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1.1       noro        1: \chapter{$B%$%G%"%k$NJ,2r(B}
                      2: $B%$%G%"%k(B $I \subset R=K[X]$ $B$KBP$7(B, $I=I_1\cap I_2$ $B$H=q$1$k;~(B,
                      3: $V(I) = V(I_1) \cup V(I_2)$ $B$,@.$jN)$D(B. $B$9$J$o$A(B, $B%$%G%"%k$NJ,2r(B
                      4: $B$ONmE@$NJ,2r$rM?$($k(B. $B$h$j>\$7$/8@$($P(B, $BNmE@$NJ,2r$O(B radical $B$N(B
                      5: $BJ,2r$KBP1~$9$k(B. $BNmE@$r2DG=$J8B$jJ,2r$9$k$3$H$O(B, $BBe?tE*=89g$r(B
                      6: $B4{LsJ,2r$9$k$3$H$KBP1~$9$k(B. $BJ}Dx<0$K$D$$$F8@$($P(B, $B0lHL$K2r$N(B
                      7: $BJ,2r$K$h$j(B, $B$h$j07$$$d$9$$2r$NI=8=$rM?$($k$3$H$,$G$-$k(B.
                      8:
                      9: \section{$BAG%$%G%"%k(B, $B=`AG%$%G%"%k(B, $B=`AG%$%G%"%kJ,2r(B}
                     10:
                     11: $B0J2<(B, $R=K[X]$ $B$r8GDj$7$F9M$($k(B.
                     12:
                     13: \begin{df}
                     14: $B%$%G%"%k(B $I \subset R$ $B$,AG%$%G%"%k(B (prime ideal)$B$H$O(B,
                     15: \begin{center}
                     16: $ab \in I$ $B$+$D(B $a\notin I$ $B$J$i$P(B $b \in I$
                     17: \end{center}
                     18: $B$J$k$3$H(B. $B$3$l$O(B $R/I$ $B$,@00h$H$J$k$3$H$HF1CM(B.
                     19: \end{df}
                     20:
                     21: \begin{df}
                     22: $B%$%G%"%k(B $I \subset R$ $B$,=`AG%$%G%"%k(B (primary ideal)$B$H$O(B,
                     23: \begin{center}
                     24: $ab \in I$ $B$+$D(B $a\notin I$ $B$J$i$P(B $b \in \sqrt{I}$
                     25: \end{center}
                     26: $B$J$k$3$H(B.
                     27: \end{df}
                     28:
                     29: \begin{df}
                     30: $B%$%G%"%k(B $I \subset R$ $B$,(B $I=\sqrt{I}$ $B$rK~$?$9$H$-(B $I$ $B$O(B radical $B%$%G%"%k(B
                     31: $B$G$"$k$H$$$&(B. $\sqrt{I}$ $B$O(B radical $B%$%G%"%k$G$"$k(B.
                     32: \end{df}
                     33:
                     34: \begin{lm}
                     35: $\sqrt{I} = \cap_{I\subset P:prime}P$
                     36: \end{lm}
                     37: \proof $I \subset P$ $B$J$i$P(B $\sqrt{I} \subset \sqrt{P}=P$ $B$h$j(B
1.2     ! noro       38: $B:8JU(B $\subset$ $B1&JU(B. $B1&JU$,:8JU$r??$K4^$`$H$9$l$P(B, $B$"$k(B
1.1       noro       39: $f \in \cap_{I\subset P:prime}P \setminus \sqrt{I}$ $B$,B8:_$9$k(B. $B$3$N$H$-(B,
                     40: $S=\{f,f^2,\cdots,\}$ $B$H$*$1$P(B,
                     41: $S \cap \sqrt{I} = \emptyset$.
                     42: $F = \{J:$ $B%$%G%"%k(B $\mid \sqrt{I} \subset J$ $B$+$D(B $S \cap J = \emptyset \}$ $B$H(B
                     43: $B$*$/$H(B, $F \neq \emptyset$ $B$G(B, $BJq4^4X78$K4X$7$F5"G<E*(B. $B$h$C$F6KBg85(B $J_0$
                     44: $B$,B8:_$9$k(B.
                     45:
                     46: \noi
                     47: \underline{$B<gD%(B\,} $J_0$ $B$OAG(B.
