version 1.2, 2002/12/06 09:23:42 |
version 1.4, 2002/12/09 04:23:05 |
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% $OpenXM: OpenXM/doc/Papers/rims-2002-noro-ja.tex,v 1.1 2002/12/04 08:57:21 noro Exp $ |
% $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} |
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\topmargin -0.5in |
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\oddsidemargin 0in |
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\evensidemargin 0in |
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% |
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\textwidth 6in |
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\textheight 9in |
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\columnsep 0.33in |
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\def\HT{{\rm HT}} |
\def\HT{{\rm HT}} |
\def\HC{{\rm HC}} |
\def\HC{{\rm HC}} |
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\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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\section{$B$O$8$a$K(B} |
\section{$B$O$8$a$K(B} |
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$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. |
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\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} |
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\> $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$$$h$&$KA*$VI |
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Line 155 $x$ $B$O(B, $x$ $B$K4X$9$kHyJ,$,>C$($J$$$h$&$KA*$VI |
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\subsection{2 $BJQ?t$N0x?tJ,2r(B} |
\subsection{2 $BJQ?t$N0x?tJ,2r(B} |
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$x$, $y$ $B$,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. |
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$B<B:]$K$O(B, $B$3$NA`:n$O(B 2 $BJQ?t$N0x?tJ,2r$G(B, $y$ $B$X$NBeF~CM$rC5$9(B |
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$B2>Dj$G9T$o$l$F$$$k$?$a(B, $B$=$N7k2L$r$=$N$^$^JV$9$h$&$JFbIt(B |
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$B%5%V%k!<%A%s$r8F$S=P$7$F$$$k(B. |
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\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)} |
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$f_a(x,y)$ $B$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$$$o$f$k<g78?tLdBj$r2sHr$9$k$?$a$K(B, $B??$N0x;R$N<g78?t(B |
$B7h$aJ}$G$"$k(B. $BC1$K(B, $BAPJ}$N78?t$r(B $f$ $B$N(B $x$ $B$K4X$9$k<g78?t(B |
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$\lc_x(f)$ $B$K9g$o$;$k$H$$$&J}K!(B |
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$B$G$O(B, Hensel $B9=@.$NCJ?t$,ITI,MW$KA}$($k(B, $B$$$o$f$k<g78?tLdBj(B |
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$B$r5/$3$9(B. $B$3$l$r2sHr$9$k$?$a$K(B, $B??$N0x;R$N<g78?t(B |
$B$H$J$k$Y$/6a$$$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$$(B. $B>/$J$/$H$b(B, |
$B$=$l$O(B, $f$ $B$N(B, $x$ $B$K4X$9$k<g78?t(B $\lc_x(f)$ $B$N0x;R$G$O$"$k$,(B, |
$B$=$l$O(B $\lc_x(f)$ $B$N0x;R$G$O$"$k(B. |
$\lc_x(f)$ $B$=$N$b$N$r$H$k$3$H$O0lHL$K(B overestimate $B$G$"$k(B. |
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$BM-M}?tBN>e$N>l9g(B, P. S. Wang $B$K$h$j<g78?t$N7hDjJ}K!$,Ds0F$5$l$F(B |
$BM-M}?tBN>e$N>l9g(B, P. S. Wang $B$K$h$j<g78?t$N7hDjJ}K!$,Ds0F$5$l$F(B |
$B$$$k$,(B, $B$3$3$G$O<!$N$h$&$K8+@Q$b$k(B: |
$B$$$k$,(B, $B$3$3$G$O<!$N$h$&$K8+@Q$b$k(B: |
