\documentclass{slides}
%\documentclass[pdf,distiller,slideColor,colorBG,azure]{prosper}
\usepackage{color}
\usepackage{rgb}
\usepackage{graphicx}
\usepackage{epsfig}
\newcommand{\qed}{$\Box$}
\newcommand{\mred}[1]{\smash{\mathop{\hbox{\rightarrowfill}}\limits_{\scriptstyle #1}}}
\newcommand{\tmred}[1]{\smash{\mathop{\hbox{\rightarrowfill}}\limits_{\scriptstyle #1}\limits^{\scriptstyle *}}}
\newtheorem{prop}{\redc $BL?Bj(B}
\def\gr{Gr\"obner basis }
\def\st{\, s.t. \,}
\def\ni{\noindent} 
\def\init{{\rm in}} 
\def\Q{{\bf Q}} 
\def\Z{{\bf Z}} 
\def\Spoly{{\rm Spoly}}
\def\Span{{\rm Span}}
\def\Supp{{\rm Supp}}
\def\StdMono{{\rm StdMono}}
\def\Im{{\rm Im}}
\def\Ker{{\rm Ker}}
\def\NF{{\rm NF}}
\def\HT{{\rm HT}}
\def\LT{{\rm LT}}
\def\ini{{\rm in}}
\def\rem{{\rm rem}}
\def\Id#1{\langle #1 \rangle}
\def\ve{\vfill\eject} 
\textwidth 9.2in
\textheight 7.2in
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\def\tc{\color{orange}}
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\def\itc{\color{DarkGreen}}
%\def\urlc{\bf\color{DarkGreen}}
\def\urlc{\bf\color{LimeGreen}}
\def\goldc{\color{goldenrod3}}
\def\redc{\color{orange}}
\def\vs{\vskip 1cm}
\def\vsh{\vskip 0.5cm}
\def\ns{\itc\LARGE}
\title{\tc\bf\ns $BBe?tBN>e$N%$%G%"%k$N%0%l%V%J!<4pDl7W;;$K$D$$$F(B}

%\slideCaption{$BBe?tBN>e$N%$%G%"%k$N%0%l%V%J!<4pDl7W;;$K$D$$$F(B}
\author{{\bf\Large $BLnO$(B $B@59T(B\\ $B?@8MBg3XM}3XIt(B}}
\date{\bf\Large Dec. 16, 2004}
\begin{document}
\setlength{\parskip}{20pt}
\maketitle

%\itc: item color
%\fbc: fbox color
%\urlc: URL color
%\goldc: bold color a
%\redc: bold color b

\large
\bf
\setlength{\parskip}{0pt}

\begin{slide}{\ns $BBe?tBN$N85$NI=8=J}K!(B}

$B86;O85$K$h$kI=8=$O2DG=(B : $F=\Q[t]/(m(t))$
$\Rightarrow$ $B78?t$NA}Bg$r>7$/(B
$\Rightarrow$ $BC`<!3HBg$,8=<BE*(B
$$F_0=\Q,\quad F_i = F_{i-1}(\alpha_i)\quad (i=1,\ldots,n),\quad F=F_m$$
$m_i(t,\alpha_1,\ldots,\alpha_{i-1})$ : $\alpha_i$ $B$N:G>.B?9`<0(B $/F_{i-1}$

$m_i$ $B$N4{Ls@-%A%'%C%/$O$+$D$F:$Fq(B
$\Rightarrow$ knapsack factorization $B$K$h$j$=$&$G$b$J$/$J$C$?(B. $B0J2<(B,
$$I=\langle m_1(x_1),m_2(x_1,x_2),\ldots,m_n(x_1,\ldots,x_n)\rangle$$
$B$G(B, $I$ $B$,(B $\Q[X]=\Q[x_1,\ldots,x_n]$ $B$N6KBg%$%G%"%k$H$9$k(B.
\end{slide}

\begin{slide}{\ns $BBe?tE*?t$N4JLs(B}

$m_i$ : $B<gJQ?t(B $x_i$ $B$K4X$9$k<g78?t$OM-M}?t$H$7$F$h$$(B.
$\Rightarrow$ 
$G=\{m_1,\ldots,m_n\}$ $B$O(B, 
$x_n > x_{n-1} > \cdots > x_1$ $B$J$k<-=q<0=g=x$K4X$9$k(B $I$ $B$N%0%l%V%J!<4pDl(B.

