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Annotation of OpenXM/doc/compalg/gr.tex, Revision 1.3

1.3     ! noro        1: %$OpenXM$
1.1       noro        2: \chapter{$B%0%l%V%J4pDl(B}
                      3: \label{chapgr}
                      4: \section{$BBe?tJ}Dx<0$N2r$H%$%G%"%k(B}
                      5:
                      6: $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,
                      7: $(x_1,\cdots,x_n)$ $B$r(B $X$ $B$HN,5-$9$k(B. $R$ $B$N85(B $f_1,\cdots,f_m$ $B$KBP$7(B,
                      8: \begin{equation}
                      9: \label{system}
                     10:  f_1 = 0, \cdots, f_m =0
                     11: \end{equation}
                     12: $B$rBe?tJ}Dx<07O(B, $B$"$k$$$OC1$KJ}Dx<0$H8F$V(B. (\ref{system}) $B$rK~$?$9(B
                     13: $K^n$ $B$N85$r(B $(1)$ $B$N2r$H8F$V(B. $B$3$N$h$&$JJ}Dx<0$r2r$$$F2r$r5a$a$h$&$H(B
                     14: $B$9$k>l9g$K:G$b4pK\E*$JJ}K!$O(B, $BCf3X0JMh$*$J$8$_$N>C5nK!$G$"$k(B.
                     15:
                     16: \begin{ex}
                     17: $f_1(x,y) = x^2+y^2 - 2 = 0, f_2(x,y) = xy - 1 = 0$
                     18: $B$r2r$1(B
                     19: \end{ex}
                     20: {\bf $B2r(B} $y^2f_1 - (xy+1)f_2 = y^4-2y^2+1 = 0$
                     21: $B$h$j(B $y=x=1$ $B$^$?$O(B $y=x=-1.$ $B$3$l$i$O<B:]$K2r$G$"$k(B. \qed\\
                     22: $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
                     23: $B$h$jJQ?t$N>/$J$$B?9`<0$r:n$j=P$9$b$N$G$"$k(B.
                     24:
                     25: \begin{df}
                     26: $f_1, \cdots, f_m \in R$ $B$KBP$7(B,
                     27: $$Id(f_1,\cdots,f_m) = \{\sum_{i=1}^n g_if_i \mid i \in R\}$$
                     28: $B$r(B $f_1,\cdots,f_m$ $B$G@8@.$5$l$k%$%G%"%k$H8F$V(B. $f_1, \cdots, f_m$ $B$r(B
                     29: $I$ $B$N@8@.7O$"$k$$$O(B{\bf $B4pDl(B}$B$H8F$V(B.
                     30: \end{df}
                     31: $B0lHL$K(B, $BJ}Dx<0(B (\ref{system}) $B$,M?$($i$l$?>l9g(B, $I=Id(f_1,\cdots,f_m)$
                     32: $B$r9M$($l$P(B, $B>C5nK!$H$O(B $I$ $B$NCf$+$i(B, $B4^$^$l$kJQ?t$N8D?t$,>/$J$$$b$N$rA*$S=P(B
                     33: $B$9J}K!$H8@$($k(B.
                     34: $I$ $B$N85A4$F$N6&DLNmE@$O(B $(\ref{system})$ $B$N2r$K0lCW$9$k(B. $B%$%G%"%k$N4p(B
                     35: $BDl$O0lAH$H$O8B$i$J$$$,(B, $BF10l$N%$%G%"%k$r@8@.$9$k4pDl$N6&DLNmE@$O0lCW$9(B
                     36: $B$k$+$i(B, $B%$%G%"%k$r9M$($kJ}$,(B, $BJ}Dx<0$N2r$r9M$($k>e$G$h$j<+A3$G$"$k$H8@$((B
                     37: $B$k(B.
                     38:
                     39: \begin{ex}
                     40: $Id(x^2+y^2 - 2,xy - 1) = Id(-y^4+2y^2-1,x+y^3-2y)$
                     41: \end{ex}
                     42:
                     43: \begin{ex} ($B@~7AJ}Dx<0(B)
                     44: $Id(2a+3b-4c+d-1,3a-2c-5d-4,a-b+4d-5,3a+2b+2c-2d)$
                     45: $=Id(-185d+78,-185c-94,-185b-299,-185a+314)$
                     46: \end{ex}
                     47: $B$3$l$i$NNc$G$O(B, $B1&JU$N4pDl$O3N$+$K2r$rMF0W$K5a$a$i$l$k7A$K$J$C$F$$$k(B.
                     48: $B$3$3$GBg;v$JE@$O(B, $BN>JU$,Ey$7$$%$%G%"%k$rM?$($F$$$k$+$I$&$+(B, $B$H$$$&E@(B
                     49: $B$G$"$k(B.
                     50:
                     51: \begin{df}
                     52: $B%$%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
                     53: \begin{center}
                     54: $V_K(I) = \{a \in K^n \mid$ $B$9$Y$F$N(B $f \in I$ $B$KBP$7(B $f(a)=0 \}$
                     55: \end{center}
                     56: $B$GDj5A$9$k(B. $B:.Mp$N$J$$>l9g$K$O(B $V(I)$ $B$H=q$/(B.
                     57: \end{df}
                     58:
                     59: \begin{co}
                     60: $I \subset J \Rightarrow V(J) \subset V(I)$
                     61: \end{co}
                     62: $B$9$J$o$A(B, $B>C5nK!$K$h$jF@$i$l$?B?9`<0$O(B, $B%$%G%"%k$N85$G$"$k$3$H$OJ]>Z(B
                     63: $B$5$l$k$,(B, $B$=$l$i$O$"$/$^$G2r$NK~$?$9$Y$-I,MW>r7o$G$"$j(B, $B<B:]$K$b$H$N(B
                     64: $BJ}Dx<0$N2r$rI=$9$+H]$+$OJL$N%A%'%C%/$,I,MW$H$J$k(B. $B$b$7?7$?$KF@$i$l$?(B
                     65: $BB?9`<0=89g$,$b$H$N%$%G%"%k$r@8@.$7$F$$$l$P(B, $B2r$,Ey$7$$$3$H$OJ]>Z$5$l(B
                     66: $B$F$$$k(B.
                     67:
                     68: \section{$B9`=g=x(B, $B%b%N%$%G%"%k(B, $B%0%l%V%J4pDl(B}
                     69:
                     70: $BA0@a$G(B, $BBe?tJ}Dx<0$N2r$r9M$($k>e$G(B, $BB?9`<0%$%G%"%k$r9M$((B, $B$=$N4pDl$r<h(B
                     71: $B$j49$($FJ}Dx<0$r2r$-$d$9$$4pDl$K<h$j49$($k$3$H$,M-8z$G$"$k$3$H$r<($7$?(B.
