\chapter{Change of ordering}

$BA0@a$G$O(B, $B<g$H$7$F(B Buchberger $B%"%k%4%j%:%`$N8zN(2=$K$D$$$F=R$Y$?(B. 
$B$7$+$7(B, $B%0%l%V%J4pDl$N7W;;K!$O(B Buchberger $B%"%k%4%j%:%`$@$1$H$O8B$i$J$$(B. 
$BK\@a$G$O(B, $B4{$K2?$i$+$N(B order $B$K4X$7$F%0%l%V%J4pDl$K$J$C$F$$$kB?9`<0(B
$B=89g$rF~NO$H$7$F(B, $BB>$N(B order $B$N%0%l%V%J4pDl$r5a$a$kJ}K!$K$D$$$F=R$Y$k(B. 

\section{FGLM $B%"%k%4%j%:%`(B}

$I \subset K[X]$ $B$r(B 0 $B<!85%$%G%"%k$H$7(B, $I$ $B$N$"$k(B order $<_1$ $B$K4X$9$k(B
$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 $<$
$B$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
$B%"%k%4%j%:%`$G$"$k(B. 

\begin{lm}
$<$ $B$r(B admissible order, $F = GB_<(I)$ $B$H$9$k(B. 
$T = \{t_1,\cdots,t_l\} \subset T(X)$ $B$r9`$N=89g$H$9$k(B.
$a_i$ $B$rL$Dj78?t$H$7(B, 
$$E = \displaystyle{\sum_{i=1}^l a_i NF_<(t_i,F)}$$
$B$H$*$/(B. $E$ $B$N(B $X$ $B$K4X$9$k78?t$N=89g$r(B $C$ $B$H$9$l$P(B, 
$Eq = \{ f = 0 \mid f \in C\}$ $B$O(B $a_i$ $B$K4X$9$k@~7AJ}Dx<0$H$J$k(B. $B$3$N;~(B
\begin{center}
$Eq$ $B$,<+L@$G$J$$2r$r;}$D(B $\Leftrightarrow$ $T$ $B$,(B $K[X]/I$ $B$K$*$$$F(B
$K$-$B@~7A=>B0(B
\end{center}
\end{lm}

\begin{al} (FGLM $B%"%k%4%j%:%`(B\cite{FGLM})
\label{fglm}
\begin{tabbing}
FGLM$(F,<_1,<)$\\
Input : \= order $<_1$, $<$; $F \subset K[X]$ \st $F = GB_{<_1}(I)$ $B$+$D(B 
$\dim(I)=0$\\
Output : $F$ $B$N(B $<$ $B$K4X$9$k%0%l%V%J4pDl(B\\
($C_2$)\= \kill
$G \leftarrow \emptyset$\\
$h \leftarrow 1$\\
$B \leftarrow \{h\}$\\
$H \leftarrow \emptyset$\\
do \{\\
\> $N \leftarrow \{u \mid u > h$ $B$+$D$9$Y$F$N(B $m \in H$ $B$KBP$7(B $m {\not|} u\}$\\
(0)\> if $N = \emptyset$ then return $G$\\
(1)\> $h_1 \leftarrow \min(N)$\\
\> $a_t$ : $t \in B$ $B$KBP1~$9$kL$Dj78?t(B\\
\> $a_{h_1} \leftarrow 1$\\
(2)\> $E \leftarrow \displaystyle{NF_{<_1}(h_1,F)+\sum_{t \in B} a_t NF_{<_1}(t,F)}$\\
\> $C \leftarrow$ $E$ $B$N(B $X$ $B$K4X$9$k78?t$N=89g(B\\
\> if \= $B@~7AJ}Dx<0(B $\{ f = 0 \mid f \in C\}$ $B$,2r(B
    $\{a_t = c_t \mid c_t \in K\}$ $B$r;}$D(B\\
\> then \\
\> \>$G \leftarrow G \cup \{\displaystyle{h_1+\sum_{t \in B} c_t t}\}$\\
\> \>$H \leftarrow H \cup \{h_1\}$\\
\> else $B \leftarrow \{h_1\} \cup B$\\
\> $h \leftarrow h_1$\\
\}\\
\end{tabbing}
\end{al}

\begin{lm}
\label{fglml1}
(0) $B$K$*$$$F(B, $B= \{u \mid u \le h$ $B$+$D$9$Y$F$N(B $m \in H$ $B$KBP$7(B $m {\not|} u\}$.
\end{lm}
\proof $B%k!<%W$K4X$9$k5"G<K!$G<($9(B. $B:G=i$K%k!<%W$KF~$C$?;~E@$G$O(B
$B@.N)$7$F$$$k(B. $B$"$k;~E@$G@.N)$7$F$$$k$H$9$k(B. $B$=$N;~E@$+$i%k!<%W$,0l2s(B
$B2s$C$?;~E@$N(B (0) $B$K$*$$$F@.N)$9$k$3$H$r<($9(B. 
$B<($=$&$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. 
$h = \min(N_0)$ $B$G(B, $B = B_0 \cup \{h\}$ $B$^$?$O(B $H = H_0 \cup \{h\}$ $B$G$"$k(B. \\
\underline{$B = B_0 \cup \{h\}$ $B$N;~(B}\, $H = H_0$ $B$h$j(B,
$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
$BBP$7(B $m{\not|} u\}= B_0 \cup \{h\}$.\\
\underline{$H = H_0 \cup \{h\}$ $B$N;~(B}\,
$B1&JU(B $= B_0 \cup 
(\{u \mid h_0 < u \le h$ $B$+$D$9$Y$F$N(B $m \in H_0$ $B$KBP$7(B $m {\not|} u\}
\cap \{u \mid h {\not|} u \}) = B_0 \cup (\{h\} \cap \{u \mid h {\not|} u \}) = B_0$.\\
$B$h$C$F(B, $B$$$:$l$N>l9g$K$b1&JU(B = $B$ $B$H$J$k(B. \qed

\begin{lm}
\label{fglml2}
(1) $B$K$*$$$F(B, $h_1 \in x_1B \cup \cdots \cup x_nB$.
\end{lm}
\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. 
$B$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

\begin{pr}
$B%"%k%4%j%:%`(B \ref{fglm} $B$O(B $GB_<(F)$ $B$r=PNO$9$k(B. 
\end{pr}
\proof $BJdBj(B \ref{fglml2}$B$h$j(B, (1) $B$K$*$1$k(B $h$ $B$N8uJd$O(B $BM-8B=89g(B 
$x_1B \cup \cdots \cup x_nB$ $B$N85$@$+$i(B $\min(N)$ $B$rM?$($kA*Br%"%k%4%j(B
$B%:%`$,B8:_$9$k(B. \\
\underline{$BDd;_@-(B} $B$^$:(B, (2) $B$,2r$r;}$?$J$$$3$H$H(B, $\{h_1\} \cup B$ $B$,(B
$K[X]/I$ $B$G(B $K$ $B>e0l<!FHN)$G$"$k$3$H$,F1CM$G$"$k$3$H$KCm0U$9$k(B. $B$h$C$F(B,
$|B|$ $B$O(B$\dim_K K[X]/I$ $B$r1[$($i$l$J$$(B. $B$^$?(B, $H$ $B$N85$O(B, $B$I$N85$bB>$N(B
$B85$r3d$i$J$$$+$i(B, $B7O(B \ref{noether} $B$h$j$d$O$jM-8B=89g(B. $B$h$C$F%"%k%4%j(B
$B%:%`$ODd;_$9$k(B.\\
\underline{$G = GB_<(F)$} $f \in I$ $B$H$7(B, $t=HT_<(F)$ $B$H$9$k(B. 
$f$ $B$O(B $G$ $B$K4X$7$FHoLs$H$7$F$h$$(B. $H = \{h_1,\cdots,h_m\}$ \quad
($h_1<\cdots<h_m$) $B$H$9$k(B. $B$^$?(B, $(1)$ $B$K$h$jA*Br$5$l$k85$r=g$KJB$Y$?(B
$B$b$N$r(B $t_1<t_2<\cdots$$B$H$9$k(B. $B$b$7(B, $B$9$Y$F$N(B $h \in H$ $B$KBP$7(B $h
{\not|} t$ $B$J$i$P(B, $BDd;_$N>r7o$h$j$"$k(B $k$$B$,$"$C$F(B, $t_k<t\le t_{k+1}$. 
$B2>Dj$K$h$j(B, $k+1$ $BHVL\$KA*$P$l$k85$O(B$t$ $B$G$J$1$l$P$J$i$J$$$+$i(B 
$t=t_{k+1}$. $t' \in T(f) \setminus \{t\}$$B$H$9$l$P(B, $t'<t$ $B$+$DA4$F$N(B 
$h \in H$ $B$KBP$7(B $h {\not|} t'$. $B$3$l$h$j(B$t' \le t_k$. $B$h$C$FJdBj(B 
\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
$B$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