                     48:
                     49: \noi
                     50: \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,
                     51: $S \cap (J_0+Id(a)) \neq \emptyset$ $B$+$D(B $S \cap (J_0+Id(b)) \neq \emptyset$.
                     52: $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,
                     53: $f^s=c+ae$, $f^t=d+bf$ $B$H=q$1$k(B. $B$9$k$H(B $f^{s+t}=abef+cd+aed+cbf \in J_0$
                     54: $B$H$J$jL7=b(B. $B$h$C$F(B $J_0$ $B$OAG(B. \qed
                     55:
                     56: \noi
                     57: $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
                     58: $BB8:_$9$k(B. $B$3$l$OL7=b(B. \qed
                     59:
                     60: \begin{lm}
                     61: $B%$%G%"%k(B $I \subset R$ $B$,=`AG$J$i$P(B, $\sqrt{I}$ $B$OAG%$%G%"%k(B.
                     62: \end{lm}
                     63:
                     64: \begin{df}
                     65: $B%$%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$$$&(B.
                     66: $B$^$?(B $P$ $B$rIUB0AG%$%G%"%k(B (associated prime ideal)$B$H8F$V(B.
                     67: \end{df}
                     68:
                     69: \begin{df}
                     70: $B%$%G%"%k(B $I \subset R$ $B$,(B
                     71: \begin{center}
                     72: $I = I_1 \cap I_2 \Rightarrow I = I_1$ $B$^$?$O(B $I = I_2$
                     73: \end{center}
                     74: $B$rK~$?$9$H$-(B, $I$ $B$O4{Ls$H$$$&(B.
                     75: \end{df}
                     76:
                     77: \begin{lm}
                     78: $B4{Ls%$%G%"%k$O=`AG(B.
                     79: \end{lm}
                     80: \proof $I$ $B$,4{Ls$H$7(B, $fg \in I$ $B$+$D(B $f\notin I$ $B$H$9$k(B.
                     81: $I:g^\infty = I:g^s$ $B$J$k(B $s$ $B$r$H$k$H(B,
                     82: $$I = (I+Id(g^s)) \cap (I+Id(f)).$$
                     83: $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,
                     84: $$ bg^{s+1} = (c-a)g+dfg \in I \Rightarrow b \in I:g^{s+1}=I:g^s \Rightarrow
                     85: bg^s \in I \Rightarrow h \in I$$
                     86: $B$+$i$o$+$k(B. $I \neq I+Id(f)$ $B$@$+$i(B $I=I+Id(g^s)$. $B$9$J$o$A(B $g \in \sqrt{I}$.
                     87: \qed
                     88:
                     89: \begin{th}
                     90: $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.
                     91: \end{th}
                     92: \proof $I$ $B$,M-8B8D$N4{Ls%$%G%"%k$NM-8B8D$N8r$o$j$G=q$1$k$3$H$r(B
                     93: $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
                     94: $B4^$`%$%G%"%k$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
                     95: $B8r$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
                     96: $BL58BA}BgNs$,B8:_$9$k$3$H$K$J$j(B, $R$ $B$N(B Noether $B@-$KH?$9$k(B. \qed
                     97:
                     98: \begin{df}
                     99: $B%$%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
                    100: $B<($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
                    101: $\sqrt{Q_i}$ $B$,Aj0[$J$j(B, $\cap_{j\neq i} Q_j {\not \subset} Q_i$ $B$J$k(B
                    102: $B$3$H(B.
                    103: \end{df}
                    104:
                    105: \begin{lm}
                    106: $B=`AG%$%G%"%k(B $I$, $J$ $B$K$D$$$F(B, $\sqrt{I} = \sqrt{J}$ $B$J$i$P(B $I\cap J$ $B$b=`AG(B.
                    107: \end{lm}
                    108: \proof $ab \in I\cap J$ $B$+$D(B $a\notin I\cap J$ $B$H$9$k(B. $a\notin I$
                    109: $B$N$H$-(B, $b \in \sqrt{I}$, $a\notin J$ $B$N$H$-(B $b\in \sqrt{J}$ $B$,@.$jN)$D(B.
                    110: $\sqrt{I\cap J} = \sqrt{I} \cap \sqrt{J}$ $B$h$j(B, $b \in \sqrt{I\cap J}$. \qed
                    111:
                    112: \begin{th}
                    113: $BG$0U$N%$%G%"%k(B $I \subset R$ $B$O(B minimal $B$J=`AGJ,2r$r;}$D(B.