                     72: $B$7$+$7(B, $B%$%G%"%k$N35G0$O(B, $BJ}Dx<0$r2r$/$?$a$@$1$KMQ$$$i$l$k$b$N$G$O$J$$(B.
                     73: $BFC$K(B, $B2r$,L58B8D$K$J$k>l9g$O(B, $B$=$N2rA4BN$OBe?tE*=89g$H$7$F07$o$l$k$Y$-(B
                     74: $B$b$N$G$"$j(B, $BJ}Dx<0$N2r$H$7$FI=8=$9$k$3$H$O0lHL$K$OFq$7$$(B. $B$`$7$m(B, $B%$%G(B
                     75: $B%"%k(B $I$ $B$"$k$$$O4D(B $R/I$ $B$N@-<A$+$i$=$NBe?tE*=89g$N@.J,(B, $B<!85$J$I$rCN(B
                     76: $B$k$3$H$,=EMW$H$J$k(B. $B$=$N$?$a$K%$%G%"%k$N4pDl$,K~$?$9$Y$->r7o$H$7$F<!$N(B
                     77: $B$h$&$J$b$N$r5s$2$k$3$H$,$G$-$k(B.
                     78:
                     79: \begin{itemize}
                     80: \item
                     81: $B$"$kB?9`<0$,(B, $B%$%G%"%k$KB0$9$k$+H]$+(B ($B%a%s%P%7%C%W(B) $B$,%"%k%4%j%:%`$K(B
                     82: $B$h$jH=Dj$G$-$k(B.
                     83:
                     84: $I$ $B$KB0$9$kB?9`<0$r6qBNE*$KGD0.$9$k$?$a$KI,MW$G$"$k(B.
                     85:
                     86: \item
                     87: $B%$%G%"%k$r(B, $BO"N)J}Dx<07O$H$_$?$H$-(B, $B2r$r5a$a$d$9$$7A$r$7$F$$$k(B.
                     88:
                     89: $B>C5nK!$HF1MM$N7k2L$rM?$($k$3$H$,$G$-$k(B.
                     90:
                     91: \item
                     92: $B%$%G%"%k$N@-<A(B ($B<!85(B, $B0x;R$J$I(B) $B$rI=$7$F$$$k(B.
                     93: \end{itemize}
                     94: $B$3$l$i$N@-<A$N$&$A(B, $BBh0l$N$b$N$KCeL\$9$k(B.
                     95:
                     96: \begin{ex}
                     97: $n=1$ $B$N>l9g(B\\
                     98: $R=K[x]$ $B$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
                     99: $B$"$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
                    100: $B$K$h$j(B, $I$ $B$KBP$9$k%a%s%P%7%C%W$,(B,
                    101: $$ g \in I \Leftrightarrow f \mid g $$
                    102: $B$GH=Dj$G$-$k$3$H$r0UL#$9$k(B. $I$ $B$N@8@.85$,4v$D$+M?$($i$l$F$$$k>l9g(B,
                    103: $f$ $B$O(B, $B$=$l$i$N@8@.85$N(B GCD $B$r5a$a$k$3$H$GF@$i$l$k(B.
                    104: \end{ex}
                    105:
                    106: $B0lJQ?t$N>l9g(B, $B%$%G%"%k$N@8@.85(B
                    107: $B$O(B, $B$=$N%$%G%"%k$KB0$9$k85$N$&$A(B, $B:G$b<!?t$N>.$5$$$b$N$r$H$l$P$h$+$C$?(B.
                    108: $B$3$l$O(B, $B<!$N$h$&$K$$$$$+$($i$l$k(B.
                    109:
                    110: \begin{itemize}
                    111: \item
                    112: $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.
                    113: $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.
                    114: \end{itemize}
                    115: $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
                    116: $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
                    117: $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
                    118: $B$rI=$9(B.
                    119:
                    120: \begin{df}
                    121: $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
                    122: 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}
                    123: $B$G$"$k$H$O(B,
                    124: \begin{enumerate}
                    125: \item $B$9$Y$F$N(B $t \in T$ $B$KBP$7(B $1 \le t$
                    126: \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)$
                    127: \end{enumerate}
                    128: $B$rK~$?$9$3$H$r8@$&(B.
                    129: \end{df}
                    130:
                    131: \begin{df}
                    132: $B;X?t$r(B $\N^n$ $B$N85$H9M$($F(B, $\N^n$ $B$K$*$1$k(B term order $<$ $B$r(B
                    133:
                    134: \begin{enumerate}
                    135: \item $B$9$Y$F$N(B $\alpha \in \N^n$ $B$KBP$7(B $0 = (0,\cdots,0) \le \alpha$
                    136: \item $B$9$Y$F$N(B $\alpha_1, \alpha_2, s \in \N$ $B$KBP$7(B
                    137: $(\alpha_1 \le \alpha_2 \Rightarrow \alpha_1 + s \le \alpha_2 + s)$
                    138: \end{enumerate}
                    139: $B$rK~$?$9$b$N$H$7$FDj5A$G$-$k(B.
                    140: \end{df}
                    141:
                    142: \begin{df}
                    143: $L \subset \N^n$ $B$,%b%N%$%G%"%k$H$O(B,
                    144: $BG$0U$N(B $\alpha \in L, \beta \in \N^n$ $B$KBP$7(B $\alpha+\beta \in L$
                    145: $B$,@.$jN)$D$3$H$r$$$&(B. $B$^$?(B, $S \subset \N^n$ $B$KBP$7(B,
                    146: $$mono(S) = \{\alpha+\beta \mid \alpha \in S, \beta \in \N^n\}$$
                    147: $B$r(B $S$ $B$G@8@.$5$l$k%b%N%$%G%"%k$H8F$V(B.
                    148: \end{df}
                    149:
                    150: \begin{df}
                    151: term $B4V$NA4=g=x$rB?9`<0$NH>=g=x$K<+A3$K3HD%$9$k(B. \\
                    152: $f, g \in R$ $B$G(B, $f = \sum_{i\ge 0}c_i f_i$, $g = \sum_{i\ge 0}d_i g_i$
                    153: ( $c_i, d_i \in K, f_i, g_i \in T, i > j \Rightarrow f_i > f_j, g_i > g_j$)
                    154: $B$H$9$k;~(B, $f > g$ $B$r(B
                    155: \begin{center}
                    156: $f > g \Leftrightarrow$ $B$"$k(B $i_0$ $B$,B8:_$7$F(B $(i<i_0 \Rightarrow f_i = g_i, f_{i_0} > g_{i_0})$
                    157: \end{center}
                    158: $B$GDj5A$9$k(B.
