\chapter{$B%0%l%V%J4pDl$N1~MQ(B}

$B%0%l%V%J4pDl$O>C5nK!0J30$K$b$5$^$6$^$J1~MQ$r;}$D(B. $BK\@a$G$O(B, $B$=$l$i$N$$(B
$B$/$D$+$K$D$$$F2r@b$9$k(B.

\begin{nt}
$B0J2<$G(B, $B<!$N$h$&$J5-K!$rMQ$$$k(B. \\
$K$ : $BBN(B.\\
$X$ = $\{x_1,\cdots,x_n\}$ : $BITDj85(B\\
$R$ : $K[X]$\\
$T$ : $R$ $B$N9`A4BN(B\\
$HT_<(f)$ : $f$ $B$N(B $<$ $B$K4X$9$kF,9`(B.\\
$HC_<(f)$ : $f$ $B$N(B $<$ $B$K4X$9$kF,78?t(B.\\
$GB_<(S)$ : $S$ $B$N(B $<$ $B$K4X$9$kHoLs%0%l%V%J4pDl(B.\\
$NF_<(f,G)$ : $f$ $B$N(B $G$ $B$K4X$9$k@55,7A$N0l$D(B. $G$ $B$,%0%l%V%J4pDl$J$i$P0l0UE*(B
$B$KDj$^$k(B.
\end{nt}

\section{$B%$%G%"%k$K4X$9$k1i;;(B}

\begin{pr}($B%$%G%"%k$NAjEy(B)\\
$B%$%G%"%k(B $I, J \subset R$ $B$K4X$7(B
$I = J \Leftrightarrow GB(I) = GB(J).$ 
\end{pr}

\begin{pr} ($B%$%G%"%k$rK!$H$9$k9gF1(B, $B%a%s%P%7%C%W(B)\\
$B%$%G%"%k(B $I$, $f, g\in R$ $B$K4X$7(B
$f \equiv g \bmod I \Leftrightarrow NF(f,GB(I)) = NF(g,GB(I)).$ 
$BFC$K(B 
$f \in I \Leftrightarrow NF(f,GB(I)) = 0.$ 
\end{pr}

\begin{pr}($B<+L@$J%$%G%"%k(B)\\
$B%$%G%"%k(B $I \subset R$ $B$K4X$7(B
$I = R \Leftrightarrow GB(I) = \{1\}.$
\end{pr}

\begin{pr}(elimination $B%$%G%"%k(B)\\
$I$ $B$r%$%G%"%k$H$9$k(B. 
$X = (X \setminus U) \cup U$ $B$H$7(B, $B$3$NJ,3d$K$h$j(B
$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 
order $<$ $B$rMQ$$$k$H(B, 
$GB(I \cap K[U]) = GB(I) \cap K[U]$
\end{pr}
\proof
$f \in J=I \cap K[U]$ $B$H$9$k(B. $f \in I$ $B$h$j$"$k(B $g \in GB(I)$ $B$,B8:_$7$F(B 
$HT(g) \mid HT(f)$. $HT(f) \in T(U)$$B$h$j(B $HT(g) \in T(U)$.  $B$3$3$G(B, $<$ 
$B$N@-<A$h$j(B, $HT(g) \in T(U)$ $B$J$i$P(B $g \in K[U]$. $B$h$C$F(B $g \in GB(I) \cap
K[U]$ $B$G(B, $GB(I) \cap K[U]$ $B$O(B $J$ $B$N%0%l%V%J4pDl(B. \qed

\begin{pr} ($B%$%G%"%k$N8r$o$j(B)
$I = Id(f_1,\cdots,f_l)$, $J = Id(g_1,\cdots,g_m)$ $B$H$9$k$H(B, 
$I\cap J = (yIR[y] + (1-y)JR[y])\cap R$
\end{pr}
\proof $f \in I\cap J$ $B$H$9$k$H(B, $f = yf+(1-y)f \in (yI + (1-y)J)\cap R$.
$B5U$K(B $f=yg + (1-y)h$ ($g \in IR[y], h \in JR[y]$) $B$H$7(B, $f \in R$
$B$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
$BBeF~$7$F(B, $f=g|_{y=1} \in I$ $B$h$j(B OK. \qed

\begin{co}
\label{intersect}
$I=Id(f_1,\cdots,f_m)$, $J=Id(g_1,\cdots,g_l)$ $B$KBP$7(B
$GB(I\cap J) = GB(\{yf_1,\cdots,yf_m,(1-y)g_1,\cdots,(1-y)g_l\}) \cap R$
$B$K$h$j(B $I\cap J$ $B$,7W;;$G$-$k(B. ($B:8JU$O(B $X < y$ $B$J$k(B 
elimination order $B$G7W;;$9$k(B. )
\end{co}

