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1.4     ! noro        1: %$OpenXM: OpenXM/doc/compalg/fglm.tex,v 1.3 2000/03/28 02:02:30 noro Exp $
1.1       noro        2: \chapter{Change of ordering}
                      3:
                      4: $BA0@a$G$O(B, $B<g$H$7$F(B Buchberger $B%"%k%4%j%:%`$N8zN(2=$K$D$$$F=R$Y$?(B.
                      5: $B$7$+$7(B, $B%0%l%V%J4pDl$N7W;;K!$O(B Buchberger $B%"%k%4%j%:%`$@$1$H$O8B$i$J$$(B.
                      6: $BK\@a$G$O(B, $B4{$K2?$i$+$N(B order $B$K4X$7$F%0%l%V%J4pDl$K$J$C$F$$$kB?9`<0(B
                      7: $B=89g$rF~NO$H$7$F(B, $BB>$N(B order $B$N%0%l%V%J4pDl$r5a$a$kJ}K!$K$D$$$F=R$Y$k(B.
                      8:
                      9: \section{FGLM $B%"%k%4%j%:%`(B}
                     10:
                     11: $I \subset K[X]$ $B$r(B 0 $B<!85%$%G%"%k$H$7(B, $I$ $B$N$"$k(B order $<_1$ $B$K4X$9$k(B
                     12: $BHoLs%0%l%V%J4pDl(B $G_1$ $B$,4{$KF@$i$l$F$$$k$H$9$k(B. $B$3$N$H$-(B, $BB>$N(B order $<$
                     13: $B$K4X$9$k(B $I$ $B$N%0%l%V%J4pDl(B $G$ $B$r(B, $B<g$H$7$F@~7ABe?t$K$h$j5a$a$k$N$,(B FGLM
                     14: $B%"%k%4%j%:%`$G$"$k(B.
                     15:
                     16: \begin{lm}
                     17: $<$ $B$r(B admissible order, $F = GB_<(I)$ $B$H$9$k(B.
                     18: $T = \{t_1,\cdots,t_l\} \subset T(X)$ $B$r9`$N=89g$H$9$k(B.
                     19: $a_i$ $B$rL$Dj78?t$H$7(B,
                     20: $$E = \displaystyle{\sum_{i=1}^l a_i NF_<(t_i,F)}$$
                     21: $B$H$*$/(B. $E$ $B$N(B $X$ $B$K4X$9$k78?t$N=89g$r(B $C$ $B$H$9$l$P(B,
                     22: $Eq = \{ f = 0 \mid f \in C\}$ $B$O(B $a_i$ $B$K4X$9$k@~7AJ}Dx<0$H$J$k(B. $B$3$N;~(B
                     23: \begin{center}
                     24: $Eq$ $B$,<+L@$G$J$$2r$r;}$D(B $\Leftrightarrow$ $T$ $B$,(B $K[X]/I$ $B$K$*$$$F(B
                     25: $K$-$B@~7A=>B0(B
                     26: \end{center}
                     27: \end{lm}
                     28:
                     29: \begin{al} (FGLM $B%"%k%4%j%:%`(B\cite{FGLM})
                     30: \label{fglm}
                     31: \begin{tabbing}
                     32: FGLM$(F,<_1,<)$\\
                     33: Input : \= order $<_1$, $<$; $F \subset K[X]$ \st $F = GB_{<_1}(I)$ $B$+$D(B
                     34: $\dim(I)=0$\\
                     35: Output : $F$ $B$N(B $<$ $B$K4X$9$k%0%l%V%J4pDl(B\\
                     36: ($C_2$)\= \kill
                     37: $G \leftarrow \emptyset$\\
                     38: $h \leftarrow 1$\\
                     39: $B \leftarrow \{h\}$\\
                     40: $H \leftarrow \emptyset$\\
                     41: do \{\\
                     42: \> $N \leftarrow \{u \mid u > h$ $B$+$D$9$Y$F$N(B $m \in H$ $B$KBP$7(B $m {\not|} u\}$\\
                     43: (0)\> if $N = \emptyset$ then return $G$\\
                     44: (1)\> $h_1 \leftarrow \min(N)$\\
                     45: \> $a_t$ : $t \in B$ $B$KBP1~$9$kL$Dj78?t(B\\
                     46: \> $a_{h_1} \leftarrow 1$\\
                     47: (2)\> $E \leftarrow \displaystyle{NF_{<_1}(h_1,F)+\sum_{t \in B} a_t NF_{<_1}(t,F)}$\\
                     48: \> $C \leftarrow$ $E$ $B$N(B $X$ $B$K4X$9$k78?t$N=89g(B\\
                     49: \> if \= $B@~7AJ}Dx<0(B $\{ f = 0 \mid f \in C\}$ $B$,2r(B
                     50:     $\{a_t = c_t \mid c_t \in K\}$ $B$r;}$D(B\\
                     51: \> then \\
                     52: \> \>$G \leftarrow G \cup \{\displaystyle{h_1+\sum_{t \in B} c_t t}\}$\\
                     53: \> \>$H \leftarrow H \cup \{h_1\}$\\
                     54: \> else $B \leftarrow \{h_1\} \cup B$\\
                     55: \> $h \leftarrow h_1$\\
                     56: \}\\
                     57: \end{tabbing}
                     58: \end{al}
                     59:
                     60: \begin{lm}
                     61: \label{fglml1}
                     62: (0) $B$K$*$$$F(B, $B= \{u \mid u \le h$ $B$+$D$9$Y$F$N(B $m \in H$ $B$KBP$7(B $m {\not|} u\}$.
                     63: \end{lm}
                     64: \proof $B%k!<%W$K4X$9$k5"G<K!$G<($9(B. $B:G=i$K%k!<%W$KF~$C$?;~E@$G$O(B
                     65: $B@.N)$7$F$$$k(B. $B$"$k;~E@$G@.N)$7$F$$$k$H$9$k(B. $B$=$N;~E@$+$i%k!<%W$,0l2s(B
                     66: $B2s$C$?;~E@$N(B (0) $B$K$*$$$F@.N)$9$k$3$H$r<($9(B.
                     67: $B<($=$&$H$9$k;~E@$N0l$DA0$N;~E@$G$N(B $N$, $B$, $h$ $B$r(B $N_0$, $B_0$, $h_0$ $B$H=q$/(B.
                     68: $h = \min(N_0)$ $B$G(B, $B = B_0 \cup \{h\}$ $B$^$?$O(B $H = H_0 \cup \{h\}$ $B$G$"$k(B. \\
                     69: \underline{$B = B_0 \cup \{h\}$ $B$N;~(B}\, $H = H_0$ $B$h$j(B,
                     70: $B1&JU(B $= B_0 \cup \{u \mid$ $h_0 < u \le h$ $B$+$D$9$Y$F$N(B $m \in H_0$ $B$K(B
                     71: $BBP$7(B $m{\not|} u\}= B_0 \cup \{h\}$.\\
                     72: \underline{$H = H_0 \cup \{h\}$ $B$N;~(B}\,
                     73: $B1&JU(B $= B_0 \cup
                     74: (\{u \mid h_0 < u \le h$ $B$+$D$9$Y$F$N(B $m \in H_0$ $B$KBP$7(B $m {\not|} u\}
                     75: \cap \{u \mid h {\not|} u \}) = B_0 \cup (\{h\} \cap \{u \mid h {\not|} u \}) = B_0$.\\
                     76: $B$h$C$F(B, $B$$$:$l$N>l9g$K$b1&JU(B = $B$ $B$H$J$k(B. \qed
                     77:
                     78: \begin{lm}
                     79: \label{fglml2}
                     80: (1) $B$K$*$$$F(B, $h_1 \in x_1B \cup \cdots \cup x_nB$.
                     81: \end{lm}
                     82: \proof $h_1>1$ $B$h$j$"$k(B $k$ $B$,B8:_$7$F(B $h_1=x_kh'$ $B$H=q$1$k(B.
                     83: $B$b$7(B $h' \in N$ $B$J$i$P(B $h_1 = \min(N)$ $B$KH?$9$k$+$i(B $h' \in B$. \qed
                     84:
                     85: \begin{pr}
                     86: $B%"%k%4%j%:%`(B \ref{fglm} $B$O(B $GB_<(F)$ $B$r=PNO$9$k(B.
                     87: \end{pr}
                     88: \proof $BJdBj(B \ref{fglml2}$B$h$j(B, (1) $B$K$*$1$k(B $h$ $B$N8uJd$O(B $BM-8B=89g(B
                     89: $x_1B \cup \cdots \cup x_nB$ $B$N85$@$+$i(B $\min(N)$ $B$rM?$($kA*Br%"%k%4%j(B
                     90: $B%:%`$,B8:_$9$k(B. \\
                     91: \underline{$BDd;_@-(B} $B$^$:(B, (2) $B$,2r$r;}$?$J$$$3$H$H(B, $\{h_1\} \cup B$ $B$,(B
                     92: $K[X]/I$ $B$G(B $K$ $B>e0l<!FHN)$G$"$k$3$H$,F1CM$G$"$k$3$H$KCm0U$9$k(B. $B$h$C$F(B,
                     93: $|B|$ $B$O(B$\dim_K K[X]/I$ $B$r1[$($i$l$J$$(B. $B$^$?(B, $H$ $B$N85$O(B, $B$I$N85$bB>$N(B
                     94: $B85$r3d$i$J$$$+$i(B, $B7O(B \ref{noether} $B$h$j$d$O$jM-8B=89g(B. $B$h$C$F%"%k%4%j(B
                     95: $B%:%`$ODd;_$9$k(B.\\
                     96: \underline{$G = GB_<(F)$} $f \in I$ $B$H$7(B, $t=HT_<(F)$ $B$H$9$k(B.