                    159: \end{df}
                    160:
                    161: \begin{df}
                    162: $M = \{c\cdot t\mid c \in K, t \in T\}$ $B$H$7(B, $M$ $B$N85$r(B {\bf monomial} $B$H8F$V(B.
                    163: \end{df}
                    164:
                    165: \begin{df}
                    166: term order $B$r0l$D8GDj$7$?;~(B, $BB?9`<0(B $f$ $B$KI=$l$k(B term $B$NCf$G(B, $B$=(B
                    167: $B$N(B order $B$K$*$$$F:GBg$N$b$N$r(B {\bf $BF,9`(B (head term)} $B$H8F$S(B, $HT(f)$ $B$H(B
                    168: $B=q$/(B. \\
                    169: $HT(f)$ $B$N78?t$r(B, $HC(f)$ $B$H=q$/(B. \\
                    170: $HC(f)\cdot HT(f)$ $B$r(B $HM(f)$ $B$H=q$/(B. \\
                    171: $HT(f)$ $B$N;X?t$r(B $HE(f)$ $B$H=q$/(B. $HE(f) \in \N^n$ $B$G$"$k(B. \\
                    172: $B$5$i$K(B, $f-HM(f)$ $B$r(B $red(f)$ ({\bf reductum} of $f$) $B$H=q$/(B.
                    173: \end{df}
                    174:
                    175: \begin{lm}
                    176: $\N^n$ $B$NG$0U$N%b%N%$%G%"%k(B $L$ $B$OM-8B@8@.(B.
                    177: \end{lm}
                    178: \proof $n$ $B$K4X$9$k5"G<K!$K$h$j<($9(B. $n=1$ $B$N$H$-(B, $L$ $B$N(B $\N$ $BCf$G$N(B
                    179: $B:G>.85(B $\alpha$ $B$r$H$l$P(B $L$ $B$O(B $\alpha$ $B$G@8@.$5$l$k(B. $n-1$ $B$^$G8@$($?(B
                    180: $B$H$9$k(B. $B3F(B $j \in \N$ $B$KBP$7(B,
1.2       noro      181: $$L_j=\{ (\alpha_1,\cdots,\alpha_{n-1}) \in
1.1       noro      182: N^{n-1}\mid (\alpha_1,\cdots,\alpha_{n-1},j)\in L \}$$$B$H$*$/$H(B $\{L_j\}$
                    183: $B$O%b%N%$%G%"%k$NA}BgNs(B. $L_\infty = \cup L_j$ $B$H$*$/$H(B$L_\infty$ $B$b%b(B
                    184: $B%N%$%G%"%k$G(B, $B5"G<K!$N2>Dj$K$h$j(B $L_\infty$ $B$OM-8B@8@.(B. $B$h$C$F$"$k(B
                    185: $j_0$ $B$,B8:_$7$F(B $L_\infty=L_{j_0}$. $B$3$N$H$-(B, $L$ $B$O(B,
                    186: $j=1,\cdots,j_0$ $B$KBP$9$k(B $L_j$ $B$N@8@.85(B ($B$3$l$O5"G<K!$N2>Dj$K$h$j$=$l(B
                    187: $B$>$lM-8B=89g(B) $B$NOB=89g$G@8@.$5$l$k(B. \qed
                    188:
                    189: \begin{co}
                    190: \label{noether}
                    191: $\{L_i\} (i=1,2,\cdots)$ $B$r%b%N%$%G%"%k$NA}BgNs$H$9$l$P(B, $B$"$k(B $i_0$ $B$,(B
                    192: $BB8:_$7$F(B, $i \ge i_0 \Rightarrow L_i=L_{i_0}$.
                    193: \end{co}
                    194:
                    195: \begin{co}
                    196: $\N^n$ $B$NG$0U$NItJ,=89g$O(B term order $<$ $B$K4X$7$F:G>.85$r;}$D(B.
                    197: \end{co}
                    198:
                    199: \begin{co}
                    200: $\le$ $B$r(B $T$ $B$N(B term order $B$H$9$l$P(B, $T$ $B$N??$N9_2<Ns$OM-8B$G@Z$l$k(B.
                    201: $\le$ $B$N(B $R$ $B$X$N<+A3$J3HD%$K$D$$$F$bF1MM$G$"$k(B. $B0J2<(B, $B$3$N@-<A$r(B, term
                    202: order $B$N(B Noether $B@-$H$7$F0zMQ$9$k(B.
                    203: \end{co}
                    204: \proof $T$ $B$K$D$$$F$O9_2<Ns$,:G>.85$r;}$D$3$H$+$i@.$jN)$D(B. $R$ $B$K(B
                    205: $B$D$$$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
                    206: $B9_2<Ns$G$"$kL58BNs$r:n$j=P$;$k$+$iL7=b(B. \qed
                    207:
                    208: \begin{df}
                    209: $S \subset R$ $B$*$h$S(B term order $<$ $B$KBP$7(B,
                    210: $$E_<(S) = \{HE(f) \mid f \in S\} \subset \N^n$$
                    211: $B$HDj5A$9$k(B. $B0J2<:.Mp$N$J$$>l9g$K$O(B $<$ $B$r>JN,$7$F(B $E(S)$ $B$H=q$/(B.
                    212: \end{df}
                    213:
                    214: \begin{lm}
                    215: $B%$%G%"%k(B $I$ $B$KBP$7(B, $E(I)$ $B$O%b%N%$%G%"%k(B.
                    216: \end{lm}
                    217:
                    218: $B0lJQ?tB?9`<04D$K$*$1$k%$%G%"%k(B $I$ $B$N@8@.7O(B $G = \{f\}$ $B$N@-<A$O(B,
                    219: $$E(I) = mono(E(G))$$
                    220: $B$H=q$/$3$H$,$G$-$k(B. $B0lHL$N>l9g$K$b$3$N$h$&$J@-<A$rK~$?$9(B $G$ $B$r9M$($k$3$H(B
                    221: $B$OM-MQ$G$"$k(B.
