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1.1     ! noro        1: \chapter{$B%0%l%V%J4pDl$N1~MQ(B}
        !             2:
        !             3: $B%0%l%V%J4pDl$O>C5nK!0J30$K$b$5$^$6$^$J1~MQ$r;}$D(B. $BK\@a$G$O(B, $B$=$l$i$N$$(B
        !             4: $B$/$D$+$K$D$$$F2r@b$9$k(B.
        !             5:
        !             6: \begin{nt}
        !             7: $B0J2<$G(B, $B<!$N$h$&$J5-K!$rMQ$$$k(B. \\
        !             8: $K$ : $BBN(B.\\
        !             9: $X$ = $\{x_1,\cdots,x_n\}$ : $BITDj85(B\\
        !            10: $R$ : $K[X]$\\
        !            11: $T$ : $R$ $B$N9`A4BN(B\\
        !            12: $HT_<(f)$ : $f$ $B$N(B $<$ $B$K4X$9$kF,9`(B.\\
        !            13: $HC_<(f)$ : $f$ $B$N(B $<$ $B$K4X$9$kF,78?t(B.\\
        !            14: $GB_<(S)$ : $S$ $B$N(B $<$ $B$K4X$9$kHoLs%0%l%V%J4pDl(B.\\
        !            15: $NF_<(f,G)$ : $f$ $B$N(B $G$ $B$K4X$9$k@55,7A$N0l$D(B. $G$ $B$,%0%l%V%J4pDl$J$i$P0l0UE*(B
        !            16: $B$KDj$^$k(B.
        !            17: \end{nt}
        !            18:
        !            19: \section{$B%$%G%"%k$K4X$9$k1i;;(B}
        !            20:
        !            21: \begin{pr}($B%$%G%"%k$NAjEy(B)\\
        !            22: $B%$%G%"%k(B $I, J \subset R$ $B$K4X$7(B
        !            23: $I = J \Leftrightarrow GB(I) = GB(J).$
        !            24: \end{pr}
        !            25:
        !            26: \begin{pr} ($B%$%G%"%k$rK!$H$9$k9gF1(B, $B%a%s%P%7%C%W(B)\\
        !            27: $B%$%G%"%k(B $I$, $f, g\in R$ $B$K4X$7(B
        !            28: $f \equiv g \bmod I \Leftrightarrow NF(f,GB(I)) = NF(g,GB(I)).$
        !            29: $BFC$K(B
        !            30: $f \in I \Leftrightarrow NF(f,GB(I)) = 0.$
        !            31: \end{pr}
        !            32:
        !            33: \begin{pr}($B<+L@$J%$%G%"%k(B)\\
        !            34: $B%$%G%"%k(B $I \subset R$ $B$K4X$7(B
        !            35: $I = R \Leftrightarrow GB(I) = \{1\}.$
        !            36: \end{pr}
        !            37:
        !            38: \begin{pr}(elimination $B%$%G%"%k(B)\\
        !            39: $I$ $B$r%$%G%"%k$H$9$k(B.
        !            40: $X = (X \setminus U) \cup U$ $B$H$7(B, $B$3$NJ,3d$K$h$j(B
        !            41: $B$9$Y$F$N(B $u \in T(U)$, $B$9$Y$F$N(B $x \in T(X \setminus U)$ $B$KBP$7(B $u < x$ $B$J$k(B
        !            42: order $<$ $B$rMQ$$$k$H(B,
        !            43: $GB(I \cap K[U]) = GB(I) \cap K[U]$
        !            44: \end{pr}
        !            45: \proof
        !            46: $f \in J=I \cap K[U]$ $B$H$9$k(B. $f \in I$ $B$h$j$"$k(B $g \in GB(I)$ $B$,B8:_$7$F(B
        !            47: $HT(g) \mid HT(f)$. $HT(f) \in T(U)$$B$h$j(B $HT(g) \in T(U)$.  $B$3$3$G(B, $<$
        !            48: $B$N@-<A$h$j(B, $HT(g) \in T(U)$ $B$J$i$P(B $g \in K[U]$. $B$h$C$F(B $g \in GB(I) \cap
        !            49: K[U]$ $B$G(B, $GB(I) \cap K[U]$ $B$O(B $J$ $B$N%0%l%V%J4pDl(B. \qed
        !            50:
        !            51: \begin{pr} ($B%$%G%"%k$N8r$o$j(B)
        !            52: $I = Id(f_1,\cdots,f_l)$, $J = Id(g_1,\cdots,g_m)$ $B$H$9$k$H(B,
        !            53: $I\cap J = (yIR[y] + (1-y)JR[y])\cap R$
        !            54: \end{pr}
        !            55: \proof $f \in I\cap J$ $B$H$9$k$H(B, $f = yf+(1-y)f \in (yI + (1-y)J)\cap R$.