                     97: $f$ $B$O(B $G$ $B$K4X$7$FHoLs$H$7$F$h$$(B. $H = \{h_1,\cdots,h_m\}$ \quad
                     98: ($h_1<\cdots<h_m$) $B$H$9$k(B. $B$^$?(B, $(1)$ $B$K$h$jA*Br$5$l$k85$r=g$KJB$Y$?(B
                     99: $B$b$N$r(B $t_1<t_2<\cdots$$B$H$9$k(B. $B$b$7(B, $B$9$Y$F$N(B $h \in H$ $B$KBP$7(B $h
                    100: {\not|} t$ $B$J$i$P(B, $BDd;_$N>r7o$h$j$"$k(B $k$$B$,$"$C$F(B, $t_k<t\le t_{k+1}$.
                    101: $B2>Dj$K$h$j(B, $k+1$ $BHVL\$KA*$P$l$k85$O(B$t$ $B$G$J$1$l$P$J$i$J$$$+$i(B
                    102: $t=t_{k+1}$. $t' \in T(f) \setminus \{t\}$$B$H$9$l$P(B, $t'<t$ $B$+$DA4$F$N(B
                    103: $h \in H$ $B$KBP$7(B $h {\not|} t'$. $B$3$l$h$j(B$t' \le t_k$. $B$h$C$FJdBj(B
                    104: \ref{fglml1} $B$h$j(B $t=t_{k+1}$ $B$,A*Br$5$l$F$$$k;~E@$G(B $t'\in B$.  $B$h$C(B
                    105: $B$F$3$N;~E@$G(B $f$ $B$O@~7AJ}Dx<0$N2r$H$J$j(B, $t\in H$ $B$H$J$k$,$3$l$OL7=b(B.\qed
                    106:
                    107: FGLM $B%"%k%4%j%:%`$r7W;;5!>e$G<BAu$9$k>l9g(B, $BFC$K(B (2)$B$NItJ,$N<BAu$K(B
                    108: $B9)IW$,I,MW$H$J$k(B. $BMWE@$r$^$H$a$k$H(B,
                    109:
                    110: \begin{enumerate}
                    111: \item $B@55,7A$N7W;;$O(B, $B3F9`$K$D$-$?$@0lEY$@$19T$$(B, $B7k2L$OI=$K$7$FJ];}$9$k(B.
                    112: \item $BKh2sFHN)$J@~7AJ}Dx<0$H$7$F2r$/$N$G$O$J$/(B, $B7k2L$,8e$G;H$($k$h$&$J(B
                    113: $B9)IW$r$9$k(B.
                    114: \end{enumerate}
                    115:
                    116: 1. $B$K4XO"$7$F(B, $B<!$N$h$&$J@~7A<LA|$r9M$($k$3$H$G(B, $B@55,7A7W;;$N8zN($r>e(B
                    117: $B$2$k$3$H$,$G$-$k(B.
                    118:
                    119: \begin{df}
                    120: $B3F(B $i (1\le i\le n)$ $B$KBP$7(B, $\phi_i \in End(K[X]/I)$ $B$r(B
                    121: $$\phi_i : f \bmod I \mapsto x_if \bmod I$$
                    122: $B$GDj5A$9$k(B. $H_1 = \{HT_{<_1}(g)\mid g \in G_1\}$,
                    123: $MB_1 = \{u \in T \mid$  $B$9$Y$F$N(B $m \in H_1$ $B$KBP$7(B $m {\not|} u\}$
                    124: $B$H$*$1$P(B $MB_1$ $B$O(B $K[X]/I$ $B$N(B $K$-$B4pDl$h$j(B,
                    125: $\{NF_{<_1}(x_iu,G_1)\mid u \in MB_1\}$ $B$rA4$F7W;;$9$k$3$H$G(B,
                    126: $\phi_i$ $B$,I=8=$G$-$k(B.
                    127: \end{df}
                    128:
                    129: $B$"$i$+$8$a(B, $\phi_i$ $B$r7W;;$7$F$*$1$P(B, $NF(x_it,G_1) = \phi_i(NF(t,G_1))$
                    130: $B$h$j(B, $B4{$KF@$i$l$F$$$k$O$:$N(B $NF(t,G_1)$ $B$NA|$H$7$F$"$i$?$J9`$N@55,7A$,(B
                    131: $B7W;;$G$-$k(B.
                    132:
                    133: \begin{re}
                    134: $B0lHL$K$O(B, FGLM $B%"%k%4%j%:%`$O(B 0 $B<!85%$%G%"%k$N>l9g$K$N$_E,MQ2DG=$@$,(B,
                    135: $BL\E*$N(B order $B$,A4<!?tHf3S$r4^$`>l9g$J$I(B, $BG$0U$N(B $s \in T$ $B$KBP$7(B
                    136: $\{t \in T \mid t < s\}$ $B$,M-8B=89g$N>l9g$K$O(B, $BG$0U$N%$%G%"%k$KE,MQ$G$-$k(B.
                    137: $B$7$+$7(B, $B8zN($O0lHL$K(B 0 $B<!85$N>l9g$KHf$Y$F4|BT$G$-$J$$(B.
                    138: \end{re}
                    139:
                    140: \section{Modular change of ordering}
                    141:
                    142: FGLM $B$O(B, $BL\E*$N9`$KE~C#$9$k$^$G9TNs$N(B Gauss $B>C5n$r7+$jJV$9J}K!$H$$$($k(B.
                    143: $B$3$N(B Gauss $B>C5n$OM-M}?tBN>e$G9T$J$o$l$k$?$a(B, $B7k2L$N%0%l%V%J4pDl$N85$N(B
                    144: $B78?t$KHf$Y$FESCf$N78?tKDD%$,7c$7$/$J$k>l9g$,$7$P$7$P@8$:$k(B. $B$3$l$O(B,
                    145: $B<!$NNc$G<($5$l$k(B.
                    146:
                    147: \begin{ex}
                    148: $A \in GL(n,\Q)$ $B$H$9$k(B. $V \in \Q^n$ $B$KBP$7(B $B=AV$ $B$H$9$k$H(B,
                    149: $B@~7AJ}Dx<0(B $AX = B$ $B$O(B $X = V$ $B$rM#0l$N2r$H$9$k(B. $B$3$NJ}Dx<0(B
                    150: $B$r(B Gauss $B>C5n$G2r$/>l9g(B, $B$=$l$O(B $A$ $B$K$N$_CmL\$7$F9T$J$o$l(B,
                    151: $B$ $B$NCM$K:81&$5$l$J$$(B. $B$9$J$o$A(B, $B2r(B $V$ $B$N@.J,$,>.$5$$@0?t(B
                    152: $B$N>l9g$G$bBg$-$$>l9g$G$b2r$/<j4V$OJQ$o$i$J$$$3$H$K$J$k(B.
                    153: \end{ex}
                    154:
                    155: \subsection{Modular $B7W;;$H@~7ABe?t$K$h$k%0%l%V%J4pDl8uJd@8@.(B}
                    156: $B$3$3$G>R2p$9$k%"%k%4%j%:%`$O(B, modular $B7W;;$r1~MQ$7$F(B, $B7k2L$N78?t$NBg$-(B
                    157: $B$5$NDxEY$N%3%9%H$G%0%l%V%J4pDl$r7W;;$9$k$b$N$G$"$k(B\cite{NY}\cite{NY2}.
                    158: $B%"%k%4%j%:%`(B \ref{mfglm} $B$G$O(B, $BM-8BBN>e$N%0%l%V%J4pDl7W;;$K$h$j(B, $BM-M}(B
                    159: $B?tBN>e$N%0%l%V%J4pDl$N3F85$K8=$l$k9`$r?dB,$7(B, $BL$Dj78?tK!$G(B, $B$=$l$i$N9`(B
                    160: $B$r<B:]$K$b$D(B $I=Id(F)$ $B$N85$r5a$a$k(B.
                    161:
                    162: \begin{al}
                    163: \label{mfglm}
                    164: \begin{tabbing}
                    165: \\
                    166: candidate\_by\_linear\_algebra$(F,p,<_1,<)$\\
                    167: Input : \= order $<_1$, $<$\\
                    168: \> $F \subset \Z[X]$ \st $F = GB_{<_1}(Id(F))$\\
                    169: \> $F$ $B$N3F85$N(B $<_1$ $B$K4X$9$k<g78?t$r3d$i$J$$(B $p$ \\
                    170: Output : $F$ $B$N(B $<$ $B$K4X$9$k%0%l%V%J4pDl8uJd$^$?$O(B {\bf nil}\\
                    171: ($C_2$)\= \kill
                    172: $\overline{G} \leftarrow$ $GB_<(Id(\phi_p(F)))$ ($BHoLs%0%l%V%J4pDl(B) \\
                    173: $G \leftarrow \emptyset$\\
                    174: for \= each $h \in \overline{G}$ do \{\\
                    175: \> $a_t$ : $t \in T(h)$ $B$KBP1~$9$kL$Dj78?t(B\\
                    176: \> $a_t \leftarrow 1$ ($t = ht_{<}(h)$ $B$KBP$7$F(B)\\
                    177: \> $H \leftarrow \displaystyle{\sum_{t \in T(h)} a_t NF_{<_1}(t,F)}$\\
                    178: \> $C \leftarrow$ $H$ $B$N(B $X$ $B$K4X$9$k78?t$N=89g(B\\
                    179: \> if \= $B@~7AJ}Dx<0(B $E_h = \{ f = 0 \mid f \in C\}$ $B$,2r(B
                    180:     $S_h = \{a_t = c_t \mid c_t \in \Q\}$ $B$r;}$D(B\\
                    181: \> then $G \leftarrow G \cup \{\displaystyle{d\sum_{t \in T(h)} c_t t}\}$\\
                    182: \> {\rm ($d$ : $c_t$ $B$NJ,Jl$N(B LCM)}\\
                    183: \> else return {\bf nil}\\
                    184: \}\\
                    185: return $G$
                    186: \end{tabbing}
                    187: \end{al}
                    188:
                    189: \begin{pr}
                    190: $B%"%k%4%j%:%`(B \ref{mfglm} $B$O(B, $BM-8B8D$N(B $p$ $B$r=|$$$F(B $GB_<(F)$ $B$r(B
                    191: $BM?$($k(B.