$h(x) \bmod I \in Q[X]/I$ $B$KBP$7(B,

$h_0 \equiv h \bmod I$, $\deg_{x_i}(h_0) < d_i$ 
$\Rightarrow$ $h=h\NF_G(h)$

$h_0=\rem_{x_1}(\rem_{x_2}(\cdots \rem_{x_n}(h,m_n)\cdots),m_2),m_1)$

$B$G$b$"$k$,(B, \underline{$B$3$N=g$G7W;;$7$F$O$$$1$J$$(B!!}

($f(a) \bmod b$ $B$r(B $c=f(a)$ $\Rightarrow$ $c \bmod b$ $B$H7W;;$9$k$h$&$J$b$N(B)
\end{slide}

\begin{slide}{\ns $BC19`4JLs$K$h$kBe?tE*?t$N4JLs(B}

$B$+$H$$$C$F(B, $B=g=x$rJQ$($l$P$h$$(B, $B$H$$$&$o$1$G$b$J$$(B

\underline{$BB?9`<0>jM><+BN$,4m81(B} : $BB>$NJQ?t$N<!?t$,5^$K>e$,$k(B

$\Rightarrow$ $G$ $B$K$h$kC19`4JLs$rMQ$$$k(B

$B3F4JLs%9%F%C%W$KMQ$$$k(B $m_i$ : $i$ $B$N>.$5$$$b$NM%@h(B
\end{slide}

\begin{slide}{\ns $B4JLs$NNc(B}

$m_1(x_1)=x_1^7-7x_1+3$,\\
$m_2(x_1,x_2)=x_2^6+x_1x_2^5+x_1^2x_2^4+x_1^3x_2^3+x_1^4x_2^2+\cdots$\\
$m_3(x_1,x_2,x_3)=63x_3^4+\cdots$

$B$GDj5A$5$l$k(B $F=\Q(\alpha_1,\alpha_2,\alpha_3)$ $B$O(B $m_1(x_1)$ $B$N(B
$B:G>.J,2rBN(B.

$(\alpha_1+\alpha_2+\alpha_3)^{20}$ $B$N4JLs(B

$i$ $B$,>.$5$$(B $m_i$ $BM%@h$G4JLs(B : 0.1 $BIC(B

$i$ $B$,Bg$-$$(B $m_i$ $BM%@h$G4JLs(B : 260 $BIC(B
\end{slide}

\begin{slide}{\ns $B5U857W;;(B}

$B5U857W;;$O%\%H%k%M%C%/$N0l$D(B

$B3HBg<!?t$r(B $d$ $B$H$9$k(B.

$BC13HBg(B $\Rightarrow$ $O(d^2)$ ($B3HD%(B Euclid $B8_=|K!(B)

$BC`<!3HBg$K$b:F5"E*$KE,MQ2DG=(B

$h(x_1,\ldots,x_n) \bmod I$ $B$N5U85$r7W;;(B

$x_n$ $B$K4X$7(B $h$, $m_n$ $B$K3HD%(B Euclid $B8_=|K!$rE,MQ$9$k(B

$\Rightarrow$ $\exists a,\exists b, \exists r$, $ah+bm_n=r(x_1,\ldots,x_{n-1})$

$\Rightarrow$ $r$ $B$N5U85$r7W;;$9$l$P(B, $h$ $B$N5U85$,5a$^$k(B
\end{slide}

\begin{slide}{\ns $B$3$NJ}K!$NLdBjE@(B}

\begin{itemize}
\item $BCf4V<0KDD%(B

$B:G=*7k2L$KHf3S$7$F(B, $r$ $B$N5U85$,5pBg$K$J$k>l9g$,$"$k(B.
($B7W;;J}K!$K0MB8(B)

\item $B4JLs2=$H$N4X78(B ($BItJ,=*7k<0;;K!$r;H$&>l9g(B)

\begin{itemize}
\item $h \in (\Q[x_1,\ldots,x_{n-1}])[x_n]$ $B$H8+$J$9(B

$B78?t$N=|;;$OB?9`<0$N@0=|$H$J$k$,(B, $B$3$l$i$K(B
$B8=$o$l$kJQ?t$KBP$9$k4JLs2=$,9T$o$l$J$$$N$G(B, $B0lHL$KBg$-$JB?9`<0(B
$B$,78?t$K8=$o$l$k(B. 