        !            56: $B5U$K(B $f=yg + (1-y)h$ ($g \in IR[y], h \in JR[y]$) $B$H$7(B, $f \in R$
        !            57: $B$H$9$k(B. $B$3$N;~(B, $y=0$ $B$rBeF~$7$F(B, $f = h|_{y=0} \in J$. $y=1$ $B$r(B
        !            58: $BBeF~$7$F(B, $f=g|_{y=1} \in I$ $B$h$j(B OK. \qed
        !            59:
        !            60: \begin{co}
        !            61: \label{intersect}
        !            62: $I=Id(f_1,\cdots,f_m)$, $J=Id(g_1,\cdots,g_l)$ $B$KBP$7(B
        !            63: $GB(I\cap J) = GB(\{yf_1,\cdots,yf_m,(1-y)g_1,\cdots,(1-y)g_l\}) \cap R$
        !            64: $B$K$h$j(B $I\cap J$ $B$,7W;;$G$-$k(B. ($B:8JU$O(B $X < y$ $B$J$k(B
        !            65: elimination order $B$G7W;;$9$k(B. )
        !            66: \end{co}
        !            67:
        !            68: \begin{df} ($B%$%G%"%k>&(B)
        !            69: $B%$%G%"%k(B $I$, $R$ $B$NItJ,=89g(B $S$ $B$KBP$7(B, $B%$%G%"%k>&(B $I:S$ $B$r(B
        !            70: $$I:S = \{f \in R \mid fS \subset I\}$$
        !            71: $B$GDj5A$9$k(B. $J=Id(S)$ $B$H$9$l$P(B, $I:S = I:J$ $B$G(B,
        !            72: $J=Id(f_1,\cdots,f_m)$ $B$J$i$P(B,
        !            73: $$I:S = \bigcap_{i=1}^m I:Id(f_i)$$
        !            74: \end{df}
        !            75:
        !            76: \begin{pr}
        !            77: $I:Id(f) = {1\over f}(I \cap Id(f))$
        !            78: \end{pr}
        !            79: \proof $g \in I:Id(f)$ $B$J$i$P(B $gf \in I$ $B$h$j(B $gf \in I \cap Id(f)$.
        !            80: $B$h$C$F(B, $g \in {1 \over f}(I \cap Id(f))$.\\
        !            81: $B5U$K(B, $g \in {1\over f}(I \cap Id(f))$ $B$J$i$P(B $gf \in I \cap Id(f)$ $B$h$j(B
        !            82: $g \in I:Id(f)$. \qed
        !            83:
        !            84: \begin{co}
        !            85: $I \cap Id(f)$ $B$N@8@.85$,5a$^$l$P(B, $B$=$l$i$O(B $f$ $B$r0x;R$K;}$D$N$G(B, $B$=$l(B
        !            86: $B$>$l(B $f$ $B$G3d$k$3$H$K$h$j(B $I:Id(f)$ $B$,5a$^$k(B. $B0lHL$N>l9g(B $I:S$ $B$O$=$l(B
        !            87: $B$i$N8r$o$j$H$J$k$,(B, $I \cap Id(f)$ $B$r4^$a$F%$%G%"%k$N8r$o$j$N7W;;$O7O(B
        !            88: \ref{intersect} $B$K$h$j7W;;$G$-$k$N$G(B, $B%$%G%"%k>&$b7W;;$G$-$k$3$H$K(B
        !            89: $B$J$k(B.