                    192: \end{pr}
                    193: \proof trace lifting $B$N>l9g$HF1MM$K(B, $BM-8BBN>e$G$N%0%l%V%J4pDl$N3F(B
                    194: $B85$K8=$l$k9`$,(B, $BM-M}?tBN>e$G$N$=$l$H0lCW$7$F$$$l$P(B, $BA4$F$N@~7AJ}Dx<0$O(B
                    195: $B2r$r;}$A(B, $G = GB_<(F)$ $B$H$J$k(B. $B$=$&$G$J$$(B $p$ $B$OM-8B8D$7$+$J$$$?$a(B,
                    196: $B$=$l$i$r=|$$$F$O(B, $B$3$N%"%k%4%j%:%`$O(B $GB_<(F)$ $B$rM?$($k(B. \qed
                    197:
                    198: \begin{re}
                    199: $B@~7AJ}Dx<0$,A4It2r$1$?$H$7$F$b(B, $B7k2L$,(B $Id(F)$ $B$N%0%l%V%J4pDl$K$J$C$F(B
                    200: $B$$$kJ]>Z$O8=;~E@$G$O$J$$$N$G(B, $B$3$NL?Bj$OIT==J,$G$"$k(B. $B<B$O(B, $B<!$G=R$Y$k(B
                    201: $B7k2L$K$h$j(B, $B%"%k%4%j%:%`(B \ref{mfglm} $B$,(B {\bf nil} $B$G$J$$B?9`<0=89g$r(B
                    202: $BJV$;$P(B, $B$=$l$O$?$@$A$K(B $Id(F)$ $B$N%0%l%V%J4pDl$H$J$C$F$$$k$3$H$,J,$+$k(B.
                    203: $B$3$l$K$D$$$F$O(B, $B@~7AJ}Dx<0$N(B, $B7k2L$NBg$-$5$K1~$8$?7W;;NL$rI,MW$H$9$k(B
                    204: $B5a2rK!$H$H$b$K(B, $B8e$G=R$Y$k(B.
                    205: \end{re}
                    206:
                    207: \subsection{$B%0%l%V%J4pDl8uJd$,%0%l%V%J4pDl$H$J$k>r7o(B}
                    208: $B$3$3$G$O(B, change of ordering $B$N>l9g$K$O(B,
                    209: trace lifting $B$N>l9g$KI,MW$@$C$?%0%l%V%J4pDl%F%9%H$H%a%s%P%7%C%W%F%9%H(B
                    210: $B$,ITI,MW$K$J$k$3$H$r<($9(B. $F \subset \Z[X]$ $B$H$9$k(B.
                    211:
                    212: \begin{as}
                    213: \label{nf}
                    214: $BM-M}?tBN>e$HM-8BBN>e$N7W;;$,0[$i$J$$$h$&<!$N2>Dj$r$*$/(B.
                    215: \begin{enumerate}
                    216: \item $BITI,MWBP$N8!=P4p=`$OF,9`$N$_$G9T$&(B.
                    217: \item $B@55,7A7W;;$K$*$$$F(B, $B@55,2=$KMQ$$$k85(B (reducer) $B$NA*Br$O(B, $B@55,2=$5$l$k(B
                    218: $BB?9`<0$N9`$*$h$S(B reducer $B$NF,9`$N=89g$K$N$_0MB8$9$k(B.
                    219: \end{enumerate}
                    220: \end{as}
                    221:
                    222: \begin{df}
                    223: {\rm (compatibile $B$JAG?t(B)}
                    224: $BAG?t(B $p$ $B$,(B $F$ $B$K4X$7(B compatible $B$H$O(B,
                    225: $$\phi_p(Id(F)\cap\Z[X]) (=\phi_p(Id(F)\cap\Z_{(p)}[X])) = Id(\phi_p(F))$$
                    226: $B$J$k$3$H(B.
                    227: \end{df}
                    228:
                    229: \begin{df}
                    230: $BAG?t(B $p$ $B$,(B $(F,<)$ $B$K4X$7$F(B strongly compatible $B$H$O(B
                    231: $p$ $B$,(B $F$ $B$K4X$7$F(B compatible $B$G(B
                    232: $$E_<(Id(F)) = E_<(Id(\phi_p(F))$$
                    233: $B$J$k$3$H(B.
                    234: \end{df}
                    235:
                    236: \begin{df}(permissible $B$JAG?t(B)
                    237: $BAG?t(B $p$ $B$,(B $(F,<)$ $B$K$D$$$F(B permissible $B$H$O(B
                    238: $B3F(B $f \in F$ $B$KBP$7(B \valid{p}{f}{<} $B$J$k$3$H(B.
                    239: \end{df}
                    240:
                    241: \begin{df}
                    242: $f \in \Q[X]$ $B$,(B $p$ $B$K4X$7(B stable $B$H$O(B $f \in \Z_{(p)}[X]$ $B$J$k$3$H(B.
                    243: \end{df}
                    244:
                    245: \begin{df}
                    246: {\rm (modular $B%0%l%V%J4pDl$N5UA|(B)}
                    247: $G \subset Id(F)\cap \Z[X]$ $B$,(B $F$ $B$N(B $<$ $B$K4X$9$k(B
                    248: $p$-compatible $B$J%0%l%V%J4pDl8uJd$H$O(B
                    249: $p$ $B$,(B $(G,<)$ $B$K$D$$$F(B permissible $B$G(B $\phi_p(G)$ $B$,(B$Id(\phi_p(F))$ $B$N(B $<$
                    250: $B$K4X$9$k%0%l%V%J4pDl$J$k$H$-$r$$$&(B.
                    251: \end{df}
                    252:
                    253: \begin{re}
                    254: compatibility $B$O(B order $B$KFHN)$J35G0$G$"$k(B.
                    255: \end{re}
                    256:
                    257: \begin{lm}
                    258: \label{valid}
                    259: $G \subset \Z[X]$, $p$ $B$r(B $(G,<)$ $B$K$D$$$F(B permissible $B$JAG?t(B,
                    260: $f \in \Z[X]$ $B$H$9$k(B. $B$3$N$H$-(B
                    261: $B2>Dj(B \ref{nf} $B$N$b$H$G(B
                    262: $$NF(\phi_p(f),\phi_p(G)) = \phi_p(NF(f,G)).$$
                    263: \end{lm}
                    264: \proof $NF(f,G)$ $B$O<!$N(B recurrence $B$G7W;;$5$l$k(B.
                    265: $$f_0 \leftarrow f,
                    266: f_i \leftarrow f_{i-1} - \alpha_i t_i g_{k_i}$$
                    267: $B$3$3$G(B, $\alpha_i\in {\bf \Q}$, $t_i$ : a term, $g_{k_i}\in G$.
                    268: $B$9$k$H(B, $p$ $B$,(B $(G,<)$ $B$KBP$7$F(B permissible $B$h$j(B $\alpha_i \in \Z_{(p)}$.
                    269: $B$h$C$F(B $BA4$F$N(B $i$ $B$KBP$7(B $f_i \in \Z_{(p)}[X]$ $B$G(B, $B$3$N(B recurrence $B$K(B
                    270: $\phi_p$ $B$rE,MQ$7$F<!$rF@$k(B.
                    271: $$\phi_p(f_i) = \phi_p(f_{i-1}) - \phi_p(\alpha_i) t_i \phi_p(g_{k_i}).$$
                    272: $B$b$7(B $\phi_p(\alpha_i) \neq 0$ $B$J$i$P(B $\phi_p(f_{i-1}) \neq 0$ $B$G(B,
                    273: $$\phi_p(f_i) \leftarrow \phi_p(f_{i-1}) - \phi_p(\alpha_i) t_i \phi_p(g_{k_i})$$
                    274: $B$O(B $\phi_p(G)$ $B$K$h$kF,9`>C5n$G$"$k(B.
                    275: $B$b$7(B $\phi_p(\alpha_i) = 0$ $B$J$i$P(B $\phi_p(f_i) = \phi_p(f_{i-1})$ $B$G(B
                    276: $$\{\phi_p(f_i)\mid i=0\, {\rm or}\, \phi_p(\alpha_i)\neq 0\}$$
                    277: $B$J$kNs$O(B
                    278: $NF(\phi_p(f),\phi_p(G))$ $B$N7W;;$KBP1~$9$k(B.
                    279: $B$b$7(B$\phi_p(f_i) = 0$ $B$J$k(B $i$ $B$,B8:_$9$l$PA|$OESCf$G@Z$l$k$,(B
                    280: $$NF(\phi_p(f),\phi_p(G)) = \phi_p(NF(f,G)) = 0$$
                    281: $B$@$+$i<gD%$,@.N)$9$k(B.
                    282: \qed
                    283: \medskip
                    284:
                    285: \begin{th}
                    286: \label{comp}
                    287: $G \subset Id(F)\cap\Z[X]$ $B$r(B $Id(F)$ $B$N(B $<$ $B$K4X$9$k%0%l%V%J4pDl$H$9$k(B.
                    288: $B$b$7(B $p$ $B$,(B $(G,<)$ $B$K4X$7$F(B permissible $B$+$D(B
                    289: $\phi_p(G) \subset Id(\phi_p(F))$ $B$J$i$P(B $p$ $B$O(B $F$ $B$K4X$7$F(B compatible $B$G$"$k(B.
                    290: $B99$K(B, $\phi_p(G)$ $B$O(B $Id(\phi_p(F))$ $B$N%0%l%V%J4pDl$G(B
                    291: $p$ $B$O(B $(F,<)$ $B$K$D$$$F(B strongly compatible $B$G$"$k(B.
