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version 1.1, 2002/12/04 08:57:21 version 1.4, 2002/12/09 04:23:05
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 % $OpenXM$  % $OpenXM: OpenXM/doc/Papers/rims-2002-noro-ja.tex,v 1.3 2002/12/09 02:09:23 noro Exp $
 \documentclass[theorem]{jarticle}  \documentclass[theorem]{jarticle}
 \usepackage{jssac}  %\usepackage{jssac}
   \topmargin -0.5in
   \oddsidemargin 0in
   \evensidemargin 0in
   %
   \textwidth 6in
   \textheight 9in
   \columnsep 0.33in
   
 \def\HT{{\rm HT}}  \def\HT{{\rm HT}}
 \def\HC{{\rm HC}}  \def\HC{{\rm HC}}
   \def\GF{{\rm GF}}
 \def\GCD{{\rm GCD}}  \def\GCD{{\rm GCD}}
 \def\tdeg{{\rm tdeg}}  \def\tdeg{{\rm tdeg}}
 \def\pp{{\rm pp}}  \def\pp{{\rm pp}}
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Line 28 
 \section{$B$O$8$a$K(B}  \section{$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  $BK\9F$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  $B$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$$$F=R$Y$k(B.  $B%`$*$h$S$=$N<BAu$K$D$$$F=R$Y$k(B.
   
 \section{$BB?JQ?tB?9`<0$NL5J?J}J,2r$H(B GCD}  \section{$BB?JQ?tB?9`<0$NL5J?J}J,2r$H(B GCD}
   
Line 75  do \= \\
Line 83  do \= \\
    \> $a \leftarrow $ $BL$;HMQ$N(B $K$ $B$N85(B\\     \> $a \leftarrow $ $BL$;HMQ$N(B $K$ $B$N85(B\\
    \> $g_a \leftarrow \GCD(f_1|_{y=a},\ldots,f_m|_{y=a})$\\     \> $g_a \leftarrow \GCD(f_1|_{y=a},\ldots,f_m|_{y=a})$\\
    \> if \= $g \neq 0$ $B$+$D(B $\HT_<(g) = \HT_<(g_a)$ then \\     \> if \= $g \neq 0$ $B$+$D(B $\HT_<(g) = \HT_<(g_a)$ then \\
    \>    \> $adj \leftarrow \cdot h_g(a)/\HC_<(g_a)\cdot g_a - g(a))$\\     \>    \> $adj \leftarrow h_g(a)/\HC_<(g_a)\cdot g_a - g(a)$\\
    \>    \> if \= $adj = 0$ $B$+$D(B, $B$9$Y$F$N(B $f_i$ $B$KBP$7(B $g | hg\cdot f_i$  then \\     \>    \> 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 $\pp(g)$\\     \>    \>    \> return $\pp(g)$\\
    \>    \> endif\\     \>    \> endif\\
    \>    \> $g \leftarrow g+adj \cdot M(a)^{-1} \cdot M$; $M \leftarrow M\cdot (y-a)$\\     \>    \> $g \leftarrow g+adj \cdot M(a)^{-1} \cdot M$; $M \leftarrow M\cdot (y-a)$\\
    \> else if $\tdeg(\HT_<(g)) > \tdeg(\HT_<(g_a)$ then \\     \> else if $\tdeg(\HT_<(g)) > \tdeg(\HT_<(g_a))$ then \\
    \>    \> $g \leftarrow g_a$; $M \leftarrow  y-a$\\     \>    \> $g \leftarrow g_a$; $M \leftarrow  y-a$\\
    \> else if $\tdeg(\HT_<(g)) = \tdeg(\HT_<(g_a))$ then \\     \> else if $\tdeg(\HT_<(g)) = \tdeg(\HT_<(g_a))$ then \\
    \>    \> $g \leftarrow 0$; $M \leftarrow 1$\\     \>    \> $g \leftarrow 0$; $M \leftarrow 1$\\
Line 147  $x$ $B$O(B, $x$ $B$K4X$9$kHyJ,$,>C$($J$$$h$&$KA*$VI
Line 155  $x$ $B$O(B, $x$ $B$K4X$9$kHyJ,$,>C$($J$$$h$&$KA*$VI
   
 \subsection{2 $BJQ?t$N0x?tJ,2r(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,  $x$, $y$ $B$,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  $BL5J?J}$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.  $a = (a_1,\ldots,a_{n-2}) \in K^{n-2}$ $B$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$3$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, $B$+$D(B  ($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,  $f_a|_{y=0}$ $B$,L5J?J}$G$"$k$h$&(B,
 $BI,MW$,$"$l$P(B $y\rightarrow y+c$ $B$H$$$&J?9T0\F0$r9T$&(B.  $BI,MW$,$"$l$P(B $y\rightarrow y+c$ $B$H$$$&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)}  \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  $f_a(x,y)$ $B$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  $f_a(x,y) = g_0(x,y)h_0(x,y)$ $B$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  $B9T$&(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. $B$$$o$f$k<g78?tLdBj$r2sHr$9$k$?$a$K(B, $B??$N0x;R$N<g78?t(B  $B7h$aJ}$G$"$k(B. $BC1$K(B, $BAPJ}$N78?t$r(B $f$ $B$N(B $x$ $B$K4X$9$k<g78?t(B
   $\lc_x(f)$ $B$K9g$o$;$k$H$$$&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$$(B. $B>/$J$/$H$b(B,  $B$H$J$k$Y$/6a$$$b$N$r$"$i$+$8$a8GDj$7$F$*$/$N$,$h$$(B. $B>/$J$/$H$b(B,
 $B$=$l$O(B, $f$ $B$N(B, $x$ $B$K4X$9$k<g78?t(B $\lc_x(f)$ $B$N0x;R$G$O$"$k$,(B,  $B$=$l$O(B $\lc_x(f)$ $B$N0x;R$G$O$"$k(B.
 $\lc_x(f)$ $B$=$N$b$N$r$H$k$3$H$O0lHL$K(B overestimate $B$G$"$k(B.  
 $BM-M}?tBN>e$N>l9g(B, P. S. Wang $B$K$h$j<g78?t$N7hDjJ}K!$,Ds0F$5$l$F(B  $BM-M}?tBN>e$N>l9g(B, P. S. Wang $B$K$h$j<g78?t$N7hDjJ}K!$,Ds0F$5$l$F(B
 $B$$$k$,(B, $B$3$3$G$O<!$N$h$&$K8+@Q$b$k(B:  $B$$$k$,(B, $B$3$3$G$O<!$N$h$&$K8+@Q$b$k(B:
   
Line 201  $f \leftarrow \lc_g\cdot \lc_h/\lc_x(f) \cdot f$
Line 214  $f \leftarrow \lc_g\cdot \lc_h/\lc_x(f) \cdot f$
 $g_0$ $B$,@5$7$$0x;R$N<M1F$J$i$P(B, $\lc_x(g_0)$ $B$O(B  $g_0$ $B$,@5$7$$0x;R$N<M1F$J$i$P(B, $\lc_x(g_0)$ $B$O(B
 $B4{$K??$N0x;R$N<g78?t$KEy$7$$(B. $B$3$3$G$O(B, $g_0$, $h_0$ $B$+$i(B $K[y]$ $B>e$N(B  $B4{$K??$N0x;R$N<g78?t$KEy$7$$(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  Hensel $B9=@.$K$h$j(B, $f=g_kh_k \bmod I^{k+1}$, $B$?$@$7(B
 $I = <z_1-a_1,\ldots,z_{n-2}-a_{n-2}>$, $B$H$J$k(B $g_k$, $h_k$ $B$r(B EZ $BK!$K(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,  $B$h$j7W;;$9$k(B. $B$^$:(B, $z_i \rightarrow z_i+a_i$ $B$J$kJ?9T0\F0$K$h$j(B,
 $I=<z_1,\ldots,z_{n-2}>$ $B$H$7$F$*$/(B. $BDL>o$N(B EZ $BK!$G$O(B, $B78?t$KJ,?t$,(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,  $B8=$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.  $B$"$kBg$-$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,  $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,
Line 246  $g_k$ $B$^$?$O(B $h_k$ $B$G(B $f$ $B$r3d$C$F$_$k$
Line 259  $g_k$ $B$^$?$O(B $h_k$ $B$G(B $f$ $B$r3d$C$F$_$k$
 $B$+$i$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  $B$+$i$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.  $B$N0x?tJ,2r$,9T$($k$h$&$K$J$C$?(B.
   
 \subsection{$B78?t4D$H$7$F$N(B $R[y]/(y^d)$ $B$K$D$$$F(B}  \subsection{$B78?t4D$H$7$F$N(B $K[y]/(y^d)$ $B$K$D$$$F(B}
   
 Hensel $B9=@.$K$*$$$F$O(B, $R[y]/(y^d)$ $B$r(B, $BB?9`<0$N78?t4D$H$7$F(B  Hensel $B9=@.$K$*$$$F$O(B, $K[y]/(y^d)$ $B$r(B, $BB?9`<0$N78?t4D$H$7$F(B
 $B07$&I,MW$,$"$k(B. $B$3$N$?$a(B, $B?7$?$J?t$N7?$H$7$F(B, $R[y]/(y^d)$ $B$r(B  $B07$&I,MW$,$"$k(B. Asir $B$K$*$$$F$O(B, $B4{$K(B, $B>.0L?tM-8BBN(B $K$ $B$NBe?t3HBg(B
 $BI=$97?$rDj5A$7$?(B. $BB?JQ?tB?9`<0$O(B, Hensel $B9=@.$N:G=i$G(B, $B$3$N(B  $B$rI=8=$9$k7?$,$"$k$,(B, $B$3$l$O(B, $K[y]/(m(y))$ ($m(y)$ $B$O:G>.B?9`<0(B)
   $B$H$7$FI=8=$5$l$F$$$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  $B7?$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,  $B$*$/$3$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  $BDL>o$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.  $BB?9`<01i;;$,<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}  \begin{thebibliography}{99}
 \bibitem{B97-2}  \bibitem{B97-2}
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