        !            90: \end{co}
        !            91:
        !            92: \begin{df} (saturation)\\
        !            93: $I$ $B$r%$%G%"%k(B, $f\in R$ $B$H$9$l$P(B, $I:f^i$ $B$O%$%G%"%k$NA}BgNs$@$,$i(B, $B$"(B
        !            94: $B$k(B $s \in \N$ $B$,B8:_$7$F(B
        !            95: $$i\ge s \Rightarrow I:f^i = I:f^s$$
        !            96: $B$,@.$jN)$D(B. $B$3$N$H$-(B
        !            97: $$I:f^\infty = I:f^s$$
        !            98: $B$HDj5A$7(B, $I$ $B$N(B $f$ $B$K4X$9$k(B saturation $B$H8F$V(B.
        !            99: \end{df}
        !           100:
        !           101: \begin{pr}
        !           102: $I$ $B$r%$%G%"%k(B, $f \in R$ $B$KBP$7(B,
        !           103: $$I:f^\infty = (IR[y]+(1-yf)R[y]) \cap R$$
        !           104: $B$9$J$o$A(B $I:f^\infty$ $B$O(B elimination $B%$%G%"%k$K$h$j7W;;$G$-$k(B.
        !           105: \end{pr}
        !           106: \proof \\
        !           107: \underline{$B1&JU(B $\subset$ $B:8JU(B} $g \in$ $B1&JU$H$9$k$H(B, $g = ah+(1-yf)b$ ($a,b \in R[y]$)
        !           108: $B$H=q$1$k(B. $B$3$N<0$G(B, $y=1/f$ $B$H$*$$$F(B, $BN>JU$K(B $f^d$ ($d$:$B==J,Bg(B) $B$r3]$1$l$P(B
        !           109: $f^dg \in I$ $B$9$J$o$A(B $g \in I:f^d$. $B$h$C$F(B $g \in$ $B:8JU(B.\\
        !           110: \underline{$B:8JU(B $\subset$ $B1&JU(B} $g \in$ $B:8JU(B, $B$9$J$o$A$"$k(B $d$ $B$KBP$7(B $f^dg \in I$ $B$H(B
        !           111: $B$9$k(B. $B$3$N$H$-(B
        !           112: $$g \equiv (yf)^d g \equiv 0 \bmod IR[y]+(1-yf)R[y]$$
        !           113: $B$^$?(B, $g \in R$ $B$h$j(B $g \in$ $B1&JU(B. \qed
        !           114:
        !           115:
        !           116:
        !           117:
        !           118: %*****************************************************
        !           119:
        !           120: \section{$B>jM>4D(B, $B<!85(B}
        !           121:
        !           122: \begin{pr}($B>jM>4D$NI=8=(B)\\
        !           123: $B%$%G%"%k(B $I$ $B$KBP$7(B, $B>jM>4D(B $R/I$ $B$O(B,$B@55,7A$r85$H$7$FDj5A$5$l$kBe?t(B
        !           124: $B9=B$$KF17A$G$"$k!%$9$J$o$A(B, $R/I$ $B$O(B, $B85$N=89g$H$7$F(B
        !           125: $$\{NF(f,GB(I))\mid f\in R \}$$ $B$HF10l;k$G$-(B, $B$=$N2CK!(B($\oplus$) $\cdot$ $B>hK!(B
        !           126: ($\odot$)$B$H$7$F<!<0$GDj5A$5$l$k$b$N$KF17A$H$J$k(B.
        !           127: $$f\oplus g = NF(f+g,GB(I))$$
        !           128: $$f\odot g = NF(fg,GB(I))$$
        !           129: \end{pr}
        !           130:
        !           131: \begin{pr}($B>jM>4D$N@~7A6u4V$H$7$F$N4pDl(B)\\
        !           132: \label{mbase}
        !           133: $B%$%G%"%k(B $I$ $B$KBP$7(B, $B>jM>4D(B $R/I$ $B$O(B $K$-$B@~7A6u4V$H$_$J$;$k$,(B,
        !           134: $B$=$N@~7A6u4V$N4pDl$H$7$F%$%G%"%k$N(B $B%0%l%V%J4pDl$KBP$7$F@55,7A$G$"$k(B
        !           135: $B9`A4BN$,$H$l$k(B.  $B$9$J$o$A(B, $R/I$ $B$N4pDl$H$7$F(B
        !           136: \begin{center}
        !           137: $\{ u \in T \mid$ $B$9$Y$F$N(B $f \in GB(I)$ $B$KBP$7(B $HT(f) {\not|} u \}$
        !           138: \end{center}
        !           139: $B$,$H$l$k(B.
