\chapter{$B%0%l%V%J4pDl(B}
\label{chapgr}
\section{$BBe?tJ}Dx<0$N2r$H%$%G%"%k(B}

$BBN(B $K$ $B>e$N(B $n$ $BJQ?tB?9`<04D(B $R = K[x_1,\cdots,x_n]$ $B$r9M$($k(B. $B0J2<(B, 
$(x_1,\cdots,x_n)$ $B$r(B $X$ $B$HN,5-$9$k(B. $R$ $B$N85(B $f_1,\cdots,f_m$ $B$KBP$7(B, 
\begin{equation}
\label{system}
 f_1 = 0, \cdots, f_m =0
\end{equation}
$B$rBe?tJ}Dx<07O(B, $B$"$k$$$OC1$KJ}Dx<0$H8F$V(B. (\ref{system}) $B$rK~$?$9(B 
$K^n$ $B$N85$r(B $(1)$ $B$N2r$H8F$V(B. $B$3$N$h$&$JJ}Dx<0$r2r$$$F2r$r5a$a$h$&$H(B
$B$9$k>l9g$K:G$b4pK\E*$JJ}K!$O(B, $BCf3X0JMh$*$J$8$_$N>C5nK!$G$"$k(B. 

\begin{ex}
$f_1(x,y) = x^2+y^2 - 2 = 0, f_2(x,y) = xy - 1 = 0$
$B$r2r$1(B
\end{ex}
{\bf $B2r(B} $y^2f_1 - (xy+1)f_2 = y^4-2y^2+1 = 0$
$B$h$j(B $y=x=1$ $B$^$?$O(B $y=x=-1.$ $B$3$l$i$O<B:]$K2r$G$"$k(B. \qed\\
$B$3$3$G9T$C$?7W;;$O(B, $f_1$, $f_2$ $B$KE,Ev$JB?9`<0$r3]$1$?$b$N$NOB$r:n$C$F(B
$B$h$jJQ?t$N>/$J$$B?9`<0$r:n$j=P$9$b$N$G$"$k(B. 

\begin{df}
$f_1, \cdots, f_m \in R$ $B$KBP$7(B, 
$$Id(f_1,\cdots,f_m) = \{\sum_{i=1}^n g_if_i \mid i \in R\}$$
$B$r(B $f_1,\cdots,f_m$ $B$G@8@.$5$l$k%$%G%"%k$H8F$V(B. $f_1, \cdots, f_m$ $B$r(B 
$I$ $B$N@8@.7O$"$k$$$O(B{\bf $B4pDl(B}$B$H8F$V(B.
\end{df}
$B0lHL$K(B, $BJ}Dx<0(B (\ref{system}) $B$,M?$($i$l$?>l9g(B, $I=Id(f_1,\cdots,f_m)$
$B$r9M$($l$P(B, $B>C5nK!$H$O(B $I$ $B$NCf$+$i(B, $B4^$^$l$kJQ?t$N8D?t$,>/$J$$$b$N$rA*$S=P(B
$B$9J}K!$H8@$($k(B. 
$I$ $B$N85A4$F$N6&DLNmE@$O(B $(\ref{system})$ $B$N2r$K0lCW$9$k(B. $B%$%G%"%k$N4p(B
$BDl$O0lAH$H$O8B$i$J$$$,(B, $BF10l$N%$%G%"%k$r@8@.$9$k4pDl$N6&DLNmE@$O0lCW$9(B
$B$k$+$i(B, $B%$%G%"%k$r9M$($kJ}$,(B, $BJ}Dx<0$N2r$r9M$($k>e$G$h$j<+A3$G$"$k$H8@$((B
$B$k(B.

\begin{ex}
$Id(x^2+y^2 - 2,xy - 1) = Id(-y^4+2y^2-1,x+y^3-2y)$
\end{ex}

\begin{ex} ($B@~7AJ}Dx<0(B)
$Id(2a+3b-4c+d-1,3a-2c-5d-4,a-b+4d-5,3a+2b+2c-2d)$
$=Id(-185d+78,-185c-94,-185b-299,-185a+314)$
\end{ex}
$B$3$l$i$NNc$G$O(B, $B1&JU$N4pDl$O3N$+$K2r$rMF0W$K5a$a$i$l$k7A$K$J$C$F$$$k(B. 
$B$3$3$GBg;v$JE@$O(B, $BN>JU$,Ey$7$$%$%G%"%k$rM?$($F$$$k$+$I$&$+(B, $B$H$$$&E@(B
$B$G$"$k(B. 

\begin{df}
$B%$%G%"%k(B $I$ $B$KBP$7(B $I$ $B$N(B $K^n$ $B$K$*$1$k(B variety $V_K(I)$ $B$r(B
\begin{center}
$V_K(I) = \{a \in K^n \mid$ $B$9$Y$F$N(B $f \in I$ $B$KBP$7(B $f(a)=0 \}$
\end{center}
$B$GDj5A$9$k(B. $B:.Mp$N$J$$>l9g$K$O(B $V(I)$ $B$H=q$/(B. 
\end{df}

\begin{co}
$I \subset J \Rightarrow V(J) \subset V(I)$
\end{co}
$B$9$J$o$A(B, $B>C5nK!$K$h$jF@$i$l$?B?9`<0$O(B, $B%$%G%"%k$N85$G$"$k$3$H$OJ]>Z(B
$B$5$l$k$,(B, $B$=$l$i$O$"$/$^$G2r$NK~$?$9$Y$-I,MW>r7o$G$"$j(B, $B<B:]$K$b$H$N(B
$BJ}Dx<0$N2r$rI=$9$+H]$+$OJL$N%A%'%C%/$,I,MW$H$J$k(B. $B$b$7?7$?$KF@$i$l$?(B
$BB?9`<0=89g$,$b$H$N%$%G%"%k$r@8@.$7$F$$$l$P(B, $B2r$,Ey$7$$$3$H$OJ]>Z$5$l(B
$B$F$$$k(B. 

