$\cap \bar{Q_i} \leftarrow I^e$ (0 $B<!85(B) $B$N(B $K(Y)$ $B>e$N=`AGJ,2r(B\\
$(f,s) \leftarrow$ $I^{ec} = I:f^\infty = I:f^s$ $B$J$k(B $f$ $B$*$h$S(B $s$\\
$\cap R_j \leftarrow I+Id(f^s)$ $B$N=`AGJ,2r(B\\
return $\bar{Q_i}^c \cap (\cap R_j)$
\end{tabbing}
\end{al}

\noi
$I+Id(f^s)$ $B$O(B $I$ $B$r??$K4^$`$+$i(B, $BDd;_@-$OJ]>Z$5$l$k(B. $B$3$N%"%k%4%j%:%`(B
$B$r<B8=$9$k$?$a$K$O(B, 

\begin{itemize}
\item 0 $B<!85%$%G%"%k$N=`AGJ,2r(B

\item maximally independent set $B$NA*$SJ}(B
\end{itemize}

\noi
$B$N%"%k%4%j%:%`$rM?$($kI,MW$,$"$k(B. $B$3$NFb(B, maximally independent set
$B$K4X$7$F$O(B, $B<!$NL?Bj$,$"$k(B. 

\begin{pr}
$I$ $B$r%$%G%"%k$H$7(B, $MB_<$ $B$r(B, $R/I$ $B$N(B,
$BA4<!?t$D$-(B order  $<$ $B$K4X$9$k(B monomial $B$K$h$k(B
$K$-$B4pDl$H$9$k(B. 
$B$3$N$H$-(B, $\dim(I) = =max(|U| \mid T(U) \subset MB_<)$.
\end{pr}

\noi
$B$3$NL?Bj$K4p$E$$$F(B, $B0l$D$N%0%l%V%J4pDl$+$i(B, $\dim(I)$ $B$r5a$a$k%"%k%4%j%:%`(B
$B$,9=@.$G$-$k(B. $B0J2<$G(B, 0 $B<!85%$%G%"%k$N=`AGJ,2r$K$D$$$F35N,$r=R$Y$k(B. 

\section{0 $B<!85%$%G%"%k$N=`AGJ,2r(B}

$I \subset R=K[X]$ $B$r(B 0 $B<!85%$%G%"%k$H$9$k(B. 

\begin{df}
$f \in K[X]$ $B$,(B $I$ $B$N(B separating element $B$H$O(B, 
$I$ $B$N(B $\overline{K}$ $B>e$NAj0[$kNmE@(B $a,b$ $B$KBP$7(B, $f(a) \neq f(b)$
$B$J$k$3$H(B. 
\end{df}

\begin{df}
$f\in K[X]$ $B$H$9$k(B. $t \notin X$ $B$J$kITDj85$r$H$j(B, $R[t]$ $B$N%$%G%"%k(B
$J = IR[t]+Id(t-f)$ $B$r9M$($l$P(B, $J$ $B$b(B 0 $B<!85%$%G%"%k$h$j(B, 
$(I+Id(t-f)) \cap K[t] = Id(g(t))$ $B$J$k%b%K%C%/$J(B $g \in K[t]$ $B$,B8:_$9$k(B. 
$g$ $B$r(B $f$ $B$N(B $I$ $B$K4X$9$k:G>.B?9`<0$H8F$V(B. 
\end{df}

\begin{pr}
$f$ $B$r(B $I$ $B$N(B separating element $B$H$7(B, $g$ $B$r(B $f$ $B$N:G>.B?9`<0$H$9$k(B. 
$B$3$N$H$-(B, 
$$g = g_1^{e_1}\cdots g_r^{e_r}$$
($g_i$ $B$O(B$K$ $B>e4{Ls(B) $B$H0x?tJ,2r$9$l$P(B, 
$$I = (I+g_1^{e_1}) \cap \cdots \cap (I+g_r^{e_r})$$
$B$O(B $I$ $B$N=`AGJ,2r$G(B, 
$$\sqrt{I} = \sqrt{I+g_1} \cap \cdots \cap \sqrt{I+g_r}$$
$B$O(B $\sqrt{I}$ $B$NAG%$%G%"%kJ,2r$H$J$k(B. $B$9$J$o$A(B, $\sqrt{I+g_i}$ $B$O(B
$I+g_i^{e_i}$ $B$NIUB0AG%$%G%"%k(B. 
\end{pr}