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Line 201 $f \leftarrow \lc_g\cdot \lc_h/\lc_x(f) \cdot f$ |
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Line 214 $f \leftarrow \lc_g\cdot \lc_h/\lc_x(f) \cdot f$ |
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$g_0$ $B$,@5$7$$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$$(B. $B$3$3$G$O(B, $g_0$, $h_0$ $B$+$i(B $K[y]$ $B>e$N(B |
Hensel $B9=@.$K$h$j(B, $f=g_kh_k \bmod I^{k+1}$, $B$?$@$7(B |
Hensel $B9=@.$K$h$j(B, $f=g_kh_k \bmod I^{k+1}$, $B$?$@$7(B |
$I = <z_1-a_1,\ldots,z_{n-2}-a_{n-2}>$, $B$H$J$k(B $g_k$, $h_k$ $B$r(B EZ $BK!$K(B |
$I = \langle z_1-a_1,\ldots,z_{n-2}-a_{n-2} \rangle$, $B$H$J$k(B $g_k$, $h_k$ $B$r(B EZ $BK!$K(B |
$B$h$j7W;;$9$k(B. $B$^$:(B, $z_i \rightarrow z_i+a_i$ $B$J$kJ?9T0\F0$K$h$j(B, |
$B$h$j7W;;$9$k(B. $B$^$:(B, $z_i \rightarrow z_i+a_i$ $B$J$kJ?9T0\F0$K$h$j(B, |
$I=<z_1,\ldots,z_{n-2}>$ $B$H$7$F$*$/(B. $BDL>o$N(B EZ $BK!$G$O(B, $B78?t$KJ,?t$,(B |
$I=\langle z_1,\ldots,z_{n-2} \rangle$ $B$H$7$F$*$/(B. $BDL>o$N(B EZ $BK!$G$O(B, $B78?t$KJ,?t$,(B |
$B8=$l$k$N$rHr$1$k$?$a(B, $B0x;R$N78?t$NBg$-$5$NI>2A$+$iDj$a$i$l$k(B, |
$B8=$l$k$N$rHr$1$k$?$a(B, $B0x;R$N78?t$NBg$-$5$NI>2A$+$iDj$a$i$l$k(B, |
$B$"$kBg$-$JAG?t6R(B $p^l$ $B$rK!$H$7$F(B $\Z/(p^l)$ $B>e$G7W;;$9$k(B. |
$B$"$kBg$-$JAG?t6R(B $p^l$ $B$rK!$H$7$F(B $\Z/(p^l)$ $B>e$G7W;;$9$k(B. |
$B$3$3$G$O(B, $f$ $B$N(B $y$ $B$K4X$9$k<!?t$r1[$($k@0?t(B $d$ $B$KBP$7(B, |
$B$3$3$G$O(B, $f$ $B$N(B $y$ $B$K4X$9$k<!?t$r1[$($k@0?t(B $d$ $B$KBP$7(B, |
Line 269 Hensel $B9=@.$K$*$$$F$O(B, $K[y]/(y^d)$ $B$r(B, $ |
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Line 282 Hensel $B9=@.$K$*$$$F$O(B, $K[y]/(y^d)$ $B$r(B, $ |
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$BI.<T$NCN$kM#0l$N$b$N$O(B Maple $B$J$N$G(B, Maple $B$HHf3S$r9T$&(B. Maple $B$O(B, |
$BI.<T$NCN$kM#0l$N$b$N$O(B Maple $B$J$N$G(B, Maple $B$HHf3S$r9T$&(B. Maple $B$O(B, |
$B$3$N5!G=$K4X$7$F$O(B, kernel $B$K$*$$$F@lMQ$NFC<l$J%G!<%?7?$rB?MQ$7$F(B |
$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$$$H9M$($k(B. |
$B8zN($r>e$2$F$$$k$?$a(B, $BHf3S$9$k$3$H$O%"%s%U%'%"$G$O$J$$$H9M$($k(B. |
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$BNc$H$7$F$O(B, P.S. Wang $B$K$h$k(B, $B<g78?tLdBj$r5/$3$7$d$9$$B?9`<0$N(B |
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$BNc(B (Asir $B$N(B {\tt lib/fctrdata} $B$N(B {\tt Wang[1]} $B$+$i(B |
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{\tt Wang[15]} $B$rMQ$$$?(B. $B%^%7%s$O(B Athlon 1900+ $B$GC10L$OIC(B, |
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$p$ $B$O(B, $B4pACBN$NI8?t$rI=$9(B. $BI=$G(B $N$ $B$O(B 60 $BICBT$C$F$bEz$($,(B |
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$B$G$J$$$b$N(B, $F$ $B$O(B, Maple $B$,2?$i$+$NM}M3$G%(%i!<$r=P$7$F(B |
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$B7W;;$G$-$J$+$C$?$b$N$rI=$9(B. Asir $B$K$*$$$F(B $p(n)$ $B$O(B, $n$ $B<!3HBg(B, |
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$B$9$J$o$A(B $\GF(p^n)$ $B>e$G0x?tJ,2r$r9T$C$?$3$H$r<($9(B. 8 $BHV$NB?9`<0(B |
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$B$N$h$&$KNc30E*$K;~4V$,$+$+$k$b$N$O$"$k$,(B, $B$*$*$`$M(B, Asir $B$,(B |
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$BNI9%$J%Q%U%)!<%^%s%9$r<($7$F$$$k(B. |
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\begin{table}[hbtp] |
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\begin{center} |
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% & & & & & & & & & & & & & & & \\ \hline |
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{\normalsize |