                    114: \end{th}
                    115: \proof $BJdBj$h$j(B, $BIUB0AG%$%G%"%k$,0lCW$9$k=`AG@.J,$O8r$o$j$r$H$k$3$H$K(B
                    116: $B$h$j0l$D$K$^$H$a$k$3$H$,$G$-$k(B. $B$=$N8e(B, $\cap_{j\neq i} Q_j \subset Q_i$
                    117: $B$J$k(B $Q_i$ $B$r<h$j=|$$$F$b;D$j$O(B $I$ $B$N=`AGJ,2r$H$J$k$+$i(B, $B$3$l$r(B
                    118: $B7+$jJV$7$F(B minimal $B$J=`AGJ,2r$rF@$k(B. \qed
                    119:
                    120: \noi
                    121: $B$5$i$K(B, $B<!$,@.$jN)$D(B.
                    122: \begin{th}
                    123: minimal $B$J=`AGJ,2r$ND9$5$O0l0UE*$G(B, $BIUB0AG%$%G%"%k$O=89g$H$7$F0lCW$9$k(B.
                    124: \end{th}
                    125:
                    126: \begin{df}
                    127: $B%$%G%"%k(B $I$ $B$N=`AG@.J,(B $Q$ $B$NIUB0AG%$%G%"%k$,6K>.$N;~8IN)(B (isolated) $B$H(B
                    128: $B$$$&(B. $B$=$&$G$J$$$H$-KdKW(B (embedded) $B$H$$$&(B.
                    129: \end{df}
                    130:
                    131: \begin{re}
                    132: $B8IN)(B, $BKdKW$H$$$&L>>N$O(B, $B$=$N(B variety $B$NMM;R$K$h$k(B. $B=`AG@.J,(B $Q$ $B$NIUB0(B
                    133: $BAG%$%G%"%k$r(B $P$ $B$H$9$k(B. $Q$ $B$,8IN)$N$H$-(B, $P$ $B$OB>$NIUB0AG%$%G%"%k$r(B
                    134: $B4^$^$J$$$+$i(B, variety $B$G$_$l$P(B, $V(P)$ $B$OB>$N@.J,$,Dj5A$9$k(B variety $B$K(B
                    135: $B4^$^$l$J$$(B. $B0lJ}(B, $Q$ $B$,KdKW$J$i$P(B, $P$ $B$O$"$kIUB0AG%$%G%"%k(B $P'$ $B$r4^(B
                    136: $B$`$+$i(B, variety $B$G$_$l$P(B $V(P) \subset V(P')$ $B$9$J$o$AKdKW$7$F$$$k(B.
                    137: \end{re}
                    138:
                    139: \section{$B=`AGJ,2r$N35N,(B}
                    140:
                    141: $B=`AGJ,2r$N$?$a$N<g$J<jCJ$O(B, {\bf $B%$%G%"%k>&(B},
                    142: {\bf extension}, {\bf contraction} $B$*$h$SB?9`<0$N0x?tJ,2r$G$"$k(B.
                    143:
                    144: \begin{lm}
                    145: $B%$%G%"%k(B $I$, $f \in R \setminus I$ $B$KBP$7(B, $I:f^m = I:f^\infty$ $B$J$k(B
                    146: $m$ $B$r$H$l$P(B,
                    147: $$I = I:f^\infty \cap (I+Id(f^m))$$
                    148: \end{lm}
                    149: \proof $h \in$ $B1&JU$H$9$k$H(B, $hf^m \in I$ $B$+$D(B $h=a+bf^m$ ($a \in I$)
                    150: $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$
                    151: $B$3$l$+$i(B $h \in I$. \qed
                    152:
                    153: \begin{df}(extension)\\
                    154: $Y \subset X$ $B$KBP$7(B, $I^e=K(Y)[X\setminus Y]I$ $B$HDj5A$9$k(B.
                    155: \end{df}
                    156:
                    157: \begin{df}
                    158: $B%$%G%"%k(B $\dim(I)=d$ $B$H$9$k$H(B, $B<!85$NDj5A$K$h$j(B,
                    159: $|Y|=d$ $B$J$k(B independent set $Y \subset X$ $B$,$H$l$k(B.