FGLM $B%"%k%4%j%:%`$r7W;;5!>e$G<BAu$9$k>l9g(B, $BFC$K(B (2)$B$NItJ,$N<BAu$K(B
$B9)IW$,I,MW$H$J$k(B. $BMWE@$r$^$H$a$k$H(B, 

\begin{enumerate}
\item $B@55,7A$N7W;;$O(B, $B3F9`$K$D$-$?$@0lEY$@$19T$$(B, $B7k2L$OI=$K$7$FJ];}$9$k(B. 
\item $BKh2sFHN)$J@~7AJ}Dx<0$H$7$F2r$/$N$G$O$J$/(B, $B7k2L$,8e$G;H$($k$h$&$J(B
$B9)IW$r$9$k(B. 
\end{enumerate}

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
$B$2$k$3$H$,$G$-$k(B.

\begin{df}
$B3F(B $i (1\le i\le n)$ $B$KBP$7(B, $\phi_i \in End(K[X]/I)$ $B$r(B
$$\phi_i : f \bmod I \mapsto x_if \bmod I$$
$B$GDj5A$9$k(B. $H_1 = \{HT_{<_1}(g)\mid g \in G_1\}$, 
$MB_1 = \{u \in T \mid$  $B$9$Y$F$N(B $m \in H_1$ $B$KBP$7(B $m {\not|} u\}$
$B$H$*$1$P(B $MB_1$ $B$O(B $K[X]/I$ $B$N(B $K$-$B4pDl$h$j(B, 
$\{NF_{<_1}(x_iu,G_1)\mid u \in MB_1\}$ $B$rA4$F7W;;$9$k$3$H$G(B, 
$\phi_i$ $B$,I=8=$G$-$k(B. 
\end{df}

$B$"$i$+$8$a(B, $\phi_i$ $B$r7W;;$7$F$*$1$P(B, $NF(x_it,G_1) = \phi_i(NF(t,G_1))$
$B$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
$B7W;;$G$-$k(B. 

\begin{re}
$B0lHL$K$O(B, FGLM $B%"%k%4%j%:%`$O(B 0 $B<!85%$%G%"%k$N>l9g$K$N$_E,MQ2DG=$@$,(B, 
$BL\E*$N(B order $B$,A4<!?tHf3S$r4^$`>l9g$J$I(B, $BG$0U$N(B $s \in T$ $B$KBP$7(B 
$\{t \in T \mid t < s\}$ $B$,M-8B=89g$N>l9g$K$O(B, $BG$0U$N%$%G%"%k$KE,MQ$G$-$k(B. 
$B$7$+$7(B, $B8zN($O0lHL$K(B 0 $B<!85$N>l9g$KHf$Y$F4|BT$G$-$J$$(B. 
\end{re}

\section{Modular change of ordering}

FGLM $B$O(B, $BL\E*$N9`$KE~C#$9$k$^$G9TNs$N(B Gauss $B>C5n$r7+$jJV$9J}K!$H$$$($k(B. 
$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
$B78?t$KHf$Y$FESCf$N78?tKDD%$,7c$7$/$J$k>l9g$,$7$P$7$P@8$:$k(B. $B$3$l$O(B, 
$B<!$NNc$G<($5$l$k(B. 

\begin{ex}
$A \in GL(n,\Q)$ $B$H$9$k(B. $V \in \Q^n$ $B$KBP$7(B $B=AV$ $B$H$9$k$H(B, 
$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
$B$r(B Gauss $B>C5n$G2r$/>l9g(B, $B$=$l$O(B $A$ $B$K$N$_CmL\$7$F9T$J$o$l(B,
$B$ $B$NCM$K:81&$5$l$J$$(B. $B$9$J$o$A(B, $B2r(B $V$ $B$N@.J,$,>.$5$$@0?t(B
$B$N>l9g$G$bBg$-$$>l9g$G$b2r$/<j4V$OJQ$o$i$J$$$3$H$K$J$k(B. 
\end{ex}

\subsection{Modular $B7W;;$H@~7ABe?t$K$h$k%0%l%V%J4pDl8uJd@8@.(B}
$B$3$3$G>R2p$9$k%"%k%4%j%:%`$O(B, modular $B7W;;$r1~MQ$7$F(B, $B7k2L$N78?t$NBg$-(B
$B$5$NDxEY$N%3%9%H$G%0%l%V%J4pDl$r7W;;$9$k$b$N$G$"$k(B\cite{NY}\cite{NY2}.
$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
$B?tBN>e$N%0%l%V%J4pDl$N3F85$K8=$l$k9`$r?dB,$7(B, $BL$Dj78?tK!$G(B, $B$=$l$i$N9`(B
$B$r<B:]$K$b$D(B $I=Id(F)$ $B$N85$r5a$a$k(B.

\begin{al}
\label{mfglm}
\begin{tabbing}
\\
candidate\_by\_linear\_algebra$(F,p,<_1,<)$\\
Input : \= order $<_1$, $<$\\
\> $F \subset \Z[X]$ \st $F = GB_{<_1}(Id(F))$\\
\> $F$ $B$N3F85$N(B $<_1$ $B$K4X$9$k<g78?t$r3d$i$J$$(B $p$ \\
Output : $F$ $B$N(B $<$ $B$K4X$9$k%0%l%V%J4pDl8uJd$^$?$O(B {\bf nil}\\
($C_2$)\= \kill
$\overline{G} \leftarrow$ $GB_<(Id(\phi_p(F)))$ ($BHoLs%0%l%V%J4pDl(B) \\
$G \leftarrow \emptyset$\\
for \= each $h \in \overline{G}$ do \{\\
\> $a_t$ : $t \in T(h)$ $B$KBP1~$9$kL$Dj78?t(B\\
\> $a_t \leftarrow 1$ ($t = ht_{<}(h)$ $B$KBP$7$F(B)\\
\> $H \leftarrow \displaystyle{\sum_{t \in T(h)} a_t NF_{<_1}(t,F)}$\\
\> $C \leftarrow$ $H$ $B$N(B $X$ $B$K4X$9$k78?t$N=89g(B\\
\> if \= $B@~7AJ}Dx<0(B $E_h = \{ f = 0 \mid f \in C\}$ $B$,2r(B
    $S_h = \{a_t = c_t \mid c_t \in \Q\}$ $B$r;}$D(B\\
\> then $G \leftarrow G \cup \{\displaystyle{d\sum_{t \in T(h)} c_t t}\}$\\
\> {\rm ($d$ : $c_t$ $B$NJ,Jl$N(B LCM)}\\
\> else return {\bf nil}\\
\}\\
return $G$
\end{tabbing}
\end{al}

\begin{pr}
$B%"%k%4%j%:%`(B \ref{mfglm} $B$O(B, $BM-8B8D$N(B $p$ $B$r=|$$$F(B $GB_<(F)$ $B$r(B
$BM?$($k(B. 
\end{pr}
\proof trace lifting $B$N>l9g$HF1MM$K(B, $BM-8BBN>e$G$N%0%l%V%J4pDl$N3F(B
$B85$K8=$l$k9`$,(B, $BM-M}?tBN>e$G$N$=$l$H0lCW$7$F$$$l$P(B, $BA4$F$N@~7AJ}Dx<0$O(B
$B2r$r;}$A(B, $G = GB_<(F)$ $B$H$J$k(B. $B$=$&$G$J$$(B $p$ $B$OM-8B8D$7$+$J$$$?$a(B, 
$B$=$l$i$r=|$$$F$O(B, $B$3$N%"%k%4%j%:%`$O(B $GB_<(F)$ $B$rM?$($k(B. \qed

\begin{re}
$B@~7AJ}Dx<0$,A4It2r$1$?$H$7$F$b(B, $B7k2L$,(B $Id(F)$ $B$N%0%l%V%J4pDl$K$J$C$F(B
$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
$B7k2L$K$h$j(B, $B%"%k%4%j%:%`(B \ref{mfglm} $B$,(B {\bf nil} $B$G$J$$B?9`<0=89g$r(B
$BJV$;$P(B, $B$=$l$O$?$@$A$K(B $Id(F)$ $B$N%0%l%V%J4pDl$H$J$C$F$$$k$3$H$,J,$+$k(B.
$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
$B5a2rK!$H$H$b$K(B, $B8e$G=R$Y$k(B. 
\end{re}