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\begin{tabular}{c|ccccccccc} \hline |
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$p$ & 1 & 2 & 3 & 4 & 5 & 6 & 7 & 8 & 9 \\ \hline |
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2 & N & F & F & F & N & N & 0.01 & 1 & 0.01 \\ \hline |
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3 & 0.07 & 0.1 & 0.07 & N & 0.4 & N & 0.01 & 0.02 & 0.06 \\ \hline |
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5 & N & 0.05 & 0.08 & 3.5 & 0.2 & 0.4 & 0.01 & 0.6 & 0.1 \\ \hline |
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7 & 0.08 & 0.1 & 0.1 & 0.25 & 0.6 & 0.5 & 0.02 & 1 & F \\ \hline |
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\end{tabular} |
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\begin{tabular}{c|cccccc} \hline |
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$p$ & 10 & 11 & 12 & 13 & 14 & 15 \\ \hline |
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2 & F & N & 0.005 & 0.006 & 0.008 & F \\ \hline |
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3 & 4 & N & 0.004 & 0.007 & 0.14 & 0.02 \\ \hline |
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5 & 0.2 & F & 0.005 & 0.006 & 0.03 & 0.4 \\ \hline |
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7 & 0.6 & 14 & 0.005 & 0.16 & 0.04 & 0.6 \\ \hline |
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\end{tabular} |
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\begin{tabular}{c|ccccccccc} \hline |
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$p$ & 1 & 2 & 3 & 4 & 5 & 6 & 7 & 8 & 9 \\ \hline |
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547 & 0.2& 0.2& 0.1& 0.3& 1& 1.2& 0.02& 6& F\\ \hline |
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32003& 0.2& 0.2& 0.2& 0.4 & 1 & 1 & 0.02 & 4.2 & F \\ \hline |
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99981793 & 0.5 & 0.6 & 0.5 & 3 & 3 & 4.5 & 0.02 & N & F\\ \hline |
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\end{tabular} |
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\begin{tabular}{c|cccccc} \hline |
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$p$ & 10 & 11 & 12 & 13 & 14 & 15 \\ \hline |
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547 & 0.9 &3.3 & 0.005 & 0.2 & 0.1 & 0.4 \\ \hline |
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32003 & 1.8 & 4.9 &0.006 & 0.3 & 0.1 & 0.4 \\ \hline |
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99981793 & 2.6 & 11 & 0.006 & 0.9 & 0.5 & 1.4 \\ \hline |
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\end{tabular} |
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} |
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\end{center} |
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\caption{$B0x?tJ,2r(B (Maple7)} |
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\end{table} |
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\begin{table}[hbtp] |
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\begin{center} |
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% & & & & & & & & & & & & & & & \\ \hline |
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{\normalsize |
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\begin{tabular}{c|ccccccccc} \hline |
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$p$ & 1 & 2 & 3 & 4 & 5 & 6 & 7 & 8 & 9 \\ \hline |
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2(5) & 0.003 & 0.003 & 0.004 & 0.01 & 0.02 & 0.05 & 0.001 & 0.01 & 0.0003 \\ \hline |