\begin{df} ($B%$%G%"%k>&(B)
$B%$%G%"%k(B $I$, $R$ $B$NItJ,=89g(B $S$ $B$KBP$7(B, $B%$%G%"%k>&(B $I:S$ $B$r(B
$$I:S = \{f \in R \mid fS \subset I\}$$
$B$GDj5A$9$k(B. $J=Id(S)$ $B$H$9$l$P(B, $I:S = I:J$ $B$G(B, 
$J=Id(f_1,\cdots,f_m)$ $B$J$i$P(B, 
$$I:S = \bigcap_{i=1}^m I:Id(f_i)$$
\end{df}

\begin{pr}
$I:Id(f) = {1\over f}(I \cap Id(f))$
\end{pr}
\proof $g \in I:Id(f)$ $B$J$i$P(B $gf \in I$ $B$h$j(B $gf \in I \cap Id(f)$.
$B$h$C$F(B, $g \in {1 \over f}(I \cap Id(f))$.\\
$B5U$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
$g \in I:Id(f)$. \qed

\begin{co}
$I \cap Id(f)$ $B$N@8@.85$,5a$^$l$P(B, $B$=$l$i$O(B $f$ $B$r0x;R$K;}$D$N$G(B, $B$=$l(B
$B$>$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
$B$i$N8r$o$j$H$J$k$,(B, $I \cap Id(f)$ $B$r4^$a$F%$%G%"%k$N8r$o$j$N7W;;$O7O(B
\ref{intersect} $B$K$h$j7W;;$G$-$k$N$G(B, $B%$%G%"%k>&$b7W;;$G$-$k$3$H$K(B
$B$J$k(B.
\end{co}

\begin{df} (saturation)\\
$I$ $B$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
$B$k(B $s \in \N$ $B$,B8:_$7$F(B
$$i\ge s \Rightarrow I:f^i = I:f^s$$
$B$,@.$jN)$D(B. $B$3$N$H$-(B
$$I:f^\infty = I:f^s$$
$B$HDj5A$7(B, $I$ $B$N(B $f$ $B$K4X$9$k(B saturation $B$H8F$V(B. 
\end{df}

\begin{pr}
$I$ $B$r%$%G%"%k(B, $f \in R$ $B$KBP$7(B, 
$$I:f^\infty = (IR[y]+(1-yf)R[y]) \cap R$$
$B$9$J$o$A(B $I:f^\infty$ $B$O(B elimination $B%$%G%"%k$K$h$j7W;;$G$-$k(B. 
\end{pr}
\proof \\
\underline{$B1&JU(B $\subset$ $B:8JU(B} $g \in$ $B1&JU$H$9$k$H(B, $g = ah+(1-yf)b$ ($a,b \in R[y]$)
$B$H=q$1$k(B. $B$3$N<0$G(B, $y=1/f$ $B$H$*$$$F(B, $BN>JU$K(B $f^d$ ($d$:$B==J,Bg(B) $B$r3]$1$l$P(B
$f^dg \in I$ $B$9$J$o$A(B $g \in I:f^d$. $B$h$C$F(B $g \in$ $B:8JU(B.\\
\underline{$B:8JU(B $\subset$ $B1&JU(B} $g \in$ $B:8JU(B, $B$9$J$o$A$"$k(B $d$ $B$KBP$7(B $f^dg \in I$ $B$H(B
$B$9$k(B. $B$3$N$H$-(B
$$g \equiv (yf)^d g \equiv 0 \bmod IR[y]+(1-yf)R[y]$$
$B$^$?(B, $g \in R$ $B$h$j(B $g \in$ $B1&JU(B. \qed