                    292: \end{th}
                    293: \proof
                    294: $h \in Id(F) \cap \Z[X]$ $B$H$9$k(B. $G$ $B$,(B $Id(F)$ $B$N%0%l%V%J4pDl$@$+$i(B,
                    295: $NF(h,G)=0$ $B$h$j(B $h = \sum_{g\in G} a_g g$ $B$H=q$1$k(B. $B$3$3$G(B $a_g \in \Q[X]$.
                    296: $B99$K(B $p$ $B$,(B $(G,<)$ $B$K$D$$$F(B permissible $B$h$j(B,
                    297: $a_g$ $B$O(B $p$ $B$K$D$$$F(B stable $B$G(B, $B2>Dj(B $\phi_p(g)\in Id(\phi_d(F))$ $B$h$j(B
                    298: $$\phi_p(h)=\sum_{g\in G} \phi_p(a_g) \phi_p(g).$$
                    299: $B$h$C$F(B $\phi_p(h)\in Id(\phi_p(F))$.
                    300: $B8N$K(B $\phi_p(Id(F)\cap \Z[X])\subset Id(\phi_p(F))$. \\
                    301: $B5U$K(B $\overline{h} \in Id(\phi_p(F))$ $B$O(B
                    302: $$\overline{h} = \sum_{f\in F} \overline{a}_f \phi_p(f)$$
                    303: $B$H=q$1$k(B. $B$3$3$G(B $\overline{a}_f \in GF(p)[X]$.
                    304: $B$9$k$H(B, $\phi_p(a_f)=\overline{a}_f$ $B$J$k(B $a_f$ $B$rA*$s$G(B
                    305: $h=\sum_{f\in F} a_f f$ $B$H$*$1$P(B $\phi_p(h)=\overline{h}$.
                    306: $B$3$l$+$i(B $\phi_p(Id(F)\cap \Z[X])=Id(\phi_p(F))$. \\
                    307: $B:G8e$K(B $\phi(G)$ $B$,(B $Id(\phi_p(F))$ $B$N%0%l%V%J4pDl$H$J$k$3$H$r<($9(B.
                    308: $B>e$G=R$Y$?$3$H$+$i(B,
                    309: $\overline{h} \in Id(\phi_p(F))$ $B$KBP$7(B,
                    310: $h \in Id(F)\cap \Z[X]$ $B$,B8:_$7$F(B $\overline{h}=\phi_p(h)$.
                    311: $B$9$k$H(B, $BJdBj(B \ref{valid} $B$h$j(B
                    312: $$NF(\overline{h},\phi_p(G))=\phi_p(NF(h,G))=0.$$
                    313: $B=>$C$F(B, $\phi_p(G)$ $B$O(B $Id(\phi_p(F))$ $B$N%0%l%V%J4pDl(B.
                    314: strong compatibility $B$O(B $E_<(Id(F))$ $B$,(B $E_<(G)$
                    315: $B$G@8@.$5$l$k$3$H$+$iJ,$+$k(B.
                    316: \qed \\
                    317: $B$3$NDjM}$G(B, $\phi_p(G) \subset Id(\phi_p(F))$ $B$O(B
                    318: $Id(\phi_p(F))$ $B$N%0%l%V%J4pDl$K$h$j%A%'%C%/$G$-$k(B.
                    319: $B$h$C$F(B, $p$ $B$N(B compatibility $B$N%A%'%C%/$OM-M}?tBN>e(B, $BM-8BBN>e$N(B
                    320: $BG$0U$N(B order $B$G$N%0%l%V%J4pDl$r7W;;$9$k$3$H$G9T$&$3$H$,$G$-$k(B.
                    321: $B$b$7(B, $BF~NO$,4{$K$"$k(B order $B$G$N%0%l%V%J4pDl$J$i(B compatibility $B$N%A%'%C%/(B
                    322: $B$O6K$a$F4JC1$G$"$k(B.
                    323:
                    324: \begin{co}
                    325: \label{compco}
                    326: $G \subset \Z[X]$ $B$,(B $<$ $B$K4X$9$k(B $Id(G)$ $B$N%0%l%V%J4pDl$H$9$k(B.
                    327: $B$b$7(B $p$ $B$,(B $(G,<)$ $B$KBP$7(B permissible $B$J$i$P(B
                    328: $\phi_p(G)$ $B$O(B $Id(\phi_p(G))$ $B$N%0%l%V%J4pDl$G(B
                    329: $p$ $B$O(B $(G,<)$ $B$KBP$7(B strongly compatible.
                    330: \end{co}
                    331: $B<!$NDjM}$O(B, $B%0%l%V%J4pDl8uJd$,<B:]$K%0%l%V%J4pDl$K$J$k$?$a$N==J,>r7o$rM?$($k(B.
                    332: $B$9$J$o$A(B, $B2f!9$,5a$a$k$b$N$G$"$k(B.
                    333:
                    334: \begin{th}
                    335: \label{candi}
                    336: $p$ $B$,(B $F$ $B$K$D$$$F(B compatible $B$G(B $G$ $B$,(B $<$ $B$K4X$7$F(B $p$-compatible $B$J%0%l%V%J(B
                    337: $B4pDl8uJd$J$i$P(B, $G$ $B$O(B $<$ $B$K4X$9$k(B $Id(F)$ $B$N%0%l%V%J4pDl$G$"$k(B.
                    338: \end{th}
                    339: \proof $BA4$F$N(B $f \in Id(F)$ $B$,(B $<$ $B$K4X$7$F(B $G$ $B$K$h$j(B 0 $B$K@55,2=(B
                    340: $B$5$l$k$3$H$r<($;$P$h$$(B. $f$ $B$O(B $G$ $B$K$D$$$FHoLs$H$7$F$h$$(B. $B$b$7(B $f
                    341: \neq 0$ $B$J$i$P(B, $BE,Ev$JM-M}?t$r$+$1$F(B, $f \in Id(F)\backslash \{0\}$ $B$,(B
                    342: $G$ $B$K$D$$$FHoLs$G(B $f$ $B$N78?t$N@0?t(B GCD ($cont(f)$ $B$H=q$/(B) $B$,(B 1 $B$KEy$7(B
                    343: $B$$(B, $B$H$7$F$h$$(B. $B$9$k$H(B $\phi_p(f) \neq 0$. $B$5$b$J$/$P(B
                    344: $cont(f)$ $B$O0x;R(B $p$ $B$r;}$D$3$H$K$J$k(B.
                    345: $p$ $B$O(B $F$ $B$K$D$-(B compatible $B$@$+$i(B $\phi_p(f) \in Id(\phi_p(F))$.
                    346: $B$9$k$H(B $\phi_p(f)$ $B$O(B $<$ $B$K4X$7(B, $\phi_p(G)$ $B$K$h$j(B 0 $B$K@55,2=$5$l$J$1$l$P(B
                    347: $B$J$i$J$$(B. $B$7$+$7(B $f$ $B$O(B $G$ $B$K$D$$$FHoLs$@$+$i(B
                    348: $\phi_p(G)$ $B$NF,9`$N=89g$O(B $G$ $B$N$=$l$HEy$7$$(B. $B$h$C$F(B
                    349: $\phi_p(f)$ $B$O(B $\phi_p(G)$ $B$K$D$$$FHoLs$H$J$j(B, $\phi_p(f) = 0$. $B$3$l$O(B
1.4     ! noro      350: $BL7=b(B. \qed\\
1.1       noro      351: $B<!$NDjM}$OA0DjM}$N@:L)2=$G$"$k(B. $B$9$J$o$A(B, $B>:=g$K7W;;$5$l$?ItJ,E*$J(B
                    352: $p$-compatible $B$J%0%l%V%J4pDl8uJd$,<B:]$K%0%l%V%J4pDl$N0lIt$H$J$C$F(B
                    353: $B$$$k$3$H$rJ]>Z$9$k(B. $B$3$l$O(B, $BESCf$^$G$N7k2L$r:FMxMQ$G$-$k$H$$$&E@$G(B
                    354: $BM-MQ$G$"$k(B. $B$^$?(B, $B8e$G=R$Y$k$h$&$K(B, $B%0%l%V%J4pDl$N$"$kFCDj$N85(B,
                    355: $BNc$($P(B, $B=g=x:G>.$N85$N$_$r5a$a$?$$(B, $B$"$k$$$O(B elimination $B8e$N7k2L$N$_(B
                    356: $B$r5a$a$?$$>l9g$K$bM-MQ$G$"$k(B.
                    357:
                    358: \begin{th}
                    359: $p$ $B$,(B $F$ $B$K$D$$$F(B compatible $B$H$9$k(B.
                    360: $\overline{G}\subset GF(p)[X], \overline{G} = GB_<(Id(\phi_p(F))$ $B$H$7(B
                    361: $\overline{g}_1<\cdots<\overline{g}_s$ $B$J$k(B $\overline{g}_i$ $B$K$h$j(B
                    362: $\overline{G}=\{\overline{g}_1,\cdots,\overline{g}_s\}$ $B$H=q$/(B.
                    363: $B99$K(B, $B$"$k@5?t(B $t\leq s$ $B$KBP$7(B,
                    364: $g_i \in Id(F) \cap \Z_{(p)}[X]$ ($1 \le i \le t$) $B$,B8:_$7$F(B
                    365: $\phi_p(g_i) = \overline{g}_i$ $B$+$D(B $g_i$ $B$O(B $\{g_1,\cdots,g_{i-1}\}$ $B$K$D$$$F(B
                    366: $BHoLs$H$9$k(B.
                    367: $B$3$N$H$-(B, $g_1,\cdots,g_t$ $B$O(B $GB_<(Id(F))$ $B$N:G=i$N(B $t$ $B8D$N85$K0lCW$9$k(B.