                    160: $B$3$l$r(B maximally independent set $B$H$h$V(B.
                    161: \end{df}
                    162:
                    163: \begin{lm}
                    164: $Y$ $B$,(B maximally independent set $B$N$H$-(B
                    165: $I^e$ $B$O(B $K(Y)$ $B>e(B 0 $B<!85%$%G%"%k(B.
                    166: \end{lm}
                    167: \proof $BDj5A$K$h$j(B, $BA4$F$N(B $x \in X\setminus Y$ $B$KBP$7(B,
                    168: $I \cap K[\{x\} \cup Y] \neq 0$ $B$@$+$i(B, $K(Y)$ $B>e$G9M$($l$P(B
                    169: $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
                    170: 0 $B<!85(B.\qed
                    171:
                    172: \begin{lm}
                    173: $<$ $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
                    174: $<$ $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
                    175: $B4X$9$k%0%l%V%J4pDl(B.
                    176: \end{lm}
                    177:
                    178: \begin{df}(contraction)\\
                    179: $B%$%G%"%k(B $J \subset K(Y)[X\setminus Y]$ $B$KBP$7(B, $J\cap K[X]$ $B$r(B
                    180: $J$ $B$N(B contraction $B$H$h$S(B, $J^c$ $B$H=q$/(B.
                    181: \end{df}
                    182:
                    183: \begin{pr}
                    184: $I$ $B$r%$%G%"%k(B, $<$ $B$r(B $Y < (X\setminus Y)$ $B$J$k(B block order $B$H$7(B, $G$
                    185: $B$r(B $<$ $B$K4X$9$k(B $I$ $B$N%0%l%V%J4pDl$H$9$k(B. $B$3$N$H$-(B,
                    186: $$f = \LCM\{HC(g) \mid g \in G\}$$
                    187: ($B$?$@$7(B, $HC(g)$ $B$O(B $K(Y)[X\setminus Y]$ $B$N85$H$7$F$H$k(B) $B$H$9$l$P(B,
                    188: $I^{ec} = I:f^\infty$
                    189: \end{pr}
                    190:
                    191: $B$3$l$i$H(B, $B8e$G=R$Y$k(B 0 $B<!85%$%G%"%k$N=`AGJ,2r$rMQ$$$F(B, $B<!$N%"%k%4%j%:%`(B
                    192: $B$,F@$i$l$k(B.
                    193:
                    194: \begin{al} \cite{GTZ}
                    195: \label{gtz}
                    196: \begin{tabbing}
                    197: Input : $B%$%G%"%k(B $I \in R=K[X]$\\
                    198: Output : \= $I = \cap Q_i$ ($I$ $B$N(B minimal $B$J=`AGJ,2r(B)\\
                    199: \> $P_i = \sqrt{Q_i}$ ($Q_i$ $B$NIUB0AG%$%G%"%k(B)\\
                    200:
                    201: $Y \leftarrow$ maximally independent set modulo $I$\\
                    202: $\cap \bar{Q_i} \leftarrow I^e$ (0 $B<!85(B) $B$N(B $K(Y)$ $B>e$N=`AGJ,2r(B\\
                    203: $(f,s) \leftarrow$ $I^{ec} = I:f^\infty = I:f^s$ $B$J$k(B $f$ $B$*$h$S(B $s$\\
                    204: $\cap R_j \leftarrow I+Id(f^s)$ $B$N=`AGJ,2r(B\\
                    205: return $\bar{Q_i}^c \cap (\cap R_j)$
                    206: \end{tabbing}
                    207: \end{al}
                    208:
                    209: \noi
                    210: $I+Id(f^s)$ $B$O(B $I$ $B$r??$K4^$`$+$i(B, $BDd;_@-$OJ]>Z$5$l$k(B. $B$3$N%"%k%4%j%:%`(B
                    211: $B$r<B8=$9$k$?$a$K$O(B,
                    212:
                    213: \begin{itemize}
                    214: \item 0 $B<!85%$%G%"%k$N=`AGJ,2r(B
                    215:
                    216: \item maximally independent set $B$NA*$SJ}(B
                    217: \end{itemize}
                    218:
                    219: \noi
                    220: $B$N%"%k%4%j%:%`$rM?$($kI,MW$,$"$k(B. $B$3$NFb(B, maximally independent set
                    221: $B$K4X$7$F$O(B, $B<!$NL?Bj$,$"$k(B.