\section{$B9`=g=x(B, $B%b%N%$%G%"%k(B, $B%0%l%V%J4pDl(B}

$BA0@a$G(B, $BBe?tJ}Dx<0$N2r$r9M$($k>e$G(B, $BB?9`<0%$%G%"%k$r9M$((B, $B$=$N4pDl$r<h(B
$B$j49$($FJ}Dx<0$r2r$-$d$9$$4pDl$K<h$j49$($k$3$H$,M-8z$G$"$k$3$H$r<($7$?(B. 
$B$7$+$7(B, $B%$%G%"%k$N35G0$O(B, $BJ}Dx<0$r2r$/$?$a$@$1$KMQ$$$i$l$k$b$N$G$O$J$$(B. 
$BFC$K(B, $B2r$,L58B8D$K$J$k>l9g$O(B, $B$=$N2rA4BN$OBe?tE*=89g$H$7$F07$o$l$k$Y$-(B
$B$b$N$G$"$j(B, $BJ}Dx<0$N2r$H$7$FI=8=$9$k$3$H$O0lHL$K$OFq$7$$(B. $B$`$7$m(B, $B%$%G(B
$B%"%k(B $I$ $B$"$k$$$O4D(B $R/I$ $B$N@-<A$+$i$=$NBe?tE*=89g$N@.J,(B, $B<!85$J$I$rCN(B
$B$k$3$H$,=EMW$H$J$k(B. $B$=$N$?$a$K%$%G%"%k$N4pDl$,K~$?$9$Y$->r7o$H$7$F<!$N(B
$B$h$&$J$b$N$r5s$2$k$3$H$,$G$-$k(B.

\begin{itemize}
\item
$B$"$kB?9`<0$,(B, $B%$%G%"%k$KB0$9$k$+H]$+(B ($B%a%s%P%7%C%W(B) $B$,%"%k%4%j%:%`$K(B
$B$h$jH=Dj$G$-$k(B.

$I$ $B$KB0$9$kB?9`<0$r6qBNE*$KGD0.$9$k$?$a$KI,MW$G$"$k(B. 

\item
$B%$%G%"%k$r(B, $BO"N)J}Dx<07O$H$_$?$H$-(B, $B2r$r5a$a$d$9$$7A$r$7$F$$$k(B. 

$B>C5nK!$HF1MM$N7k2L$rM?$($k$3$H$,$G$-$k(B. 

\item
$B%$%G%"%k$N@-<A(B ($B<!85(B, $B0x;R$J$I(B) $B$rI=$7$F$$$k(B. 
\end{itemize}
$B$3$l$i$N@-<A$N$&$A(B, $BBh0l$N$b$N$KCeL\$9$k(B. 

\begin{ex}
$n=1$ $B$N>l9g(B\\
$R=K[x]$ $B$O(B PID ($BC19`%$%G%"%k@00h(B) $B$G$"$k(B. $B$9$J$o$AG$0U$N%$%G%"%k(B $I$ $B$O(B
$B$"$k(B $f \in R$ $B$K$h$j(B $I = Id(f)$ $B$H=q$1$k(B. $B$3$l$O(B, $f$ $B$r4pDl$H$9$k$3$H(B
$B$K$h$j(B, $I$ $B$KBP$9$k%a%s%P%7%C%W$,(B, 
$$ g \in I \Leftrightarrow f \mid g $$
$B$GH=Dj$G$-$k$3$H$r0UL#$9$k(B. $I$ $B$N@8@.85$,4v$D$+M?$($i$l$F$$$k>l9g(B, 
$f$ $B$O(B, $B$=$l$i$N@8@.85$N(B GCD $B$r5a$a$k$3$H$GF@$i$l$k(B. 
\end{ex}

$B0lJQ?t$N>l9g(B, $B%$%G%"%k$N@8@.85(B
$B$O(B, $B$=$N%$%G%"%k$KB0$9$k85$N$&$A(B, $B:G$b<!?t$N>.$5$$$b$N$r$H$l$P$h$+$C$?(B. 
$B$3$l$O(B, $B<!$N$h$&$K$$$$$+$($i$l$k(B. 

\begin{itemize}
\item
$BB?9`<0$N3F9`$r9_QQ$N=g$KJB$Y$?$H$-(B, $B@hF,$N9`$r(B {\bf $BF,9`(B} $B$H8F$V(B. 
$B$3$N;~(B, $BF,9`$,(B, $B%$%G%"%k$N$9$Y$F$N85$NF,9`$r3d$j@Z$k$h$&$J85$,@8@.85$H$J$k(B. 
\end{itemize}
$B$3$l$rB?JQ?t$K3HD%$9$k$?$a$K(B, $B0lJQ?t$N>l9g$HF1MM$K(B, $BB?JQ?tB?9`<0$N9`(B
$B$N4V$K!V<+A3$J!WA4=g=x$rF~$l$k(B. $B0J2<(B, $BBN(B $K$ $B>e$N(B $n$ $BJQ?tB?9`<04D(B
$R = K[x_1,\cdots,x_n]$ $B$r8GDj$7$F9M$($k(B. $B<+A3?t(B $\N$ $B$O(B, 0 $B0J>e$N@0?t(B
$B$rI=$9(B. 

\begin{df}
$T = \{x_1^{i_1}\cdots x_n^{i_n}\mid i_1,\cdots,i_n \in \N \}$ $B$H$7(B, $T$ $B$N85$r(B
term {\bf ($B9`(B)} $B$H8F$V(B. $B$3$N;~(B, T $B$K$*$1$kA4=g=x(B $\le$ $B$,(B {\bf term order}
$B$G$"$k$H$O(B, 
\begin{enumerate}
\item $B$9$Y$F$N(B $t \in T$ $B$KBP$7(B $1 \le t$
\item $B$9$Y$F$N(B $t_1, t_2, s \in T$ $B$KBP$7(B $(t_1 \le t_2 \Rightarrow t_1 \cdot s \le t_2 \cdot s)$
\end{enumerate}
$B$rK~$?$9$3$H$r8@$&(B. 
\end{df}

\begin{df}
$B;X?t$r(B $\N^n$ $B$N85$H9M$($F(B, $\N^n$ $B$K$*$1$k(B term order $<$ $B$r(B

\begin{enumerate}
\item $B$9$Y$F$N(B $\alpha \in \N^n$ $B$KBP$7(B $0 = (0,\cdots,0) \le \alpha$
\item $B$9$Y$F$N(B $\alpha_1, \alpha_2, s \in \N$ $B$KBP$7(B
$(\alpha_1 \le \alpha_2 \Rightarrow \alpha_1 + s \le \alpha_2 + s)$
\end{enumerate}
$B$rK~$?$9$b$N$H$7$FDj5A$G$-$k(B. 
\end{df}