\item $h\in (\Q(\alpha_1,\ldots,\alpha_{n-1})[x_n]$ $B$H8+$J$9(B

$B78?t=|;;$,BN1i;;$H$J$j(B, $B5U857W;;$,I,MW$H$J$k(B.
\end{itemize}
\end{itemize}
\end{slide}

\begin{slide}{\ns $B%b%8%e%i!<7W;;$K$h$k5U857W;;(B}

$BBe?tE*?t$N7W;;$K$O%b%8%e%i!<7W;;$,M-8z(B 

[NORO96], [HOEIJ02]).

\begin{itemize}
\item $BCf9q>jM>DjM}(B

$B==J,B?$/$NK!(B $p$ $B$KBP$7(B, $BK!(B $p$ $B$G$N5U85$r7W;;(B

$BCf9q>jM>DjM}(B,  $B@0?t(B-$BM-M}?tJQ49$K$h$j5U85$rF@$k(B

$BM-8B8D$N(B $p$ $B$r=|$$$FK!(B $p$ $B$G$N5U85$O7W;;$G$-$k(B. 

\item $BL$Dj78?tK!(B
$$M=\{x_1^{e_1}\cdots x_n^{e_n} \bmod I \,|\, 0 \le e_i \le d_i-1
(i=1,\ldots,n)\}$$ $B$K$h$j5U85$r(B $u = \sum_{t \in M} a_t t$ $B$HI=$7(B, $hu
\equiv 1 \bmod I$ $B$+$i(B $a_t$ $B$N@~7AJ}Dx<07O$r:n$C$F2r$/(B
\end{itemize}
\end{slide}

\begin{slide}{\ns $BL$Dj78?tK!(B + Hensel lifting}

$BL$Dj78?tK!(B $B$O(B $O(d^3)$ $B$@$,(B, $B@~7AJ}Dx<07O$r(B Hensel lifting+$B@0?t(B-$BM-M}?tJQ49(B
$B$G2r$1$k(B

$\Rightarrow$ 

\begin{itemize}
\item $O(d^3)$ $B$OM-8BBN>e$N(B LU $BJ,2r$N$_(B

$B7k2L$,Bg$-$$78?t$r$b$D$J$i$P(B, $B7W;;;~4V$O(B
Hensel lifting (1 step $B$"$?$j(B $O(d^2)$) $B$,(B dominant

\item $\NF_G(th)$ ($t \in M$) $B$N7W;;$N$_$G@~7AJ}Dx<0$,$G$-$k(B

$BCf4V<0KDD%$dBN=|;;$K$h$kLdBj$O8=$o$l$J$$(B.
\end{itemize}
\end{slide}

\begin{slide}{\ns $BBe?tBN>e$N%0%l%V%J!<4pDl7W;;(B}

\underline{$BDjM}(B}

$F = \Q[\alpha_1,\ldots,\alpha_l] = \Q[T]/I$

$I=\langle m_1(t_1),\ldots,m_l(t_1,\ldots,t_l)\rangle$

$J =\langle B \rangle \subset R = F[x_1,\ldots,x_n]$ : $R$ $B$N??$N%$%G%"%k(B

$<$ : $R$ $B$N9`=g=x(B

$<_F$ : $\Q[X]$ $B>e$G(B $<$ $B$KEy$7$/(B, $X >> T$ $B$G$"$k%V%m%C%/=g=x(B

$B_F = B \cup \{m_1,\ldots,m_l\}$ 

$G_F = \langle B_F\rangle$ $B$N(B  $<_F$ $B$K4X$9$k%0%l%V%J!<4pDl(B

$\Rightarrow$ $G=(G_F \setminus \Q[T]) \bmod I$ $B$O(B $J$ $B$N(B $<$ $B$K4X$9$k%0%l%V%J!<4pDl(B
\end{slide}

\begin{slide}{\ns monic $B2=$N$?$a$K(B S-$BB?9`<0$,A}$($k(B}

$BDjM}$K$h$j(B, $F$ $B>e$N%0%l%V%J!<4pDl7W;;$O(B $\Q$ $B>e$N$=$l$K5"Ce(B

$B<B9T$r4Q;!$9$k$H(B $\cdots$ $B@8@.$5$l$kCf4V4pDl$NF,9`$N(B $t$ $BJQ?t$,$@$s$@$s>CLG(B 
= $BF,78?t$N5U857W;;$r(B S-$BB?9`<0$HC19`4JLs$G<B9T(B