                    368: \end{th}
                    369: $B>ZL@N,(B ($B5"G<K!$K$h$k(B.)\\
                    370: $B0J>e=R$Y$?$3$H$K$h$j(B, $B<!$N$h$&$J0lHLE*$J(B change of ordering $B%"%k%4%j%:%`(B
                    371: $B$,F@$i$l$k(B.
                    372:
                    373: \begin{pro}
                    374: \begin{tabbing}
                    375: \\
                    376: candidate$(F,p,<)$\\
                    377: Input : \= $F \subset Z[X]$\\
                    378: \> $BAG?t(B p\\
                    379: \> order $<$\\
                    380: Output : $F$ $B$N(B $p$-compatible $B$J%0%l%V%J4pDl8uJd$^$?$P(B {\bf nil}\\
                    381: {\rm ($B3F(B $F$ $B$KBP$7(B, {\bf nil} $B$rJV$9(B $p$ $B$N8D?t$OM-8B8D$G$J$1$l$P$J$i$J$$(B.)}\\
                    382: \end{tabbing}
                    383: \end{pro}
                    384:
                    385: \begin{al}(compatibility check $B$K$h$k%F%9%H$N>JN,(B)
                    386: \label{bconv}
                    387: \begin{tabbing}
                    388: gr\"obner\_by\_change-of-ordering$(F,<)$\\
                    389: Input : \= $F \subset \Z[X]$, order $<$\\
                    390: Output : $Id(F)$ $B$N(B $<$ $B$K4X$9$k%0%l%V%J4pDl(B $G$\\
                    391: $G_0 \leftarrow$ $F$ $B$N(B, $B$"$k(B order $<_0$ $B$K4X$9$k%0%l%V%J4pDl(B; $G_0 \subset \Z[X]$\\
                    392: {\bf again:}\\
                    393: for\= \kill
                    394: \> $p \leftarrow (G_0,<_0)$ $B$K4X$7$F(B permissible $B$JL$;HMQ$NAG?t(B \\
                    395: \> $G \leftarrow$ candidate($G_0$,$p$,$<$)\\
                    396: \> If $G$ = {\bf nil} goto {\bf again:}\\
                    397: \> else return $G$
                    398: \end{tabbing}
                    399: \end{al}
                    400: candidate() $B$K$*$$$F$O(B,  $p$-compatible $B$J%0%l%V%J4pDl8uJd$rJV$9(B
                    401: $BG$0U$N%"%k%4%j%:%`$,;HMQ2DG=$G$"$k(B. $B$3$l$^$G=R$Y$?$b$N$G$O(B,
                    402: \begin{itemize}
                    403: \item tl\_guess()
                    404: \item $B@F<!2=(B + tl\_guess() + $BHs@F<!2=(B
                    405: \item candidate\_by\_linear\_algebra()
                    406: \end{itemize}
                    407: $B$,E,9g$9$k(B. $B$3$l$i$N$&$A(B, $BA0<T(B 2 $B$D$K$D$$$F$OL@$i$+$@$,(B, $B:G8e$N$b$N$K(B
                    408: $B$D$$$F$O8!>Z$rMW$9$k(B. $B$3$l$K$D$$$F<!@a$G=R$Y$k(B.
                    409:
                    410: \subsection{candidate\_by\_linear\_algebra()}
                    411:
                    412: \begin{lm}
                    413: \label{munique}
                    414: $B%"%k%4%j%:%`(B \ref{mfglm} $B$K$*$$$F(B $C$ $B$KB0$9$kB?9`<0$O(B $p$ $B$K$D$$$F(B stable
                    415: $B$G(B, $E_{h,p}=\{\phi_p(c)=0 \mid c \in C\}$ $B$O0l0U2r$r;}$D(B.
                    416: \end{lm}
                    417: \proof
                    418: $p$ $B$,(B $(F,<_1)$ $B$K4X$7(B permissible $B$h$j(B $NF_{<_1}(t,F)$ $B$O(B $p$ $B$K$D$$$F(B
                    419: stable. $B$h$C$F(B $c \in C$ $B$b(B $p$ $B$K$D$$$F(B stable.
                    420: $$S_{h,p}=\{a_t=\overline{c}_t \mid \overline{c}_t \in GF(p)\}$$
                    421: $B$,(B $E_{h,p}$ $B$N2r$H$9$k(B.
                    422: $$\overline{h}=\displaystyle{\sum_{t \in T(h)} \overline{c}_t t}$$
                    423: $B$H$*$/(B. $B$9$k$H(B,
                    424: $$0=\displaystyle{\sum_{t \in T(h)} \overline{c}_t \phi_p(NF_{<_1}(t,F)})=
                    425: NF_{<_1}(\overline{h},\phi_p(F)).$$
                    426: $B$h$C$F(B $\overline{h} \in Id(\phi_p(F))$ $B$h$j(B $NF_<(\overline{h},\overline{G})=0$.
                    427: $B$9$k$H(B $\overline{G}$ $B$,HoLs%0%l%V%J4pDl$G(B $T(\overline{h})\subset T(h)$
                    428: $B$h$j(B $\overline{h}=h$ $B$,@.$jN)$D(B.
                    429: $B$3$l$O(B, $B2r$,0l0UE*$G(B $h$ $B$K0lCW$9$k$3$H$r0UL#$9$k(B. \qed
                    430: \medskip
                    431:
                    432: \begin{co}
                    433: \label{sqmat}
                    434: $n$ $B$rITDj85(B $a_t$ $B$N8D?t$H$9$k$H(B, $E_h$ $B$+$i<!$N@-<A$r$b$D(B
                    435: subsystem $E'_h$ $B$rA*$V$3$H$,$G$-$k(B.
                    436: \begin{itemize}
                    437: \item $E'_h$ $B$O(B $n$ $B8D$NJ}Dx<0$+$i$J$k(B.
                    438: \item $\phi_p(E'_h)$ $B$O(B $GF(p)$ $B>e$G0l0U2r$r$b$D(B.
                    439: \end{itemize}
                    440: $B$3$l$+$i<!$N$3$H$,J,$+$k(B.
                    441: \begin{itemize}
                    442: \item $E'_h$ $B$O(B $\Q$\, $B>e0l0U2r$r;}$A(B, $B2r$O(B $p$ $B$K$D$$$F(B stable.
                    443: \item $E_h$ $B$,2r$r$b$F$P(B, $B$=$l$O(B $E'_h$ $B$N0l0U2r$K0lCW$9$k(B.
                    444: \end{itemize}
                    445: \end{co}
                    446:
                    447: \begin{th}
                    448: $B%"%k%4%j%:%`(B \ref{mfglm} $B$,B?9`<0=89g(B $G$ $B$rJV$;$P(B,
                    449: $G$ $B$O(B $F$ $B$N(B $<$ $B$K4X$7$F(B $p$-compatible $B$J%0%l%V%J4pDl8uJd$G$"$k(B.
                    450: \end{th}
                    451: \proof
                    452: $B3F(B $g \in G$ $B$KBP$7(B
                    453: $$g = \displaystyle{\sum_{t \in T(h)} c_t t}$$
                    454: $B$H=q$1(B, $\{c_t/c\}$\, ($c = c_{hc_<(g)}$) $B$,(B $E_h$ $B$N2r$H$J$k$h$&$J(B
                    455: $h \in \overline{G}$ $B$,B8:_$9$k(B. $B$9$k$H(B,
                    456: $$0 = c\displaystyle{\sum_{t \in T(h)} c_t/c NF_{<_1}(t,F)}
                    457: = NF_{<_1}(g,F).$$
                    458: $B8N$K(B $g \in Id(F)$.
                    459: $B7O(B \ref{sqmat} $B$K$h$j(B $p$ $B$O(B $(G,<)$ $B$K$D$$$F(B permissible $B$G(B
                    460: $\phi_p(g)$ = $\phi_p(c) h$ $B$h$j(B
                    461: $\phi_p(G)$ $B$O(B $\overline{G}$ $B$N%0%l%V%J4pDl$G$"$k(B.
                    462: \qed\\
                    463: \medskip
                    464: $B>e$NJdBj$rMQ$$$F(B $E_h$ $B$r<!$N<j=g$G2r$/(B.
                    465:
                    466: \begin{enumerate}
                    467: \item $E'_h$ $B$rA*$V(B.
                    468: \item $S \leftarrow$ $E'_h$ $B$N0l0U2r(B.
                    469: \item $B$b$7(B $S$ $B$,(B $E_h$ $B$rK~$?$;$P(B $S$ $B$O(B $E_h$ $B$N0l0U2r(B, $B$5$b$J$/$P(B
                    470: $E_h$ $B$O2r$r;}$?$J$$(B.
                    471: \end{enumerate}
                    472: $E_h$ $B$O(B $E'_h$ $B$r(B $GF(p)$ $B>e$G2r$/2>Dj$GF@$i$l$k(B.
                    473: $B0J2<$G$O(B, $E'_h$ $B$r2r$/J}K!$K$D$$$F=R$Y$k(B. $B$3$l$O<!$N$h$&$K(B
                    474: $BDj<02=$G$-$k(B.
                    475:
                    476: \begin{prob}
                    477: $M$, $B$ $B$r$=$l$>$l(B $n\times n$, $n\times 1$ $B@0?t9TNs$H$7(B,
                    478: $X$ $B$r(B, $BL$Dj78?t$r@.J,$H$9$k(B $n\times 1$ $B9TNs$H$9$k(B.
                    479: $\det(\phi_p(M))\neq 0$ $B$N$b$H$G(B,
                    480: $MX=B$ $B$r2r$1(B.