\begin{df}
$L \subset \N^n$ $B$,%b%N%$%G%"%k$H$O(B, 
$BG$0U$N(B $\alpha \in L, \beta \in \N^n$ $B$KBP$7(B $\alpha+\beta \in L$
$B$,@.$jN)$D$3$H$r$$$&(B. $B$^$?(B, $S \subset \N^n$ $B$KBP$7(B, 
$$mono(S) = \{\alpha+\beta \mid \alpha \in S, \beta \in \N^n\}$$
$B$r(B $S$ $B$G@8@.$5$l$k%b%N%$%G%"%k$H8F$V(B. 
\end{df}

\begin{df}
term $B4V$NA4=g=x$rB?9`<0$NH>=g=x$K<+A3$K3HD%$9$k(B. \\
$f, g \in R$ $B$G(B, $f = \sum_{i\ge 0}c_i f_i$, $g = \sum_{i\ge 0}d_i g_i$
( $c_i, d_i \in K, f_i, g_i \in T, i > j \Rightarrow f_i > f_j, g_i > g_j$)
$B$H$9$k;~(B, $f > g$ $B$r(B
\begin{center}
$f > g \Leftrightarrow$ $B$"$k(B $i_0$ $B$,B8:_$7$F(B $(i<i_0 \Rightarrow f_i = g_i, f_{i_0} > g_{i_0})$
\end{center}
$B$GDj5A$9$k(B. 
\end{df}

\begin{df}
$M = \{c\cdot t\mid c \in K, t \in T\}$ $B$H$7(B, $M$ $B$N85$r(B {\bf monomial} $B$H8F$V(B. 
\end{df}

\begin{df}
term order $B$r0l$D8GDj$7$?;~(B, $BB?9`<0(B $f$ $B$KI=$l$k(B term $B$NCf$G(B, $B$=(B
$B$N(B order $B$K$*$$$F:GBg$N$b$N$r(B {\bf $BF,9`(B (head term)} $B$H8F$S(B, $HT(f)$ $B$H(B
$B=q$/(B. \\
$HT(f)$ $B$N78?t$r(B, $HC(f)$ $B$H=q$/(B. \\
$HC(f)\cdot HT(f)$ $B$r(B $HM(f)$ $B$H=q$/(B. \\
$HT(f)$ $B$N;X?t$r(B $HE(f)$ $B$H=q$/(B. $HE(f) \in \N^n$ $B$G$"$k(B. \\
$B$5$i$K(B, $f-HM(f)$ $B$r(B $red(f)$ ({\bf reductum} of $f$) $B$H=q$/(B.
\end{df}

\begin{lm}
$\N^n$ $B$NG$0U$N%b%N%$%G%"%k(B $L$ $B$OM-8B@8@.(B. 
\end{lm}
\proof $n$ $B$K4X$9$k5"G<K!$K$h$j<($9(B. $n=1$ $B$N$H$-(B, $L$ $B$N(B $\N$ $BCf$G$N(B
$B:G>.85(B $\alpha$ $B$r$H$l$P(B $L$ $B$O(B $\alpha$ $B$G@8@.$5$l$k(B. $n-1$ $B$^$G8@$($?(B
$B$H$9$k(B. $B3F(B $j \in \N$ $B$KBP$7(B, 
$$L_j=\{ (\alpha_1,\cdots,\alpha_{n-1} \in
N^{n-1}\mid (\alpha_1,\cdots,\alpha_{n-1},j)\in L \}$$$B$H$*$/$H(B $\{L_j\}$ 
$B$O%b%N%$%G%"%k$NA}BgNs(B. $L_\infty = \cup L_j$ $B$H$*$/$H(B$L_\infty$ $B$b%b(B
$B%N%$%G%"%k$G(B, $B5"G<K!$N2>Dj$K$h$j(B $L_\infty$ $B$OM-8B@8@.(B. $B$h$C$F$"$k(B 
$j_0$ $B$,B8:_$7$F(B $L_\infty=L_{j_0}$. $B$3$N$H$-(B, $L$ $B$O(B,
$j=1,\cdots,j_0$ $B$KBP$9$k(B $L_j$ $B$N@8@.85(B ($B$3$l$O5"G<K!$N2>Dj$K$h$j$=$l(B
$B$>$lM-8B=89g(B) $B$NOB=89g$G@8@.$5$l$k(B. \qed

\begin{co}
\label{noether}
$\{L_i\} (i=1,2,\cdots)$ $B$r%b%N%$%G%"%k$NA}BgNs$H$9$l$P(B, $B$"$k(B $i_0$ $B$,(B
$BB8:_$7$F(B, $i \ge i_0 \Rightarrow L_i=L_{i_0}$.
\end{co}

\begin{co}
$\N^n$ $B$NG$0U$NItJ,=89g$O(B term order $<$ $B$K4X$7$F:G>.85$r;}$D(B. 
\end{co}

\begin{co}
$\le$ $B$r(B $T$ $B$N(B term order $B$H$9$l$P(B, $T$ $B$N??$N9_2<Ns$OM-8B$G@Z$l$k(B.
$\le$ $B$N(B $R$ $B$X$N<+A3$J3HD%$K$D$$$F$bF1MM$G$"$k(B. $B0J2<(B, $B$3$N@-<A$r(B, term
order $B$N(B Noether $B@-$H$7$F0zMQ$9$k(B. 
\end{co}
\proof $T$ $B$K$D$$$F$O9_2<Ns$,:G>.85$r;}$D$3$H$+$i@.$jN)$D(B. $R$ $B$K(B
$B$D$$$F$O(B, $B$b$7(B $R$ $B$N??$N9_2<Ns$G$"$kL58BNs$,$"$l$P(B, $BF,9`$,(B $T$ $B$N??$N(B
$B9_2<Ns$G$"$kL58BNs$r:n$j=P$;$k$+$iL7=b(B. \qed

\begin{df}
$S \subset R$ $B$*$h$S(B term order $<$ $B$KBP$7(B, 
$$E_<(S) = \{HE(f) \mid f \in S\} \subset \N^n$$
$B$HDj5A$9$k(B. $B0J2<:.Mp$N$J$$>l9g$K$O(B $<$ $B$r>JN,$7$F(B $E(S)$ $B$H=q$/(B. 
\end{df}

\begin{lm}
$B%$%G%"%k(B $I$ $B$KBP$7(B, $E(I)$ $B$O%b%N%$%G%"%k(B.
\end{lm}

$B0lJQ?tB?9`<04D$K$*$1$k%$%G%"%k(B $I$ $B$N@8@.7O(B $G = \{f\}$ $B$N@-<A$O(B, 
$$E(I) = mono(E(G))$$
$B$H=q$/$3$H$,$G$-$k(B. $B0lHL$N>l9g$K$b$3$N$h$&$J@-<A$rK~$?$9(B $G$ $B$r9M$($k$3$H(B
$B$OM-MQ$G$"$k(B. 