$\Rightarrow$ S-$BB?9`<0$N?t$,A}Bg$7$F$$$k(B

$BJ@32(B : $BITE,@Z$J=g=x$G(B S-$BB?9`<0$,=hM}$5$l$k2DG=@-$bA}$($k(B.
 ($BITI,MW$J78?tKDD%$N2DG=@-(B)
\end{slide}

\begin{slide}{\ns $B@55,7A$r(B monic $B2=(B}

\begin{itemize}
\item $BDL>o$N=hM}(B

$S(f,g) \tmred{G} h \neq 0$ $B$J$i$P(B, $G \leftarrow G \cup \{h\}$

\item $BJQ998e$N=hM}(B

$S(f,g) \tmred{G} h(x,t) \neq 0$ $B$J$i$P(B, 
$h(x,\alpha)$ $B$r(B monic $B2=$7$?$b$N(B
$\tilde{h}(x,\alpha)$ $B$r:n$j(B,
$G \leftarrow G \cup \{\tilde{h}(x,t)\}$
\end{itemize}

$B$3$N$h$&$JJQ99$r9T$C$F$b(B $G_F$ $B$,7W;;$G$-$k$3$H$O$"$-$i$+(B

\end{slide}

\begin{slide}{\ns trace $B%"%k%4%j%:%`(B}

$\bmod p$ $B$G$N(Btrace $B%"%k%4%j%:%`$NB39T$KI,MW$J$3$H(B

\begin{itemize}
\item $\cdots$ $\Q$ $B>e$N4JLs2=$K$"$i$o$l$kJ,Jl$,(B $p$ $B$G3d$j@Z$l$J$$(B

\item $B@55,7A$NF,78?t$,(B $p$ $B$G3d$j@Z$l$J$$(B
\end{itemize}

$\Rightarrow$ monic $B2=$K(B $p$ $B$G3d$l$kJ,Jl$,$"$i$o$l$J$$(B, $B$H4K$a$i$l$k(B

($B$"$i$+$8$a%$%G%"%k$N@8@.85$N0l$D$G$"$C$?$H9M$($l$P$h$$(B)

\underline{$BB>$N<B8=J}K!(B}

$\overline{I}=I \bmod p$ $B$,:,4p%$%G%"%k(B

$\Rightarrow$ $GF(p)[t]/\overline{I}$ $B$OM-8BBN$ND>OB(B

$I_p$ : $I$ $B$N78?t$r(B $\Q_{<p>}=\{a/b\,|\, a, b \in Z, p \not{|} b\}$
$B$K@)8B(B

$\phi$ : $I_p$ $B$+$i$"$kD>OB@.J,$X$N<M1F(B

$B$H$7$F$b$h$$(B

$\Rightarrow$ $BM-8BBN>e$N(B trace $B7W;;$N<j4V$r8:$i$;$k2DG=@-$,$"$k(B.
\end{slide}

\begin{slide}{\ns Risa/Asir $B>e$G$N<BAu(B : $BC`<!Be?t3HBg(B}

{\tt Alg} : $B%\%G%#It$,:F5"I=8=B?9`<0(B ($B4{B8(B)

{\tt DAlg} : $B%\%G%#It$,J,;6I=8=B?9`<0(B ($B?75,(B) -- Alg $B$+$iJQ49(B
\begin{verbatim}
typedef struct oDAlg {
        short id;
        char nid;
        char pad;
        struct oDP *nm;  /* $B<B:]$K$O@0?t78?t(B */
        struct oQ *dn;   /* $B<B:]$K$O@0?t(B */
} *DAlg;
\end{verbatim}
$BBe?tE*?t$H$7$F$O(B {\tt nm/dn} : $BJ,Jl$rDLJ,$7$F(B {\tt dn} $B$H$9$k(B.

$BB>$K(B, $B4JLsMQ$K3HBgBN%G!<%?$r9=B$BN$H$7$FJ];}(B

{\tt set\_field()} $B$G@_Dj$G$-$k(B.
\end{slide}

\begin{slide}{\ns Risa/Asir $B>e$G$N<BAu(B : $B%0%l%V%J!<4pDl7W;;(B}

\begin{itemize}
\item {\tt nd\_gr} $B$*$h$S(B {\tt nd\_gr\_trace} $B$r2~B$(B

$BF~NO$K:G>.B?9`<0$rDI2C$7$F<B9T(B + $B@55,7A$N(B monic $B2=(B
$BF~NO$O(B {\tt Alg} $B7?$r78?t$K4^$`B?9`<0=89g$G$h$$(B. 

\item $BFbItI=8=(B

$BBe?tE*?t$O(B, $B85$NB?9`<0JQ?t$HF1Ey(B, $B@55,2=7W;;$OM-M}?tBN>e$G(B
$\Rightarrow$ $B78?t$N(B content $B=|5n$,<+F0E*$KE,MQ$5$l$k(B.

\item monic $B2=(B

$B$N:]$K$N$_(B, $BK\Mh$N78?t$,Be?tE*?t(B ({\tt DAlg $B7?(B}) $B$H(B
$B$7$F<h$j=P$5$l(B, $B5U857W;;$J$I$,9T$o$l$k(B.

\item weight

$BBe?tE*?t$KBP1~$9$k(Bweight $B$r(B 0 $B$K@_Dj(B $\Rightarrow$ sugar $B$r(B
$BE,@5$K$9$k$?$a(B
\end{itemize}
\end{slide}

\begin{slide}{\ns $BB>$N<BAuK!(B}

\underline{$BBe?tE*?t$r40A4$K78?t$H$7$F07$&(B}

$B78?t$K4X$7$F4JLs2=$*$h$S5U857W;;$r9T$&(B

$B<+A3$J<BAu$H8@$($k(B.

content $B=|5n$KAjEv$9$kJ}K!$r?7$?$K9M0F$9$kI,MW(B
$B$,$"$j(B, $B:#8e$N2]Bj(B.
\end{slide}

\begin{slide}{\ns $B7W;;$NNc(B}

$\langle \sqrt{2}x^2+(\sqrt{2}+\sqrt{3})xy+\sqrt{3}y^2-\sqrt{3},(\sqrt{2}-2\sqrt{3})x^2+2\sqrt{3}xy+2\sqrt{2}x^2+\sqrt{2}+\sqrt{3}\rangle$ $B$N%0%l%V%J!<7W;;(B
\begin{verbatim}
[0] S2=newalg(x^2-2);
(#0)
[1] S3=newalg(x^2-3);
(#1)
[2] F1=S2*x^2+(S2+S3)*x*y+S3*y^2-S3$
F2=(S2-2*S3)*x^2+2*S3*x*y+2*S2*x^2+S2+S3]$
[3] nd_gr_trace([F1,F2],[x,y],1,1,2);
[90*y^4+(-21*#0*#1-246)*y^2+(16*#0*#1+144),
20*x+(15*#0*#1-60)*y^3+(-7*#0*#1+83)*y]
\end{verbatim}
\end{slide}

\begin{slide}{\ns $B<B83(B : $BC1:,E:2C(B}

{\small
\begin{eqnarray*}
Cyc&=&\{f_1,f_2,f_3,f_4,f_5,f_6,f_7\}\\
f_1&=&\omega c_5c_4c_3c_2c_1c_0-1\\
f_2&=&(((((c_5+\omega )c_4+\omega c_5)c_3+\omega c_5c_4)c_2+\omega c_5c_4c_3)c_1+\omega c_5c_4c_3c_2)c_0+\omega c_5c_4c_3c_2c_1\\
f_3&=&((((c_4+\omega )c_3+\omega c_5)c_2+\omega c_5c_4)c_1+\omega c_5c_4c_3)c_0+c_5c_4c_3c_2c_1+\omega c_5c_4c_3c_2\\
f_4&=&(((c_3+\omega )c_2+\omega c_5)c_1+\omega c_5c_4)c_0+c_4c_3c_2c_1+c_5c_4c_3c_2+\omega c_5c_4c_3\\
f_5&=&((c_2+\omega )c_1+\omega c_5)c_0+c_3c_2c_1+c_4c_3c_2+c_5c_4c_3+\omega c_5c_4\\
f_6&=&(c_1+\omega )c_0+c_2c_1+c_3c_2+c_4c_3+c_5c_4+\omega c_5\\
f_7&=&c_0+c_1+c_2+c_3+c_4+c_5+\omega
\end{eqnarray*}}
$Cyc$: cyclic-7 $B$N(B $c_6$ $B$K(B 1 $B$N86;O(B 7 $B>h:,$rBeF~(B

\underline{$\Q(\omega)$ $B>e$G$N(B GB $B7W;;(B}: $B@F<!2=(B trace $B%"%k%4%j%:%`$K$h$j(B 22 $BIC(B
(monic $B2=$K(B 2.2 $BIC(B ($B5U857W;;(B 0.2$BIC(B))