                    481: \end{prob}
                    482: $M$, $B$ $B$O(B, $B0lHL$K(B, $BD9Bg$J@0?t$r@.J,$K;}$DL)9TNs$H$J$k(B. $B$7$+$7(B,
                    483: $B$b$H$b$H$NF~NOB?9`<0$N78?t$,>.$5$$>l9g(B, $B%0%l%V%J4pDl$N85$N78?t(B
                    484: $B$9$J$o$A(B $MX=B$ $B$N2r(B $X$ $B$OHf3SE*>.$5$$>l9g$,B?$$(B. $B$3$N$h$&$J>l9g$K(B
                    485: $B$3$NLdBj$r(B Gauss $B>C5n$G2r$/$3$H$O$3$N@a$N:G=i$NNc$G=R$Y$?$h$&$K(B
                    486: $BHs8zN(E*$G$"$k(B. $B$3$N$h$&$J>l9g$KM-8z$J$N$,(B, Hensel $B9=@.(B, $BCf9q>jM>DjM}(B
                    487: $B$J$I$K$h$k(B modular $B7W;;$G$"$k(B. $B$3$3$G$O(B Hensel $B9=@.$K$h$kJ}K!$r(B
                    488: $B>R2p$9$k(B.
                    489:
                    490: \begin{al}
                    491: \label{lineq}
                    492: \begin{tabbing}
                    493: \\
                    494: solve\_linear\_equation\_by\_hensel$(M,B,p)$\\
                    495: ($C_2$)\= \kill
                    496: Input : \= $n\times n$ $B9TNs(B $M$, $n\times 1$ $B9TNs(B $B$\\
                    497: \> $\phi_p(\det(M))\neq 0$ $B$J$kAG?t(B\\
                    498: Output : $MX=B$ $B$J$k(B $n\times 1$ matrix $X$\\
                    499: $R \leftarrow \phi_p(M)^{-1}$\\
                    500: $c \leftarrow B$\\
                    501: $x \leftarrow 0$\\
                    502: $q \leftarrow 1$\\
                    503: $count \leftarrow 0$\\
                    504: do \{\\
                    505: (1)\> $t \leftarrow \phi_p^{-1}(R \phi_p(c))$\\
                    506: (2)\> $x \leftarrow x + qt$\\
                    507: \> $c \leftarrow (c-Mt)/p$\\
                    508: \> $q \leftarrow qp$\\
                    509: \> $count \leftarrow count+1$\\
                    510: \> {\rm ($\phi_p^{-1}$ $B$O(B $[-p/2,p/2]$ $B$K@55,2=$5$l$?5UA|(B, $(c-Mt)/p$ $B$O@0=|(B.)}\\
                    511: \> if \= $count = {\bf Predetermined\_Constant}$ then \{\\
                    512: \> \> $count \leftarrow 0$\\
                    513: \> \> $X \leftarrow$ inttorat$(x,q)$\\
                    514: \> \> if \= $X \neq$ {\bf nil} $B$+$D(B $MX=B$ then return $X$\\
                    515: \> \}\\
                    516: \}
                    517: \end{tabbing}
                    518: \end{al}
                    519:
                    520: \begin{al}
                    521: \label{intrat}
                    522: \begin{tabbing}
                    523: \\
                    524: inttoat$(x,q)$\\
                    525: ($C_2$)\= \kill
                    526: Input : $q>x$, $\GCD(x,q)=1$ $B$J$k@5@0?t(B $x$, $q$ \\
                    527: Output : $bx \equiv a \bmod q$ $B$+$D(B
                    528: $|a|,|b| \le \sqrt{q\over 2}$ $B$J$kM-M}?t(B $a/b$ $B$^$?$O(B {\bf nil}\\
                    529: if $x \le \sqrt{q\over 2}$ then return $x$\\
                    530: $f_1 \leftarrow q$, $f_2 = x$,
                    531: $a_1 \leftarrow 0$, $a_2 \leftarrow 1$\\
                    532: $i \leftarrow 1$\\
                    533: do \= \{\\
                    534: \> if \= $|f_i| \le \sqrt{q\over 2}$ then \\
                    535: \>\> if $|a_i| \le \sqrt{q\over 2}$ then return $f_i/a_i$\\
                    536: \>\> else return {\bf nil}\\
                    537: \> $f_i - q_i f_{i+1} < f_{i+1}$ $B$J$k(B $q_i$ $B$r5a$a$k(B\\
                    538: \> $f_{i+2} \leftarrow f_i - q_i f_{i+1}$\\
                    539: \> $a_{i+2} \leftarrow a_i - q_i a_{i+1}$\\
                    540: \> $i \leftarrow i+1$\\
                    541: \}
                    542: \end{tabbing}
                    543:
                    544: \end{al}
                    545: \begin{lm}(\cite{WANG2}\cite{DIXON})
                    546: \label{intratlm}
                    547: $x$, $q$ $B$r(B $q>x>1$, $\GCD(x,q)=1$ $B$J$k@5@0?t$H$9$k(B. $B$3$N$H$-(B,
                    548: $|a|,|b|\le\sqrt{q/2}$, $\GCD(a,b)=1$ $B$J$k@0?t(B $a,b$ $(b>0)$ $B$*$h$S@0?t(B
                    549: $c$ $B$,$"$C$F(B $ax+cq=b$ $B$H$J$k$J$i$P(B, $(a,b,c)$ $B$O(B $(q,x)$ $B$KBP$9$k3HD%(B
                    550: Euclid $B8_=|K!$K$h$jF@$i$l$k(B.
                    551: \end{lm}
                    552: [$BN,>Z(B]
                    553: $x>1$ $B$h$j(B $a \neq b$ $B$H$7$F$h$$(B.
                    554: $ax+cq=b$ $B$h$j(B ${x \over q}+{c \over a} = {b \over aq}.$ $B$3$3$G(B
                    555: $$|{b \over {aq}}| = |{{ab} \over {a^2q}}| < {1 \over {2a^2}}$$
                    556: $B$h$j(B
                    557: $$|{x \over q}+{c \over a}| < {1 \over {2a^2}}.$$$B$h$C$F(B, $-c/a$ $B$O(B
                    558: $x/q$ $B$N<g6a;wJ,?t$N0l$D(B. ($BO"J,?t$N@-<A$K$h$k(B. \cite[Section 24]{TAKAGI} $B;2>H(B.)
                    559: $\GCD(a,b)=1$ $B$h$j(B $\GCD(a,c)=1$ $B$G$"$j(B, $b>0$ $B$KBP$7$F(B $(a,c)$ $B$O0l0U(B
                    560: $BE*$K7h$^$k(B. $B$h$C$F(B, $(a,b,c)$ $B$O(B $(q,x)$ $B$KBP$9$k3HD%(B Euclid $B8_=|K!$N(B
                    561: $B78?t$H$7$FF@$i$l$k(B. \qed
                    562:
                    563: \begin{lm}
                    564: \label{lineqlm}
                    565: (1) $B$K$*$$$F(B $Mx \equiv B \bmod q$ $B$+$D(B $c = (B-Mx)/q$.
                    566: \end{lm}
                    567: \proof
                    568: $B5"G<K!$K$h$j<($9(B. $count = 0$ $B$N$H$-L@$i$+(B. $count = m$
                    569: $B$^$G8@$($?$H$9$k(B. $B$3$N$H$-(B, (2) $B$G(B
                    570: $Mt \equiv c \bmod p$ $B$h$j(B $M(x+qt) \equiv Mx+q(B-Mx)/q \bmod qp$.
                    571: $B$9$J$o$A(B $M(x+qt) \equiv B \bmod qp$. $B$^$?(B,
                    572: $(c-Mt)/p = ((B-Mx)-qMt)/qp = (B-M(x+qt))/qp$ $B$h$j(B $count = m+1$
                    573: $B$G$b8@$($k(B. \qed
                    574:
                    575: \begin{pr}
                    576: $B%"%k%4%j%:%`(B \ref{lineq} $B$O(B $MX=B$ $B$N2r(B $X$ $B$r=PNO$9$k(B.
                    577: \end{pr}
                    578: \proof
                    579: $BJdBj(B \ref{lineqlm} $B$h$j(B, $count=m$ $B$N$H$-(B (1) $B$K$*$$$F(B $Mx \equiv B
                    580: \bmod p^m$. $\det(M) \neq 0$ $B$h$j(B $x$ $B$OK!(B $p^m$ $B$G0l0UE*$G$"$k(B. $B0lJ}(B
                    581: $MX=B$ $B$OM-M}?tBN>e0l0UE*$K2r$r;}$D(B. $B$3$N2r$r(B$X=N/D$ ($D$ $B$O@0?t(B, $N$
                    582: $B$O@0?t%Y%/%H%k(B) $B$H=q$/$H$-(B, $Dr \equiv 1 \bmod p^m$$B$J$k@0?t(B $r$ $B$r$H$l(B
                    583: $B$P(B $M(rN) \equiv B \bmod p^n$. $B$h$C$F(B $rN \equiv x \bmod p^m$.
                    584: $D$, $N$ $B$N3F@.J,$,(B
                    585: $A$ $B$r1[$($J$$$H$-(B, $p^m > 2A^2$ $B$J$k(B $m$ $B$r$H$l$P(B, $BJdBj(B \ref{intratlm}
                    586: $B$h$j(B $rN$ $B$9$J$o$A(B $x$ $B$+$i%"%k%4%j%:%`(B \ref{intrat} $B$K$h$j(B
                    587: $N/D$ $B$,I|85$5$l$k(B.