\subsection{$B%0%l%V%J4pDl8uJd$,%0%l%V%J4pDl$H$J$k>r7o(B}
$B$3$3$G$O(B, change of ordering $B$N>l9g$K$O(B, 
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
$B$,ITI,MW$K$J$k$3$H$r<($9(B. $F \subset \Z[X]$ $B$H$9$k(B. 

\begin{as}
\label{nf}
$BM-M}?tBN>e$HM-8BBN>e$N7W;;$,0[$i$J$$$h$&<!$N2>Dj$r$*$/(B. 
\begin{enumerate}
\item $BITI,MWBP$N8!=P4p=`$OF,9`$N$_$G9T$&(B.
\item $B@55,7A7W;;$K$*$$$F(B, $B@55,2=$KMQ$$$k85(B (reducer) $B$NA*Br$O(B, $B@55,2=$5$l$k(B
$BB?9`<0$N9`$*$h$S(B reducer $B$NF,9`$N=89g$K$N$_0MB8$9$k(B. 
\end{enumerate}
\end{as}

\begin{df}
{\rm (compatibile $B$JAG?t(B)}
$BAG?t(B $p$ $B$,(B $F$ $B$K4X$7(B compatible $B$H$O(B, 
$$\phi_p(Id(F)\cap\Z[X]) (=\phi_p(Id(F)\cap\Z_{(p)}[X])) = Id(\phi_p(F))$$
$B$J$k$3$H(B. 
\end{df}

\begin{df}
$BAG?t(B $p$ $B$,(B $(F,<)$ $B$K4X$7$F(B strongly compatible $B$H$O(B
$p$ $B$,(B $F$ $B$K4X$7$F(B compatible $B$G(B 
$$E_<(Id(F)) = E_<(Id(\phi_p(F))$$
$B$J$k$3$H(B. 
\end{df}

\begin{df}(permissible $B$JAG?t(B)
$BAG?t(B $p$ $B$,(B $(F,<)$ $B$K$D$$$F(B permissible $B$H$O(B
$B3F(B $f \in F$ $B$KBP$7(B \valid{p}{f}{<} $B$J$k$3$H(B. 
\end{df}

\begin{df}
$f \in \Q[X]$ $B$,(B $p$ $B$K4X$7(B stable $B$H$O(B $f \in \Z_{(p)}[X]$ $B$J$k$3$H(B.
\end{df}

\begin{df}
{\rm (modular $B%0%l%V%J4pDl$N5UA|(B)}
$G \subset Id(F)\cap \Z[X]$ $B$,(B $F$ $B$N(B $<$ $B$K4X$9$k(B
$p$-compatible $B$J%0%l%V%J4pDl8uJd$H$O(B
$p$ $B$,(B $(G,<)$ $B$K$D$$$F(B permissible $B$G(B $\phi_p(G)$ $B$,(B$Id(\phi_p(F))$ $B$N(B $<$
$B$K4X$9$k%0%l%V%J4pDl$J$k$H$-$r$$$&(B. 
\end{df}

\begin{re}
compatibility $B$O(B order $B$KFHN)$J35G0$G$"$k(B. 
\end{re}

\begin{lm}
\label{valid}
$G \subset \Z[X]$, $p$ $B$r(B $(G,<)$ $B$K$D$$$F(B permissible $B$JAG?t(B,
$f \in \Z[X]$ $B$H$9$k(B. $B$3$N$H$-(B
$B2>Dj(B \ref{nf} $B$N$b$H$G(B
$$NF(\phi_p(f),\phi_p(G)) = \phi_p(NF(f,G)).$$
\end{lm}
\proof $NF(f,G)$ $B$O<!$N(B recurrence $B$G7W;;$5$l$k(B. 
$$f_0 \leftarrow f, 
f_i \leftarrow f_{i-1} - \alpha_i t_i g_{k_i}$$
$B$3$3$G(B, $\alpha_i\in {\bf \Q}$, $t_i$ : a term, $g_{k_i}\in G$.
$B$9$k$H(B, $p$ $B$,(B $(G,<)$ $B$KBP$7$F(B permissible $B$h$j(B $\alpha_i \in \Z_{(p)}$. 
$B$h$C$F(B $BA4$F$N(B $i$ $B$KBP$7(B $f_i \in \Z_{(p)}[X]$ $B$G(B, $B$3$N(B recurrence $B$K(B
$\phi_p$ $B$rE,MQ$7$F<!$rF@$k(B. 
$$\phi_p(f_i) = \phi_p(f_{i-1}) - \phi_p(\alpha_i) t_i \phi_p(g_{k_i}).$$
$B$b$7(B $\phi_p(\alpha_i) \neq 0$ $B$J$i$P(B $\phi_p(f_{i-1}) \neq 0$ $B$G(B, 
$$\phi_p(f_i) \leftarrow \phi_p(f_{i-1}) - \phi_p(\alpha_i) t_i \phi_p(g_{k_i})$$
$B$O(B $\phi_p(G)$ $B$K$h$kF,9`>C5n$G$"$k(B.
$B$b$7(B $\phi_p(\alpha_i) = 0$ $B$J$i$P(B $\phi_p(f_i) = \phi_p(f_{i-1})$ $B$G(B
$$\{\phi_p(f_i)\mid i=0\, {\rm or}\, \phi_p(\alpha_i)\neq 0\}$$
$B$J$kNs$O(B
$NF(\phi_p(f),\phi_p(G))$ $B$N7W;;$KBP1~$9$k(B.
$B$b$7(B$\phi_p(f_i) = 0$ $B$J$k(B $i$ $B$,B8:_$9$l$PA|$OESCf$G@Z$l$k$,(B
$$NF(\phi_p(f),\phi_p(G)) = \phi_p(NF(f,G)) = 0$$
$B$@$+$i<gD%$,@.N)$9$k(B. 
\qed
\medskip