                    222:
                    223: \begin{df}
                    224: $B%$%G%"%k(B $I$ $B$KBP$7(B, $B$=$NM-8BItJ,=89g(B $G$ $B$G(B,
                    225: $$E(I) = mono(E(G))$$
                    226: $B$rK~$?$9$b$N$r(B $I$ $B$N(B $<$ $B$K4X$9$k%0%l%V%J4pDl$H8F$V(B.
                    227: \end{df}
                    228: $B$3$NDj5A$O(B, $B%$%G%"%k$N$9$Y$F$N85$NF,9`$,(B,
                    229: $G$ $B$N$$$:$l$+$N85$NF,9`$G3d$j@Z$l$k$3$H$r0UL#$9$k(B.
                    230:
                    231: \begin{pr}
                    232: $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.
                    233: \end{pr}
                    234: \proof $B%b%N%$%G%"%k$NM-8B@8@.@-$h$jL@$i$+(B. \qed
                    235:
                    236: \begin{pr}
                    237: $B%$%G%"%k(B $I$ $B$N(B $B%0%l%V%J4pDl(B $G$ $B$O(B $I$ $B$r@8@.$9$k(B.
                    238: \end{pr}
                    239: \proof
                    240: $f \in I$ $B$H$9$k(B. $B2>Dj$K$h$j(B, $B$"$k(B $g \in G$ $B$,B8:_$7$F(B, $HT(g)|HT(f)$.
                    241: $B$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
                    242: \in I$$B$@$+$i(B, $B$3$NA`:n$r7+$jJV$9$3$H$,$G$-$k(B. term order $B$N(B Noether
                    243: $B@-$h$j(B, $B$3$NA`:n$OM-8B2s$G=*N;$9$k(B. $B$9$J$o$A(B, $BM-8B2s$NA`:n$N8e(B, 0 $B$H$J$k(B.
                    244: $B$3$l$O(B, $G$ $B$,(B $I$ $B$r@8@.$9$k$3$H$r0UL#$9$k(B.  \qed\\
                    245: $B$3$NL?Bj$K$*$$$F(B, $f -s\cdot g$ ($f, g \in R; s \in M$) $B$J$k1i;;$,8=$l$?(B.
                    246: $BL?Bj$K$*$$$F$O(B, $f$ $B$NF,9`$r>C5n$9$k$?$a$K9T$J$o$l$?$,(B, $B0lHL$K(B, $f$ $B$N(B
                    247: $B9`$r(B, $B$3$N$h$&$K>C5n$9$k1i;;$,(B, $B%0%l%V%J4pDl7W;;$K$*$$$F4pK\E*$J1i;;$H$J$k(B.
                    248:
                    249: \begin{df}
                    250: $f, g \in R$ $B$H$7(B, $f$ $B$K8=$l$k$"$k(B monomial $m$ $B$,(B $HT(g)$ $B$G3d$j@Z$l(B
                    251: $B$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$$(B
                    252: $B$&(B. $B$3$N$H$-(B $h = f-m/HT(g)\cdot g$ $B$KBP$7(B, $ f \mred{g} h$ $B$H=q$/(B. \\
                    253: $G \subset R$ $B$K$D$$$F$b(B, $G$ $B$N$"$k85$K$h$j(B $f$ $B$,(B $h$ $B$K4JLs$5$l$k$H$-(B
                    254: $f \mred{G} h$ $B$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
                    255: $BF@$i$l$k$H$-(B, $f\tmred{G} h$ $B$H=q$/(B. \\
                    256: $f$ $B$N$I$N9`$b(B, $G$ $B$G4JLs$G$-$J$$$H$-(B, $f$ $B$O(B $G$ $B$K4X$7$F(B {\bf $B@55,(B
                    257: $B7A(B (normal form)} $B$G$"$k$H$$$&(B.
                    258: \end{df}
                    259:
                    260:
                    261: \begin{pr}
                    262: $B%$%G%"%k(B $I$ $B$N(B $B%0%l%V%J4pDl(B $G$ $B$K$D$$$F(B, $B0J2<$N$3$H$,@.$jN)$D(B.
                    263: \begin{enumerate}
                    264: \item
                    265: $f \in I \Leftrightarrow f \tmred{G} 0$
                    266: \item
                    267: $f \tmred{G} f_1, f \tmred{G} f_2$ $B$+$D(B $f_1, f_2$ $B$,@55,7A(B $\Rightarrow f_1 = f_2$
                    268: \item
                    269: $f \in G$ $B$G(B, $B$"$k(B $h \in G$ $B$,B8:_$7$F(B $HT(h) | HT(f)$
                    270: $\Rightarrow G \setminus \{f\}$ $B$O(B $I$ $B$N%0%l%V%J4pDl(B
                    271: \end{enumerate}
                    272: \end{pr}
                    273: \proof
                    274: 1. $B$O(B, $BA0L?Bj$N>ZL@$h$jL@$i$+(B. 3. $B$bDj5A$h$jL@$i$+(B. 2. $B$r<($9(B. $f -
                    275: f_1, f - f_2 \in I$ $B$h$j(B $f_1 - f_2 \in I$. $B$h$C$F(B $f_1 - f_2 \tmred{G} 0$.
                    276: $B$H$3$m$,(B, $f_1$, $f_2$ $B$H$b(B, $G$ $B$K$D$$$F@55,7A$h$j(B $f_1 - f_2$ $B$b@55,7A(B.
                    277: $B$h$C$F(B, $f_1 - f_2 = 0$. \qed
                    278:
                    279: \begin{co}
                    280: $G = \{g_1, \cdots, g_l\}$ $B$r%$%G%"%k(B $I$ $B$N(B $B%0%l%V%J4pDl$H$9$k(B. $B$3$N$H$-(B,
                    281: $B$"$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 $HT(g_i) (g_i \in H)$
                    282: $B$N$I$NFs$D$b8_$$$KB>$r3d$i$J$$(B.
                    283: \end{co}
                    284: $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
                    285: $BDl$NB>$N85$KBP$7$F@55,7A$H$J$k$h$&4JLs$r9T$J$$(B, $BF,9`$N78?t$r(B 1 $B$H$J$k(B
                    286: $B$h$&$K$7$?4pDl$r(B{\bf $BHoLs(B (reduced) $B%0%l%V%J4pDl(B} $B$H8F$V(B. $B$3$l$,<B:]$K(B
                    287: $B85$N%$%G%"%k$N(B $B%0%l%V%J4pDl$K$J$C$F$$$k$3$H$O(B, $BF,9`$,JQ$o$C$F$$$J$$$3(B
                    288: $B$H$h$j$o$+$k(B. $B$5$i$K<!$NL?Bj$O(B, $BDj5A$h$j$?$@$A$KF@$i$l$k(B.