                    588: \qed\\
                    589: {\bf Predetermined\_Constant} $B$O$"$k@5@0?t$G(B, $BM-M}?t$K0z$-La$7$F%A%'%C(B
                    590: $B%/$r9T$&IQEY$r@)8f$9$k(B. $B%"%k%4%j%:%`(B \ref{lineq} $B$K$*$$$F(B $c$ $B$O(B
                    591: $nMAX(\|M\|_\infty,\|B\|_\infty)$$B$G2!$($i$l$k$3$H$,$o$+$k(B. $B$3$l$O(B, $B3F(B
                    592: $B%9%F%C%W$,$"$k(B constant $B;~4VFb$G7W;;$G$-$k$3$H$r<($9(B. $B$^$?(B, $B2r$NJ,JlJ,(B
                    593: $B;R$,(B $A$ $B$G2!$($i$l$F$$$l$P(B, $q > 2A^2$ $B$K$J$C$?CJ3,$G2r$rI|85$G$-(B
                    594: $B$k(B. $B$3$l$O(B, $B2r$NBg$-$5$K1~$8$?<j4V$G7W;;$G$-$k$3$H$r0UL#$9$k(B. $B$3$l$KBP(B
                    595: $B$7$F(B, $B78?tKDD%$r2!$($?(B Gauss $B>C5nK!$G$"$k(B fraction-free $BK!$K$h$C$F$b(B,
                    596: $B2r$NBg$-$5$K$+$+$o$i$:(B, $B78?t9TNs$N9TNs<0$r7W;;$7$F$7$^$&$H$$$&E@$G(B,
                    597: Gauss $B>C5nK!$O(B, $B$3$NLdBj$r2r$/$?$a$K$OITE,@Z$G$"$k(B.
                    598:
                    599: \section{$B%?%$%_%s%0%G!<%?(B}
                    600:
                    601: $BK\@a$G$O(B, $B0J2<$N$h$&$J%Y%s%A%^!<%/LdBj$K4X$7(B, $BK\>O$G=R$Y$?MM!9$J(B
                    602: change of ordering $B%"%k%4%j%:%`$N8zN($NHf3S$r<($9(B. $B$^$?(B, $BL?Bj(B
                    603: \ref{RUR} $B$G?($l$?(B RUR $B$N%b%8%e%i7W;;$rF1$8LdBj$KE,MQ$7(B,
                    604: $B$=$NM%0L@-$r<B837k2L$K$h$j<($9(B.  $B7WB,$O(B, PC (FreeBSD, 300MHz Pentium
                    605: II, 512MB of memory) $B$G9T$C$?(B. $BC10L$OIC(B. garbage collection $B;~4V$O=|$$(B
                    606: $B$F$"$k(B.
                    607: \begin{tabbing}
                    608: $MMM\;\;$ \= \kill
                    609: $C(n)$ \> The cyclic n-roots system of n variables. (Faugere {\it et al.},1993).\\
                    610:        \> $\{f_1,\cdots,f_n\}$ where
                    611:          $f_k=
                    612: \displaystyle{\sum_{i=1}^n\prod_{j=i}^{k+j-1}c_{j \bmod n}-\delta_{k,n}}$.
                    613: ($\delta$ is the Kronecker symbol.) \\
                    614: \> The variables and ordering : $c_n \succ c_{n-1} \succ \cdots \succ c_1$\\
                    615: $K(n)$ \> The Katsura system of n+1 variables. \\
                    616:       \> $\{u_l - \sum_{i=-n}^n u_i u_{l-i} (l = 0,\cdots, n-1),
                    617:            \sum_{l=-n}^n u_l - 1\}$\\
                    618:       \> The variables and ordering : $u_0 \succ u_1 \succ \cdots \succ u_n$.\\
                    619:       \> Conditions : $u_{-l} = u_l$ and $u_l = 0 (|l| > n)$. \\
                    620: $R(n)$ \> {\tt e7} in Rouillier (1996). \\
                    621:       \> $\{-1/2+\sum_{i=1}^n(-1)^{i+1}x_i^k (k=2, \cdots, n+1) \}$\\
                    622:       \> The variables and ordering : $x_n \succ x_{n-1} \succ \cdots \succ x_1$.\\
                    623: $D(3)$ \> {\tt e8} in Rouillier (1996). \\
                    624:        \> $\{f_0,f_1,f_2,\cdots,f_7\}$\\
                    625:        \> {\scriptsize $f_0=-420y^2-280zy-168uy-140vy-120sy-210ty-105ay+12600y-13440$}\\
                    626:        \> {\scriptsize $f_1=-840zy-630z^2-420uz-360vz-315sz-504tz-280az+18900z-20160$}\\
                    627:        \> {\scriptsize $f_2=-630ty-504tz-360tu-315tv-280ts-420t^2-252at+12600t-13440$}\\
                    628:        \> {\scriptsize $f_3=-5544uy-4620uz-3465u^2-3080vu-2772su-3960tu-2520au+103950u-110880$}\\
                    629:        \> {\scriptsize $f_4=-4620vy-3960vz-3080vu-2772v^2-2520sv-3465tv-2310av+83160v-88704$}\\
                    630:        \> {\scriptsize $f_5=-51480sy-45045sz-36036su-32760sv-30030s^2-40040ts-27720as+900900s-960960$}\\
                    631:        \> {\scriptsize $f_6=-45045ay-40040az-32760au-30030av-27720as-36036at-25740a^2+772200a-823680$}\\
                    632:        \> {\scriptsize $f_7=-40040by-36036bz-30030bu-27720bv-25740bs-32760bt-24024ba+675675b-720720$}\\
                    633:        \normalsize
                    634:          \> The variables and ordering : $b \succ a \succ s \succ v \succ u \succ t \succ z \succ y$.\\
                    635: $Rose$ \> The Rose system.\\
                    636: %      \> $\{u_4^4-20/7a_{46}^2, a_{46}^2u_3^4+7/10a_{46}u_3^4+7/48u_3^4-50/27a_{46}^2-35/27a_{46}-49/216,$\\
                    637: %      \> $a_{46}^5u_4^3+7/5a_{46}^4u_4^3+609/1000a_{46}^3u_4^3+49/1250a_{46}^2u_4^3$\\
                    638: %      \> $-27391/800000a_{46}u_4^3-1029/160000u_4^3+3/7a_{46}^5u_3u_4^2+3/5a_{46}^6u_3u_4^2$\\
                    639: %      \> $+63/200a_{46}^3u_3u_4^2+147/2000a_{46}^2u_3u_4^2+4137/800000a_{46}u_3u_4^2$\\
                    640: %      \> $-7/20a_{46}^4u_3^2u_4-77/125a_{46}^3u_3^2u_4-23863/60000a_{46}^2u_3^2u_4$\\
                    641: %      \> $-1078/9375a_{46}u_3^2u_4-24353/1920000u_3^2u_4-3/20a_{46}^4u_3^3-21/100a_{46}^3u_3^3$\\
                    642: %      \> $-91/800a_{46}^2u_3^3-5887/200000a_{46}u_3^3-343/128000u_3^3 \}$\\
                    643:     \> $O_1$ : $u_3 \succ u_4 \succ a_{46}$, $O_2$ : $u_3 \succ a_{46} \succ u_4$.\\
                    644: $Liu$ \> The Liu system.\\
                    645:       \> $\{y(z-t)-x+a, z(t-x)-y+a, t(x-y)-z+a, x(y-z)-t+a\}$\\
                    646:       \> The variables and ordering : $x \succ y \succ z \succ t \succ a$.\\
                    647: $Fate$ \> The Fateman system, appeared on NetNews. \\
                    648:        \> $\{s^3+2r^3+2q^3+2p^3$, $s^5+2r^5+2q^5+2p^5$,\\
                    649:        \> $-s^5+(r+q+p)s^4+(r^2+(2q+2p)r+q^2+2pq+p^2)s^3+(r^3+q^3+p^3)s^2$\\
                    650:        \> $+(3r^4+(2q+2p)r^3+(4q^3+4p^3)r+3q^4+2pq^3+4p^3q+3p^4)s+(4q+4p)r^4$\\
                    651:        \> $+(2q^2+4pq+2p^2)r^3+(4q^3+4p^3)r^2+(6q^4+4pq^3+8p^3q+6p^4)r$\\
                    652:           \> $+4pq^4+2p^2q^3+4p^3q^2+6p^4q\}$\\
                    653:        \> The variables and ordering : $p \succ q \succ r \succ s$.\\
                    654: $hC(6)$ \> A homogenization of C(6). \\
                    655:        \> $(C_6\backslash \{c_1c_2c_3c_4c_5c_6-1\})\cup \{c_1c_2c_3c_4c_5c_6-t^6\}$\\
                    656:        \> The variables and ordering :
                    657:           $c_1 \succ c_2 \succ c_3 \succ c_4 \succ c_5 \succ c_6 \succ t$.\\
                    658: \end{tabbing}
                    659:
                    660: \subsection{Change of ordering}
                    661:
1.2       noro      662: $BM=$a7W;;$7$F$"$k(B DRL ($BA4<!?t5U<-=q<0=g=x(B)$B%0%l%V%J4pDl$+$i=PH/$7$F(B, LEX
                    663: ($B<-=q<0=g=x(B)$B%0%l%V%J4pDl$r7W;;$9$k(B. $BMQ$$$k%"%k%4%j%:%`$O(B, TL
                    664: (tl\_guess$()$; $B%"%k%4%j%:%`(B \ref{tlguess}), HTL ($B@F<!2=(B
                    665: +tl\_guess$()$+$BHs@F2=(B), LA (candidate\_by\_linear\_algebra$()$;$B%"%k%4(B
                    666: $B%j%:%`(B \ref{mfglm} (0 $B<!85%7%9%F%`$N$_(B))$B$G$"$k(B. $BI=(B
1.1       noro      667: \ref{mcotab} $B$O(B DRL $B$+$i(B LEX $B$X$NJQ49$K$+$+$k;~4V$r$7$a$9(B. {\it DRL}
                    668: $B$O(B, DRL $B$N7W;;;~4V$r<($9(B. $B%0%l%V%J4pDl%A%'%C%/$r>J$/8z2L$r<($9$?$a$K(B,
                    669: tl\_ckeck$()$ ($B%"%k%4%j%:%`(B \ref{tlcheck}) $B$N;~4V$b<($9(B.