        !           140: \end{pr}
        !           141:
        !           142: \begin{df}
        !           143: $I$ $B$r%$%G%"%k$H$9$k(B. $U \subset X$ $B$KBP$7(B, $I \cap K[U] = 0$ $B$,@.$jN)(B
        !           144: $B$D$H$-(B $U$ $B$O(B independent modulo $I$ $B$H$$$&(B.
        !           145: \end{df}
        !           146:
        !           147: \begin{df}($B%$%G%"%k$N<!85(B)\\
        !           148: $B%$%G%"%k(B $I$ $B$KBP$7(B, $B%$%G%"%k$N<!85(B $\dim(I)$ $B$r(B
        !           149: \begin{center}
        !           150: $\dim(I)$ = $\max(|U| \mid U \subset X$ independent modulo $I)$
        !           151: \end{center}
        !           152: $B$GDj5A$9$k(B.
        !           153: \end{df}
        !           154:
        !           155: \begin{re}($B4v2?3XE*0UL#(B)
        !           156: \begin{enumerate}
        !           157: \item $B%$%G%"%k(B $I$ $B$N<!85$O(B, $K$ $B$NBe?tJDJq>e$G9M$($?Be?tE*=89g(B $V(I)$
        !           158: $B$N@.J,$N:GBg<!?t$KEy$7$$(B.
        !           159: \item $B$h$j0lHL$K(B, $BAG%$%G%"%k$N8:>/Ns$ND9$5$rMQ$$$F4D$N<!85(B (Krull $B<!85(B)
        !           160: $B$,Dj5A$5$l(B, $B>e$NDj5A$H0lCW$9$k$3$H$,<($5$l$k(B.
        !           161: \end{enumerate}
        !           162: \end{re}
        !           163:
        !           164: \begin{df}(Hilbert function)\\
        !           165: $R=K[X]$ $B$N(B $s$-$B<!@F<!85A4BN$r(B $R_s$ $B$H=q$/$3$H$K$9$k(B. $B%$%G%"%k(B
        !           166: $I$ $B$KBP$7(B, $I_s = I \cap K[X]_s$ $B$H=q$/(B.
        !           167: $B@F<!%$%G%"%k(B $I$ $B$KBP$7(B, $I$ $B$N(B Hilbert function $H_{R/I}(s)$ $B$r(B
        !           168: $$H_{R/I}(s) = \dim_K R_s/I_s$$
        !           169: $B$GDj5A$9$k(B.
        !           170: \end{df}
        !           171:
        !           172: \begin{pr}
        !           173: $<$ $B$rG$0U$N(B order $B$H$7(B, $J$ $B$r(B $I$ $B$N85$NF,9`$G@8@.$5$l$k%$%G%"%k(B
        !           174: $B$H$9$k$H(B, $H_{R/I}(s) = H_{R/J}(s)$
        !           175: \end{pr}
        !           176:
        !           177: \section{$B>C5nK!(B}
        !           178:
        !           179: \begin{df}
        !           180: $B%$%G%"%k(B $I$ $B$KBP$7(B, $I$ $B$N(B radical ($B:,4p(B) $\sqrt{I}$ $B$r(B
        !           181: \begin{center}
        !           182: $\sqrt{I} = \{f \in R \mid$ $B$"$k(B $e \in \N$ $B$,B8:_$7$F(B $f^e \in I\}$
        !           183: \end{center}
        !           184: $B$GDj5A$9$k(B. $\sqrt{I}$ $B$b%$%G%"%k$H$J$k(B.
        !           185: \end{df}
        !           186:
        !           187:
        !           188: \begin{df}
        !           189: $L$ $B$r(B $K$ $B$N3HBgBN$H$7(B, $V \subset K^n$ $B$H$9$k(B. $B$3$N$H$-%$%G%"%k(B $I(V) \subset R$
        !           190: $B$r(B
        !           191: $$I(V) = \{f \in R \mid f|_V=0 \}$$
        !           192: $B$GDj5A$9$k(B.
        !           193:
        !           194: \end{df}
        !           195: $B<!$NDjM}$O>C5nK!$N4pK\$H$J$k(B.