                    222:
                    223: \begin{pr}
                    224: $I$ $B$r%$%G%"%k$H$7(B, $MB_<$ $B$r(B, $R/I$ $B$N(B,
                    225: $BA4<!?t$D$-(B order  $<$ $B$K4X$9$k(B monomial $B$K$h$k(B
                    226: $K$-$B4pDl$H$9$k(B.
                    227: $B$3$N$H$-(B, $\dim(I) = =max(|U| \mid T(U) \subset MB_<)$.
                    228: \end{pr}
                    229:
                    230: \noi
                    231: $B$3$NL?Bj$K4p$E$$$F(B, $B0l$D$N%0%l%V%J4pDl$+$i(B, $\dim(I)$ $B$r5a$a$k%"%k%4%j%:%`(B
                    232: $B$,9=@.$G$-$k(B. $B0J2<$G(B, 0 $B<!85%$%G%"%k$N=`AGJ,2r$K$D$$$F35N,$r=R$Y$k(B.
                    233:
                    234: \section{0 $B<!85%$%G%"%k$N=`AGJ,2r(B}
                    235:
                    236: $I \subset R=K[X]$ $B$r(B 0 $B<!85%$%G%"%k$H$9$k(B.
                    237:
                    238: \begin{df}
                    239: $f \in K[X]$ $B$,(B $I$ $B$N(B separating element $B$H$O(B,
                    240: $I$ $B$N(B $\overline{K}$ $B>e$NAj0[$kNmE@(B $a,b$ $B$KBP$7(B, $f(a) \neq f(b)$
                    241: $B$J$k$3$H(B.
                    242: \end{df}
                    243:
                    244: \begin{df}
                    245: $f\in K[X]$ $B$H$9$k(B. $t \notin X$ $B$J$kITDj85$r$H$j(B, $R[t]$ $B$N%$%G%"%k(B
                    246: $J = IR[t]+Id(t-f)$ $B$r9M$($l$P(B, $J$ $B$b(B 0 $B<!85%$%G%"%k$h$j(B,
                    247: $(I+Id(t-f)) \cap K[t] = Id(g(t))$ $B$J$k%b%K%C%/$J(B $g \in K[t]$ $B$,B8:_$9$k(B.
                    248: $g$ $B$r(B $f$ $B$N(B $I$ $B$K4X$9$k:G>.B?9`<0$H8F$V(B.
                    249: \end{df}
                    250:
                    251: \begin{pr}
                    252: $f$ $B$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.
                    253: $B$3$N$H$-(B,
                    254: $$g = g_1^{e_1}\cdots g_r^{e_r}$$
                    255: ($g_i$ $B$O(B$K$ $B>e4{Ls(B) $B$H0x?tJ,2r$9$l$P(B,
                    256: $$I = (I+g_1^{e_1}) \cap \cdots \cap (I+g_r^{e_r})$$
                    257: $B$O(B $I$ $B$N=`AGJ,2r$G(B,
                    258: $$\sqrt{I} = \sqrt{I+g_1} \cap \cdots \cap \sqrt{I+g_r}$$
                    259: $B$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
                    260: $I+g_i^{e_i}$ $B$NIUB0AG%$%G%"%k(B.
                    261: \end{pr}
                    262:
                    263: \noi
                    264: $B%$%G%"%k$N(B seratating element $B$O(B, $BD>@\5a$a$k$N$O:$Fq$G$"$k(B. $BDj5A$K$h$j(B,
                    265: $\sqrt{I}$ $B$N(B separating element $B$,(B $I$ $B$N(B seratating element $B$H$J$k$3$H(B
                    266: $B$rMQ$$$F(B, $\sqrt{I}$ $B$r7W;;$7(B, separating element $B$r5a$a$k$3$H$r9M$($k(B.
                    267:
                    268: \begin{df}
                    269: $BBN(B $K$ $B$,40A4BN$H$O(B, $B4{LsB?9`<0$,A4$FJ,N%E*$G$"$k$3$H(B.