\begin{df}
$B%$%G%"%k(B $I$ $B$KBP$7(B, $B$=$NM-8BItJ,=89g(B $G$ $B$G(B, 
$$E(I) = mono(E(G))$$
$B$rK~$?$9$b$N$r(B $I$ $B$N(B $<$ $B$K4X$9$k%0%l%V%J4pDl$H8F$V(B. 
\end{df}
$B$3$NDj5A$O(B, $B%$%G%"%k$N$9$Y$F$N85$NF,9`$,(B, 
$G$ $B$N$$$:$l$+$N85$NF,9`$G3d$j@Z$l$k$3$H$r0UL#$9$k(B. 

\begin{pr}
$BG$0U$N(B term order $<$ $B$K4X$7(B, $B%$%G%"%k(B $I$ $B$N%0%l%V%J4pDl$OB8:_$9$k(B. 
\end{pr}
\proof $B%b%N%$%G%"%k$NM-8B@8@.@-$h$jL@$i$+(B. \qed

\begin{pr}
$B%$%G%"%k(B $I$ $B$N(B $B%0%l%V%J4pDl(B $G$ $B$O(B $I$ $B$r@8@.$9$k(B. 
\end{pr}
\proof
$f \in I$ $B$H$9$k(B. $B2>Dj$K$h$j(B, $B$"$k(B $g \in G$ $B$,B8:_$7$F(B, $HT(g)|HT(f)$. 
$B$h$C$F(B, $B$"$k(B $s \in M$ $B$,B8:_$7$F(B, $HT(f-s\cdot g) < f$. $f-s\cdot g
\in I$$B$@$+$i(B, $B$3$NA`:n$r7+$jJV$9$3$H$,$G$-$k(B. term order $B$N(B Noether
$B@-$h$j(B, $B$3$NA`:n$OM-8B2s$G=*N;$9$k(B. $B$9$J$o$A(B, $BM-8B2s$NA`:n$N8e(B, 0 $B$H$J$k(B. 
$B$3$l$O(B, $G$ $B$,(B $I$ $B$r@8@.$9$k$3$H$r0UL#$9$k(B.  \qed\\
$B$3$NL?Bj$K$*$$$F(B, $f -s\cdot g$ ($f, g \in R; s \in M$) $B$J$k1i;;$,8=$l$?(B. 
$BL?Bj$K$*$$$F$O(B, $f$ $B$NF,9`$r>C5n$9$k$?$a$K9T$J$o$l$?$,(B, $B0lHL$K(B, $f$ $B$N(B
$B9`$r(B, $B$3$N$h$&$K>C5n$9$k1i;;$,(B, $B%0%l%V%J4pDl7W;;$K$*$$$F4pK\E*$J1i;;$H$J$k(B. 

\begin{df}
$f, g \in R$ $B$H$7(B, $f$ $B$K8=$l$k$"$k(B monomial $m$ $B$,(B $HT(g)$ $B$G3d$j@Z$l(B
$B$k$H$9$k(B. $B$3$N$H$-(B $f$ $B$O(B $g$ $B$G(B {\bf $B4JLs2DG=(B (reducible)} $B$G$"$k$H$$(B
$B$&(B. $B$3$N$H$-(B $h = f-m/HT(g)\cdot g$ $B$KBP$7(B, $ f \mred{g} h$ $B$H=q$/(B. \\
$G \subset R$ $B$K$D$$$F$b(B, $G$ $B$N$"$k85$K$h$j(B $f$ $B$,(B $h$ $B$K4JLs$5$l$k$H$-(B
$f \mred{G} h$ $B$H=q$/(B. $B$5$i$K(B, $G$ $B$K$h$k4JLs$r(B 0 $B2s0J>e7+$jJV$7$F(B $h$ $B$,(B
$BF@$i$l$k$H$-(B, $f\tmred{G} h$ $B$H=q$/(B. \\
$f$ $B$N$I$N9`$b(B, $G$ $B$G4JLs$G$-$J$$$H$-(B, $f$ $B$O(B $G$ $B$K4X$7$F(B {\bf $B@55,(B
$B7A(B (normal form)} $B$G$"$k$H$$$&(B. 
\end{df}


\begin{pr}
$B%$%G%"%k(B $I$ $B$N(B $B%0%l%V%J4pDl(B $G$ $B$K$D$$$F(B, $B0J2<$N$3$H$,@.$jN)$D(B. 
\begin{enumerate}
\item
$f \in I \Leftrightarrow f \tmred{G} 0$
\item
$f \tmred{G} f_1, f \tmred{G} f_2$ $B$+$D(B $f_1, f_2$ $B$,@55,7A(B $\Rightarrow f_1 = f_2$
\item
$f \in G$ $B$G(B, $B$"$k(B $h \in G$ $B$,B8:_$7$F(B $HT(h) | HT(f)$
$\Rightarrow G \setminus \{f\}$ $B$O(B $I$ $B$N%0%l%V%J4pDl(B
\end{enumerate}
\end{pr}
\proof
1. $B$O(B, $BA0L?Bj$N>ZL@$h$jL@$i$+(B. 3. $B$bDj5A$h$jL@$i$+(B. 2. $B$r<($9(B. $f -
f_1, f - f_2 \in I$ $B$h$j(B $f_1 - f_2 \in I$. $B$h$C$F(B $f_1 - f_2 \tmred{G} 0$. 
$B$H$3$m$,(B, $f_1$, $f_2$ $B$H$b(B, $G$ $B$K$D$$$F@55,7A$h$j(B $f_1 - f_2$ $B$b@55,7A(B. 
$B$h$C$F(B, $f_1 - f_2 = 0$. \qed