                    670:
                    671: \begin{table}[hbtp]
                    672: \caption{Modular change of ordering}
                    673: \label{mcotab}
                    674: \begin{center}
                    675: \begin{tabular}{|c||c|c|c|c|c|c|c|} \hline
                    676:        & $K(5)$ & $K(6)$ & $K(7)$ & $C(6)$ & $C(7)$ & $R(5)$ & $R(6)$ \\ \hline
                    677: {\it DRL}&0.84 &8.4    &74     &3.1    &1616   &11     &1775   \\ \hline
                    678: {\it TL}&$\infty$              &$\infty$               &$\infty$ &$\infty$     &$\infty$       &$\infty$       &$\infty$       \\ \hline
                    679: {\it HTL}      &16     &1402   &$1.6\times 10^5$       &5.6    &$2\times 10^4$ &383    &$2.1\times 10^5$       \\ \hline
                    680: {\it LA}       &4.7    &158    &6813   &4      &435    &9.5            &258            \\ \hline
                    681: tl\_check      &2.3    &177    &$1.3\times 10^4$       &1.1    &2172   &3      &40             \\ \hline
                    682: \end{tabular}
                    683:
                    684: \begin{tabular}{|c||c|c|c||c|c|c|} \hline
                    685:        & $D(3)$ & $RoseO_1$ & $RoseO_2$ & $Liu$ & $Fate$ & $hC(6)$ \\ \hline
                    686: {\it DRL}      &30     &0.19   &0.15   &0.06   &0.5    &7.2    \\ \hline
                    687: {\it TL}       & $\infty$      &1.7    &354    &$\infty$       &4      &25     \\ \hline
                    688: {\it HTL}      &$4.1\times 10^4$       &1.7    &36     &18     &4      &25     \\ \hline
                    689: {\it LA}       &585    &3.3    &12     & --- & --- & --- \\ \hline
                    690: tl\_check      &575            &0.6    &13     &17             &26     &24     \\ \hline
                    691: \end{tabular}
                    692: \end{center}
                    693: \end{table}
                    694: $B@0?t78?tB?9`<0$KBP$7(B, $B$=$N(B {\bf maginitude} $B$r(B, $B78?t$N%S%C%HD9$NOB$GDj5A$9$k(B.
                    695: {\it TL} $B$H(B {\it HTL} $B$N:9$r8+$k$?$a$K(B,
                    696: $BI=(B \ref{magnitude} $B$G(B, $B7W;;ESCf$K$*$1$k:GBg(B magnitude $B$r<($9(B.
                    697:
                    698: \begin{table}[hbtp]
                    699: \caption{Maximal magnitude}
                    700: \label{magnitude}
                    701: \begin{center}
                    702: \begin{tabular}{|c||c|c|c|c|c|c|} \hline
                    703:        & $C(6)$ & $K(5)$ & $K(6)$ & $RoseO_1$ & $RoseO_2$ & Liu \\ \hline
                    704: {\it TL}& $>$ 735380 & $> 2407737 $ & $>$ 57368231 & 69764 & 947321 & $>$ 327330 \\ \hline
                    705: {\it HTL}& 1992 & 44187 & 422732 & 37220 & 70018 & 21095 \\ \hline
                    706: \end{tabular}
                    707: \end{center}
                    708: \end{table}
                    709: $BI=$h$jL@$i$+$K(B, {\it TL} $B$OHs@F<!B?9`<0$KBP$9$k%0%l%V%J4pDl7W;;$KIT8~$-(B
                    710: $B$G$"$k$3$H$,$o$+$k(B. $B$5$i$K(B, $BI=(B \ref{mcotab} $B$O(B {\it HTL} $B$KBP$9$k(B
                    711: {\it LA} $B$NM%0L@-$r<($7$F$$$k(B. $B$3$l$O(B, Buchberger $B%"%k%4%j%:%`$,(B
                    712: Euclid $B$N8_=|K!$KBP1~$7$F$$$F(B, $BCf4V78?tKDD%$G8zN($,:81&$5$l$k$N(B
                    713: $B$KBP$7(B, modular $B%"%k%4%j%:%`$N8zN($O7k2L$NBg$-$5$N$_$K0MB8$9$k(B
                    714: $B$3$H$K$h$k(B.
                    715:
                    716: \subsection{RUR}
                    717:
                    718: RUR $B$N(B modular $B7W;;$N%?%$%_%s%0%G!<%?$r<($9(B. $B7W;;4D6-$OA0@a$HF1MM$G$"(B
                    719: $B$k(B. $B$3$3$G$O(B, $BM=$a(B modular $B7W;;$K$h$j(B separating element$B$r5a$a$F$"(B
                    720: $B$k(B. $B$3$l$i$rMQ$$$F(B, $B$=$l$>$l<!$N$h$&$JB?9`<0$rE:2C$7$?%$%G%"%k$KBP$7(B,
                    721: $w$ $B$K4X$9$k(B RUR $B7W;;$r9T$&(B.
                    722:
                    723: \begin{tabbing}
                    724: $MMM\;\;$ \= \kill
                    725: $C(6)$ \> $w-(c_1+3c_2+9c_3+27c_4+81c_5+243c_6)$\\
                    726: $C(7)$ \> $w-(c_1+3c_2+9c_3+27c_4+81c_5+243c_6+729c_7)$\\
                    727: $K(n)$ \> $w-u_n$\\
                    728: $R(5)$ \> $w-(x_1-3x_2-2x_3+3x_4+2x_5)$\\
                    729: $R(6)$ \> $w-(x_1-3x_2-2x_3+3x_4+2x_5-4x_6)$\\
                    730: $D(3)$ \> $w-y$
                    731: \end{tabbing}
                    732: %\begin{table}[hbtp]
                    733: %\label{mrurdata}
                    734: %\caption{$BF~NO%$%G%"%k$K4X$9$k%G!<%?(B}
                    735: %\begin{center}
                    736: %\begin{tabular}{|c||c|c|c|c||c|c|c|c|c|} \hline
                    737: %      & $K(5)$        & $K(6)$        & $K(7)$        & $K(8)$        & $C(6)$& $C(7)$        & $R(5)$ & $R(6)$ & $D(3)$ \\ \hline
                    738: %$\dim_{\Q} R/I$       & 32    & 64    & 128   & 256   & 156   & 924   &144    &576    & 128 \\ \hline
                    739: %DRL GB& 0.8   & 7.2   & 68    & 798   & 3.1   & 1616  & 11    & 1775  & 30    \\ \hline
                    740: %\end{tabular}
                    741: %\end{center}
                    742: %\end{table}
                    743: \begin{table}[hbtp]
                    744: \label{mrurtab}
                    745: \caption{$B7W;;;~4V(B ($BIC(B)}
                    746: \begin{center}
                    747: \begin{tabular}{|c|c|c|c||c|c|c|c|c|} \hline
                    748:        & $K(6)$& $K(7)$& $K(8)$& $C(6)$& $C(7)$& $R(5)$ & $R(6)$ & $D(3)$ \\ \hline
                    749: Total  & 7.4   & 69    & 1209  & 4.6   & 1643  & 52    & 8768  & 67    \\ \hline
                    750: Quick test& 0.4        & 3.2   & 26    & 0.5   & 57    & 6.5   & 384   & 3.1   \\ \hline
                    751: Normal form& 1.1       & 12    & 308   & 1.4   & 762   & 15    & 2861  & 7.3   \\ \hline
                    752: Linear equation& 4.1   & 43    & 775   & 1.4   & 641   & 22    & 3841  & 45    \\ \hline
                    753: Garbage collection& 1.7        & 10    & 100   & 1.2   & 181   & 7.8   & 1681  & 11    \\ \hline
                    754: \end{tabular}
                    755: \end{center}
                    756: \end{table}
                    757:
                    758: %\begin{table}[hbtp]
                    759: %\label{maxblen}
                    760: %\caption{Maximal bit length of coefficients in LEX basis and the RUR}
                    761: %\begin{center}
                    762: %\begin{tabular}{|c||c|c|c|c|c|} \hline
                    763: %& $K(5)$ & $K(6)$     & $K(7)$        & $K(8)$ & $D(3)$ \\ \hline
                    764: %LEX & 1421 & 6704 & 36181 & --- & 6589 \\ \hline
                    765: %RUR & 120 & 249 & 592 & 1258 & 821 \\ \hline
                    766: %\end{tabular}
                    767: %\end{center}
                    768: %\end{table}
                    769: $BI=$G(B, Quick Test $B$O(B modular $B7W;;$G(B $w$$B$,(B separating element $B$H$J$k$3$H(B
                    770: $B$r%A%'%C%/$9$k;~4V(B, Normal Form $B$O(B, $B@~7AJ}Dx<0$r@8@.$9$k$?$a$N(B,
                    771: monomial $B$N@55,7A$N7W;;(B, Linear Equation $B$O(B, $B@~7AJ}Dx<05a2r$N;~4V$G$"(B
                    772: $B$k(B. $BI=(B \ref{mcotab} $B$HHf3S$9$l$P(B, $B$"$kJQ?t$,(B separating element $B$H$J$C(B
                    773: $B$F$$$k$h$&$JLdBj$G$O(B, RUR $B7W;;$N8zN($,Hs>o$K$h$$$3$H$,J,$+$k(B. $B$3$NM}M3(B
                    774: $B$O(B, RUR $B$K8=$l$k78?t$N(B bit $BD9$,(B, LEX $B4pDl$N$=$l$KHf$Y$FHs>o$K>.$5$$(B($BNc(B
                    775: $B$($P(B $K(7)$ $B$G$O(B 60 $BJ,$N(B 1 $BDxEY(B) $B$3$H$H(B, $B@~7AJ}Dx<0$N5a2r$,(B, $B7k2L$NBg(B
                    776: $B$-$5$K1~$8$?<j4V$G$G$-$k$3$H$+$iJ,$+$k(B.
                    777: