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% $OpenXM: OpenXM/doc/Papers/rims-2002-noro-ja.tex,v 1.4 2002/12/09 04:23:05 noro Exp $
\documentclass[theorem]{jarticle}
%\usepackage{jssac}
\topmargin -0.5in
\oddsidemargin 0in
\evensidemargin 0in
%
\textwidth 6in
\textheight 9in
\columnsep 0.33in

\def\HT{{\rm HT}}
\def\HC{{\rm HC}}
\def\GF{{\rm GF}}
\def\GCD{{\rm GCD}}
\def\tdeg{{\rm tdeg}}
\def\pp{{\rm pp}}
\def\lc{{\rm lc}}
\def\Z{{\bf Z}}

\title{$BM-8BBN>e$NB?JQ?tB?9`<0$N0x?tJ,2r$K$D$$$F(B ($B$=$N(B 2)}

\author{$BLnO$(B $B@59T(B ($B?@8MBg!&M}(B)}

\begin{document}
\maketitle

\section{$B$O$8$a$K(B}

$BK\9F$G$O(B, \cite{funny01} $B$G=R$Y$?(B, $BM-8BBN>e$G$N(B 2 $BJQ?tB?9`<0$N0x?tJ,2r(B
$B$r4pAC$H$7$F(B, $B0lHL$NB?JQ?tB?9`<0$N(B GCD, $BL5J?J}J,2r(B, $B0x?tJ,2r%"%k%4%j%:(B
$B%`$*$h$S$=$N<BAu$K$D$$$F=R$Y$k(B.

\section{$BB?JQ?tB?9`<0$NL5J?J}J,2r$H(B GCD}

$B8=:_(B Risa/Asir $B$GMQ$$$F$$$k%"%k%4%j%:%`$O(B, 
$B0J2<$K=R$Y$k$h$&$K(B, 
Bernardin $B$NL5J?J}J,2r%"%k%4%j%:%`(B \cite{B97-2} $B$r(B
modify $B$7$?$b$N$G$"$k(B.

$F$ $B$rI8?t(B $p$ $B$NM-8BBN$H$7(B, $f \in F[x_1,\ldots,x_n]$ $B$H(B
$B$9$k(B. $'$ $B$,(B $d/dx_1$ $B$rI=$9$H$9$k(B. 
$$f = FGH, F=\prod f_i^{a_i}, G=\prod g_j^{b_j}, 
H=\prod h_k^{c_k}$$ 
($f_i, g_j, h_k$ $B$OL5J?J}(B, $B8_$$$KAG$G(B, 
$f_i' \neq 0$, $p \not{|}a_j$, $p | b_j$, $h_k' = 0$)
$B$H=q$/$H(B $f' = F'GH$ $B$,@.$jN)$D(B. 
$B$9$k$H(B $$\GCD(f,f') = \GCD(F,F')GH$$ $B$G(B, 
$\GCD(F,F') = \prod f_i^{a_i-1}$ $B$@$+$i(B 
$$f/\GCD(f,f')=\prod f_i.$$
$\prod f_i$ $B$G(B $f$ $B$r7+$jJV$73d$j(B, $B3d$j@Z$l$J$/$J$C$?(B
$BCJ3,$G(B, $B$=$N>&$H(B $\prod f_i$ $B$N(B GCD $B$r7W;;$9$k$3$H$G(B, 
$F$ $BCf$N=EJ#EY:G>.$N0x;R(B $f_1$ $B$,5a$^$k(B. $B$3$l$r7+$jJV$9$H(B
$F$ $B$,A4$FL5J?J}J,2r$G$-$k(B. 
$B;D$j$N(B $f$ $B$O(B $f = GH$ $B$H=q$1$k(B. $B$3$3$G(B $f' = 0$ $B$,(B
$B@.$jN)$D$3$H$KCm0U$9$k(B. $B0J>e$N<jB3$-$r(B
$B3F(B $x_i$ $B$K$D$$$F7+$jJV$7$F;D$C$?(B $f$ $B$O(B, 
$$df/dx_1 = \ldots = df/dx_n = 0$$
$B$rK~$?$9(B. $B$3$l$O(B, $BA4$F$N;X?t$,(B $p$ 
$B$G3d$j@Z$l$k$3$H$r0UL#$9$k(B. $B$9$k$H(B, $F$ $B$OI8?t(B $p$ $B$N(B
$BM-8BBN$@$+$i(B, $f = g^p$ $B$H=q$1$k$3$H$K$J$k(B. 
$B$3$N(B $g$ $B$KBP$7$F(B, $B0J>e$N<jB3$-$r:F5"E*$KE,MQ$9$k(B
$B$3$H$G(B, $f$ $B$NL5J?J}J,2r$,F@$i$l$k(B. 

$B$3$N%"%k%4%j%:%`$K$*$$$F(B, $g=\GCD(f,f')$ $B$N7W;;$,I,MW$H$J$k(B. 
$BI8?t$,(B 0 $B$N>l9g(B, $\GCD(g,f'/g)=1$ $B$,(B
$BJ]>Z$5$l$k$?$a(B, $B$3$N7W;;$KB?JQ?t$N(B Hensel $B9=@.$rMQ$$$k$3$H$,$G$-$k(B. 
$B$7$+$7(B, $B@5I8?t$N>l9g$K$O(B, 
$\GCD(g,f'/g) = \GCD(\GCD(F,F')GH,F'/\GCD(F,F'))$ $B$H$J$j(B, 
$B0x;R(B $GH$ $B$NB8:_$N$?$a(B, $B$3$N(B GCD $B$,(B 1 $B$H$O8B$i$J$$(B. $B$3$N$?$a(B, 
$B$d$`$J$/(B Brown $B$N%"%k%4%j%:%`(B ($BCf9q>jM>DjM}$K$h$k(B GCD $B$N7W;;(B)
$B$rMQ$$$F$$$k(B. 

\begin{tabbing}
GCD $B$N7W;;(B \\
$BF~NO(B : $f_1,\ldots,f_m \in K[X]$ ($K$ $B$OBN(B, $X$ $B$OJQ?t$N=89g(B)\\
$B=PNO(B : $\GCD(f_1,\ldots,f_m)$\\
$y \leftarrow$ $BE,Ev$JJQ?t(B; $Z \leftarrow X\setminus \{y\}$\\
$< \leftarrow K[Z]$ $B$NE,Ev$J9`=g=x(B; $B0J2<(B $f_i \in K[y][Z]$ $B$H$_$J$9(B\\
$h_i(y) \leftarrow \HT_<(f_i)$; $h_g(y) \leftarrow \GCD(h_1,\ldots,h_m)$\\
$g \leftarrow 0$; $M \leftarrow 1$\\
do \= \\
   \> $a \leftarrow $ $BL$;HMQ$N(B $K$ $B$N85(B\\
   \> $g_a \leftarrow \GCD(f_1|_{y=a},\ldots,f_m|_{y=a})$\\
   \> if \= $g \neq 0$ $B$+$D(B $\HT_<(g) = \HT_<(g_a)$ then \\
   \>    \> $adj \leftarrow h_g(a)/\HC_<(g_a)\cdot g_a - g(a)$\\
   \>    \> if \= $adj = 0$ $B$+$D(B, $B$9$Y$F$N(B $f_i$ $B$KBP$7(B $g | h_g\cdot f_i$  then \\
   \>    \>    \> return $\pp(g)$\\
   \>    \> endif\\
   \>    \> $g \leftarrow g+adj \cdot M(a)^{-1} \cdot M$; $M \leftarrow M\cdot (y-a)$\\
   \> else if $\tdeg(\HT_<(g)) > \tdeg(\HT_<(g_a))$ then \\
   \>    \> $g \leftarrow g_a$; $M \leftarrow  y-a$\\
   \> else if $\tdeg(\HT_<(g)) = \tdeg(\HT_<(g_a))$ then \\
   \>    \> $g \leftarrow 0$; $M \leftarrow 1$\\
   \> endif\\
end do
\end{tabbing}

$\HT$ $B$OF,9`(B, $\HC$ $B$OF,78?t(B, $\tdeg$ $B$OA4<!?t(B, $\pp$ $B$O86;OE*ItJ,(B
$B$rI=$9(B. 

$B$3$N%"%k%4%j%:%`$G$O(B, $f_1, \ldots, f_m$ $B$KBP$7(B, $B$"$kJQ?t(B $y$ $B$K$5$^$6$^$J(B
$BCM(B $a$ 
$B$rBeF~$7$F(BGCD $g_a$ $B$r7W;;$7(B, $B$=$l$i$rCf9q>jM>DjM}$G7k9g$9$k(B. $B$=$N:](B, 
$B;D$j$NJQ?t(B $Z$ $B$K4X$7$FE,Ev$J9`=g=x(B $<$ $B$r@_Dj$7(B, $B$=$N9`=g=x$K4X$9$kF,(B
$B9`$,Ey$7$$4V$O(B, $B??$N(B GCD $B$N@5$7$$<M1F$K$J$C$F$$$k$H(B
$B2>Dj$9$k(B. $B$=$l$^$G$H0[$J$kF,9`$r;}$D(B $g_a$ $B$,F@$i$l$?>l9g(B, 
$B$=$NF,9`$NA4<!?t$,(B, $B$=$l$^$G$NF,9`$NA4<!?t$h$j??$KBg$-$$>l9g$K$O(B, 
$BL@$i$+$K@5$7$/$J$$$N$G(B, $B<N$F$k(B. $B$^$?(B, $B??$K>.$5$$>l9g$K$O(B, $B$3$l$^$G(B
$B$N7k2L$O@5$7$/$J$$$3$H$K$J$j(B, $B?7$?$K(B $g_a$ $B$+$i%j%9%?!<%H$9$k(B. 
$B$b$7(B, $BA4<!?t$,Ey$7$1$l$P(B, $BN>J}$N7k2L$,@5$7$/$J$$$3$H$K$J$j(B, 
$BN>J}$r<N$F$F%j%9%?!<%H$9$k(B. 

$B$^$?(B, $B??$N(B GCD $B$N(B $<$ $B$K4X$9$kF,78?t$,(B
$f_i$ $B$NF,78?t$N(B GCD $B$G$"$k(B $h_g$ $B$N0x;R$K$J$C$F$$$k$3$H$rMQ$$(B, $BCf9q>jM>DjM}$r(B
$BE,MQ$9$k:](B, $BF,78?t$,(B $h_g$ $B$K$J$k$h$&$KD4@a$7$F$$$k(B. 

$B?7$?$J(B $a$ $B$GF@$?(B $g_a$ $B$H(B, $B$3$l$^$GF@$?(B $g$ $B$N(B $a$ $B$G$NCM$,(B
$BEy$7$$>l9g$K(B, $B<B:]$K(B $g$ $B$,(B $h_gf_i$ $B$r$9$Y$F3d$j@Z$k$+%A%'%C%/$r(B
$B9T$C$F$$$k(B. $B<B:]$N<BAu$K$*$$$F$O(B, $BE,Ev$J(B $f_i$ $B$rA*$s$G$*$-(B, 
cofactor $B$N$[$&$bCf9q>jM>DjM}$G9=@.$7$F$$$-(B, $B$=$A$i$G$b3d;;%A%'%C%/(B
$B$r9T$&$3$H$G(B, GCD, cofactor $B$$$:$l$+$,I|85$G$-$?;~E@$G%"%k%4%j%:%`(B
$B$,=*N;$G$-$k(B. 

$B$3$N%"%k%4%j%:%`$G$O(B, GCD $B$N(B $y$ $B$K4X$9$k<!?t$h$jB?$$(B $K$ $B$N85$,I,MW(B
$B$K$J$k(B. $K$ $B$N0L?t$,$3$l$KK~$?$J$$>l9g$K$O(B, $K$ $B$r3HBg$9$k(B. $B$3$l$K(B
$B$D$$$F$O8e$G=R$Y$k(B. 

\section{$BL5J?J}B?9`<0$N0x?tJ,2r(B}

$B0J2<(B, $f \in K[X]$ $B$OL5J?J}$H$9$k(B. $f$ $B$N0x?tJ,2r$O<!$N$h$&$K9T$&(B. 

\subsection{$B<gJQ?t(B $x$ $B$NA*Br(B}

$f$ $B$N0x?tJ,2r$O(B, $B$"$kJQ?t(B $x$ $B0J30$NJQ?t$KE,Ev$JCM$rBeF~$7$FF@$i$l$?(B
$x$ $B$N0lJQ?tB?9`<0$N0x?tJ,2r$r%?%M$+$i=g<!(B Hensel $B9=@.$K$h$j7W;;$9$k(B (EZ $BK!(B).
$x$ $B$O(B, $x$ $B$K4X$9$kHyJ,$,>C$($J$$$h$&$KA*$VI,MW$,$"$k(B. 
$B$^$?(B, $x$ $B$K4X$9$k<!?t$,Bg$-$$Dx(B, $B%?%M$H$J$k0x;R$N?t$,Bg$-$/$J$k2DG=@-$,(B
$B$"$k$?$a(B, $B$J$k$Y$/<!?t$,>.$5$$$h$&$J(B $x$ $B$rA*$s$G$$$k(B. 

\subsection{$B=>JQ?t(B $y$ $B$NA*Br(B}

$B=>JQ?t$H$OL/$JMQ8l$G$"$k$,(B, $B$3$3$G$O(B, $BB?JQ?t$N0x?tJ,2r$r(B, 2 $BJQ?t$N0x?t(B
$BJ,2r$+$i(B Hensel $B9=@.$K$h$jF@$k$?$a(B, $B$=$N$?$a$NJQ?t$r$b$&0l$DA*$s$G$*$/(B
$BI,MW$,$"$k(B. $B$9$J$o$A(B, $B$"$kJQ?t(B $y$ $B$rA*$S(B, $x$, $y$ $B0J30$NJQ?t$K(B
$BE,Ev$JCM$rBeF~$7$?(B 2 $BJQ?tB?9`<0$N0x?tJ,2r$r(B \cite{funny01} $B$G=R$Y$?(B
$BJ}K!$K$h$j7W;;$9$k(B. $B$=$l$r4p$K(B, $B;D$j$NJQ?t$K4X$7$F(B Hensel $B9=@.$r(B
$B9T$&(B. $B$3$3$G(B, Hensel $B9=@.$O(B $K[X] = K[y][x,Z]$ 
($Z=X\setminus\{x,y\} = \{z_1,\ldots,z_{n-2}\}$) $B$H$_$J$7$F(B
$B9T$&(B. $B$9$J$o$A(B, $B0lJQ?tB?9`<04D(B $K[y]$ $B$r(B, $BM-M}?tBN>e$NB?JQ?tB?9`<0(B
$B$N0x?tJ,2r$K$*$1$k(B, $B@0?t4D$NN`;w$H$_$J$9$o$1$G$"$k(B. $B$3$&$9$k$3$H$K(B
$B$h$j(B, 1 $BJQ?t$NJ,2r$+$i=PH/$7$?>l9g$K@8$:$kBgNL$N%K%;0x;R$K$h$k:$Fq$r(B
$BHr$1$k$3$H$,$G$-$k(B. $B$"$H$G<($9$h$&$K(B, Hensel $B9=@.<+BN$b(B, $B@0?t>e$N(B
$B>l9g$NN`;w$NJ}K!$K$h$j9T$&$3$H$,$G$-$k(B. 

\subsection{2 $BJQ?t$N0x?tJ,2r(B}

$x$, $y$ $B$,7h$^$C$?$i(B, $f_a(x,y) = f(x,y,a)$ $B$,(B 
$BL5J?J}$K$J$k$h$&$K(B $Z$ $B$KBeF~$9$kCM$N%Y%/%H%k(B
$a = (a_1,\ldots,a_{n-2}) \in K^{n-2}$ $B$rA*$S(B, $f_a$ $B$r0x?tJ,2r$9$k(B. 
$B$3$3$G(B, $f_a$ $B$N(B $x$ $B$K4X$9$k<g78?t(B 
($B$3$l$O(B $y$ $B$NB?9`<0(B) $B$NDj?t9`$,(B 0 $B$G$J$/(B, $B$+$D(B
$f_a|_{y=0}$ $B$,L5J?J}$G$"$k$h$&(B, 
$BI,MW$,$"$l$P(B $y\rightarrow y+c$ $B$H$$$&J?9T0\F0$r9T$&(B. 
$B<B:]$K$O(B, $B$3$NA`:n$O(B 2 $BJQ?t$N0x?tJ,2r$G(B, $y$ $B$X$NBeF~CM$rC5$9(B
$B2>Dj$G9T$o$l$F$$$k$?$a(B, $B$=$N7k2L$r$=$N$^$^JV$9$h$&$JFbIt(B
$B%5%V%k!<%A%s$r8F$S=P$7$F$$$k(B. 

\subsection{$K[y]$ $B>e$G$N(B Hensel $B9=@.(B ($BA0=hM}(B)}

$f_a(x,y)$ $B$N0x?tJ,2r$N7k2L$h$j(B, $B0x;R$r(B 2 $BAH$K$o$1(B
$f_a(x,y) = g_0(x,y)h_0(x,y)$ $B$H$7$?>e$G(B, $K[y]$ $B>e$G(B Hensel $B9=@.$r(B
$B9T$&(B. $B$3$N:](B, $BLdBj$H$J$k$N$,(B $g_0$, $h_0$ $B$N(B $x$ $B$K4X$9$k<g78?t$N(B
$B7h$aJ}$G$"$k(B. $BC1$K(B, $BAPJ}$N78?t$r(B $f$ $B$N(B $x$ $B$K4X$9$k<g78?t(B
$\lc_x(f)$ $B$K9g$o$;$k$H$$$&J}K!(B
$B$G$O(B, Hensel $B9=@.$NCJ?t$,ITI,MW$KA}$($k(B, $B$$$o$f$k<g78?tLdBj(B
$B$r5/$3$9(B. $B$3$l$r2sHr$9$k$?$a$K(B, $B??$N0x;R$N<g78?t(B
$B$H$J$k$Y$/6a$$$b$N$r$"$i$+$8$a8GDj$7$F$*$/$N$,$h$$(B. $B>/$J$/$H$b(B, 
$B$=$l$O(B $\lc_x(f)$ $B$N0x;R$G$O$"$k(B. 
$BM-M}?tBN>e$N>l9g(B, P. S. Wang $B$K$h$j<g78?t$N7hDjJ}K!$,Ds0F$5$l$F(B
$B$$$k$,(B, $B$3$3$G$O<!$N$h$&$K8+@Q$b$k(B:

\begin{enumerate}
\item $\lc_x(f) = \prod u_i^{n_i}$ $B$H0x?tJ,2r$9$k(B ($u_i \in K[y,Z]$ : $B4{Ls(B).

\item $B3F(B $i$ $B$KBP$7(B, $u_i(a) \in K[y]$ $B$,(B $\lc_x(g_0)$ $B$r3d$j@Z$k2s?t$r(B
$B?t$($k(B. $B$=$l$r(B $m_i$ $B$H$7$?$H$-(B, $\lc_g = \prod u_i^{m_i}$ $B$H$9$k(B. 
$BF1MM$K(B $h_0$ $B$KBP$7$F$b(B $\lc_h$ $B$r5a$a$k(B. 

$B$b$7(B, 
$\lc_x(g_0) \not{|}\, \lc_g(a)$ $B$^$?$O(B $\lc_x(h_0) \not{|}\, \lc_h(a)$
$B$^$?$O(B, 
$\lc_x(f) \not{|}\, \lc_g \cdot \lc_h$
$B$J$i$P(B, $B$=$l$O(B, $f_a$ $B$N0x;R$NAH9g$;$,@5$7$/$J$$$3$H$r0UL#$9$k$N$G(B, 
$g_0$, $h_0$ $B$r$H$jD>$9(B. 

\item

$g_0 \leftarrow \lc_g(a)/\lc_x(g_0)\cdot g_0$ $B$N<g78?t$r(B $\lc_g$ $B$GCV$-49$($?$b$N(B

$h_0 \leftarrow \lc_h(a)/\lc_x(h_0)\cdot h_0$ $B$N<g78?t$r(B $\lc_h$ $B$GCV$-49$($?$b$N(B

$f \leftarrow \lc_g\cdot \lc_h/\lc_x(f) \cdot f$

$B$H$9$k(B. $B$3$N;~(B, $f = g_0h_0$ $B$H$J$C$F$$$k(B. 
\end{enumerate}
$B$3$NCJ3,$G(B, $B$b$7(B $g_0$ $B$,??$N0x;R$N<M1F$H$J$C$F$$$l$P(B, Hensel $B9=@.(B
$B$K$h$j(B, $f$ $B$N0x;R$K;}$A>e$k$O$:$G$"$j(B, $\lc_x(g_0)$ $B$O4{$K(B, $B??$N0x;R(B
$B$N<g78?t$KEy$7$/$J$C$F$$$k(B. $B$3$N>l9g(B, $h_0$ $B$bF1MM$N@-<A$rK~$?$9(B. 

\subsection{$K[y]$ $B>e$G$N(B Hensel $B9=@.(B}

$BA09`$K$h$j(B, $f=g_0h_0$ $B$K$*$$$F(B, 
$g_0$ $B$,@5$7$$0x;R$N<M1F$J$i$P(B, $\lc_x(g_0)$ $B$O(B
$B4{$K??$N0x;R$N<g78?t$KEy$7$$(B. $B$3$3$G$O(B, $g_0$, $h_0$ $B$+$i(B $K[y]$ $B>e$N(B
Hensel $B9=@.$K$h$j(B, $f=g_kh_k \bmod I^{k+1}$, $B$?$@$7(B 
$I = \langle z_1-a_1,\ldots,z_{n-2}-a_{n-2} \rangle$, $B$H$J$k(B $g_k$, $h_k$ $B$r(B EZ $BK!$K(B
$B$h$j7W;;$9$k(B. $B$^$:(B, $z_i \rightarrow z_i+a_i$ $B$J$kJ?9T0\F0$K$h$j(B, 
$I=\langle z_1,\ldots,z_{n-2} \rangle$ $B$H$7$F$*$/(B. $BDL>o$N(B EZ $BK!$G$O(B, $B78?t$KJ,?t$,(B
$B8=$l$k$N$rHr$1$k$?$a(B, $B0x;R$N78?t$NBg$-$5$NI>2A$+$iDj$a$i$l$k(B, 
$B$"$kBg$-$JAG?t6R(B $p^l$ $B$rK!$H$7$F(B $\Z/(p^l)$ $B>e$G7W;;$9$k(B. 
$B$3$3$G$O(B, $f$ $B$N(B $y$ $B$K4X$9$k<!?t$r1[$($k@0?t(B $d$ $B$KBP$7(B, 
$K[y]/(y^d)$ $B$G$N1i;;$rF3F~$9$k$3$H$G(B, $K[y]$ $B$G$N>&BN$G$N1i;;$rHr$1$k(B. 
$B$9$J$o$A(B, $u g_0(a)+v h_0(a)=1 \bmod y^d$ $B$H$J$k(B $u, v \in K[y]$ $B$r(B
$B7W;;$7$F$*$-(B, Hensel $B9=@.$N78?t$N7W;;$O(B $u, v$ $B$rMQ$$$F(B $\bmod\,  y^d$
$B$G9T$&$N$G$"$k(B. $B$3$N$H$-(B, $f = g_kh_k \bmod (I^{k+1},y^d)$ $B$@$,(B, 
$d$ $B$,==J,Bg$-$$$?$a(B, $g_0$ $B$,??$N0x;R$N<M1F$J$i$P(B, Hensel $B9=@.$N0l0U(B
$B@-$K$h$j(B, $B==J,Bg$-$$(B $k$ $B$KBP$7(B $f = g_kh_k$ $B$H$J$k(B. $u$, $v$ $B$N7W;;$O(B, 
$g_0(a)|_{y=0}$, $h_0(a)_{y=0}$ $B$,8_$$$KAG$G$"$k$3$H$rMxMQ$7$F(B, 
$B$d$O$j(B Hensel $B9=@.$K$h$j7W;;$G$-$k(B. 

$K[y]$ $B>e$N(B Hensel $B9=@.$O<!$N$h$&$K9T$&(B. 

\begin{enumerate}
\item $f-g_{k-1}h_{k-1} = \sum_t F_t t \bmod (I^k,y^d)$ $B$H=q$/(B. 
$B$3$3$G(B $t \in I^k$ $B$OC19`<0(B, $F_t \in K[y][x]$.
\item $G_th_0+H_tg_0 = F_t \bmod y^d$ $B$H$J$k(B $G_t, H_t \in K[y][x]$ $B$r7W;;$9$k(B. 
$B$3$l$O(B $u$, $v$ $B$r;H$C$F:n$l$k(B. 
\item $g_{k+1} \leftarrow g_k + \sum_t G_t t$,
$h_{k+1} \leftarrow h_k + \sum_t H_t t$ $B$H$9$l$P(B $f = g_{k+1}h_{k+1} \bmod (I^{k+1},y^d)$.
\end{enumerate}

$B$3$NA`:n$K$*$$$F(B, $\sum_t G_t t$ $B$^$?$O(B $\sum_t H_t t$ $B$,(B 0 $B$H$J$C$?>l9g$K(B
$g_k$ $B$^$?$O(B $h_k$ $B$G(B $f$ $B$r3d$C$F$_$k$3$H$G(B, $B<!?t$N>e8B$^$G(B Hensel $B9=@.(B
$B$;$:$K(B, $B??$N0x;R$r8!=P$9$k$3$H$,$G$-$k(B. 


\section{$B<BAu$K$D$$$F(B}

\subsection{$BM-8BBN$NI=8=$K$D$$$F(B}

$B$3$3$G=R$Y$?3F%"%k%4%j%:%`$K$*$$$F$O(B, $B78?tBN$N0L?t$,==J,Bg$-$$I,MW$,$"(B
$B$k(B. \cite{funny01} $B$G=R$Y$?$h$&$K(B, $BBeF~$9$kE@$N?t$,ITB-$9$k>l9g$N$?$a(B
$B$K(B, $BBe?t3HBg$7$F$b7W;;8zN($,Mn$A$J$$$h$&$J(B, $B86;O:,$rMQ$$$?I=8=$r<BAu$7(B,
$B$=$NM-8z@-$r<($7$?(B. $B7gE@$H$7$F$O(B, $B$3$NI=8=$K$*$$$F(B, $B<BMQE*$J(B
$B0L?t$,(B $2^{16}$ $BDxEY$K8B$i$l$k$3$H$G$"$C$?(B. $B:#2s(B, $B$=$N2~NI$H$7$F(B, $BI8?t(B
$B$,(B $2^{14}$ $B0J2<$N>l9g$K$O(B, $B86;O:,I=8=$r5v$7(B, $B$=$l0J>e$N>l9g$K$O(B, $BDL>o$N(B
$BI=8=$r$H$k$3$H$K$7$?(B. $B$3$l$O(B, $B8e<T$G$O<BMQ>e==J,$KBeF~$9$kE@$,F@$i$l$k(B
$B$+$i$G$"$k(B. $B$3$l$K$h$j(B, $B0L?t$,(B $2^{29}$ $BDxEY$^$G$NAGBN>e$G(B, $BB?JQ?tB?9`<0(B
$B$N0x?tJ,2r$,9T$($k$h$&$K$J$C$?(B. 

\subsection{$B78?t4D$H$7$F$N(B $K[y]/(y^d)$ $B$K$D$$$F(B}

Hensel $B9=@.$K$*$$$F$O(B, $K[y]/(y^d)$ $B$r(B, $BB?9`<0$N78?t4D$H$7$F(B
$B07$&I,MW$,$"$k(B. Asir $B$K$*$$$F$O(B, $B4{$K(B, $B>.0L?tM-8BBN(B $K$ $B$NBe?t3HBg(B
$B$rI=8=$9$k7?$,$"$k$,(B, $B$3$l$O(B, $K[y]/(m(y))$ ($m(y)$ $B$O:G>.B?9`<0(B)
$B$H$7$FI=8=$5$l$F$$$k$?$a(B, $m(y)=y^d$ $B$H$9$l$P(B, $B2C8:>h;;$ON.MQ(B
$B$G$-$k(B. $B$^$?(B, $B=|;;$K8=$l$k5U857W;;$K$D$$$F$O(B, 0 $B$G$J$$Dj?t9`$r;}$D(B
$BB?9`<0$GI=8=$5$l$k85$K8B$l$P(B, $B$=$l$O(B $y^d$ $B$H8_$$$KAG$J$N$G(B
$B5U85$r;}$A(B, $B8_=|K!$G7W;;$G$-$k(B. $B4{$K=R$Y$?$h$&$K(B, $x$ $B$K4X$9$k<g78?t(B
$B$,(B 0 $B$G$J$$Dj?t9`$r;}$D$h$&$KJ?9T0\F0$7$F$"$k$N$G(B, $K[y]/(y^d)$ $B$N(B
$B7W;;$r(B, $B$3$3$G=R$Y$?J}K!$G9T$&$3$H$,$G$-$k(B. 
$BB?JQ?tB?9`<0$O(B, Hensel $B9=@.$N:G=i$G(B, $B$3$N(B
$B7?$N78?t$r;}$DB?9`<0$KJQ49$5$l$k(B. $B$"$i$+$8$a(B $d$ $B$r%;%C%H$7$F(B
$B$*$/$3$H$K$h$j(B, $B1i;;$O<+F0E*$K(B $\bmod \, y^d$ $B$5$l$k$?$a(B, 
$BDL>o$NB?9`<01i;;$N8F$S=P$7$r9T$&$@$1$G(B, $K[y]/(y^d)$ $B78?t$N(B
$BB?9`<01i;;$,<B9T$G$-$k(B. 

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

$BM-8BBN>e$NB?JQ?tB?9`<0$N0x?tJ,2r$rDs6!$7$F$$$k%7%9%F%`$O?t>/$J$$(B. 
$BI.<T$NCN$kM#0l$N$b$N$O(B Maple $B$J$N$G(B, Maple $B$HHf3S$r9T$&(B. Maple $B$O(B, 
$B$3$N5!G=$K4X$7$F$O(B, kernel $B$K$*$$$F@lMQ$NFC<l$J%G!<%?7?$rB?MQ$7$F(B
$B8zN($r>e$2$F$$$k$?$a(B, $BHf3S$9$k$3$H$O%"%s%U%'%"$G$O$J$$$H9M$($k(B. 
$BNc$H$7$F$O(B, P.S. Wang $B$K$h$k(B, $B<g78?tLdBj$r5/$3$7$d$9$$B?9`<0$N(B
$BNc(B (Asir $B$N(B {\tt lib/fctrdata} $B$N(B {\tt Wang[1]} $B$+$i(B
{\tt Wang[15]} $B$rMQ$$$?(B. $B%^%7%s$O(B Athlon 1900+ $B$GC10L$OIC(B, 
$p$ $B$O(B, $B4pACBN$NI8?t$rI=$9(B. $BI=$G(B $N$ $B$O(B 60 $BICBT$C$F$bEz$($,(B
$B$G$J$$$b$N(B, $F$ $B$O(B, Maple $B$,2?$i$+$NM}M3$G%(%i!<$r=P$7$F(B
$B7W;;$G$-$J$+$C$?$b$N$rI=$9(B. Asir $B$K$*$$$F(B $p(n)$ $B$O(B, $n$ $B<!3HBg(B,
$B$9$J$o$A(B $\GF(p^n)$ $B>e$G0x?tJ,2r$r9T$C$?$3$H$r<($9(B. 8 $BHV$NB?9`<0(B
$B$N$h$&$KNc30E*$K;~4V$,$+$+$k$b$N$O$"$k$,(B, $B$*$*$`$M(B, Asir $B$,(B
$BNI9%$J%Q%U%)!<%^%s%9$r<($7$F$$$k(B. 

\begin{table}[hbtp]
\begin{center}
% &  &  &  &  &  &  &  &  &  &  &  &  &  &  &  \\ \hline
{\normalsize
\begin{tabular}{c|ccccccccc} \hline
$p$ & 1 & 2 & 3 & 4 & 5 & 6 & 7 & 8 & 9 \\ \hline
2 & N & F & F & F & N & N & 0.01 & 1 & 0.01 \\ \hline
3 & 0.07 & 0.1 & 0.07 & N & 0.4 & N & 0.01 & 0.02 & 0.06 \\ \hline
5 & N & 0.05 & 0.08 & 3.5 & 0.2 & 0.4 & 0.01 & 0.6 & 0.1 \\ \hline
7 & 0.08  & 0.1  & 0.1  & 0.25 & 0.6  & 0.5  & 0.02  & 1  & F \\ \hline
\end{tabular}

\begin{tabular}{c|cccccc} \hline
$p$ & 10 & 11 & 12 & 13 & 14 & 15 \\ \hline
2 & F & N & 0.005 & 0.006 & 0.008 & F \\ \hline
3 & 4 & N & 0.004 & 0.007 & 0.14  & 0.02 \\ \hline
5 & 0.2  & F  & 0.005 & 0.006 & 0.03  &  0.4 \\ \hline
7 & 0.6  & 14  & 0.005  & 0.16  & 0.04 & 0.6 \\ \hline
\end{tabular}

\begin{tabular}{c|ccccccccc} \hline
$p$ & 1 & 2 & 3 & 4 & 5 & 6 & 7 & 8 & 9 \\ \hline
547 & 0.2& 0.2& 0.1& 0.3& 1& 1.2& 0.02& 6& F\\ \hline
32003&  0.2& 0.2&  0.2& 0.4 & 1 & 1 & 0.02 & 4.2 & F  \\ \hline
99981793 & 0.5 & 0.6 & 0.5 & 3 & 3 & 4.5 & 0.02 & N &  F\\ \hline
\end{tabular}

\begin{tabular}{c|cccccc} \hline
$p$ & 10 & 11 & 12 & 13 & 14 & 15 \\ \hline
547 & 0.9 &3.3  & 0.005 & 0.2 & 0.1 & 0.4  \\ \hline
32003 & 1.8 & 4.9 &0.006  & 0.3 & 0.1  & 0.4 \\ \hline
99981793 & 2.6  & 11  & 0.006 & 0.9 & 0.5  & 1.4  \\ \hline
\end{tabular}
}
\end{center}
\caption{$B0x?tJ,2r(B (Maple7)}
\end{table}

\begin{table}[hbtp]
\begin{center}
% &  &  &  &  &  &  &  &  &  &  &  &  &  &  &  \\ \hline
{\normalsize
\begin{tabular}{c|ccccccccc} \hline
$p$ & 1 & 2 & 3 & 4 & 5 & 6 & 7 & 8 & 9 \\ \hline
2(5) & 0.003 & 0.003 & 0.004 & 0.01 & 0.02 & 0.05 & 0.001 & 0.01 & 0.0003 \\ \hline
3(5) & 0.003 & 0.002 & 0.005 & 0.003 & 0.003 & 0.1 & 0.002 & 0.001 & 0.003  \\ \hline
5(2) & 0.004 & 0.003 & 0.004 & 0.02 & 0.06 & 0.4 & 0.002 & 0.4 & 0.005 \\ \hline
7(2) & 0.004  & 0.004  & 0.005 & 0.03 & 0.1  & 0.1  & 0.004  & 1.8  & 0.2 \\ \hline
\end{tabular}

\begin{tabular}{c|cccccc} \hline
$p$ & 10 & 11 & 12 & 13 & 14 &  15 \\ \hline
2(5) & 0.03 & 0.07 & 0.0006 & 0.001 & 0.002 & 0.001  \\ \hline
3(5) & 0.04 & 0.2 & 0.0001  & 0.0005 & 0.02  & 0.001 \\ \hline
5(2) & 0.01  & 0.2  & 0.001 & 0.001 & 0.004  & 0.01  \\ \hline
7(2) & 0.02  & 0.6  & 0.001  & 0.007  & 0.005 & 0.01 \\ \hline
\end{tabular}

\begin{tabular}{c|ccccccccc} \hline
$p$ & 1 & 2 & 3 & 4 & 5 & 6 & 7 & 8 & 9 \\ \hline
547 & 0.004 & 0.004 & 0.005 & 0.03 & 0.05 & 0.2 & 0.02& 2& 0.2\\ \hline
32003 & 0.004  &  0.004 & 0.005  &0.04  &0.07  & 0.2 & 0.004 & 2 & 0.2 \\ \hline
99981793& 0.004 & 0.004& 0.005 & 0.03 & 0.03 & 0.2 & 0.004 & 4 & 0.2  \\ \hline
\end{tabular}

\begin{tabular}{c|cccccc} \hline
$p$ & 10 & 11 & 12 & 13 & 14 &  15 \\ \hline
547 & 0.04 & 0.3 & 0.001 &0.006  & 0.006 & 0.01  \\ \hline
32003 & 0.04 &0.2  &0.001  &0.007  & 0.006  & 0.03 \\ \hline
99981793 & 0.04  & 0.3  &0.001  & 0.008 & 0.008  & 0.01  \\ \hline
\end{tabular}
}
\end{center}
\caption{$B0x?tJ,2r(B (Asir)}
\end{table}
\section{$B$*$o$j$K(B}

$B:#8e$NM=Dj$H$7$F(B, $B<!$N$h$&$J$3$H$r9M$($F$$$k(B. 

\begin{itemize}
\item $B@5I8?t$N=`AGJ,2r$N<BAu(B. 

\item $BBN$N0L?t$,B-$j$J$$>l9g$K(B, $B<+F0E*$K4pACBN$r3HBg$9$k(B. 

\item $BBN$NI8?t$,==J,Bg$-$$>l9g$K(B, $BL5J?J}J,2r$rI8?t(B 0 $B$H(B
$BF1MM$N(B Hensel $B9=@.$G9T$&$h$&$K$9$k(B. 

\item 2 $BJQ?t$N0x?tJ,2r$K$*$$$F(B, \cite{funny01} $B$G=R$Y$?(B, $BB?9`<0(B
$B;~4V%"%k%4%j%:%`$r<+F0E*$KA*Br$7$F<B9T$9$k(B. 
\end{itemize}

\begin{thebibliography}{99}
\bibitem{B97-2}
Bernardin, L. (1997).
On square-free factorization of multivariate polynomials over a finite
field.
{\em Theoret.\ Comput.\ Sci.\/} {\bf 187}, 105--116. 

\bibitem{funny01}
M. Noro and K. Yokoyama (2002).
Yet Another Practical Implementation of Polynomial Factorization
 over Finite Fields.
Proceedings of ISSAC2002, ACM Press, 200--206.
\end{thebibliography}
\end{document}