\begin{co}
$G = \{g_1, \cdots, g_l\}$ $B$r%$%G%"%k(B $I$ $B$N(B $B%0%l%V%J4pDl$H$9$k(B. $B$3$N$H$-(B, 
$B$"$k(B $H \subset G$ $B$,B8:_$7$F(B $H$ $B$O(B $I$ $B$N(B $B%0%l%V%J4pDl$+$D(B $HT(g_i) (g_i \in H)$ 
$B$N$I$NFs$D$b8_$$$KB>$r3d$i$J$$(B. 
\end{co}
$B$3$N7O$K$h$j(B, $B>iD9$J85$r$9$Y$F=|5n$7$?(B $B%0%l%V%J4pDl$KBP$7(B, $B3F85$r(B, $B4p(B
$BDl$NB>$N85$KBP$7$F@55,7A$H$J$k$h$&4JLs$r9T$J$$(B, $BF,9`$N78?t$r(B 1 $B$H$J$k(B
$B$h$&$K$7$?4pDl$r(B{\bf $BHoLs(B (reduced) $B%0%l%V%J4pDl(B} $B$H8F$V(B. $B$3$l$,<B:]$K(B
$B85$N%$%G%"%k$N(B $B%0%l%V%J4pDl$K$J$C$F$$$k$3$H$O(B, $BF,9`$,JQ$o$C$F$$$J$$$3(B
$B$H$h$j$o$+$k(B. $B$5$i$K<!$NL?Bj$O(B, $BDj5A$h$j$?$@$A$KF@$i$l$k(B.

\begin{pr}
$BHoLs%0%l%V%J4pDl$O=89g$H$7$F0l0UE*$KDj$^$k(B. 
\end{pr}
$B0J>e$,(B $B%0%l%V%J4pDl$NDj5A$+$i$?$@$A$KF3$+$l$k4pK\E*$J@-<A$G$"$k$,(B, $B<B:]$K(B $B%0%l%V%J4pDl$r(B
$B9=@.$9$k%"%k%4%j%:%`$rF@$k$?$a$K$O(B, $B%0%l%V%J4pDl$NDj5A$N8@$$$+$($r$$$/$D$+9T$J$&(B
$BI,MW$,$"$k(B.

\begin{df}
$G = \{g_1,\cdots,g_l\}$ $B$r(B ($B%0%l%V%J4pDl$H$O8B$i$J$$(B) R $B$NM-8BItJ,=89g$H$9$k(B.
$B$3$l$KBP$7(B, $B<LA|(B $d_1$ $B$r<!$GDj5A$9$k(B. 

\begin{tabbing}
aaaa \= aaaaaaaaaa        \= aaa       \= aaaa\kill
$d_1 :$\> $R^l$              \> $\longrightarrow$ \> $R$\\
       \> $(f_1,\cdots,f_l)$ \> $\longmapsto$     \> $\sum f_i\cdot HT(g_i)$
\end{tabbing}
\end{df}

\begin{df}
$f = (f_1,\cdots,f_l) \in R^l$ $B$,(B {\bf $T$-$B@F<!(B} $B$H$O(B, $B$"$k(B term $t$ 
$B$,B8:_$7$F(B, $B$9$Y$F$N(B $i$ $B$KBP$7(B, $f_i = 0$ $B$^$?$O(B $t = f_i\cdot
HT(g_i)$ $B$H=q$1$k$H$-$r8@$&(B.
\end{df}

\begin{df}
$e_i \in R^l$ $B$r(B $e_i = (0,\cdots,1,\cdots,0)$ ($BBh(B $i$ $B@.J,$N$_(B 1) $B$HDj5A(B
$B$9$k(B. \\
$i_1,\cdots,i_k$ $B$KBP$7(B, $T_{i_1,\cdots, i_k}$ $B$r(B
$$T_{i_1,\cdots, i_k} = \LCM(HT(g_{i_1}),\cdots,HT(g_{i_k}))$$
$B$HDj5A$9$k(B.
$BFC$K(B, $T_i = HT(g_i)$ $B$G$"$k(B.
\end{df}