\noi
$B%$%G%"%k$N(B seratating element $B$O(B, $BD>@\5a$a$k$N$O:$Fq$G$"$k(B. $BDj5A$K$h$j(B, 
$\sqrt{I}$ $B$N(B separating element $B$,(B $I$ $B$N(B seratating element $B$H$J$k$3$H(B
$B$rMQ$$$F(B, $\sqrt{I}$ $B$r7W;;$7(B, separating element $B$r5a$a$k$3$H$r9M$($k(B. 

\begin{df}
$BBN(B $K$ $B$,40A4BN$H$O(B, $B4{LsB?9`<0$,A4$FJ,N%E*$G$"$k$3$H(B. 
\end{df}

\begin{re}
$B0J2<$NL?Bj(B, $B%"%k%4%j%:%`$G$O(B, $B4pACBN$,40A4BN$G$"$k$3$H$rMW5a$9$k$b$N$,(B
$B$$$/$D$+$"$k(B. $BI8?t(B 0 $B$NBN$OA4$F40A4BN$G$"$k(B. $B$^$?(B, $BM-8BBN$b40A4BN$G$"(B
$B$k$,(B, $BM-8BBN>e$NM-M}4X?tBN$O40A4BN$G$J$$(B. $B=`AGJ,2r$K$*$$$F$O4pACBN>e$N(B
$BM-M}4X?tBN$r78?t$H$9$kB?9`<04D$G$N7W;;$r9T$&$?$a(B, $B4pACBN$NI8?t$O(B 0
$B$K8B$i$l$k(B. 
\end{re}

\begin{pr}
$K$ $B$,40A4BN$J$i(B, 
\begin{center}
0 $B<!85%$%G%"%k(B $I$ $B$,(B radical $\Leftrightarrow$ $I$ $B$,(B, $B3FJQ?t$K$D$$$F(B
$B0lJQ?tL5J?J}B?9`<0$r4^$`(B. 
\end{center}
\end{pr}

\begin{pr}
$K$ $B$,40A4BN$H$9$k$H(B, 0 $B<!85(B radical $B%$%G%"%k(B $I$ $B$NNmE@$N8D?t$O(B
$\dim_K K[X]/I$ $B$KEy$7$$(B. 
\end{pr}

\begin{df}
$BB?9`<0(B $f$ $B$,(B $f = f_1^{e_1}\cdots f_m^{e_m}$ ($f_i$ $B$OL5J?J}(B) $B$H=q$1$?(B
$B$H$-(B, $f_1\cdots f_m$ $B$r(B $f$ $B$NL5J?J}ItJ,$H8F$V(B. 
\end{df}

\begin{co}
$I \cap K[x_i] = Id(f_i(x_i))$ $B$H$9$k(B. 
$$\sqrt{I} = I+Id(h_1,\cdots,h_n)$$
($h_i$ $B$O(B, $f_i$ $B$NL5J?J}ItJ,(B)
\end{co}

\noi
radical $B%$%G%"%k$KBP$7$F$O(B, separating element $B$NH=Dj$O<!$N$h$&$K(B
$B=R$Y$i$l$k(B. 

\begin{pr}
$K$ $B$r40A4BN$H$7(B, $B%$%G%"%k(B $I$ $B$,(B 0 $B<!85(B radical $B$H$9$k(B. $f$ $B$N:G>.(B
$BB?9`<0$r(B $g$ $B$H$9$k$H(B,
\begin{center}
$f$ $B$,(B separating element $\Leftrightarrow$ $\deg(f)=\dim_K R/I$
\end{center}
$B$3$N$h$&$J(B $f$ $B$OB8:_$9$k(B. $K$ $B$,L58BBN$J$i$P(B, $f$ $B$H$7$F(B $X$ $B$N85$N@~(B
$B7AOB$+$iA*$Y$k(B.
\end{pr}

\noi
$B0J>e$,(B, $B40A4BN>e$N(B 0 $B<!85%$%G%"%k$N=`AGJ,2r$N35N,$G$"$k(B. $BD>A0$N(B
$BL?Bj$K4XO"$7$F(B, $B<!$N$3$H$,@.$jN)$D(B. 