\begin{th}
\label{comp}
$G \subset Id(F)\cap\Z[X]$ $B$r(B $Id(F)$ $B$N(B $<$ $B$K4X$9$k%0%l%V%J4pDl$H$9$k(B. 
$B$b$7(B $p$ $B$,(B $(G,<)$ $B$K4X$7$F(B permissible $B$+$D(B 
$\phi_p(G) \subset Id(\phi_p(F))$ $B$J$i$P(B $p$ $B$O(B $F$ $B$K4X$7$F(B compatible $B$G$"$k(B.
$B99$K(B, $\phi_p(G)$ $B$O(B $Id(\phi_p(F))$ $B$N%0%l%V%J4pDl$G(B
$p$ $B$O(B $(F,<)$ $B$K$D$$$F(B strongly compatible $B$G$"$k(B.
\end{th}
\proof
$h \in Id(F) \cap \Z[X]$ $B$H$9$k(B. $G$ $B$,(B $Id(F)$ $B$N%0%l%V%J4pDl$@$+$i(B, 
$NF(h,G)=0$ $B$h$j(B $h = \sum_{g\in G} a_g g$ $B$H=q$1$k(B. $B$3$3$G(B $a_g \in \Q[X]$. 
$B99$K(B $p$ $B$,(B $(G,<)$ $B$K$D$$$F(B permissible $B$h$j(B,
$a_g$ $B$O(B $p$ $B$K$D$$$F(B stable $B$G(B, $B2>Dj(B $\phi_p(g)\in Id(\phi_d(F))$ $B$h$j(B
$$\phi_p(h)=\sum_{g\in G} \phi_p(a_g) \phi_p(g).$$
$B$h$C$F(B $\phi_p(h)\in Id(\phi_p(F))$. 
$B8N$K(B $\phi_p(Id(F)\cap \Z[X])\subset Id(\phi_p(F))$. \\
$B5U$K(B $\overline{h} \in Id(\phi_p(F))$ $B$O(B
$$\overline{h} = \sum_{f\in F} \overline{a}_f \phi_p(f)$$
$B$H=q$1$k(B. $B$3$3$G(B $\overline{a}_f \in GF(p)[X]$. 
$B$9$k$H(B, $\phi_p(a_f)=\overline{a}_f$ $B$J$k(B $a_f$ $B$rA*$s$G(B
$h=\sum_{f\in F} a_f f$ $B$H$*$1$P(B $\phi_p(h)=\overline{h}$. 
$B$3$l$+$i(B $\phi_p(Id(F)\cap \Z[X])=Id(\phi_p(F))$. \\
$B:G8e$K(B $\phi(G)$ $B$,(B $Id(\phi_p(F))$ $B$N%0%l%V%J4pDl$H$J$k$3$H$r<($9(B. 
$B>e$G=R$Y$?$3$H$+$i(B, 
$\overline{h} \in Id(\phi_p(F))$ $B$KBP$7(B, 
$h \in Id(F)\cap \Z[X]$ $B$,B8:_$7$F(B $\overline{h}=\phi_p(h)$.
$B$9$k$H(B, $BJdBj(B \ref{valid} $B$h$j(B
$$NF(\overline{h},\phi_p(G))=\phi_p(NF(h,G))=0.$$
$B=>$C$F(B, $\phi_p(G)$ $B$O(B $Id(\phi_p(F))$ $B$N%0%l%V%J4pDl(B. 
strong compatibility $B$O(B $E_<(Id(F))$ $B$,(B $E_<(G)$ 
$B$G@8@.$5$l$k$3$H$+$iJ,$+$k(B. 
\qed \\
$B$3$NDjM}$G(B, $\phi_p(G) \subset Id(\phi_p(F))$ $B$O(B
$Id(\phi_p(F))$ $B$N%0%l%V%J4pDl$K$h$j%A%'%C%/$G$-$k(B.
$B$h$C$F(B, $p$ $B$N(B compatibility $B$N%A%'%C%/$OM-M}?tBN>e(B, $BM-8BBN>e$N(B
$BG$0U$N(B order $B$G$N%0%l%V%J4pDl$r7W;;$9$k$3$H$G9T$&$3$H$,$G$-$k(B. 
$B$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
$B$O6K$a$F4JC1$G$"$k(B. 

\begin{co}
\label{compco}
$G \subset \Z[X]$ $B$,(B $<$ $B$K4X$9$k(B $Id(G)$ $B$N%0%l%V%J4pDl$H$9$k(B.
$B$b$7(B $p$ $B$,(B $(G,<)$ $B$KBP$7(B permissible $B$J$i$P(B
$\phi_p(G)$ $B$O(B $Id(\phi_p(G))$ $B$N%0%l%V%J4pDl$G(B
$p$ $B$O(B $(G,<)$ $B$KBP$7(B strongly compatible.
\end{co}
$B<!$NDjM}$O(B, $B%0%l%V%J4pDl8uJd$,<B:]$K%0%l%V%J4pDl$K$J$k$?$a$N==J,>r7o$rM?$($k(B. 
$B$9$J$o$A(B, $B2f!9$,5a$a$k$b$N$G$"$k(B. 

\begin{th}
\label{candi}
$p$ $B$,(B $F$ $B$K$D$$$F(B compatible $B$G(B $G$ $B$,(B $<$ $B$K4X$7$F(B $p$-compatible $B$J%0%l%V%J(B
$B4pDl8uJd$J$i$P(B, $G$ $B$O(B $<$ $B$K4X$9$k(B $Id(F)$ $B$N%0%l%V%J4pDl$G$"$k(B. 
\end{th}
\proof $BA4$F$N(B $f \in Id(F)$ $B$,(B $<$ $B$K4X$7$F(B $G$ $B$K$h$j(B 0 $B$K@55,2=(B
$B$5$l$k$3$H$r<($;$P$h$$(B. $f$ $B$O(B $G$ $B$K$D$$$FHoLs$H$7$F$h$$(B. $B$b$7(B $f
\neq 0$ $B$J$i$P(B, $BE,Ev$JM-M}?t$r$+$1$F(B, $f \in Id(F)\backslash \{0\}$ $B$,(B 
$G$ $B$K$D$$$FHoLs$G(B $f$ $B$N78?t$N@0?t(B GCD ($cont(f)$ $B$H=q$/(B) $B$,(B 1 $B$KEy$7(B
$B$$(B, $B$H$7$F$h$$(B. $B$9$k$H(B $\phi_p(f) \neq 0$. $B$5$b$J$/$P(B 
$cont(f)$ $B$O0x;R(B $p$ $B$r;}$D$3$H$K$J$k(B. 
$p$ $B$O(B $F$ $B$K$D$-(B compatible $B$@$+$i(B $\phi_p(f) \in Id(\phi_p(F))$.
$B$9$k$H(B $\phi_p(f)$ $B$O(B $<$ $B$K4X$7(B, $\phi_p(G)$ $B$K$h$j(B 0 $B$K@55,2=$5$l$J$1$l$P(B
$B$J$i$J$$(B. $B$7$+$7(B $f$ $B$O(B $G$ $B$K$D$$$FHoLs$@$+$i(B  
$\phi_p(G)$ $B$NF,9`$N=89g$O(B $G$ $B$N$=$l$HEy$7$$(B. $B$h$C$F(B
$\phi_p(f)$ $B$O(B $\phi_p(G)$ $B$K$D$$$FHoLs$H$J$j(B, $\phi_p(f) = 0$. $B$3$l$O(B
$BL7=b(B. \qed
\medskip
$B<!$NDjM}$OA0DjM}$N@:L)2=$G$"$k(B. $B$9$J$o$A(B, $B>:=g$K7W;;$5$l$?ItJ,E*$J(B 
$p$-compatible $B$J%0%l%V%J4pDl8uJd$,<B:]$K%0%l%V%J4pDl$N0lIt$H$J$C$F(B
$B$$$k$3$H$rJ]>Z$9$k(B. $B$3$l$O(B, $BESCf$^$G$N7k2L$r:FMxMQ$G$-$k$H$$$&E@$G(B
$BM-MQ$G$"$k(B. $B$^$?(B, $B8e$G=R$Y$k$h$&$K(B, $B%0%l%V%J4pDl$N$"$kFCDj$N85(B, 
$BNc$($P(B, $B=g=x:G>.$N85$N$_$r5a$a$?$$(B, $B$"$k$$$O(B elimination $B8e$N7k2L$N$_(B
$B$r5a$a$?$$>l9g$K$bM-MQ$G$"$k(B. 

\begin{th}
$p$ $B$,(B $F$ $B$K$D$$$F(B compatible $B$H$9$k(B. 
$\overline{G}\subset GF(p)[X], \overline{G} = GB_<(Id(\phi_p(F))$ $B$H$7(B
$\overline{g}_1<\cdots<\overline{g}_s$ $B$J$k(B $\overline{g}_i$ $B$K$h$j(B
$\overline{G}=\{\overline{g}_1,\cdots,\overline{g}_s\}$ $B$H=q$/(B. 
$B99$K(B, $B$"$k@5?t(B $t\leq s$ $B$KBP$7(B, 
$g_i \in Id(F) \cap \Z_{(p)}[X]$ ($1 \le i \le t$) $B$,B8:_$7$F(B
$\phi_p(g_i) = \overline{g}_i$ $B$+$D(B $g_i$ $B$O(B $\{g_1,\cdots,g_{i-1}\}$ $B$K$D$$$F(B
$BHoLs$H$9$k(B. 
$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. 
\end{th}
$B>ZL@N,(B ($B5"G<K!$K$h$k(B.)\\
$B0J>e=R$Y$?$3$H$K$h$j(B, $B<!$N$h$&$J0lHLE*$J(B change of ordering $B%"%k%4%j%:%`(B
$B$,F@$i$l$k(B. 