                    270: \end{df}
                    271:
                    272: \begin{re}
                    273: $B0J2<$NL?Bj(B, $B%"%k%4%j%:%`$G$O(B, $B4pACBN$,40A4BN$G$"$k$3$H$rMW5a$9$k$b$N$,(B
                    274: $B$$$/$D$+$"$k(B. $BI8?t(B 0 $B$NBN$OA4$F40A4BN$G$"$k(B. $B$^$?(B, $BM-8BBN$b40A4BN$G$"(B
                    275: $B$k$,(B, $BM-8BBN>e$NM-M}4X?tBN$O40A4BN$G$J$$(B. $B=`AGJ,2r$K$*$$$F$O4pACBN>e$N(B
                    276: $BM-M}4X?tBN$r78?t$H$9$kB?9`<04D$G$N7W;;$r9T$&$?$a(B, $B4pACBN$NI8?t$O(B 0
                    277: $B$K8B$i$l$k(B.
                    278: \end{re}
                    279:
                    280: \begin{pr}
                    281: $K$ $B$,40A4BN$J$i(B,
                    282: \begin{center}
                    283: 0 $B<!85%$%G%"%k(B $I$ $B$,(B radical $\Leftrightarrow$ $I$ $B$,(B, $B3FJQ?t$K$D$$$F(B
                    284: $B0lJQ?tL5J?J}B?9`<0$r4^$`(B.
                    285: \end{center}
                    286: \end{pr}
                    287:
                    288: \begin{pr}
                    289: $K$ $B$,40A4BN$H$9$k$H(B, 0 $B<!85(B radical $B%$%G%"%k(B $I$ $B$NNmE@$N8D?t$O(B
                    290: $\dim_K K[X]/I$ $B$KEy$7$$(B.
                    291: \end{pr}
                    292:
                    293: \begin{df}
                    294: $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
                    295: $B$H$-(B, $f_1\cdots f_m$ $B$r(B $f$ $B$NL5J?J}ItJ,$H8F$V(B.
                    296: \end{df}
                    297:
                    298: \begin{co}
                    299: $I \cap K[x_i] = Id(f_i(x_i))$ $B$H$9$k(B.
                    300: $$\sqrt{I} = I+Id(h_1,\cdots,h_n)$$
                    301: ($h_i$ $B$O(B, $f_i$ $B$NL5J?J}ItJ,(B)
                    302: \end{co}
                    303:
                    304: \noi
                    305: radical $B%$%G%"%k$KBP$7$F$O(B, separating element $B$NH=Dj$O<!$N$h$&$K(B
                    306: $B=R$Y$i$l$k(B.
                    307:
                    308: \begin{pr}
                    309: $K$ $B$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
                    310: $BB?9`<0$r(B $g$ $B$H$9$k$H(B,
                    311: \begin{center}
                    312: $f$ $B$,(B separating element $\Leftrightarrow$ $\deg(f)=\dim_K R/I$
                    313: \end{center}
                    314: $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
                    315: $B7AOB$+$iA*$Y$k(B.
                    316: \end{pr}
                    317:
                    318: \noi
                    319: $B0J>e$,(B, $B40A4BN>e$N(B 0 $B<!85%$%G%"%k$N=`AGJ,2r$N35N,$G$"$k(B. $BD>A0$N(B
                    320: $BL?Bj$K4XO"$7$F(B, $B<!$N$3$H$,@.$jN)$D(B.
                    321:
                    322: \begin{pr}(shape lemma)\\
                    323: $I$ $B$r40A4BN(B $K$ $B>e$N(B 0 $B<!85(B radical $B%$%G%"%k(B $B$H$7(B,
                    324: $f$ $B$r(B separating element $B$H$9$k(B.
                    325: $z << X$ $B$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
                    326: $$\{x_1-f_1(z),\cdots,x_n-f_n(z),z-f_z(z),m(z)\}$$
                    327: $B$H$$$&7A$N%0%l%V%J4pDl$r$b$D(B. $B$3$N7A$N4pDl$r(B shape basis $B$H8F$V(B.