\begin{pr}
\label{'thomo'}
$B%$%G%"%k(B $I$ $B$K$D$$$F(B, $B<!$OF1CM(B.
\begin{enumerate}
\item
$G = \{g_1,\cdots,g_l\}$ $B$O(B $I$ $B$N%0%l%V%J4pDl(B
\item
$f \in I \Leftrightarrow f \tmred{G} 0$
\item
$f \in I \Leftrightarrow$ $B$"$k(B $f_i (i = 1, \cdots, l)$ $B$,B8:_$7$F(B
$f = \sum_i f_i g_i$ $B$+$D(B $HT(f_i g_i) \le HT(f)$
\item
L $B$r(B $T$-$B@F<!(B $B$J(B $\Ker(d_1)$ $B$N4pDl$H$9$k$H$-(B, $BG$0U$N(B $h =
(h_1,\cdots,h_l) \in L$ $B$KBP$7(B, $\sum h_i\cdot g_i \tmred{G} 0$
\end{enumerate}
\end{pr}
\proof\\
1. $\Leftrightarrow$ 2.)
$BL@$i$+(B. \\
2. $\Rightarrow$ 3.) 
$G$ $B$N85$K$h$k4JLsA`:n$r0l$D$K$^$H$a$k$HF@$i$l$k(B. \\
3. $\Rightarrow$ 2.)
$B$"$k(B $i$ $B$,B8:_$7$F(B $HT(f_i g_i) = HT(f)$ $B$H$J$j(B, 
$HT(f)$ $B$,(B $HT(f_i)$ $B$G3d$j@Z$l$k$3$H$+$i$o$+$k(B. \\
4. $\Rightarrow$ 1.)
$f = \sum f_i\cdot g_i \in I$ $B$H$7(B, $HT(f)$ $B$,(B $HT(g_i)$ $B$N$$$:$l$+$G(B
$B3d$j@Z$l$k$3$H$r8@$($P$h$$(B. $B4JC1$N$?$a(B, $B$9$Y$F$N(B $i$ $B$KBP$7(B, $HC(g_i)
= 1$ $B$H$9$k(B. $L = \{b_1,\cdots,b_l\}$$B$H$9$k(B.
$m = \max_i(HT(f_i g_i))$ $B$H$*$/(B. \\
\underline{$HT(f) = m$ $B$N>l9g(B}
\quad $HT(f)$ $B$O(B, $HT(g_i)$ $B$N$$$:$l$+$G3d$j@Z$l$k(B.\\
\underline{$HT(f) < m$ $B$N>l9g(B}
\quad $A = \{i \mid HT(f_i g_i) = m\}$ $B$H$*$/$H(B, $B2>Dj$h$j(B, 
$$\sum_{i\in A}HM(f_i)\cdot HT(g_i) = 0.$$ 
$B$h$C$F(B, $h = (h_1,\cdots,h_l) \in R^l$ $B$r(B, 
$h_i = HM(f_i) (i \in A), h_i = 0 (i \notin A)$
$B$HDj5A$9$l$P(B, $h \in \Ker(d_1)$. $B$h$C$F2>Dj$h$j(B, 
\begin{center}
$B$"$k(B $c_i \in M$ $B$,B8:_$7$F(B $h = \sum_i c_i b_i.$
\end{center}
$B$3$l$h$j(B
$$\sum_k h_k g_k = \sum_i c_i \sum_k g_k b_{ik}.$$
$\displaystyle{G_i = \sum_k g_k b_{ik}}$ $B$H$*$/$H(B, $G_i \tmred{G} 0$ $B$h$j(B, 
2. $\Rightarrow$ 3. $B$HF1MM$K(B, 
$B$"$k(B $b'_{ik}$ $B$,B8:_$7$F(B $G_i = \sum_k g_k b'_{ik}$ $B$+$D(B $HT(g_k b'_{ik}) \le HT(G_i).$
$B0lJ}(B, $b_i \in \Ker(d_1)$ $B$h$j(B, $HT(G_i) < \max_k(HT(g_k b_{ik}))$.\\
$B$5$F(B, $\displaystyle{f'_k = \sum_i c_i b'_{ik}}$ $B$H$*$/$H(B, $\displaystyle{\sum_k h_k g_k = \sum_k f'_k g_k}$ $B$G(B, 
$$f = \sum_{i \notin A} f_k g_k + \sum_{i \in A} (red(f)+f'_k)g_k$$
 $B$H=q$1(B, 
$$HT(f'_k g_k) \le \max_i(HT(c_i b'_{ik} g_k)) \le \max_i(HT(c_i G_i))$$
$$< \max_i(\max_k(HT(c_i g_k b_{ik}))) = \max_k(HT(h_k g_k)) = m$$
$B$h$j(B, $m$ $B$,$h$jDc$$=g=x$N>l9g$K5"Ce$G$-$k(B. $B$h$C$F(B, order $B$N(B Noether $B@-$K$h$j(B
$BM-8B2s$NA`:n$N$N$A(B, $m = HT(f)$ $B$N>l9g$K5"Ce$G$-$k(B. \qed
\begin{pr}
$G = \{g_1,\cdots,g_l\}$ $B$r(B $HC(g_i)=1$ $B$J$k(B R $B$NM-8BItJ,=89g$H$9$k(B.
$i,j \in \{1,\cdots,l\}$ $B$KBP$7(B, $S_{ij} \in R^l$ $B$r(B
$S_{ij} = T_{ij}/T_i e_i - T_{ij}/T_j e_j$ $B$GDj5A$9$k(B. $B$3$N;~(B, 
$L = \{S_{ij}\mid i < j\}$ $B$O(B $\Ker(d_1)$ $B$N(B $T$-$B@F<!(B $B$J4pDl$H$J$k(B. $B$3$N(B
$B4pDl$r(B {\bf Taylor $B4pDl(B} $B$H8F$V(B. 
\end{pr}
\proof
$f \in \Ker(d_1)$ $B$H$9$k(B. $f$ $B$r(B $T$-$B@F<!@.J,$KJ,2r$9$k$3$H$K$h$j(B, 
$f$ $B<+?H(B $T$-$B@F<!(B $B$H$7$F$h$$(B. $f = \sum_i f_i e_i$ $B$H$9$k$H(B, 
$f \in \Ker(d_1)$ $B$h$j(B, $f \neq 0$ $B$J$i$P(B, $B>/$J$/$H$b(B 2 $B$D$N@.J,$,(B 0
$B$G$J$$(B. $B$=$l$i$r(B $f_k, f_l (k < l)$ $B$H$9$k$H(B, $HT(f_k g_k) = HT(f_l g_l)$
$B$h$j(B, $T_{kl} | HT(f_k g_k)$. $B$3$l$h$j(B $f' = f - (HM(f_k g_k)/T_{kl}) S_{kl}$
$B$H$*$/$H(B, $f'$ $B$OBh(B $k$ $B@.J,$,(B 0 $B$H$J$j(B, 0 $B$G$J$$@.J,$,0l$D8:$k(B. $B$3$N(B
$BA`:n$r7+$jJV$7$F(B, $f$ $B$r(B $\{S_{ij}\}$ $B$G@8@.$G$-$k(B. \qed

\begin{df}
$f, g \in R$ $B$KBP$7(B, {\bf S $BB?9`<0(B} $Sp(f,g)$ $B$r(B,
$$Sp(f,g) = {HC(g)T_{fg}\over HT(f)}\cdot f - {HC(f)T_{fg}\over HT(g)}\cdot g$$\\
($T_{fg} = \LCM(HT(f),HT(g))$) $B$HDj5A$9$k(B. 
\end{df}
$B0J>e$K$h$j(B, $B?7$?$J(B $B%0%l%V%J4pDl$NH=Dj>r7o$,F@$i$l$k(B. 
\begin{pr}
$B%$%G%"%k(B $I$ $B$K$D$$$F(B, $B<!$OF1CM(B.
\begin{enumerate}
\item
$G = \{g_1,\cdots,g_l\}$ $B$O(B $I$ $B$N%0%l%V%J4pDl(B
\item
$BG$0U$NBP(B $\{f,g\} (f, g \in G; f \neq g)$ $B$KBP$7(B, $Sp(f,g) \tmred{G} 0$
\end{enumerate}
\end{pr}

\section{Buchberger $B%"%k%4%j%:%`(B}
$BA0@a$N:G8e$NL?Bj$K$h$j(B, $B<!$N%"%k%4%j%:%`$,F3$+$l$k(B. 

\begin{al}(Buchberger{\rm\cite{BUCH}})
\label{buch}
\begin{tabbing}
Input : $R$ $B$NM-8BItJ,=89g(B $F = {f_1,\cdots,f_l}$\\
Output : $F$ $B$G@8@.$5$l$k%$%G%"%k$N(B $B%0%l%V%J4pDl(B$G$\\