%*****************************************************

\section{$B>jM>4D(B, $B<!85(B}

\begin{pr}($B>jM>4D$NI=8=(B)\\
$B%$%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
$B9=B$$KF17A$G$"$k!%$9$J$o$A(B, $R/I$ $B$O(B, $B85$N=89g$H$7$F(B 
$$\{NF(f,GB(I))\mid f\in R \}$$ $B$HF10l;k$G$-(B, $B$=$N2CK!(B($\oplus$) $\cdot$ $B>hK!(B
($\odot$)$B$H$7$F<!<0$GDj5A$5$l$k$b$N$KF17A$H$J$k(B.
$$f\oplus g = NF(f+g,GB(I))$$
$$f\odot g = NF(fg,GB(I))$$
\end{pr}

\begin{pr}($B>jM>4D$N@~7A6u4V$H$7$F$N4pDl(B)\\
\label{mbase}
$B%$%G%"%k(B $I$ $B$KBP$7(B, $B>jM>4D(B $R/I$ $B$O(B $K$-$B@~7A6u4V$H$_$J$;$k$,(B, 
$B$=$N@~7A6u4V$N4pDl$H$7$F%$%G%"%k$N(B $B%0%l%V%J4pDl$KBP$7$F@55,7A$G$"$k(B 
$B9`A4BN$,$H$l$k(B.  $B$9$J$o$A(B, $R/I$ $B$N4pDl$H$7$F(B 
\begin{center}
$\{ u \in T \mid$ $B$9$Y$F$N(B $f \in GB(I)$ $B$KBP$7(B $HT(f) {\not|} u \}$
\end{center}
$B$,$H$l$k(B.
\end{pr}

\begin{df}
$I$ $B$r%$%G%"%k$H$9$k(B. $U \subset X$ $B$KBP$7(B, $I \cap K[U] = 0$ $B$,@.$jN)(B
$B$D$H$-(B $U$ $B$O(B independent modulo $I$ $B$H$$$&(B.
\end{df}

\begin{df}($B%$%G%"%k$N<!85(B)\\
$B%$%G%"%k(B $I$ $B$KBP$7(B, $B%$%G%"%k$N<!85(B $\dim(I)$ $B$r(B
\begin{center}
$\dim(I)$ = $\max(|U| \mid U \subset X$ independent modulo $I)$
\end{center}
$B$GDj5A$9$k(B.
\end{df}

\begin{re}($B4v2?3XE*0UL#(B)
\begin{enumerate}
\item $B%$%G%"%k(B $I$ $B$N<!85$O(B, $K$ $B$NBe?tJDJq>e$G9M$($?Be?tE*=89g(B $V(I)$
$B$N@.J,$N:GBg<!?t$KEy$7$$(B. 
\item $B$h$j0lHL$K(B, $BAG%$%G%"%k$N8:>/Ns$ND9$5$rMQ$$$F4D$N<!85(B (Krull $B<!85(B) 
$B$,Dj5A$5$l(B, $B>e$NDj5A$H0lCW$9$k$3$H$,<($5$l$k(B. 
\end{enumerate}
\end{re}

\begin{df}(Hilbert function)\\
$R=K[X]$ $B$N(B $s$-$B<!@F<!85A4BN$r(B $R_s$ $B$H=q$/$3$H$K$9$k(B. $B%$%G%"%k(B
$I$ $B$KBP$7(B, $I_s = I \cap K[X]_s$ $B$H=q$/(B. 
$B@F<!%$%G%"%k(B $I$ $B$KBP$7(B, $I$ $B$N(B Hilbert function $H_{R/I}(s)$ $B$r(B
$$H_{R/I}(s) = \dim_K R_s/I_s$$
$B$GDj5A$9$k(B. 
\end{df}

\begin{pr}
$<$ $B$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
$B$H$9$k$H(B, $H_{R/I}(s) = H_{R/J}(s)$
\end{pr}

\section{$B>C5nK!(B}

\begin{df}
$B%$%G%"%k(B $I$ $B$KBP$7(B, $I$ $B$N(B radical ($B:,4p(B) $\sqrt{I}$ $B$r(B
\begin{center}
$\sqrt{I} = \{f \in R \mid$ $B$"$k(B $e \in \N$ $B$,B8:_$7$F(B $f^e \in I\}$
\end{center}
$B$GDj5A$9$k(B. $\sqrt{I}$ $B$b%$%G%"%k$H$J$k(B. 
\end{df}


\begin{df}
$L$ $B$r(B $K$ $B$N3HBgBN$H$7(B, $V \subset K^n$ $B$H$9$k(B. $B$3$N$H$-%$%G%"%k(B $I(V) \subset R$
$B$r(B
$$I(V) = \{f \in R \mid f|_V=0 \}$$
$B$GDj5A$9$k(B. 