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3(5) & 0.003 & 0.002 & 0.005 & 0.003 & 0.003 & 0.1 & 0.002 & 0.001 & 0.003 \\ \hline |
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5(2) & 0.004 & 0.003 & 0.004 & 0.02 & 0.06 & 0.4 & 0.002 & 0.4 & 0.005 \\ \hline |
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7(2) & 0.004 & 0.004 & 0.005 & 0.03 & 0.1 & 0.1 & 0.004 & 1.8 & 0.2 \\ \hline |
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\end{tabular} |
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\begin{tabular}{c|cccccc} \hline |
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$p$ & 10 & 11 & 12 & 13 & 14 & 15 \\ \hline |
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2(5) & 0.03 & 0.07 & 0.0006 & 0.001 & 0.002 & 0.001 \\ \hline |
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3(5) & 0.04 & 0.2 & 0.0001 & 0.0005 & 0.02 & 0.001 \\ \hline |
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5(2) & 0.01 & 0.2 & 0.001 & 0.001 & 0.004 & 0.01 \\ \hline |
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7(2) & 0.02 & 0.6 & 0.001 & 0.007 & 0.005 & 0.01 \\ \hline |
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\end{tabular} |
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\begin{tabular}{c|ccccccccc} \hline |
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$p$ & 1 & 2 & 3 & 4 & 5 & 6 & 7 & 8 & 9 \\ \hline |
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547 & 0.004 & 0.004 & 0.005 & 0.03 & 0.05 & 0.2 & 0.02& 2& 0.2\\ \hline |
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32003 & 0.004 & 0.004 & 0.005 &0.04 &0.07 & 0.2 & 0.004 & 2 & 0.2 \\ \hline |
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99981793& 0.004 & 0.004& 0.005 & 0.03 & 0.03 & 0.2 & 0.004 & 4 & 0.2 \\ \hline |
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\end{tabular} |
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\begin{tabular}{c|cccccc} \hline |
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$p$ & 10 & 11 & 12 & 13 & 14 & 15 \\ \hline |
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547 & 0.04 & 0.3 & 0.001 &0.006 & 0.006 & 0.01 \\ \hline |
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32003 & 0.04 &0.2 &0.001 &0.007 & 0.006 & 0.03 \\ \hline |
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99981793 & 0.04 & 0.3 &0.001 & 0.008 & 0.008 & 0.01 \\ \hline |
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\end{tabular} |
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} |
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\end{center} |
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\caption{$B0x?tJ,2r(B (Asir)} |
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\end{table} |
\section{$B$*$o$j$K(B} |
\section{$B$*$o$j$K(B} |
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$B:#8e$NM=Dj$H$7$F(B, $B<!$N$h$&$J$3$H$r9M$($F$$$k(B. |
$B:#8e$NM=Dj$H$7$F(B, $B<!$N$h$&$J$3$H$r9M$($F$$$k(B. |
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\begin{itemize} |
\begin{itemize} |
\item $B@5I8?t$N=`AGJ,2r(B |
\item $B@5I8?t$N=`AGJ,2r$N<BAu(B. |
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\item $BBN$N0L?t$,B-$j$J$$>l9g$K(B, $B<+F0E*$K4pACBN$r3HBg$9$k(B |
\item $BBN$N0L?t$,B-$j$J$$>l9g$K(B, $B<+F0E*$K4pACBN$r3HBg$9$k(B. |
|
|
\item $B3FItJ,$N8zN(2=(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} |
\end{itemize} |
|
|
\begin{thebibliography}{99} |
\begin{thebibliography}{99} |