\begin{pro}
\begin{tabbing}
\\
candidate$(F,p,<)$\\
Input : \= $F \subset Z[X]$\\
\> $BAG?t(B p\\
\> order $<$\\
Output : $F$ $B$N(B $p$-compatible $B$J%0%l%V%J4pDl8uJd$^$?$P(B {\bf nil}\\
{\rm ($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.)}\\
\end{tabbing}
\end{pro}

\begin{al}(compatibility check $B$K$h$k%F%9%H$N>JN,(B)
\label{bconv}
\begin{tabbing}
gr\"obner\_by\_change-of-ordering$(F,<)$\\
Input : \= $F \subset \Z[X]$, order $<$\\
Output : $Id(F)$ $B$N(B $<$ $B$K4X$9$k%0%l%V%J4pDl(B $G$\\
$G_0 \leftarrow$ $F$ $B$N(B, $B$"$k(B order $<_0$ $B$K4X$9$k%0%l%V%J4pDl(B; $G_0 \subset \Z[X]$\\
{\bf again:}\\
for\= \kill
\> $p \leftarrow (G_0,<_0)$ $B$K4X$7$F(B permissible $B$JL$;HMQ$NAG?t(B \\
\> $G \leftarrow$ candidate($G_0$,$p$,$<$)\\
\> If $G$ = {\bf nil} goto {\bf again:}\\
\> else return $G$
\end{tabbing}
\end{al}
candidate() $B$K$*$$$F$O(B,  $p$-compatible $B$J%0%l%V%J4pDl8uJd$rJV$9(B
$BG$0U$N%"%k%4%j%:%`$,;HMQ2DG=$G$"$k(B. $B$3$l$^$G=R$Y$?$b$N$G$O(B, 
\begin{itemize}
\item tl\_guess()
\item $B@F<!2=(B + tl\_guess() + $BHs@F<!2=(B
\item candidate\_by\_linear\_algebra()
\end{itemize}
$B$,E,9g$9$k(B. $B$3$l$i$N$&$A(B, $BA0<T(B 2 $B$D$K$D$$$F$OL@$i$+$@$,(B, $B:G8e$N$b$N$K(B
$B$D$$$F$O8!>Z$rMW$9$k(B. $B$3$l$K$D$$$F<!@a$G=R$Y$k(B. 

\subsection{candidate\_by\_linear\_algebra()}

\begin{lm}
\label{munique}
$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
$B$G(B, $E_{h,p}=\{\phi_p(c)=0 \mid c \in C\}$ $B$O0l0U2r$r;}$D(B. 
\end{lm}
\proof 
$p$ $B$,(B $(F,<_1)$ $B$K4X$7(B permissible $B$h$j(B $NF_{<_1}(t,F)$ $B$O(B $p$ $B$K$D$$$F(B
stable. $B$h$C$F(B $c \in C$ $B$b(B $p$ $B$K$D$$$F(B stable.
$$S_{h,p}=\{a_t=\overline{c}_t \mid \overline{c}_t \in GF(p)\}$$
$B$,(B $E_{h,p}$ $B$N2r$H$9$k(B.  
$$\overline{h}=\displaystyle{\sum_{t \in T(h)} \overline{c}_t t}$$ 
$B$H$*$/(B. $B$9$k$H(B, 
$$0=\displaystyle{\sum_{t \in T(h)} \overline{c}_t \phi_p(NF_{<_1}(t,F)})=
NF_{<_1}(\overline{h},\phi_p(F)).$$
$B$h$C$F(B $\overline{h} \in Id(\phi_p(F))$ $B$h$j(B $NF_<(\overline{h},\overline{G})=0$.
$B$9$k$H(B $\overline{G}$ $B$,HoLs%0%l%V%J4pDl$G(B $T(\overline{h})\subset T(h)$
$B$h$j(B $\overline{h}=h$ $B$,@.$jN)$D(B. 
$B$3$l$O(B, $B2r$,0l0UE*$G(B $h$ $B$K0lCW$9$k$3$H$r0UL#$9$k(B. \qed
\medskip

\begin{co}
\label{sqmat}
$n$ $B$rITDj85(B $a_t$ $B$N8D?t$H$9$k$H(B, $E_h$ $B$+$i<!$N@-<A$r$b$D(B
subsystem $E'_h$ $B$rA*$V$3$H$,$G$-$k(B. 
\begin{itemize}
\item $E'_h$ $B$O(B $n$ $B8D$NJ}Dx<0$+$i$J$k(B. 
\item $\phi_p(E'_h)$ $B$O(B $GF(p)$ $B>e$G0l0U2r$r$b$D(B.
\end{itemize}
$B$3$l$+$i<!$N$3$H$,J,$+$k(B. 
\begin{itemize}
\item $E'_h$ $B$O(B $\Q$\, $B>e0l0U2r$r;}$A(B, $B2r$O(B $p$ $B$K$D$$$F(B stable.
\item $E_h$ $B$,2r$r$b$F$P(B, $B$=$l$O(B $E'_h$ $B$N0l0U2r$K0lCW$9$k(B. 
\end{itemize}
\end{co}

\begin{th}
$B%"%k%4%j%:%`(B \ref{mfglm} $B$,B?9`<0=89g(B $G$ $B$rJV$;$P(B, 
$G$ $B$O(B $F$ $B$N(B $<$ $B$K4X$7$F(B $p$-compatible $B$J%0%l%V%J4pDl8uJd$G$"$k(B.
\end{th}
\proof
$B3F(B $g \in G$ $B$KBP$7(B
$$g = \displaystyle{\sum_{t \in T(h)} c_t t}$$
$B$H=q$1(B, $\{c_t/c\}$\, ($c = c_{hc_<(g)}$) $B$,(B $E_h$ $B$N2r$H$J$k$h$&$J(B
$h \in \overline{G}$ $B$,B8:_$9$k(B. $B$9$k$H(B, 
$$0 = c\displaystyle{\sum_{t \in T(h)} c_t/c NF_{<_1}(t,F)}
= NF_{<_1}(g,F).$$
$B8N$K(B $g \in Id(F)$.
$B7O(B \ref{sqmat} $B$K$h$j(B $p$ $B$O(B $(G,<)$ $B$K$D$$$F(B permissible $B$G(B
$\phi_p(g)$ = $\phi_p(c) h$ $B$h$j(B
$\phi_p(G)$ $B$O(B $\overline{G}$ $B$N%0%l%V%J4pDl$G$"$k(B. 
\qed\\
\medskip
$B>e$NJdBj$rMQ$$$F(B $E_h$ $B$r<!$N<j=g$G2r$/(B. 

\begin{enumerate}
\item $E'_h$ $B$rA*$V(B.
\item $S \leftarrow$ $E'_h$ $B$N0l0U2r(B.
\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
$E_h$ $B$O2r$r;}$?$J$$(B. 
\end{enumerate}
$E_h$ $B$O(B $E'_h$ $B$r(B $GF(p)$ $B>e$G2r$/2>Dj$GF@$i$l$k(B. 
$B0J2<$G$O(B, $E'_h$ $B$r2r$/J}K!$K$D$$$F=R$Y$k(B. $B$3$l$O<!$N$h$&$K(B
$BDj<02=$G$-$k(B. 

\begin{prob}
$M$, $B$ $B$r$=$l$>$l(B $n\times n$, $n\times 1$ $B@0?t9TNs$H$7(B, 
$X$ $B$r(B, $BL$Dj78?t$r@.J,$H$9$k(B $n\times 1$ $B9TNs$H$9$k(B. 
$\det(\phi_p(M))\neq 0$ $B$N$b$H$G(B, 
$MX=B$ $B$r2r$1(B. 
\end{prob}
$M$, $B$ $B$O(B, $B0lHL$K(B, $BD9Bg$J@0?t$r@.J,$K;}$DL)9TNs$H$J$k(B. $B$7$+$7(B, 
$B$b$H$b$H$NF~NOB?9`<0$N78?t$,>.$5$$>l9g(B, $B%0%l%V%J4pDl$N85$N78?t(B
$B$9$J$o$A(B $MX=B$ $B$N2r(B $X$ $B$OHf3SE*>.$5$$>l9g$,B?$$(B. $B$3$N$h$&$J>l9g$K(B
$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
$BHs8zN(E*$G$"$k(B. $B$3$N$h$&$J>l9g$KM-8z$J$N$,(B, Hensel $B9=@.(B, $BCf9q>jM>DjM}(B
$B$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
$B>R2p$9$k(B. 