        !           196:
        !           197: \begin{th}(Nullstellensatz; Hilbert $B$NNmE@DjM}(B)\\
        !           198: $K$ $B$rBN(B, $\bar{K}$ $B$r(B $K$ $B$NBe?tJDJq$H$9$k(B. $B%$%G%"%k(B $I \subset K[X]$
        !           199: $B$KBP$7(B,
        !           200: $I(V_{\bar{K}}(I))=\sqrt{I}$
        !           201: \end{th}
        !           202:
        !           203: \begin{co}
        !           204: $B%$%G%"%k(B $I$, $J$ $B$KBP$7(B,
        !           205: $V_{\bar{K}}(I)=V_{\bar{K}}(J) \Leftrightarrow \sqrt{I} = \sqrt{J}$
        !           206: \end{co}
        !           207:
        !           208: \begin{pr}(0 $B<!85%$%G%"%k$N@-<A(B)\\
        !           209: $BBe?tJDBN(B $K$ $B>e$NB?9`<04D$N%$%G%"%k(B $I$ $B$NNmE@$N8D?t$,M-8B8D(B
        !           210: $\Leftrightarrow$ $R/I$ $B$,(B $K$ $B>eM-8B<!85$N@~7A6u4V(B
        !           211: \end{pr}
        !           212: \proof \\
        !           213: $\Rightarrow$)
        !           214: $BM-8B8D$N2r$r(B $r_k = (r_{k1}, \cdots, r_{kn})\quad (k = 1,
        !           215: \cdots, m)$$B$H$9$k(B. $f_i(x_i) = \prod_k (x_i-r_{ki})$ $B$H$*$/$H(B,
        !           216: $f_i(x_i)$ $B$O(B$I$ $B$NNmE@>e$G(B 0 $B$H$J$k$+$i(B, Hilbert $B$NNmE@DjM}$K$h$j$"(B
        !           217: $B$k(B $t$ $B$,B8:_$7$F(B $f_i(x_i)^t \in I$.  $B$h$C$F(B, $GB(I)$ $B$K$b(B, $B3F(B $i$ $B$K(B
        !           218: $BBP$7(B, $HT(g)$ $B$,(B $x_i$ $B$NQQ$H$J$k$b$N$,B8:_$9$k(B. $B$9$k$H(B, $BL?Bj(B \ref{mbase}
        !           219: $B$h$j(B $R/I$ $B$O(B $K$ $B>eM-8B<!85$H$J$k(B.\\
        !           220: $\Leftarrow$)
        !           221: $B3F(B $i$ $B$KBP$7(B, $BJQ?tCf$G(B $x_i$ $B$,:GDc$N=g=x$K$J$k$h$&$J<-=q<0=g=x$r$H$l$P(B,
        !           222: $GB(I)$ $BCf$K(B, $f_i(x_i)$ $B$J$k0lJQ?tB?9`<0$,B8:_$9$k$3$H$,$o$+$k(B.
        !           223: $B$h$C$F(B, $B2r$OM-8B8D(B. \qed
        !           224:
        !           225: $BJQ?t=g=x$H$7$F(B $x_1 < x_2 < \cdots < x_n$ $B$J$k<-=q<0=g=x$r9M$($l$P(B,
        !           226: $B%$%G%"%k(B $I$ $B$NNmE@$,M-8B8D$N>l9g$O(B, $B$9$Y$F$N(B $i$ $B$KBP$7(B, $B$"$k(B
        !           227: $f_i \in GB(I) \cap (K[x_1,\cdots,x_i]\setminus K[x_1,\cdots,x_{i-1}])$
        !           228: $B$,B8:_$9$k(B. $B$h$C$F(B
        !           229: $f_1(x_1)$$B$+$i:,(B $\alpha_1$ $B$r5a$a(B, $f_2(\alpha_1,x_2)$ $B$+$i:,(B
        !           230: $\alpha_2$ $B$r5a$a(B, $B$H$$$&A`:n$r7+$jJV$;$P(B, $F$ $B$N6&DLNmE@$r$9$Y$F5a$a(B
        !           231: $B$k$3$H$,$G$-$k(B.
        !           232:
        !           233: \section{$B2C72$N%0%l%V%J4pDl(B}
        !           234:
        !           235: $B<+M32C72(B $K[X]^l$ $B$*$h$S(B, $B$=$NItJ,2C72(B $M \subset F$ $B$K(B
        !           236: $BBP$7$F$b%0%l%V%J4pDl$,Dj5A$5$l$k(B. $B$3$N>l9g(B, $B9`$H$7$F$O(B,
        !           237: $te_i$ ($t \in T(X)$; $e_i = (0,\cdots,1,\cdots,0)$ : $BBh(B $i$ $B@.J,$N$_(B 1)
        !           238: $B$r$H$j(B,
        !           239: \begin{enumerate}
        !           240: \item $B$9$Y$F$N(B $t \in T$, $B$9$Y$F$N(B $F$ $B$N9`(B $m$ $B$KBP$7(B $m \le tm$
        !           241: \item $B$9$Y$F$N(B $t \in T$, $B$9$Y$F$N(B $F$ $B$N9`$NAH(B $m_1, m_2$ $B$KBP$7(B
        !           242: $m_1 \le m_2$ $\Rightarrow$ $tm_1 \le tm_2$
        !           243: \end{enumerate}
        !           244: $B$rK~$?$9A4=g=x$rF~$l$k(B. $B%b%N%$%G%"%k(B $E(S)$ $B$bF1MM$KDj5A$5$l(B, $B%0%l%V%J(B
        !           245: $B4pDl$b(B, $E(G)$ $B$,(B $E(M)$ $B$r@8@.$9$k$b$N$H$7$FDj5A$5$l$k(B. Buchberger $B%"(B
        !           246: $B%k%4%j%:%`$O(B, $S$-$BB?9`<0$r(B, $BF,9`$N(B $F$ $B$K$*$1$k0LCV$,Ey$7$$(B ($B$9$J$o$A(B,
        !           247: $HT(a)=t_ae_a$, $HT(b)=t_be_b$ $B$N$H$-(B $a=b$)$B$KBP$7$FDL>o$NB?9`<0$HF1MM(B
        !           248: $B$KDj5A$7(B, $B$=$l0J30$O(B 0 $B$HDj5A$9$l$PA4$/F1MM$K$G$-$k(B. $B2C72$N%0%l%V%J4p(B
        !           249: $BDl$O(B, syzygy $B$N7W;;$rDL$7$F(B, $B2C72$N<+M3J,2r(B (free resolution) $B$rM?$($k(B.
        !           250: $B$3$l$K$h$j(B, $B2C72$N%[%b%m%8!<$N7W;;$,2DG=$K$J$k$,(B, $B$3$3$G$O=R$Y$J(B
        !           251: $B$$(B.
        !           252:
        !           253: \section{$BNc(B : $BAPBP6J@~$N7W;;(B}
        !           254: elimination $B%$%G%"%k$N1~MQ$H$7$F(B, $BAPBP6J@~$N7W;;$NNc$r<($9(B.
        !           255: $f(x_1,x_2) \in \Q[x_1,x_2]$ $B$H$7(B, $F$ $B$N(B total degree $B$r(B $d$ $B$H$9$l$P(B,
        !           256: $F(x_0,x_1,x_2)=x_0^df(x_1/x_0,x_2/x_0)$
        !           257: $B$O(B $d$ $B<!F1<!B?9`<0$G(B, $F$ $B$NDj5A$9$kBe?t6J@~$NAPBP6J@~$O(B,
        !           258: $$\left\{
        !           259: \parbox[c]{8in}{
        !           260: $u_i={{\partial F}\over {\partial x_i}} (x_0,x_1,x_2)$ $(i=0,1,2)$\\
        !           261: $F(x_0,x_1,x_2)=0$
        !           262: }
        !           263: \right.$$
        !           264: $B$+$i(B $x_0, x_1, x_2$ $B$r>C5n$7$FF@$i$l$k(B. $B>C5nK!$N0l$D$H$7$F%0%l%V%J4pDl(B
        !           265: $B$K$h$k>C5n$,2DG=$G$"$k(B.
        !           266: $$I = Id(
        !           267: u_0-{{\partial F}\over {\partial x_0}},
        !           268: u_1-{{\partial F}\over {\partial x_1}},
        !           269: u_2-{{\partial F}\over {\partial x_2}},
        !           270: F)$$
        !           271: $B$H$9$k;~(B, $\{x_0, x_1, x_2\}$ $\succ$ $\{u_0, u_1, u_2\}$ $B$J$kG$0U$N>C(B
        !           272: $B5n=g=x$K$h$j(B $I$ $B$N%0%l%V%J4pDl(B $GB(I)$ $B$r7W;;$9$l$P(B,
        !           273: $$I \cap \Q[u_0,u_1,u_2] = Id(GB(I) \cap Q[u_0,u_1,u_2]).$$
        !           274: $B0J2<$NNc$G(B, $V(g_i)$ $B$O(B $V(f_i)$ $B$NAPBP6J@~$G$"$k(B.
        !           275:
        !           276: \vskip\baselineskip
        !           277: $\left\{
        !           278: \parbox[c]{6in}{
        !           279: $f_1=x^5-x^3+y^2$\\
        !           280: $g_1=108x^7-108x^5+1017y^2x^4-16y^4x^3-4250y^2x^2+1800y^4x-108y^6+3125y^2$
        !           281: }
        !           282: \right.$
        !           283:
        !           284: \vskip\baselineskip
        !           285: $\left\{
        !           286: \parbox[c]{6in}{
        !           287: $f_2=x^6+3y^2x^4+(3y^4-4y^2)x^2+y^6$\\
        !           288: $g_2=-256x^6+(64y^4-192y^2+864)x^4+(-192y^4+1620y^2-729)x^2-256y^6+864y^4-729y^2$
        !           289: }
        !           290: \right.$
        !           291:
        !           292: \vskip\baselineskip
        !           293: $\left\{
        !           294: \parbox[c]{6in}{
        !           295: $f_3=2x^4-3yx^2+y^4-2y^3+y^2$\\
        !           296: $g_3=-12x^6+(-y^2+178y-37)x^4+(12y^3-768y^2+2208y+4608)x^2-32y^4+1024y^3-7680y^2-8192y-2048$
        !           297: }
        !           298: \right.$
        !           299:
        !           300: \vskip\baselineskip
        !           301: \begin{figure}[hbtp]
        !           302: \begin{tabular}{cc}
        !           303: \begin{minipage}{.5\hsize}
        !           304: \begin{center}
        !           305: \epsfxsize=7cm
        !           306: \epsffile{ps/1.ps}
        !           307: \end{center}
        !           308: \caption{$f_1=0$}
        !           309: \label{f2}
        !           310: \end{minipage} &
        !           311:
        !           312: \begin{minipage}{.5\hsize}
        !           313: \begin{center}
        !           314: \epsfxsize=7cm
        !           315: \epsffile{ps/1d.ps}
        !           316: \end{center}
        !           317: \caption{$g_1=0$}
        !           318: \label{g2}
        !           319: \end{minipage}
        !           320: \end{tabular}
        !           321: \end{figure}
        !           322:
        !           323:
        !           324: \begin{figure}[hbtp]
        !           325: \begin{tabular}{cc}
        !           326: \begin{minipage}{.5\hsize}
        !           327: \begin{center}
        !           328: \epsfxsize=7cm
        !           329: \epsffile{ps/2.ps}
        !           330: \end{center}
        !           331: \caption{$f_2=0$}
        !           332: \label{f3}
        !           333: \end{minipage} &
        !           334:
        !           335: \begin{minipage}{.5\hsize}
        !           336: \begin{center}
        !           337: \epsfxsize=7cm
        !           338: \epsffile{ps/2d.ps}
        !           339: \end{center}
        !           340: \caption{$g_2=0$}
        !           341: \label{g3}
        !           342: \end{minipage}
        !           343: \end{tabular}
        !           344: \end{figure}
        !           345:
        !           346: \begin{figure}[hbtp]
        !           347: \begin{tabular}{cc}
        !           348: \begin{minipage}{.5\hsize}
        !           349: \begin{center}
        !           350: \epsfxsize=7cm
        !           351: \epsffile{ps/4.ps}
        !           352: \end{center}
        !           353: \caption{$f_3=0$}
        !           354: \label{f5}
        !           355: \end{minipage} &
        !           356:
        !           357: \begin{minipage}{.5\hsize}
        !           358: \begin{center}
        !           359: \epsfxsize=7cm
        !           360: \epsffile{ps/4d.ps}
        !           361: \end{center}
        !           362: \caption{$g_3=0$}
        !           363: \label{g5}
        !           364: \end{minipage}
        !           365: \end{tabular}
        !           366: \end{figure}