                    328: \end{pr}
                    329: \proof $z$ $B$N:G>.B?9`<0(B $m$ $B$O(B $f$ $B$N:G>.B?9`<0$K0l(B
                    330: $BCW$7(B, $B$=$N<!?t$O(B $\dim_K K[X]/I$ $B$HEy$7$/$J$k(B. $z << X$
                    331: $B$J$k=g=x$N$b$H$G$O(B, $B%0%l%V%J4pDl$O(B $m$ $B$r4^$_(B, $B%b%N%$%G%"%k$r(B
                    332: $B9M$($l$P(B, $m$ $B0J30$N85$NF,9`$O3FJQ?t$N(B 1 $B<!<00J30$G$O$"$j$($J$$(B. \qed\\
                    333: shape basis $B$O(B, 0 $B<!85%$%G%"%k$NNmE@$r?tCM$G5a$a$h$&$H$9$k>l9g$K(B,
                    334: $B8+3]$1>eM-8z$J7A$r$7$F$$$k(B. $B<B:](B, $I$ $B$NNmE@$O(B, $f_n(x_n)$ $B$NNmE@$K(B
                    335: $B$h$j(B,
                    336: $$\{(f_1(\alpha),\cdots,f_n(\alpha))\mid m(\alpha) = 0\}$$
                    337: $B$H=q$1$k(B. $B$7$+$7(B, $BM-M}?tBN>e$G<B:]$K(B shape basis $B$r5a$a$F8+$k$H(B,
                    338: $m$ $B$N78?t$KHf$Y$F(B $f_i$ $B$N78?t$,6K$a$FBg$-$/$J$k$3$H$,(B
                    339: $BB?$$(B. $B$3$N:$Fq$r9nI~$9$k$?$a(B, $B<!$NJ}K!$,9M0F$5$l$?(B.
                    340:
                    341: \begin{pr}(rational univariate representation; RUR)\\
                    342: \label{RUR}
                    343: $BA0L?Bj$HF1$82>Dj$N$b$H$G(B, $IR[z]+Id(z-f)$ $B$N4pDl$H$7$F(B,
                    344: $$\{ m'x_1-g_1(z), \cdots, m'x_{z}-g_n(z), m(z)\}$$
                    345: $B$H$$$&7A$N$b$N$,$H$l$k(B.
                    346: \end{pr}
                    347: \noi
                    348: $m$ $B$O(B shape basis $B$N>l9g$H0lCW$9$k(B. $B$3$N4pDl$K$h$k$NNmE@$NI=8=$O(B,
                    349: $$\{({g_1(\alpha)\over m'(\alpha)},\cdots,{g_n(\alpha)\over
                    350: m'(\alpha)}) \mid m(\alpha)=0\}$$
                    351: \noi
                    352: $B$H=q$1$k(B. $BB?$/$N<BNc$K$*$$$F(B, $g_i$ $B$N3F78?t$,(B, $m$ $B$N78?t$HF1DxEY$N(B
                    353: $BBg$-$5$K2!$($i$l$k$3$H$,J,$+$C$F$*$j(B, 0 $B<!85(B radical$B$NNmE@$NI=8=$H$7$F(B
                    354: $B$O(B RUR $B$K$h$k$b$N$,M%$l$F$$$k$H$$$C$F$h$$(B.
                    355: RUR $B$N7W;;K!$H$7$F$O(B, $BBP>N<0$K$h$kJ}K!$,:G=i$KDs0F$5$l$F$$$,(B,
                    356: modular change of ordering $B$HF1MM$N<jK!$rE,MQ$9$k$3$H$b$G$-(B, RUR $B$,(B
                    357: $B7k2L$NBg$-$5DxEY$G7W;;$G$-$k(B \cite{NY2}.
                    358:
                    359: \section{$B=`AGJ,2r$NNc(B}
                    360: $B<!$NNc$O(B, symplectic integrator $B$H8F$P$l$k0BDj$J@QJ,%9%-!<%`$N(B
                    361: $B?tCM7W;;K!$K4X$7$F8=$l$?J}Dx<07O$G$"$k(B \cite{SYMP}.