$D \leftarrow \{\{f,g\} \mid f, g \in F; f \neq g\}$\\
$G \leftarrow F$\\
while \=( $D \neq \emptyset$ ) do \{\\
\>$\{f,g\} \leftarrow D$ $B$N85(B\\
\>$D \leftarrow D \setminus \{C\}$\\
\>$h \leftarrow Sp(f,g)$ $B$N@55,7A$N0l$D(B\\
\>if \=$h \neq 0$ then \{\\
\>\>$D \leftarrow D \cup \{\{f,h\} \mid f \in G\}$\\
\>\>$G \leftarrow G \cup \{h\}$\\
\>\}\\
\}\\
return $G$
\end{tabbing}
\end{al}

$B0J2<$G(B, $D$ $B$N85$r(B{\bf $BBP(B} (pair) $B$H8F$V$3$H$K$9$k(B. 

\begin{th}
$B%"%k%4%j%:%`(B \ref{buch} $B$ODd;_$7(B, $B%0%l%V%J4pDl$r=PNO$9$k(B. 
\end{th}
\proof\\
\underline{$BDd;_@-(B}\quad $B@8@.$5$l$k@55,7A$NF,9`$,(B, $B$=$l$^$G$K@8@.$5$l$?@55,7A(B
$B$NF,9`$G3d$j@Z$l$J$$$3$H$h$j(B, $B7O(B \ref{noether} $B$+$i8@$($k(B. \\
\underline{$B=PNO$,%0%l%V%J4pDl$H$J$k$3$H(B}\quad $BA0L?Bj$K$h$j(B OK. \qed\\
$B$3$N%"%k%4%j%:%`$,(B Buchberger $B%"%k%4%j%:%`$N:G$b86;OE*$J7A$G$"$k$,(B, 
\begin{itemize}
\item
$B@55,7A$,(B 0 $B$G$J$$>l9g(B, $D$ $B$NMWAG$,(B $G$ $B$NMWAG$N8D?t$@$1A}2C$9$k(B. 
\item
$D$ $B$+$i0l$D85$rA*$VJ}K!$,L@<($5$l$F$$$J$$(B. 
\end{itemize}
$B$J$I$NE@$G<BMQE*$G$J$$(B. $B<B:]$K7W;;5!>e$K%$%s%W%j%a%s%H$9$k(B
$B>l9g(B, $B>e5-FsE@$K4X$7$F9)IW$r$9$kI,MW$,$"$k(B. $B$3$l$i$K4X$7$F$O(B, $B8e$K(B
$B>\$7$/=R$Y$k(B. 

\section{Term order $B$NNc(B}

$BMM!9$J(B term order $B$,Dj5A$G$-$k(B. $B$3$l$i$N(B order $B$O$=$l$>$l0[$C$?(B
$B@-<A$r$b$A(B, $B$=$N@-<A$K1~$8$F$5$^$6$^$JMQES$KMQ$$$i$l$k(B. 

\begin{df}
{\bf $B<-=q<0=g=x(B (lexicographical order; LEX)}\\
$x_1^{i_1}\cdots x_n^{i_n} > x_1^{j_1}\cdots x_n^{j_n} \Leftrightarrow$\\
$B$"$k(B $m$ $B$,B8:_$7$F(B $i_1 = j_1, \cdots, i_{m-1} = j_{m-1}, i_m > j_m$
\end{df}

$B$3$N=g=x$O>C5nK!$K$h$kJ}Dx<05a2r$K:G$bE,$7$?7A$N%0%l%V%J4pDl$rM?$($k(B.
$B$7$+$7(B, $B$=$ND>@\7W;;$O(B, $B;~4V(B, $B6u4V7W;;NL$,$7$P$7$P6K$a$FBg$-$/$J$k(B
$B$H$$$&$3$H$+$iITMx$G$"$k(B. 

\begin{df}
{\bf $BA4<!?t<-=q<0=g=x(B (total degree lexicographical order; DLEX)}\\
$x_1^{i_1}\cdots x_n^{i_n} > x_1^{j_1}\cdots x_n^{j_n} \Leftrightarrow$\\
$\sum_k i_k > \sum_k j_k$ $B$^$?$O(B\\
($\sum_k i_k = \sum_k j_k$ $B$+$D(B
 $B$"$k(B $m$ $B$,B8:_$7$F(B $i_1 = j_1, \cdots, i_{m-1} = j_{m-1}, i_m > j_m)$
\end{df}

$BF~NO$,@F<!$N>l9g(B, $B$3$N=g=x$N$b$H$G$N%0%l%V%J4pDl$H<-=q<0=g=x$K$h$k(B
$B%0%l%V%J4pDl$O0lCW$9$k(B. degree compatible order $B$K$h$k7W;;$O(B, $B$=$&$G$J$$(B
order $B$K$h$k7W;;$KHf$Y$F8zN($,$h$$$3$H$,7P83E*$KCN$i$l$F$*$j(B, $B@F<!(B
$B$J>l9g$K$3$N=g=x$G<-=q<0=g=x%0%l%V%J4pDl$r7W;;$9$k(B, $B$"$k$$$OHs@F<!(B
$B$JF~NO$r@F<!2=$7$F(B, $B$3$N=g=x$G%0%l%V%J4pDl$r7W;;$7(B, $BHs@F<!2=$9$k(B
$B$3$H$G$b$H$NF~NO$N<-=q<0=g=x%0%l%V%J4pDl$r5a$a$k$H$$$&$3$H$,9T$o$l$k(B. 

\begin{df}
{\bf $BA4<!?t5U<-=q<0=g=x(B (total degree reverse lexicographical order; DRL)}\\
$x_1^{i_1}\cdots x_n^{i_n} > x_1^{j_1}\cdots x_n^{j_n} \Leftrightarrow$\\
$\sum_k i_k > \sum_k j_k$ $B$^$?$O(B\\
($\sum_k i_k = \sum_k j_k$ $B$+$D(B 
$B$"$k(B $m$ $B$,B8:_$7$F(B $i_n = j_n, \cdots, i_{m+1} = j_{m+1}, i_m < j_m)$
\end{df}