\begin{pr}(shape lemma)\\
$I$ $B$r40A4BN(B $K$ $B>e$N(B 0 $B<!85(B radical $B%$%G%"%k(B $B$H$7(B, 
$f$ $B$r(B separating element $B$H$9$k(B. 
$z << X$ $B$J$kG$0U$N=g=x$N$b$H$G(B, $R[z]$ $B$N%$%G%"%k(B $IR[z]+Id(z-f)$ $B$O(B
$$\{x_1-f_1(z),\cdots,x_n-f_n(z),z-f_z(z),m(z)\}$$
$B$H$$$&7A$N%0%l%V%J4pDl$r$b$D(B. $B$3$N7A$N4pDl$r(B shape basis $B$H8F$V(B. 
\end{pr}
\proof $z$ $B$N:G>.B?9`<0(B $m$ $B$O(B $f$ $B$N:G>.B?9`<0$K0l(B
$BCW$7(B, $B$=$N<!?t$O(B $\dim_K K[X]/I$ $B$HEy$7$/$J$k(B. $z << X$
$B$J$k=g=x$N$b$H$G$O(B, $B%0%l%V%J4pDl$O(B $m$ $B$r4^$_(B, $B%b%N%$%G%"%k$r(B
$B9M$($l$P(B, $m$ $B0J30$N85$NF,9`$O3FJQ?t$N(B 1 $B<!<00J30$G$O$"$j$($J$$(B. \qed\\
shape basis $B$O(B, 0 $B<!85%$%G%"%k$NNmE@$r?tCM$G5a$a$h$&$H$9$k>l9g$K(B, 
$B8+3]$1>eM-8z$J7A$r$7$F$$$k(B. $B<B:](B, $I$ $B$NNmE@$O(B, $f_n(x_n)$ $B$NNmE@$K(B
$B$h$j(B, 
$$\{(f_1(\alpha),\cdots,f_n(\alpha))\mid m(\alpha) = 0\}$$
$B$H=q$1$k(B. $B$7$+$7(B, $BM-M}?tBN>e$G<B:]$K(B shape basis $B$r5a$a$F8+$k$H(B, 
$m$ $B$N78?t$KHf$Y$F(B $f_i$ $B$N78?t$,6K$a$FBg$-$/$J$k$3$H$,(B
$BB?$$(B. $B$3$N:$Fq$r9nI~$9$k$?$a(B, $B<!$NJ}K!$,9M0F$5$l$?(B. 

\begin{pr}(rational univariate representation; RUR)\\
\label{RUR}
$BA0L?Bj$HF1$82>Dj$N$b$H$G(B, $IR[z]+Id(z-f)$ $B$N4pDl$H$7$F(B,
$$\{ m'x_1-g_1(z), \cdots, m'x_{z}-g_n(z), m(z)\}$$
$B$H$$$&7A$N$b$N$,$H$l$k(B.
\end{pr}
\noi
$m$ $B$O(B shape basis $B$N>l9g$H0lCW$9$k(B. $B$3$N4pDl$K$h$k$NNmE@$NI=8=$O(B,
$$\{({g_1(\alpha)\over m'(\alpha)},\cdots,{g_n(\alpha)\over
m'(\alpha)}) \mid m(\alpha)=0\}$$
\noi
$B$H=q$1$k(B. $BB?$/$N<BNc$K$*$$$F(B, $g_i$ $B$N3F78?t$,(B, $m$ $B$N78?t$HF1DxEY$N(B
$BBg$-$5$K2!$($i$l$k$3$H$,J,$+$C$F$*$j(B, 0 $B<!85(B radical$B$NNmE@$NI=8=$H$7$F(B
$B$O(B RUR $B$K$h$k$b$N$,M%$l$F$$$k$H$$$C$F$h$$(B. 
RUR $B$N7W;;K!$H$7$F$O(B, $BBP>N<0$K$h$kJ}K!$,:G=i$KDs0F$5$l$F$$$,(B, 
modular change of ordering $B$HF1MM$N<jK!$rE,MQ$9$k$3$H$b$G$-(B, RUR $B$,(B
$B7k2L$NBg$-$5DxEY$G7W;;$G$-$k(B \cite{NY2}. 

\section{$B=`AGJ,2r$NNc(B}
$B<!$NNc$O(B, symplectic integrator $B$H8F$P$l$k0BDj$J@QJ,%9%-!<%`$N(B
$B?tCM7W;;K!$K4X$7$F8=$l$?J}Dx<07O$G$"$k(B \cite{SYMP}. 

\vskip\baselineskip
{\small
$\left\{
\parbox[c]{6in}{
$d_1+d_2+d_3+d_4=1, c_1+c_2+c_3+c_4=1,$\\
$(6d_1c_2+(6d_1+6d_2)c_3+(6d_1+6d_2+6d_3)c_4)c_1
 +(6d_2c_3+(6d_2+6d_3)c_4)c_2+6d_3c_4c_3=1,$\\
$(3d_1^2+(6d_2+6d_3+6d_4)d_1+3d_2^2+(6d_3+6d_4)d_2+3d_3^2+6d_4d_3+3d_4^2)c_1$\\
$+(3d_2^2+(6d_3+6d_4)d_2+3d_3^2+6d_4d_3+3d_4^2)c_2+(3d_3^2+6d_4d_3+3d_4^2)c_3+3d_4^2c_4=1,$\\
$(3d_1+3d_2+3d_3+3d_4)c_1^2+((6d_2+6d_3+6d_4)c_2+(6d_3+6d_4)c_3+6d_4c_4)c_1$\\
$+(3d_2+3d_3+3d_4)c_2^2+((6d_3+6d_4)c_3+6d_4c_4)c_2+(3d_3+3d_4)c_3^2+6d_4c_4c_3+3d_4c_4^2=1,$\\
$(24d_2d_1c_3+(24d_2+24d_3)d_1c_4)c_2+(24d_3d_1+24d_3d_2)c_4c_3=1,$\\
$(12d_2^2+(24d_3+24d_4)d_2+12d_3^2+24d_4d_3+12d_4^2)d_1c_2
+((12d_3^2+24d_4d_3+12d_4^2)d_1$\\
$+(12d_3^2+24d_4d_3+12d_4^2)d_2)c_3
+(12d_4^2d_1+12d_4^2d_2+12d_4^2d_3)c_4=1,$\\
$4d_1c_2^3+(12d_1c_3+12d_1c_4)c_2^2+(12d_1c_3^2+24d_1c_4c_3
+12d_1c_4^2)c_2+(4d_1+4d_2)c_3^3$\\
$+(12d_1+12d_2)c_4c_3^2+(12d_1+12d_2)c_4^2c_3+(4d_1+4d_2+4d_3)c_4^3=1$
}
\right.$}

\vskip\baselineskip
\noindent
$B=`AGJ,2r$K$h$j(B, $B$3$NJ}Dx<0$O0J2<$N$h$&$KJ,2r$5$l$k$3$H$,J,$+$k(B. 

\vskip\baselineskip
$\left\{
\parbox[c]{8in}{
$24c_4^2-6c_4+1=0$\\
$c_1=-c_4+{1\over 4}$,
$c_2=-c_4+{1\over 2}$,
$c_3=c_4+{1\over 4}$
$d_1=-2c_4+{1\over 2}$,
$d_2={1\over 2}$,
$d_3=2c_4$,
$d_4=0$}
\right.$

$\left\{
\parbox[c]{8in}{
$6c_4^3-12c_4^2+6c_4-1=0$\\
$c_1=0$,
$c_2=c_4$,
$c_3=-2c_4+1$
$d_1={1\over 2}c_4$,
$d_2=-{1\over 2}c_4+{1\over 2}$,
$d_3=-{1\over 2}c_4+{1\over 2}$,
$d_4={1\over 2}c_4$}
\right.$

$\left\{
\parbox[c]{8in}{
$48c_4^3-48c_4^2+12c_4-1=0$\\
$c_1=c_4$,
$c_2=-c_4+{1\over 2}$,
$c_3=-c_4+{1\over 2}$
$d_1=2c_4$,
$d_2=-4c_4+1$,
$d_3=2c_4$,
$d_4=0$}
\right.$

$\left\{
\parbox[c]{8in}{
$6c_4^2-3c_4+1=0$\\
$c_1=0$,
$c_2=-c_4+{1\over 2}$,
$c_3={1\over 2}$
$d_1=-{1\over 2}c_4+{1\over 4}$,
$d_2=-{1\over 2}c_4+{1\over 2}$,
$d_3={1\over 2}c_4+{1\over 4}$,
$d_4={1\over 2}c_4$}
\right.$