                    362:
                    363: \vskip\baselineskip
                    364: {\small
                    365: $\left\{
                    366: \parbox[c]{6in}{
                    367: $d_1+d_2+d_3+d_4=1, c_1+c_2+c_3+c_4=1,$\\
                    368: $(6d_1c_2+(6d_1+6d_2)c_3+(6d_1+6d_2+6d_3)c_4)c_1
                    369:  +(6d_2c_3+(6d_2+6d_3)c_4)c_2+6d_3c_4c_3=1,$\\
                    370: $(3d_1^2+(6d_2+6d_3+6d_4)d_1+3d_2^2+(6d_3+6d_4)d_2+3d_3^2+6d_4d_3+3d_4^2)c_1$\\
                    371: $+(3d_2^2+(6d_3+6d_4)d_2+3d_3^2+6d_4d_3+3d_4^2)c_2+(3d_3^2+6d_4d_3+3d_4^2)c_3+3d_4^2c_4=1,$\\
                    372: $(3d_1+3d_2+3d_3+3d_4)c_1^2+((6d_2+6d_3+6d_4)c_2+(6d_3+6d_4)c_3+6d_4c_4)c_1$\\
                    373: $+(3d_2+3d_3+3d_4)c_2^2+((6d_3+6d_4)c_3+6d_4c_4)c_2+(3d_3+3d_4)c_3^2+6d_4c_4c_3+3d_4c_4^2=1,$\\
                    374: $(24d_2d_1c_3+(24d_2+24d_3)d_1c_4)c_2+(24d_3d_1+24d_3d_2)c_4c_3=1,$\\
                    375: $(12d_2^2+(24d_3+24d_4)d_2+12d_3^2+24d_4d_3+12d_4^2)d_1c_2
                    376: +((12d_3^2+24d_4d_3+12d_4^2)d_1$\\
                    377: $+(12d_3^2+24d_4d_3+12d_4^2)d_2)c_3
                    378: +(12d_4^2d_1+12d_4^2d_2+12d_4^2d_3)c_4=1,$\\
                    379: $4d_1c_2^3+(12d_1c_3+12d_1c_4)c_2^2+(12d_1c_3^2+24d_1c_4c_3
                    380: +12d_1c_4^2)c_2+(4d_1+4d_2)c_3^3$\\
                    381: $+(12d_1+12d_2)c_4c_3^2+(12d_1+12d_2)c_4^2c_3+(4d_1+4d_2+4d_3)c_4^3=1$
                    382: }
                    383: \right.$}
                    384:
                    385: \vskip\baselineskip
                    386: \noindent
                    387: $B=`AGJ,2r$K$h$j(B, $B$3$NJ}Dx<0$O0J2<$N$h$&$KJ,2r$5$l$k$3$H$,J,$+$k(B.
                    388:
                    389: \vskip\baselineskip
                    390: $\left\{
                    391: \parbox[c]{8in}{
                    392: $24c_4^2-6c_4+1=0$\\
                    393: $c_1=-c_4+{1\over 4}$,
                    394: $c_2=-c_4+{1\over 2}$,
                    395: $c_3=c_4+{1\over 4}$
                    396: $d_1=-2c_4+{1\over 2}$,
                    397: $d_2={1\over 2}$,
                    398: $d_3=2c_4$,
                    399: $d_4=0$}
                    400: \right.$
                    401:
                    402: $\left\{
                    403: \parbox[c]{8in}{
                    404: $6c_4^3-12c_4^2+6c_4-1=0$\\
                    405: $c_1=0$,
                    406: $c_2=c_4$,
                    407: $c_3=-2c_4+1$
                    408: $d_1={1\over 2}c_4$,
                    409: $d_2=-{1\over 2}c_4+{1\over 2}$,
                    410: $d_3=-{1\over 2}c_4+{1\over 2}$,
                    411: $d_4={1\over 2}c_4$}
                    412: \right.$
                    413:
                    414: $\left\{
                    415: \parbox[c]{8in}{
                    416: $48c_4^3-48c_4^2+12c_4-1=0$\\
                    417: $c_1=c_4$,
                    418: $c_2=-c_4+{1\over 2}$,
                    419: $c_3=-c_4+{1\over 2}$
                    420: $d_1=2c_4$,
                    421: $d_2=-4c_4+1$,
                    422: $d_3=2c_4$,
                    423: $d_4=0$}
                    424: \right.$
                    425:
                    426: $\left\{
                    427: \parbox[c]{8in}{
                    428: $6c_4^2-3c_4+1=0$\\
                    429: $c_1=0$,
                    430: $c_2=-c_4+{1\over 2}$,
                    431: $c_3={1\over 2}$
                    432: $d_1=-{1\over 2}c_4+{1\over 4}$,
                    433: $d_2=-{1\over 2}c_4+{1\over 2}$,
                    434: $d_3={1\over 2}c_4+{1\over 4}$,
                    435: $d_4={1\over 2}c_4$}
                    436: \right.$