                    289:
                    290: \begin{pr}
                    291: $BHoLs%0%l%V%J4pDl$O=89g$H$7$F0l0UE*$KDj$^$k(B.
                    292: \end{pr}
                    293: $B0J>e$,(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
                    294: $B9=@.$9$k%"%k%4%j%:%`$rF@$k$?$a$K$O(B, $B%0%l%V%J4pDl$NDj5A$N8@$$$+$($r$$$/$D$+9T$J$&(B
                    295: $BI,MW$,$"$k(B.
                    296:
                    297: \begin{df}
                    298: $G = \{g_1,\cdots,g_l\}$ $B$r(B ($B%0%l%V%J4pDl$H$O8B$i$J$$(B) R $B$NM-8BItJ,=89g$H$9$k(B.
                    299: $B$3$l$KBP$7(B, $B<LA|(B $d_1$ $B$r<!$GDj5A$9$k(B.
                    300:
                    301: \begin{tabbing}
                    302: aaaa \= aaaaaaaaaa        \= aaa       \= aaaa\kill
                    303: $d_1 :$\> $R^l$              \> $\longrightarrow$ \> $R$\\
                    304:        \> $(f_1,\cdots,f_l)$ \> $\longmapsto$     \> $\sum f_i\cdot HT(g_i)$
                    305: \end{tabbing}
                    306: \end{df}
                    307:
                    308: \begin{df}
                    309: $f = (f_1,\cdots,f_l) \in R^l$ $B$,(B {\bf $T$-$B@F<!(B} $B$H$O(B, $B$"$k(B term $t$
                    310: $B$,B8:_$7$F(B, $B$9$Y$F$N(B $i$ $B$KBP$7(B, $f_i = 0$ $B$^$?$O(B $t = f_i\cdot
                    311: HT(g_i)$ $B$H=q$1$k$H$-$r8@$&(B.
                    312: \end{df}
                    313:
                    314: \begin{df}
                    315: $e_i \in R^l$ $B$r(B $e_i = (0,\cdots,1,\cdots,0)$ ($BBh(B $i$ $B@.J,$N$_(B 1) $B$HDj5A(B
                    316: $B$9$k(B. \\
                    317: $i_1,\cdots,i_k$ $B$KBP$7(B, $T_{i_1,\cdots, i_k}$ $B$r(B
                    318: $$T_{i_1,\cdots, i_k} = \LCM(HT(g_{i_1}),\cdots,HT(g_{i_k}))$$
                    319: $B$HDj5A$9$k(B.
                    320: $BFC$K(B, $T_i = HT(g_i)$ $B$G$"$k(B.
                    321: \end{df}
                    322:
                    323: \begin{pr}
                    324: \label{'thomo'}
                    325: $B%$%G%"%k(B $I$ $B$K$D$$$F(B, $B<!$OF1CM(B.
                    326: \begin{enumerate}
                    327: \item
                    328: $G = \{g_1,\cdots,g_l\}$ $B$O(B $I$ $B$N%0%l%V%J4pDl(B
                    329: \item
                    330: $f \in I \Leftrightarrow f \tmred{G} 0$
                    331: \item
                    332: $f \in I \Leftrightarrow$ $B$"$k(B $f_i (i = 1, \cdots, l)$ $B$,B8:_$7$F(B
                    333: $f = \sum_i f_i g_i$ $B$+$D(B $HT(f_i g_i) \le HT(f)$
                    334: \item
                    335: 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 $h =
                    336: (h_1,\cdots,h_l) \in L$ $B$KBP$7(B, $\sum h_i\cdot g_i \tmred{G} 0$
                    337: \end{enumerate}
                    338: \end{pr}
                    339: \proof\\
                    340: 1. $\Leftrightarrow$ 2.)
                    341: $BL@$i$+(B. \\
                    342: 2. $\Rightarrow$ 3.)
                    343: $G$ $B$N85$K$h$k4JLsA`:n$r0l$D$K$^$H$a$k$HF@$i$l$k(B. \\
                    344: 3. $\Rightarrow$ 2.)
                    345: $B$"$k(B $i$ $B$,B8:_$7$F(B $HT(f_i g_i) = HT(f)$ $B$H$J$j(B,
                    346: $HT(f)$ $B$,(B $HT(f_i)$ $B$G3d$j@Z$l$k$3$H$+$i$o$+$k(B. \\
                    347: 4. $\Rightarrow$ 1.)
                    348: $f = \sum f_i\cdot g_i \in I$ $B$H$7(B, $HT(f)$ $B$,(B $HT(g_i)$ $B$N$$$:$l$+$G(B
                    349: $B3d$j@Z$l$k$3$H$r8@$($P$h$$(B. $B4JC1$N$?$a(B, $B$9$Y$F$N(B $i$ $B$KBP$7(B, $HC(g_i)
                    350: = 1$ $B$H$9$k(B. $L = \{b_1,\cdots,b_l\}$$B$H$9$k(B.