\underline{$B:G>.B?9`<0$rE:2C$7$F(B $\Q$ $B>e$G7W;;(B} : 220 $BIC(B


\end{slide}

\begin{slide}{\ns $B<B83(B : 2 $B:,E:2C(B}
{\small
\begin{eqnarray*}
Cap&=&\{f_1,f_2,f_3,f_4\}\\
f_1&=&(2ty-2)x-(\alpha+\beta)zy^2-z\\
f_2&=&2\beta\alpha^4zx^3+(4ty+\beta)x^2+(4zy^2+4z)x+2ty^3-10y^2-10ty+2\alpha^2+\beta^2\\
f_3&=&(t^2-1)x+(\beta\alpha^4+\beta^3\alpha^3)tzy-2z\\
f_4&=&(-z^2+4t^2+\beta\alpha+2\beta^3)zx+(4tz^2+2t^3-10t)y+4z^2-10t^2+\beta\alpha^3\\
m_1&=&u^7-7u+3\\
m_2&=&u^6+\alpha u^5+\alpha^2u^4+\alpha^3u^3+\alpha^4u^2+\alpha^5u+\alpha^6-7
\end{eqnarray*}}
$Cap$ : $Caprasse$ [SYMBDATA] $B$N78?t$r%i%s%@%`$KBe?tE*?t$KCV$-49$($?(B

$\alpha$, $\beta$ : $t^7-7t+3$ $B$N(B 2 $B:,(B

\underline{$\Q(\alpha,\beta)$ $B>e$G$N(B GB $B7W;;(B} : $B@F<!2=(B trace $B%"%k%4%j%:%`$G(B 589 $BIC(B
(monic $B2=(B 36 $BIC(B)

\underline{$\Q$ $B>e$G$N7W;;(B} : 1 $B;~4VBT$C$F$b=*N;$7$J$$(B.
\end{slide}

\begin{slide}{\ns $B<B83(B : 3 $B:,E:2C(B}

$f(x)=x^7-7x+3$ $B$N:G>.J,2rBN(B $F$ $B$O(B 3 $B:,E:2C$G<B8=$5$l$k(B ($B4{=P(B).

$f(x)$ $B$N(B $F$ $B>e0x?tJ,2r$K8=$l$k(B, $F$ $B>e$N(B 2 $B$D$N(B 2 $B<!<0$N(B
GCD $B7W;;(B (GCD $B$O(B 1 $B<!<0(B) $B%0%l%V%J!<4pDl$G7W;;$9$k(B.

\underline{$F$ $B>e$N(B GB $B7W;;(B} : 0.8 $BIC(B ($B5U857W;;(B 1 $B2sJ,$,BgItJ,(B)

\underline{$\Q$ $B>e$G7W;;(B} : 60 $BIC(B
	
\end{slide}

\begin{slide}{\ns $B$*$o$j$K(B}

\begin{itemize}
\item DCGB $B$H$N4X78(B

$B:4F#$i(B [SATO01] $B$K$h$k(B DCGB $B$H$NHf3S(B

\item change of ordering, RUR 

FGLM $B$d(B RUR $B$N7W;;$NBe?tBN>e$X$N3HD%(B

($BBe?tBN>e(B? or $BM-M}?tBN>e(B?)

\item $BBe?tBN1i;;$N<BAu$N8zN(2=(B

$BBe?tBN$NI=8=$r(B {\tt DP} $B$+$i(B, $B$h$j8zN($h$$(B
$B<BAu$KJQ99$9$k(B
\end{itemize}

\end{slide}

\begin{slide}{\ns $BJ88%(B}

[HOEIJ02] M.v. Hoeij, M. Monagan, A Modular GCD algorithm over Number Fields presented with Multiple Extensions. Proc. ISSAC'02 (2002), 109-116.

[NORO96] $BLnO$(B, $BC`<!Be?t3HBgBN>e$G$N(B 1 $BJQ?tB?9`<0$N(B GCD $B$K$D$$$F(B. $B?tM}8&9V5fO?(B 920 (1996), 1-8.

[SATO01] Y. Sato, A. Suzuki,  Discrete Comprehensive Gr\"obner Bases.
Proc. ISSAC'01 (2001), 292-296.

[SYMBDATA] {\tt http://www.SymbolicData.org}.
\end{slide}

%\begin{slide}{\ns }
%\end{slide}

\end{document}