\begin{al}
\label{lineq}
\begin{tabbing}
\\
solve\_linear\_equation\_by\_hensel$(M,B,p)$\\
($C_2$)\= \kill
Input : \= $n\times n$ $B9TNs(B $M$, $n\times 1$ $B9TNs(B $B$\\
\> $\phi_p(\det(M))\neq 0$ $B$J$kAG?t(B\\
Output : $MX=B$ $B$J$k(B $n\times 1$ matrix $X$\\
$R \leftarrow \phi_p(M)^{-1}$\\
$c \leftarrow B$\\
$x \leftarrow 0$\\
$q \leftarrow 1$\\
$count \leftarrow 0$\\
do \{\\
(1)\> $t \leftarrow \phi_p^{-1}(R \phi_p(c))$\\
(2)\> $x \leftarrow x + qt$\\
\> $c \leftarrow (c-Mt)/p$\\
\> $q \leftarrow qp$\\
\> $count \leftarrow count+1$\\
\> {\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.)}\\
\> if \= $count = {\bf Predetermined\_Constant}$ then \{\\
\> \> $count \leftarrow 0$\\
\> \> $X \leftarrow$ inttorat$(x,q)$\\
\> \> if \= $X \neq$ {\bf nil} $B$+$D(B $MX=B$ then return $X$\\
\> \}\\
\}
\end{tabbing}
\end{al}

\begin{al}
\label{intrat}
\begin{tabbing}
\\
inttoat$(x,q)$\\
($C_2$)\= \kill
Input : $q>x$, $\GCD(x,q)=1$ $B$J$k@5@0?t(B $x$, $q$ \\
Output : $bx \equiv a \bmod q$ $B$+$D(B
$|a|,|b| \le \sqrt{q\over 2}$ $B$J$kM-M}?t(B $a/b$ $B$^$?$O(B {\bf nil}\\
if $x \le \sqrt{q\over 2}$ then return $x$\\
$f_1 \leftarrow q$, $f_2 = x$,
$a_1 \leftarrow 0$, $a_2 \leftarrow 1$\\
$i \leftarrow 1$\\
do \= \{\\
\> if \= $|f_i| \le \sqrt{q\over 2}$ then \\
\>\> if $|a_i| \le \sqrt{q\over 2}$ then return $f_i/a_i$\\
\>\> else return {\bf nil}\\
\> $f_i - q_i f_{i+1} < f_{i+1}$ $B$J$k(B $q_i$ $B$r5a$a$k(B\\
\> $f_{i+2} \leftarrow f_i - q_i f_{i+1}$\\
\> $a_{i+2} \leftarrow a_i - q_i a_{i+1}$\\
\> $i \leftarrow i+1$\\
\}
\end{tabbing}

\end{al}
\begin{lm}(\cite{WANG2}\cite{DIXON})
\label{intratlm}
$x$, $q$ $B$r(B $q>x>1$, $\GCD(x,q)=1$ $B$J$k@5@0?t$H$9$k(B. $B$3$N$H$-(B,
$|a|,|b|\le\sqrt{q/2}$, $\GCD(a,b)=1$ $B$J$k@0?t(B $a,b$ $(b>0)$ $B$*$h$S@0?t(B 
$c$ $B$,$"$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
Euclid $B8_=|K!$K$h$jF@$i$l$k(B. 
\end{lm}
[$BN,>Z(B]
$x>1$ $B$h$j(B $a \neq b$ $B$H$7$F$h$$(B. 
$ax+cq=b$ $B$h$j(B ${x \over q}+{c \over a} = {b \over aq}.$ $B$3$3$G(B 
$$|{b \over {aq}}| = |{{ab} \over {a^2q}}| < {1 \over {2a^2}}$$
$B$h$j(B 
$$|{x \over q}+{c \over a}| < {1 \over {2a^2}}.$$$B$h$C$F(B, $-c/a$ $B$O(B 
$x/q$ $B$N<g6a;wJ,?t$N0l$D(B. ($BO"J,?t$N@-<A$K$h$k(B. \cite[Section 24]{TAKAGI} $B;2>H(B.)
$\GCD(a,b)=1$ $B$h$j(B $\GCD(a,c)=1$ $B$G$"$j(B, $b>0$ $B$KBP$7$F(B $(a,c)$ $B$O0l0U(B
$BE*$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
$B78?t$H$7$FF@$i$l$k(B. \qed

\begin{lm}
\label{lineqlm}
(1) $B$K$*$$$F(B $Mx \equiv B \bmod q$ $B$+$D(B $c = (B-Mx)/q$.
\end{lm}
\proof 
$B5"G<K!$K$h$j<($9(B. $count = 0$ $B$N$H$-L@$i$+(B. $count = m$
$B$^$G8@$($?$H$9$k(B. $B$3$N$H$-(B, (2) $B$G(B
$Mt \equiv c \bmod p$ $B$h$j(B $M(x+qt) \equiv Mx+q(B-Mx)/q \bmod qp$.
$B$9$J$o$A(B $M(x+qt) \equiv B \bmod qp$. $B$^$?(B, 
$(c-Mt)/p = ((B-Mx)-qMt)/qp = (B-M(x+qt))/qp$ $B$h$j(B $count = m+1$
$B$G$b8@$($k(B. \qed

\begin{pr}
$B%"%k%4%j%:%`(B \ref{lineq} $B$O(B $MX=B$ $B$N2r(B $X$ $B$r=PNO$9$k(B. 
\end{pr}
\proof
$BJdBj(B \ref{lineqlm} $B$h$j(B, $count=m$ $B$N$H$-(B (1) $B$K$*$$$F(B $Mx \equiv B
\bmod p^m$. $\det(M) \neq 0$ $B$h$j(B $x$ $B$OK!(B $p^m$ $B$G0l0UE*$G$"$k(B. $B0lJ}(B 
$MX=B$ $B$OM-M}?tBN>e0l0UE*$K2r$r;}$D(B. $B$3$N2r$r(B$X=N/D$ ($D$ $B$O@0?t(B, $N$ 
$B$O@0?t%Y%/%H%k(B) $B$H=q$/$H$-(B, $Dr \equiv 1 \bmod p^m$$B$J$k@0?t(B $r$ $B$r$H$l(B
$B$P(B $M(rN) \equiv B \bmod p^n$. $B$h$C$F(B $rN \equiv x \bmod p^m$.
$D$, $N$ $B$N3F@.J,$,(B
$A$ $B$r1[$($J$$$H$-(B, $p^m > 2A^2$ $B$J$k(B $m$ $B$r$H$l$P(B, $BJdBj(B \ref{intratlm}
$B$h$j(B $rN$ $B$9$J$o$A(B $x$ $B$+$i%"%k%4%j%:%`(B \ref{intrat} $B$K$h$j(B
$N/D$ $B$,I|85$5$l$k(B.
\qed\\
{\bf Predetermined\_Constant} $B$O$"$k@5@0?t$G(B, $BM-M}?t$K0z$-La$7$F%A%'%C(B
$B%/$r9T$&IQEY$r@)8f$9$k(B. $B%"%k%4%j%:%`(B \ref{lineq} $B$K$*$$$F(B $c$ $B$O(B 
$nMAX(\|M\|_\infty,\|B\|_\infty)$$B$G2!$($i$l$k$3$H$,$o$+$k(B. $B$3$l$O(B, $B3F(B
$B%9%F%C%W$,$"$k(B constant $B;~4VFb$G7W;;$G$-$k$3$H$r<($9(B. $B$^$?(B, $B2r$NJ,JlJ,(B
$B;R$,(B $A$ $B$G2!$($i$l$F$$$l$P(B, $q > 2A^2$ $B$K$J$C$?CJ3,$G2r$rI|85$G$-(B
$B$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
$B$7$F(B, $B78?tKDD%$r2!$($?(B Gauss $B>C5nK!$G$"$k(B fraction-free $BK!$K$h$C$F$b(B, 
$B2r$NBg$-$5$K$+$+$o$i$:(B, $B78?t9TNs$N9TNs<0$r7W;;$7$F$7$^$&$H$$$&E@$G(B,
Gauss $B>C5nK!$O(B, $B$3$NLdBj$r2r$/$?$a$K$OITE,@Z$G$"$k(B.