\end{df}
$B<!$NDjM}$O>C5nK!$N4pK\$H$J$k(B. 

\begin{th}(Nullstellensatz; Hilbert $B$NNmE@DjM}(B)\\
$K$ $B$rBN(B, $\bar{K}$ $B$r(B $K$ $B$NBe?tJDJq$H$9$k(B. $B%$%G%"%k(B $I \subset K[X]$
$B$KBP$7(B, 
$I(V_{\bar{K}}(I))=\sqrt{I}$
\end{th}

\begin{co}
$B%$%G%"%k(B $I$, $J$ $B$KBP$7(B, 
$V_{\bar{K}}(I)=V_{\bar{K}}(J) \Leftrightarrow \sqrt{I} = \sqrt{J}$
\end{co}

\begin{pr}(0 $B<!85%$%G%"%k$N@-<A(B)\\
$BBe?tJDBN(B $K$ $B>e$NB?9`<04D$N%$%G%"%k(B $I$ $B$NNmE@$N8D?t$,M-8B8D(B 
$\Leftrightarrow$ $R/I$ $B$,(B $K$ $B>eM-8B<!85$N@~7A6u4V(B
\end{pr}
\proof \\
$\Rightarrow$)
$BM-8B8D$N2r$r(B $r_k = (r_{k1}, \cdots, r_{kn})\quad (k = 1,
\cdots, m)$$B$H$9$k(B. $f_i(x_i) = \prod_k (x_i-r_{ki})$ $B$H$*$/$H(B,
$f_i(x_i)$ $B$O(B$I$ $B$NNmE@>e$G(B 0 $B$H$J$k$+$i(B, Hilbert $B$NNmE@DjM}$K$h$j$"(B
$B$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
$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}
$B$h$j(B $R/I$ $B$O(B $K$ $B>eM-8B<!85$H$J$k(B.\\
$\Leftarrow$)
$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, 
$GB(I)$ $BCf$K(B, $f_i(x_i)$ $B$J$k0lJQ?tB?9`<0$,B8:_$9$k$3$H$,$o$+$k(B. 
$B$h$C$F(B, $B2r$OM-8B8D(B. \qed

$BJQ?t=g=x$H$7$F(B $x_1 < x_2 < \cdots < x_n$ $B$J$k<-=q<0=g=x$r9M$($l$P(B, 
$B%$%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 
$f_i \in GB(I) \cap (K[x_1,\cdots,x_i]\setminus K[x_1,\cdots,x_{i-1}])$
$B$,B8:_$9$k(B. $B$h$C$F(B 
$f_1(x_1)$$B$+$i:,(B $\alpha_1$ $B$r5a$a(B, $f_2(\alpha_1,x_2)$ $B$+$i:,(B 
$\alpha_2$ $B$r5a$a(B, $B$H$$$&A`:n$r7+$jJV$;$P(B, $F$ $B$N6&DLNmE@$r$9$Y$F5a$a(B
$B$k$3$H$,$G$-$k(B. 

\section{$B2C72$N%0%l%V%J4pDl(B}

$B<+M32C72(B $K[X]^l$ $B$*$h$S(B, $B$=$NItJ,2C72(B $M \subset F$ $B$K(B
$BBP$7$F$b%0%l%V%J4pDl$,Dj5A$5$l$k(B. $B$3$N>l9g(B, $B9`$H$7$F$O(B, 
$te_i$ ($t \in T(X)$; $e_i = (0,\cdots,1,\cdots,0)$ : $BBh(B $i$ $B@.J,$N$_(B 1) 
$B$r$H$j(B, 
\begin{enumerate}
\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 $m \le tm$
\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
$m_1 \le m_2$ $\Rightarrow$ $tm_1 \le tm_2$
\end{enumerate}
$B$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
$B4pDl$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
$B%k%4%j%:%`$O(B, $S$-$BB?9`<0$r(B, $BF,9`$N(B $F$ $B$K$*$1$k0LCV$,Ey$7$$(B ($B$9$J$o$A(B,
$HT(a)=t_ae_a$, $HT(b)=t_be_b$ $B$N$H$-(B $a=b$)$B$KBP$7$FDL>o$NB?9`<0$HF1MM(B
$B$KDj5A$7(B, $B$=$l0J30$O(B 0 $B$HDj5A$9$l$PA4$/F1MM$K$G$-$k(B. $B2C72$N%0%l%V%J4p(B
$BDl$O(B, syzygy $B$N7W;;$rDL$7$F(B, $B2C72$N<+M3J,2r(B (free resolution) $B$rM?$($k(B. 
$B$3$l$K$h$j(B, $B2C72$N%[%b%m%8!<$N7W;;$,2DG=$K$J$k$,(B, $B$3$3$G$O=R$Y$J(B
$B$$(B.