$B0lHL$K(B, $B:G$b9bB.$K%0%l%V%J4pDl$r7W;;$G$-$k$,(B, $B%0%l%V%J4pDl$N8D!9$N(B
$B85$N;}$D@-<A$ODO$_$K$/$/(B, $BO"N)J}Dx<0$rD>@\2r$/$3$H$O:$Fq$G$"$k(B. 
$B$7$+$7(B, $B<!85(B, Hilbert function $B$=$NB>$NITJQNL$r7W;;$9$k>l9g$J$I(B, 
$B%0%l%V%J4pDl$H$$$&@-<A$N$_$,I,MW$H$5$l$k>l9g$K(B, $B9bB.$K7W;;$G$-$k(B
$B$H$$$&FC@-$r@8$+$7$FMQ$$$i$l$k>l9g$,B?$$(B. $B$^$?(B, $B8e$K=R$Y$k(B
$B4pDlJQ49$NF~NO$H$7$FMQ$$$i$l$k$3$H$bB?$$(B. 

\begin{df}
{\bf block order}\\
$\{x_1,\cdots,x_n\} = S_1 \cup \cdots \cup S_l$ (disjoint sum) $B$H$7(B, 
$<_i$ $B$r(B $T_i = K[y_1,\cdots]$ ($y_k \in S_l$) $B>e$N(B term
order $B$H$9$k(B. $B$3$N$H$-(B, $T$ $B>e$N(B order $B$r(B, $<_i$ $B$N(B order $B$r=g$KE,MQ$7$F(B
$B7h$a$k(B. 
\end{df}

$B<-=q<0=g=x$O(B $\{x_1,\cdots,x_n\} = \{x_1\} \cup \cdots \cup \{x_n\}$
 $B$J$kJ,3d$K$h$k(B block order $B$G$"$k$,(B, $BC1$K(B, $B4v$D$+$NJQ?t$r>C5n$7$?7k2L(B
$B$r5a$a$?$$>l9g$K$O(B, $B8zN($r9M$($l$PLdBj$,$"$k(B. $B$3$N$h$&$J>l9g$K(B, 
$S_1$ $B$K>C5n$7$?$$JQ?t(B, $S_2$ $B$K;D$j$NJQ?t(B, $B$HJ,3d$7(B, $B$=$l$>$l$K(B
$BBP$7(B, $BNc$($P(B DRL order $B$r@_Dj$9$k$3$H$G(B $S_1$ $B$KB0$9$kJQ?t$r>C5n$G$-$k(B. 
$B$3$l$K$D$$$F$O8e$G=R$Y$k(B. 

\begin{df}
{\bf matrix order}\\
$M$ $B$r(B $B<!$rK~$?$9<B(B $m\times n$ $B9TNs$H$9$k(B. 
\begin{enumerate}
\item $BD9$5(B $n$ $B$N@0?t%Y%/%H%k(B $v$ $B$KBP$7(B, $Mv=0 \Leftrightarrow v=0$
\item $BHsIi@.J,$r;}$DD9$5(B $n$ $B$N@0?t%Y%/%H%k(B $v$ $B$KBP$7(B, $Mv$ $B$N(B 0 $B$G$J$$:G=i(B
$B$N@.J,$O@5(B. 
\end{enumerate}

$B$3$N;~(B, $\N^n$ $B$N%Y%/%H%k(B $u,v$ $B$KBP$7(B, 
\begin{center}
$u>v \Leftrightarrow M(u-v)$ $B$N(B 0 $B$G$J$$:G=i$N@.J,$,@5(B
\end{center}
$B$GDj5A$9$l$P(B, $B$3$N(B order $B$O(B term order $B$H$J$k(B. $B$3$l$r(B, $M$ $B$K$h(B
$B$jDj5A$5$l$k(B matrix order $B$H8F$V(B.
\end{df}

\begin{pr} (Robbiano [\cite{ROBBIANO}])
$BG$0U$N(B term order $B$O(B, matrix order $B$K$h$jDj5A$G$-$k(B. 
\end{pr}

\newfont{\bigfont}{cmr10 scaled\magstep4}
\newcommand{\bigzl}{\smash{\hbox{\bigfont 0}}}
\newcommand{\bigzu}{\smash{\lower1.0ex \hbox{\bigfont 0}}}
\def\udots{\mathinner{\mkern1mu\raise1pt\vbox{\kern7pt\hbox{.}}\mkern2mu
    \raise4pt\hbox{.}\mkern2mu\raise7pt\hbox{.}\mkern1mu}}

\begin{ex} $B$h$/CN$i$l$?(B order $B$rDj5A$9$k(B matrix $B$NNc(B
\vskip\baselineskip

$M_{DLEX}=\left(
\begin{array}{ccc}
1 & \cdots & 1 \\
1 & & \bigzu \\
  & \ \ddots & \\
\bigzl & & 1 \\
\end{array}
\right)$
$M_{DRL}=\left(
\begin{array}{ccc}
1 & \cdots & 1 \\
\bigzu & & -1 \\
  & \udots & \\
-1 & & \bigzl \\
\end{array}
\right)$
$M_{LEX}=\left(
\begin{array}{ccc}
1 & & \bigzu \\
  & \ \ddots & \\
\bigzl & & 1 \\
\end{array}
\right)$

\vskip\baselineskip
$M_{DLEX}$, $M_{DRL}$, $M_{LEX}$ $B$O$=$l$>$lA4<!?t<-=q<0(B, $BA4<!?t5U<-=q<0(B, 
$B<-=q<0=g=x$rDj5A$9$k(B. 
\end{ex}


\begin{ex}
weighted order

\vskip\baselineskip
$M_{wDRL}=\left(
\begin{array}{ccc}
w_1 & \cdots & w_n \\
\bigzu & & -1 \\
  & \udots & \\
-1 & & \bigzl \\
\end{array}
\right)$

\vskip\baselineskip
$BBh0l9T$O(B, $B;X?t%Y%/%H%k(B $(d_1,\cdots,d_n)$ $B$KBP$7$F(B, 
$\displaystyle{\sum_{i=1}^n w_id_i}$ $B$9$J$o$A(B weight $BIU$-$N(B
$BA4<!?t$G:G=i$KHf3S$r9T$&$3$H$r0UL#$9$k(B. 
\end{ex}

\begin{ex}
block order

\vskip\baselineskip
$M_{block}$={\Large $\left(
\begin{array}{ccc}
M_1 & & \bigzl \\
  & \ddots & \\
\bigzu & & M_l \\
\end{array}
\right)$}

\vskip\baselineskip
$B3F(B $M_k$ $B$O(B, $B3F%V%m%C%/$KBP$9$k(B term order $B$rDj5A$9$k(B matrix
$B$G$"$k(B. 
\end{ex}