\section{$B%?%$%_%s%0%G!<%?(B}

$BK\@a$G$O(B, $B0J2<$N$h$&$J%Y%s%A%^!<%/LdBj$K4X$7(B, $BK\>O$G=R$Y$?MM!9$J(B 
change of ordering $B%"%k%4%j%:%`$N8zN($NHf3S$r<($9(B. $B$^$?(B, $BL?Bj(B 
\ref{RUR} $B$G?($l$?(B RUR $B$N%b%8%e%i7W;;$rF1$8LdBj$KE,MQ$7(B, 
$B$=$NM%0L@-$r<B837k2L$K$h$j<($9(B.  $B7WB,$O(B, PC (FreeBSD, 300MHz Pentium
II, 512MB of memory) $B$G9T$C$?(B. $BC10L$OIC(B. garbage collection $B;~4V$O=|$$(B
$B$F$"$k(B.
\begin{tabbing}
$MMM\;\;$ \= \kill
$C(n)$ \> The cyclic n-roots system of n variables. (Faugere {\it et al.},1993).\\
	\> $\{f_1,\cdots,f_n\}$ where
	  $f_k=
\displaystyle{\sum_{i=1}^n\prod_{j=i}^{k+j-1}c_{j \bmod n}-\delta_{k,n}}$. 
($\delta$ is the Kronecker symbol.) \\ 
\> The variables and ordering : $c_n \succ c_{n-1} \succ \cdots \succ c_1$\\
$K(n)$ \> The Katsura system of n+1 variables. \\
      \> $\{u_l - \sum_{i=-n}^n u_i u_{l-i} (l = 0,\cdots, n-1),
           \sum_{l=-n}^n u_l - 1\}$\\
      \> The variables and ordering : $u_0 \succ u_1 \succ \cdots \succ u_n$.\\
      \> Conditions : $u_{-l} = u_l$ and $u_l = 0 (|l| > n)$. \\
$R(n)$ \> {\tt e7} in Rouillier (1996). \\
      \> $\{-1/2+\sum_{i=1}^n(-1)^{i+1}x_i^k (k=2, \cdots, n+1) \}$\\
      \> The variables and ordering : $x_n \succ x_{n-1} \succ \cdots \succ x_1$.\\
$D(3)$ \> {\tt e8} in Rouillier (1996). \\
	\> $\{f_0,f_1,f_2,\cdots,f_7\}$\\
	\> {\scriptsize $f_0=-420y^2-280zy-168uy-140vy-120sy-210ty-105ay+12600y-13440$}\\
	\> {\scriptsize $f_1=-840zy-630z^2-420uz-360vz-315sz-504tz-280az+18900z-20160$}\\
	\> {\scriptsize $f_2=-630ty-504tz-360tu-315tv-280ts-420t^2-252at+12600t-13440$}\\
	\> {\scriptsize $f_3=-5544uy-4620uz-3465u^2-3080vu-2772su-3960tu-2520au+103950u-110880$}\\
	\> {\scriptsize $f_4=-4620vy-3960vz-3080vu-2772v^2-2520sv-3465tv-2310av+83160v-88704$}\\
	\> {\scriptsize $f_5=-51480sy-45045sz-36036su-32760sv-30030s^2-40040ts-27720as+900900s-960960$}\\
	\> {\scriptsize $f_6=-45045ay-40040az-32760au-30030av-27720as-36036at-25740a^2+772200a-823680$}\\
	\> {\scriptsize $f_7=-40040by-36036bz-30030bu-27720bv-25740bs-32760bt-24024ba+675675b-720720$}\\
	\normalsize
	  \> The variables and ordering : $b \succ a \succ s \succ v \succ u \succ t \succ z \succ y$.\\
$Rose$ \> The Rose system.\\
%	\> $\{u_4^4-20/7a_{46}^2, a_{46}^2u_3^4+7/10a_{46}u_3^4+7/48u_3^4-50/27a_{46}^2-35/27a_{46}-49/216,$\\
%	\> $a_{46}^5u_4^3+7/5a_{46}^4u_4^3+609/1000a_{46}^3u_4^3+49/1250a_{46}^2u_4^3$\\
%	\> $-27391/800000a_{46}u_4^3-1029/160000u_4^3+3/7a_{46}^5u_3u_4^2+3/5a_{46}^6u_3u_4^2$\\
%	\> $+63/200a_{46}^3u_3u_4^2+147/2000a_{46}^2u_3u_4^2+4137/800000a_{46}u_3u_4^2$\\
%	\> $-7/20a_{46}^4u_3^2u_4-77/125a_{46}^3u_3^2u_4-23863/60000a_{46}^2u_3^2u_4$\\
%	\> $-1078/9375a_{46}u_3^2u_4-24353/1920000u_3^2u_4-3/20a_{46}^4u_3^3-21/100a_{46}^3u_3^3$\\
%	\> $-91/800a_{46}^2u_3^3-5887/200000a_{46}u_3^3-343/128000u_3^3 \}$\\
    \> $O_1$ : $u_3 \succ u_4 \succ a_{46}$, $O_2$ : $u_3 \succ a_{46} \succ u_4$.\\
$Liu$ \> The Liu system.\\
      \> $\{y(z-t)-x+a, z(t-x)-y+a, t(x-y)-z+a, x(y-z)-t+a\}$\\
      \> The variables and ordering : $x \succ y \succ z \succ t \succ a$.\\
$Fate$ \> The Fateman system, appeared on NetNews. \\
       \> $\{s^3+2r^3+2q^3+2p^3$, $s^5+2r^5+2q^5+2p^5$,\\
       \> $-s^5+(r+q+p)s^4+(r^2+(2q+2p)r+q^2+2pq+p^2)s^3+(r^3+q^3+p^3)s^2$\\
       \> $+(3r^4+(2q+2p)r^3+(4q^3+4p^3)r+3q^4+2pq^3+4p^3q+3p^4)s+(4q+4p)r^4$\\
       \> $+(2q^2+4pq+2p^2)r^3+(4q^3+4p^3)r^2+(6q^4+4pq^3+8p^3q+6p^4)r$\\
	   \> $+4pq^4+2p^2q^3+4p^3q^2+6p^4q\}$\\
       \> The variables and ordering : $p \succ q \succ r \succ s$.\\
$hC(6)$ \> A homogenization of C(6). \\
       \> $(C_6\backslash \{c_1c_2c_3c_4c_5c_6-1\})\cup \{c_1c_2c_3c_4c_5c_6-t^6\}$\\
       \> The variables and ordering : 
	   $c_1 \succ c_2 \succ c_3 \succ c_4 \succ c_5 \succ c_6 \succ t$.\\
\end{tabbing}

\subsection{Change of ordering}

$BM=$a7W;;$7$F$"$k(B DRL ($BA4<!?t5U<-=q<0=g=x(B)$B%0%l%V%J4pDl$+$i=PH/$7$F(B, LEX
($B<-=q<0=g=x(B)$B%0%l%V%J4pDl$r7W;;$9$k(B. $BMQ$$$k%"%k%4%j%:%`$O(B, TL
(tl\_guess$()$; $B%"%k%4%j%:%`(B \ref{tlguess}), HTL ($B@F<!2=(B
+tl\_guess$()$+$BHs@F2=(B), LA (candidate\_by\_linear\_algebra$()$;$B%"%k%4(B
$B%j%:%`(B \ref{mfglm} (0 $B<!85%7%9%F%`$N$_(B))$B$G$"$k(B. $BI=(B
\ref{mcotab} $B$O(B DRL $B$+$i(B LEX $B$X$NJQ49$K$+$+$k;~4V$r$7$a$9(B. {\it DRL} 
$B$O(B, DRL $B$N7W;;;~4V$r<($9(B. $B%0%l%V%J4pDl%A%'%C%/$r>J$/8z2L$r<($9$?$a$K(B,
tl\_ckeck$()$ ($B%"%k%4%j%:%`(B \ref{tlcheck}) $B$N;~4V$b<($9(B.

\begin{table}[hbtp]
\caption{Modular change of ordering}
\label{mcotab}
\begin{center}
\begin{tabular}{|c||c|c|c|c|c|c|c|} \hline
	& $K(5)$ & $K(6)$ & $K(7)$ & $C(6)$ & $C(7)$ & $R(5)$ & $R(6)$ \\ \hline
{\it DRL}&0.84	&8.4	&74	&3.1	&1616	&11	&1775	\\ \hline
{\it TL}&$\infty$		&$\infty$		&$\infty$ &$\infty$	&$\infty$	&$\infty$	&$\infty$	\\ \hline
{\it HTL}	&16	&1402	&$1.6\times 10^5$	&5.6	&$2\times 10^4$	&383	&$2.1\times 10^5$	\\ \hline
{\it LA}	&4.7	&158	&6813	&4	&435	&9.5		&258		\\ \hline
tl\_check	&2.3	&177	&$1.3\times 10^4$	&1.1	&2172	&3	&40		\\ \hline
\end{tabular}

\begin{tabular}{|c||c|c|c||c|c|c|} \hline
	& $D(3)$ & $RoseO_1$ & $RoseO_2$ & $Liu$ & $Fate$ & $hC(6)$ \\ \hline  
{\it DRL}	&30	&0.19	&0.15	&0.06	&0.5	&7.2	\\ \hline
{\it TL}	& $\infty$	&1.7	&354	&$\infty$	&4	&25	\\ \hline
{\it HTL}	&$4.1\times 10^4$	&1.7	&36	&18	&4	&25	\\ \hline
{\it LA}	&585	&3.3	&12	& --- & --- & --- \\ \hline
tl\_check	&575		&0.6	&13	&17		&26	&24	\\ \hline
\end{tabular}
\end{center}
\end{table}
$B@0?t78?tB?9`<0$KBP$7(B, $B$=$N(B {\bf maginitude} $B$r(B, $B78?t$N%S%C%HD9$NOB$GDj5A$9$k(B. 
{\it TL} $B$H(B {\it HTL} $B$N:9$r8+$k$?$a$K(B, 
$BI=(B \ref{magnitude} $B$G(B, $B7W;;ESCf$K$*$1$k:GBg(B magnitude $B$r<($9(B. 

\begin{table}[hbtp]
\caption{Maximal magnitude}
\label{magnitude}
\begin{center}
\begin{tabular}{|c||c|c|c|c|c|c|} \hline
	& $C(6)$ & $K(5)$ & $K(6)$ & $RoseO_1$ & $RoseO_2$ & Liu \\ \hline
{\it TL}& $>$ 735380 & $> 2407737 $ & $>$ 57368231 & 69764 & 947321 & $>$ 327330 \\ \hline
{\it HTL}& 1992 & 44187 & 422732 & 37220 & 70018 & 21095 \\ \hline
\end{tabular}
\end{center}
\end{table}
$BI=$h$jL@$i$+$K(B, {\it TL} $B$OHs@F<!B?9`<0$KBP$9$k%0%l%V%J4pDl7W;;$KIT8~$-(B
$B$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
{\it LA} $B$NM%0L@-$r<($7$F$$$k(B. $B$3$l$O(B, Buchberger $B%"%k%4%j%:%`$,(B
Euclid $B$N8_=|K!$KBP1~$7$F$$$F(B, $BCf4V78?tKDD%$G8zN($,:81&$5$l$k$N(B
$B$KBP$7(B, modular $B%"%k%4%j%:%`$N8zN($O7k2L$NBg$-$5$N$_$K0MB8$9$k(B
$B$3$H$K$h$k(B. 

\subsection{RUR}

RUR $B$N(B modular $B7W;;$N%?%$%_%s%0%G!<%?$r<($9(B. $B7W;;4D6-$OA0@a$HF1MM$G$"(B
$B$k(B. $B$3$3$G$O(B, $BM=$a(B modular $B7W;;$K$h$j(B separating element$B$r5a$a$F$"(B
$B$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,
$w$ $B$K4X$9$k(B RUR $B7W;;$r9T$&(B.  

\begin{tabbing}
$MMM\;\;$ \= \kill
$C(6)$ \> $w-(c_1+3c_2+9c_3+27c_4+81c_5+243c_6)$\\
$C(7)$ \> $w-(c_1+3c_2+9c_3+27c_4+81c_5+243c_6+729c_7)$\\
$K(n)$ \> $w-u_n$\\
$R(5)$ \> $w-(x_1-3x_2-2x_3+3x_4+2x_5)$\\
$R(6)$ \> $w-(x_1-3x_2-2x_3+3x_4+2x_5-4x_6)$\\
$D(3)$ \> $w-y$
\end{tabbing}
%\begin{table}[hbtp]
%\label{mrurdata}
%\caption{$BF~NO%$%G%"%k$K4X$9$k%G!<%?(B}
%\begin{center}
%\begin{tabular}{|c||c|c|c|c||c|c|c|c|c|} \hline
%	& $K(5)$	& $K(6)$	& $K(7)$	& $K(8)$	& $C(6)$& $C(7)$	& $R(5)$ & $R(6)$ & $D(3)$ \\ \hline
%$\dim_{\Q} R/I$	& 32 	& 64 	& 128	& 256	& 156	& 924	&144	&576	& 128 \\ \hline
%DRL GB& 0.8	& 7.2 	& 68	& 798 	& 3.1 	& 1616	& 11	& 1775 	& 30	\\ \hline
%\end{tabular}
%\end{center}
%\end{table}
\begin{table}[hbtp]
\label{mrurtab}
\caption{$B7W;;;~4V(B ($BIC(B)}
\begin{center}
\begin{tabular}{|c|c|c|c||c|c|c|c|c|} \hline
	& $K(6)$& $K(7)$& $K(8)$& $C(6)$& $C(7)$& $R(5)$ & $R(6)$ & $D(3)$ \\ \hline
Total	& 7.4 	& 69	& 1209	& 4.6 	& 1643	& 52 	& 8768	& 67	\\ \hline
Quick test& 0.4 	& 3.2	& 26	& 0.5	& 57 	& 6.5	& 384	& 3.1	\\ \hline
Normal form& 1.1	& 12	& 308	& 1.4	& 762 	& 15 	& 2861 	& 7.3	\\ \hline
Linear equation& 4.1	& 43	& 775	& 1.4	& 641	& 22	& 3841	& 45	\\ \hline
Garbage collection& 1.7 	& 10	& 100	& 1.2 	& 181	& 7.8 	& 1681	& 11	\\ \hline
\end{tabular}
\end{center}
\end{table}

%\begin{table}[hbtp]
%\label{maxblen}
%\caption{Maximal bit length of coefficients in LEX basis and the RUR}
%\begin{center}
%\begin{tabular}{|c||c|c|c|c|c|} \hline
%& $K(5)$ & $K(6)$	& $K(7)$	& $K(8)$ & $D(3)$ \\ \hline
%LEX & 1421 & 6704 & 36181 & --- & 6589 \\ \hline
%RUR & 120 & 249 & 592 & 1258 & 821 \\ \hline
%\end{tabular}
%\end{center}
%\end{table}
$BI=$G(B, Quick Test $B$O(B modular $B7W;;$G(B $w$$B$,(B separating element $B$H$J$k$3$H(B
$B$r%A%'%C%/$9$k;~4V(B, Normal Form $B$O(B, $B@~7AJ}Dx<0$r@8@.$9$k$?$a$N(B,
monomial $B$N@55,7A$N7W;;(B, Linear Equation $B$O(B, $B@~7AJ}Dx<05a2r$N;~4V$G$"(B
$B$k(B. $BI=(B \ref{mcotab} $B$HHf3S$9$l$P(B, $B$"$kJQ?t$,(B separating element $B$H$J$C(B
$B$F$$$k$h$&$JLdBj$G$O(B, RUR $B7W;;$N8zN($,Hs>o$K$h$$$3$H$,J,$+$k(B. $B$3$NM}M3(B
$B$O(B, RUR $B$K8=$l$k78?t$N(B bit $BD9$,(B, LEX $B4pDl$N$=$l$KHf$Y$FHs>o$K>.$5$$(B($BNc(B
$B$($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
$B$-$5$K1~$8$?<j4V$G$G$-$k$3$H$+$iJ,$+$k(B.