\section{$BNc(B : $BAPBP6J@~$N7W;;(B}
elimination $B%$%G%"%k$N1~MQ$H$7$F(B, $BAPBP6J@~$N7W;;$NNc$r<($9(B. 
$f(x_1,x_2) \in \Q[x_1,x_2]$ $B$H$7(B, $F$ $B$N(B total degree $B$r(B $d$ $B$H$9$l$P(B, 
$F(x_0,x_1,x_2)=x_0^df(x_1/x_0,x_2/x_0)$
$B$O(B $d$ $B<!F1<!B?9`<0$G(B, $F$ $B$NDj5A$9$kBe?t6J@~$NAPBP6J@~$O(B, 
$$\left\{
\parbox[c]{8in}{
$u_i={{\partial F}\over {\partial x_i}} (x_0,x_1,x_2)$ $(i=0,1,2)$\\
$F(x_0,x_1,x_2)=0$
}
\right.$$
$B$+$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
$B$K$h$k>C5n$,2DG=$G$"$k(B. 
$$I = Id(
u_0-{{\partial F}\over {\partial x_0}},
u_1-{{\partial F}\over {\partial x_1}},
u_2-{{\partial F}\over {\partial x_2}},
F)$$
$B$H$9$k;~(B, $\{x_0, x_1, x_2\}$ $\succ$ $\{u_0, u_1, u_2\}$ $B$J$kG$0U$N>C(B
$B5n=g=x$K$h$j(B $I$ $B$N%0%l%V%J4pDl(B $GB(I)$ $B$r7W;;$9$l$P(B, 
$$I \cap \Q[u_0,u_1,u_2] = Id(GB(I) \cap Q[u_0,u_1,u_2]).$$
$B0J2<$NNc$G(B, $V(g_i)$ $B$O(B $V(f_i)$ $B$NAPBP6J@~$G$"$k(B. 

\vskip\baselineskip
$\left\{
\parbox[c]{6in}{
$f_1=x^5-x^3+y^2$\\
$g_1=108x^7-108x^5+1017y^2x^4-16y^4x^3-4250y^2x^2+1800y^4x-108y^6+3125y^2$
}
\right.$

\vskip\baselineskip
$\left\{
\parbox[c]{6in}{
$f_2=x^6+3y^2x^4+(3y^4-4y^2)x^2+y^6$\\
$g_2=-256x^6+(64y^4-192y^2+864)x^4+(-192y^4+1620y^2-729)x^2-256y^6+864y^4-729y^2$
}
\right.$

\vskip\baselineskip
$\left\{
\parbox[c]{6in}{
$f_3=2x^4-3yx^2+y^4-2y^3+y^2$\\
$g_3=-12x^6+(-y^2+178y-37)x^4+(12y^3-768y^2+2208y+4608)x^2-32y^4+1024y^3-7680y^2-8192y-2048$
}
\right.$

\vskip\baselineskip
\begin{figure}[hbtp]
\begin{tabular}{cc}
\begin{minipage}{.5\hsize}
\begin{center}
\epsfxsize=7cm
\epsffile{ps/1.ps}
\end{center}
\caption{$f_1=0$}
\label{f2}
\end{minipage} &

\begin{minipage}{.5\hsize}
\begin{center}
\epsfxsize=7cm
\epsffile{ps/1d.ps}
\end{center}
\caption{$g_1=0$}
\label{g2}
\end{minipage}
\end{tabular}
\end{figure}


\begin{figure}[hbtp]
\begin{tabular}{cc}
\begin{minipage}{.5\hsize}
\begin{center}
\epsfxsize=7cm
\epsffile{ps/2.ps}
\end{center}
\caption{$f_2=0$}
\label{f3}
\end{minipage} &

\begin{minipage}{.5\hsize}
\begin{center}
\epsfxsize=7cm
\epsffile{ps/2d.ps}
\end{center}
\caption{$g_2=0$}
\label{g3}
\end{minipage}
\end{tabular}
\end{figure}

\begin{figure}[hbtp]
\begin{tabular}{cc}
\begin{minipage}{.5\hsize}
\begin{center}
\epsfxsize=7cm
\epsffile{ps/4.ps}
\end{center}
\caption{$f_3=0$}
\label{f5}
\end{minipage} &

\begin{minipage}{.5\hsize}
\begin{center}
\epsfxsize=7cm
\epsffile{ps/4d.ps}
\end{center}
\caption{$g_3=0$}
\label{g5}
\end{minipage}
\end{tabular}
\end{figure}