                    351: $m = \max_i(HT(f_i g_i))$ $B$H$*$/(B. \\
                    352: \underline{$HT(f) = m$ $B$N>l9g(B}
                    353: \quad $HT(f)$ $B$O(B, $HT(g_i)$ $B$N$$$:$l$+$G3d$j@Z$l$k(B.\\
                    354: \underline{$HT(f) < m$ $B$N>l9g(B}
                    355: \quad $A = \{i \mid HT(f_i g_i) = m\}$ $B$H$*$/$H(B, $B2>Dj$h$j(B,
                    356: $$\sum_{i\in A}HM(f_i)\cdot HT(g_i) = 0.$$
                    357: $B$h$C$F(B, $h = (h_1,\cdots,h_l) \in R^l$ $B$r(B,
                    358: $h_i = HM(f_i) (i \in A), h_i = 0 (i \notin A)$
                    359: $B$HDj5A$9$l$P(B, $h \in \Ker(d_1)$. $B$h$C$F2>Dj$h$j(B,
                    360: \begin{center}
                    361: $B$"$k(B $c_i \in M$ $B$,B8:_$7$F(B $h = \sum_i c_i b_i.$
                    362: \end{center}
                    363: $B$3$l$h$j(B
                    364: $$\sum_k h_k g_k = \sum_i c_i \sum_k g_k b_{ik}.$$
                    365: $\displaystyle{G_i = \sum_k g_k b_{ik}}$ $B$H$*$/$H(B, $G_i \tmred{G} 0$ $B$h$j(B,
                    366: 2. $\Rightarrow$ 3. $B$HF1MM$K(B,
                    367: $B$"$k(B $b'_{ik}$ $B$,B8:_$7$F(B $G_i = \sum_k g_k b'_{ik}$ $B$+$D(B $HT(g_k b'_{ik}) \le HT(G_i).$
                    368: $B0lJ}(B, $b_i \in \Ker(d_1)$ $B$h$j(B, $HT(G_i) < \max_k(HT(g_k b_{ik}))$.\\
                    369: $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,
                    370: $$f = \sum_{i \notin A} f_k g_k + \sum_{i \in A} (red(f)+f'_k)g_k$$
                    371:  $B$H=q$1(B,
                    372: $$HT(f'_k g_k) \le \max_i(HT(c_i b'_{ik} g_k)) \le \max_i(HT(c_i G_i))$$
                    373: $$< \max_i(\max_k(HT(c_i g_k b_{ik}))) = \max_k(HT(h_k g_k)) = m$$
                    374: $B$h$j(B, $m$ $B$,$h$jDc$$=g=x$N>l9g$K5"Ce$G$-$k(B. $B$h$C$F(B, order $B$N(B Noether $B@-$K$h$j(B
                    375: $BM-8B2s$NA`:n$N$N$A(B, $m = HT(f)$ $B$N>l9g$K5"Ce$G$-$k(B. \qed
                    376: \begin{pr}
                    377: $G = \{g_1,\cdots,g_l\}$ $B$r(B $HC(g_i)=1$ $B$J$k(B R $B$NM-8BItJ,=89g$H$9$k(B.
                    378: $i,j \in \{1,\cdots,l\}$ $B$KBP$7(B, $S_{ij} \in R^l$ $B$r(B
                    379: $S_{ij} = T_{ij}/T_i e_i - T_{ij}/T_j e_j$ $B$GDj5A$9$k(B. $B$3$N;~(B,
                    380: $L = \{S_{ij}\mid i < j\}$ $B$O(B $\Ker(d_1)$ $B$N(B $T$-$B@F<!(B $B$J4pDl$H$J$k(B. $B$3$N(B
                    381: $B4pDl$r(B {\bf Taylor $B4pDl(B} $B$H8F$V(B.
                    382: \end{pr}
                    383: \proof
                    384: $f \in \Ker(d_1)$ $B$H$9$k(B. $f$ $B$r(B $T$-$B@F<!@.J,$KJ,2r$9$k$3$H$K$h$j(B,
                    385: $f$ $B<+?H(B $T$-$B@F<!(B $B$H$7$F$h$$(B. $f = \sum_i f_i e_i$ $B$H$9$k$H(B,
                    386: $f \in \Ker(d_1)$ $B$h$j(B, $f \neq 0$ $B$J$i$P(B, $B>/$J$/$H$b(B 2 $B$D$N@.J,$,(B 0
                    387: $B$G$J$$(B. $B$=$l$i$r(B $f_k, f_l (k < l)$ $B$H$9$k$H(B, $HT(f_k g_k) = HT(f_l g_l)$
                    388: $B$h$j(B, $T_{kl} | HT(f_k g_k)$. $B$3$l$h$j(B $f' = f - (HM(f_k g_k)/T_{kl}) S_{kl}$
                    389: $B$H$*$/$H(B, $f'$ $B$OBh(B $k$ $B@.J,$,(B 0 $B$H$J$j(B, 0 $B$G$J$$@.J,$,0l$D8:$k(B. $B$3$N(B
                    390: $BA`:n$r7+$jJV$7$F(B, $f$ $B$r(B $\{S_{ij}\}$ $B$G@8@.$G$-$k(B. \qed
                    391:
                    392: \begin{df}
                    393: $f, g \in R$ $B$KBP$7(B, {\bf S $BB?9`<0(B} $Sp(f,g)$ $B$r(B,
                    394: $$Sp(f,g) = {HC(g)T_{fg}\over HT(f)}\cdot f - {HC(f)T_{fg}\over HT(g)}\cdot g$$\\
                    395: ($T_{fg} = \LCM(HT(f),HT(g))$) $B$HDj5A$9$k(B.
                    396: \end{df}
                    397: $B0J>e$K$h$j(B, $B?7$?$J(B $B%0%l%V%J4pDl$NH=Dj>r7o$,F@$i$l$k(B.
                    398: \begin{pr}
                    399: $B%$%G%"%k(B $I$ $B$K$D$$$F(B, $B<!$OF1CM(B.
                    400: \begin{enumerate}
                    401: \item
                    402: $G = \{g_1,\cdots,g_l\}$ $B$O(B $I$ $B$N%0%l%V%J4pDl(B
                    403: \item
                    404: $BG$0U$NBP(B $\{f,g\} (f, g \in G; f \neq g)$ $B$KBP$7(B, $Sp(f,g) \tmred{G} 0$
                    405: \end{enumerate}
                    406: \end{pr}
                    407:
                    408: \section{Buchberger $B%"%k%4%j%:%`(B}
                    409: $BA0@a$N:G8e$NL?Bj$K$h$j(B, $B<!$N%"%k%4%j%:%`$,F3$+$l$k(B.
                    410:
                    411: \begin{al}(Buchberger{\rm\cite{BUCH}})
                    412: \label{buch}
                    413: \begin{tabbing}
                    414: Input : $R$ $B$NM-8BItJ,=89g(B $F = {f_1,\cdots,f_l}$\\
                    415: Output : $F$ $B$G@8@.$5$l$k%$%G%"%k$N(B $B%0%l%V%J4pDl(B$G$\\
                    416:
                    417: $D \leftarrow \{\{f,g\} \mid f, g \in F; f \neq g\}$\\
                    418: $G \leftarrow F$\\
                    419: while \=( $D \neq \emptyset$ ) do \{\\
                    420: \>$\{f,g\} \leftarrow D$ $B$N85(B\\
                    421: \>$D \leftarrow D \setminus \{C\}$\\
                    422: \>$h \leftarrow Sp(f,g)$ $B$N@55,7A$N0l$D(B\\
                    423: \>if \=$h \neq 0$ then \{\\
                    424: \>\>$D \leftarrow D \cup \{\{f,h\} \mid f \in G\}$\\
                    425: \>\>$G \leftarrow G \cup \{h\}$\\
                    426: \>\}\\
                    427: \}\\
                    428: return $G$
                    429: \end{tabbing}
                    430: \end{al}
                    431:
                    432: $B0J2<$G(B, $D$ $B$N85$r(B{\bf $BBP(B} (pair) $B$H8F$V$3$H$K$9$k(B.
                    433:
                    434: \begin{th}
                    435: $B%"%k%4%j%:%`(B \ref{buch} $B$ODd;_$7(B, $B%0%l%V%J4pDl$r=PNO$9$k(B.
                    436: \end{th}
                    437: \proof\\
                    438: \underline{$BDd;_@-(B}\quad $B@8@.$5$l$k@55,7A$NF,9`$,(B, $B$=$l$^$G$K@8@.$5$l$?@55,7A(B
                    439: $B$NF,9`$G3d$j@Z$l$J$$$3$H$h$j(B, $B7O(B \ref{noether} $B$+$i8@$($k(B. \\
1.2       noro      440: \underline{$B=PNO$,%0%l%V%J4pDl$H$J$k$3$H(B}\quad $BA0L?Bj$K$h$j8@$($k(B. \qed\\
1.1       noro      441: $B$3$N%"%k%4%j%:%`$,(B Buchberger $B%"%k%4%j%:%`$N:G$b86;OE*$J7A$G$"$k$,(B,
                    442: \begin{itemize}
                    443: \item
                    444: $B@55,7A$,(B 0 $B$G$J$$>l9g(B, $D$ $B$NMWAG$,(B $G$ $B$NMWAG$N8D?t$@$1A}2C$9$k(B.
                    445: \item
                    446: $D$ $B$+$i0l$D85$rA*$VJ}K!$,L@<($5$l$F$$$J$$(B.
                    447: \end{itemize}
                    448: $B$J$I$NE@$G<BMQE*$G$J$$(B. $B<B:]$K7W;;5!>e$K%$%s%W%j%a%s%H$9$k(B
                    449: $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
                    450: $B>\$7$/=R$Y$k(B.
                    451:
                    452: \section{Term order $B$NNc(B}
                    453:
                    454: $BMM!9$J(B term order $B$,Dj5A$G$-$k(B. $B$3$l$i$N(B order $B$O$=$l$>$l0[$C$?(B
                    455: $B@-<A$r$b$A(B, $B$=$N@-<A$K1~$8$F$5$^$6$^$JMQES$KMQ$$$i$l$k(B.
                    456:
                    457: \begin{df}
                    458: {\bf $B<-=q<0=g=x(B (lexicographical order; LEX)}\\
                    459: $x_1^{i_1}\cdots x_n^{i_n} > x_1^{j_1}\cdots x_n^{j_n} \Leftrightarrow$\\
                    460: $B$"$k(B $m$ $B$,B8:_$7$F(B $i_1 = j_1, \cdots, i_{m-1} = j_{m-1}, i_m > j_m$
                    461: \end{df}
                    462:
                    463: $B$3$N=g=x$O>C5nK!$K$h$kJ}Dx<05a2r$K:G$bE,$7$?7A$N%0%l%V%J4pDl$rM?$($k(B.
                    464: $B$7$+$7(B, $B$=$ND>@\7W;;$O(B, $B;~4V(B, $B6u4V7W;;NL$,$7$P$7$P6K$a$FBg$-$/$J$k(B
                    465: $B$H$$$&$3$H$+$iITMx$G$"$k(B.
                    466:
                    467: \begin{df}
                    468: {\bf $BA4<!?t<-=q<0=g=x(B (total degree lexicographical order; DLEX)}\\
                    469: $x_1^{i_1}\cdots x_n^{i_n} > x_1^{j_1}\cdots x_n^{j_n} \Leftrightarrow$\\
                    470: $\sum_k i_k > \sum_k j_k$ $B$^$?$O(B\\
                    471: ($\sum_k i_k = \sum_k j_k$ $B$+$D(B
                    472:  $B$"$k(B $m$ $B$,B8:_$7$F(B $i_1 = j_1, \cdots, i_{m-1} = j_{m-1}, i_m > j_m)$
                    473: \end{df}
                    474:
                    475: $BF~NO$,@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
                    476: $B%0%l%V%J4pDl$O0lCW$9$k(B. degree compatible order $B$K$h$k7W;;$O(B, $B$=$&$G$J$$(B
                    477: order $B$K$h$k7W;;$KHf$Y$F8zN($,$h$$$3$H$,7P83E*$KCN$i$l$F$*$j(B, $B@F<!(B
                    478: $B$J>l9g$K$3$N=g=x$G<-=q<0=g=x%0%l%V%J4pDl$r7W;;$9$k(B, $B$"$k$$$OHs@F<!(B
                    479: $B$JF~NO$r@F<!2=$7$F(B, $B$3$N=g=x$G%0%l%V%J4pDl$r7W;;$7(B, $BHs@F<!2=$9$k(B
                    480: $B$3$H$G$b$H$NF~NO$N<-=q<0=g=x%0%l%V%J4pDl$r5a$a$k$H$$$&$3$H$,9T$o$l$k(B.
                    481:
                    482: \begin{df}
                    483: {\bf $BA4<!?t5U<-=q<0=g=x(B (total degree reverse lexicographical order; DRL)}\\
                    484: $x_1^{i_1}\cdots x_n^{i_n} > x_1^{j_1}\cdots x_n^{j_n} \Leftrightarrow$\\
                    485: $\sum_k i_k > \sum_k j_k$ $B$^$?$O(B\\
                    486: ($\sum_k i_k = \sum_k j_k$ $B$+$D(B
                    487: $B$"$k(B $m$ $B$,B8:_$7$F(B $i_n = j_n, \cdots, i_{m+1} = j_{m+1}, i_m < j_m)$
                    488: \end{df}
                    489:
                    490: $B0lHL$K(B, $B:G$b9bB.$K%0%l%V%J4pDl$r7W;;$G$-$k$,(B, $B%0%l%V%J4pDl$N8D!9$N(B
                    491: $B85$N;}$D@-<A$ODO$_$K$/$/(B, $BO"N)J}Dx<0$rD>@\2r$/$3$H$O:$Fq$G$"$k(B.
                    492: $B$7$+$7(B, $B<!85(B, Hilbert function $B$=$NB>$NITJQNL$r7W;;$9$k>l9g$J$I(B,
                    493: $B%0%l%V%J4pDl$H$$$&@-<A$N$_$,I,MW$H$5$l$k>l9g$K(B, $B9bB.$K7W;;$G$-$k(B
                    494: $B$H$$$&FC@-$r@8$+$7$FMQ$$$i$l$k>l9g$,B?$$(B. $B$^$?(B, $B8e$K=R$Y$k(B
                    495: $B4pDlJQ49$NF~NO$H$7$FMQ$$$i$l$k$3$H$bB?$$(B.
                    496:
                    497: \begin{df}
                    498: {\bf block order}\\
                    499: $\{x_1,\cdots,x_n\} = S_1 \cup \cdots \cup S_l$ (disjoint sum) $B$H$7(B,
                    500: $<_i$ $B$r(B $T_i = K[y_1,\cdots]$ ($y_k \in S_l$) $B>e$N(B term
                    501: 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
                    502: $B7h$a$k(B.
                    503: \end{df}
                    504:
                    505: $B<-=q<0=g=x$O(B $\{x_1,\cdots,x_n\} = \{x_1\} \cup \cdots \cup \{x_n\}$
                    506:  $B$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
                    507: $B$r5a$a$?$$>l9g$K$O(B, $B8zN($r9M$($l$PLdBj$,$"$k(B. $B$3$N$h$&$J>l9g$K(B,
                    508: $S_1$ $B$K>C5n$7$?$$JQ?t(B, $S_2$ $B$K;D$j$NJQ?t(B, $B$HJ,3d$7(B, $B$=$l$>$l$K(B
                    509: $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.
                    510: $B$3$l$K$D$$$F$O8e$G=R$Y$k(B.
                    511:
                    512: \begin{df}
                    513: {\bf matrix order}\\
                    514: $M$ $B$r(B $B<!$rK~$?$9<B(B $m\times n$ $B9TNs$H$9$k(B.
                    515: \begin{enumerate}
                    516: \item $BD9$5(B $n$ $B$N@0?t%Y%/%H%k(B $v$ $B$KBP$7(B, $Mv=0 \Leftrightarrow v=0$
                    517: \item $BHsIi@.J,$r;}$DD9$5(B $n$ $B$N@0?t%Y%/%H%k(B $v$ $B$KBP$7(B, $Mv$ $B$N(B 0 $B$G$J$$:G=i(B
                    518: $B$N@.J,$O@5(B.
                    519: \end{enumerate}
                    520:
                    521: $B$3$N;~(B, $\N^n$ $B$N%Y%/%H%k(B $u,v$ $B$KBP$7(B,
                    522: \begin{center}
                    523: $u>v \Leftrightarrow M(u-v)$ $B$N(B 0 $B$G$J$$:G=i$N@.J,$,@5(B
                    524: \end{center}
                    525: $B$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
                    526: $B$jDj5A$5$l$k(B matrix order $B$H8F$V(B.
                    527: \end{df}
                    528:
                    529: \begin{pr} (Robbiano [\cite{ROBBIANO}])
                    530: $BG$0U$N(B term order $B$O(B, matrix order $B$K$h$jDj5A$G$-$k(B.
                    531: \end{pr}
                    532:
                    533: \newfont{\bigfont}{cmr10 scaled\magstep4}
                    534: \newcommand{\bigzl}{\smash{\hbox{\bigfont 0}}}
                    535: \newcommand{\bigzu}{\smash{\lower1.0ex \hbox{\bigfont 0}}}
                    536: \def\udots{\mathinner{\mkern1mu\raise1pt\vbox{\kern7pt\hbox{.}}\mkern2mu
                    537:     \raise4pt\hbox{.}\mkern2mu\raise7pt\hbox{.}\mkern1mu}}
                    538:
                    539: \begin{ex} $B$h$/CN$i$l$?(B order $B$rDj5A$9$k(B matrix $B$NNc(B
                    540: \vskip\baselineskip
                    541:
                    542: $M_{DLEX}=\left(
                    543: \begin{array}{ccc}
                    544: 1 & \cdots & 1 \\
                    545: 1 & & \bigzu \\
                    546:   & \ \ddots & \\
                    547: \bigzl & & 1 \\
                    548: \end{array}
                    549: \right)$
                    550: $M_{DRL}=\left(
                    551: \begin{array}{ccc}
                    552: 1 & \cdots & 1 \\
                    553: \bigzu & & -1 \\
                    554:   & \udots & \\
                    555: -1 & & \bigzl \\
                    556: \end{array}
                    557: \right)$
                    558: $M_{LEX}=\left(
                    559: \begin{array}{ccc}
                    560: 1 & & \bigzu \\
                    561:   & \ \ddots & \\
                    562: \bigzl & & 1 \\
                    563: \end{array}
                    564: \right)$
                    565:
                    566: \vskip\baselineskip
                    567: $M_{DLEX}$, $M_{DRL}$, $M_{LEX}$ $B$O$=$l$>$lA4<!?t<-=q<0(B, $BA4<!?t5U<-=q<0(B,
                    568: $B<-=q<0=g=x$rDj5A$9$k(B.
                    569: \end{ex}
                    570:
                    571:
                    572: \begin{ex}
                    573: weighted order
                    574:
                    575: \vskip\baselineskip
                    576: $M_{wDRL}=\left(
                    577: \begin{array}{ccc}
                    578: w_1 & \cdots & w_n \\
                    579: \bigzu & & -1 \\
                    580:   & \udots & \\
                    581: -1 & & \bigzl \\
                    582: \end{array}
                    583: \right)$
                    584:
                    585: \vskip\baselineskip
                    586: $BBh0l9T$O(B, $B;X?t%Y%/%H%k(B $(d_1,\cdots,d_n)$ $B$KBP$7$F(B,
                    587: $\displaystyle{\sum_{i=1}^n w_id_i}$ $B$9$J$o$A(B weight $BIU$-$N(B
                    588: $BA4<!?t$G:G=i$KHf3S$r9T$&$3$H$r0UL#$9$k(B.
                    589: \end{ex}
                    590:
                    591: \begin{ex}
                    592: block order
                    593:
                    594: \vskip\baselineskip
                    595: $M_{block}$={\Large $\left(
                    596: \begin{array}{ccc}
                    597: M_1 & & \bigzl \\
                    598:   & \ddots & \\
                    599: \bigzu & & M_l \\
                    600: \end{array}
                    601: \right)$}
                    602:
                    603: \vskip\baselineskip
                    604: $B3F(B $M_k$ $B$O(B, $B3F%V%m%C%/$KBP$9$k(B term order $B$rDj5A$9$k(B matrix
                    605: $B$G$"$k(B.
                    606: \end{ex}