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version 1.1, 2003/12/12 10:45:59

version 1.4, 2003/12/13 14:59:28

Line 1

\documentclass[12pt]{jarticle}

\topmargin -0.5in

\oddsidemargin -0in

\evensidemargin -0in

\textheight 9.5in

\textwidth 6in

\IfFileExists{my.sty}{\usepackage{my}}{}

\IfFileExists{graphicx.sty}{\usepackage{graphicx}}{}

\IfFileExists{epsfig.sty}{\usepackage{epsfig}}{}

\title{Risa/Asir $B$N?7$7$$J,;6I=8=B?9`<0%Q%C%1!<%8(B}

\title{Risa/Asir $B$N?7%0%l%V%J!<4pDl7W;;%Q%C%1!<%8$K$D$$$F(B}

\author{$BLnO$(B $B@59T(B \\ ($B?@8MBgM}(B)}

\date{}

\begin{document}

Line 54 Risa/Asir $B$N%0%l%V%J!<4pDl7W;;$K$*$$$F$O!"%Z%"$NA*B

Line 59 Risa/Asir $B$N%0%l%V%J!<4pDl7W;;$K$*$$$F$O!"%Z%"$NA*B

$B$^$?!"M-M}?tBN>e$K$*$$$F$b!"B?G\D91i;;$K(B {\tt gmp} $B$r;HMQ$7$F$$$k(B

Singular $B$J$I$N%7%9%F%`$G$O!"6aG/$H$_$K9bB.2=$7$?(B

{\tt gmp} $B$N@-G=$H!"(BRisa/Asir $B$G(B

$B;HMQ$7$F$k<+<g3+H/$NB?G\D91i;;5!G=$H$N@-G=:9$K$h$j!"I,$:$7$b(B Risa/Asir $B$N(B

$B;HMQ$7$F$$$k<+<g3+H/$NB?G\D91i;;5!G=$H$N@-G=:9$K$h$j!"I,$:$7$b(B Risa/Asir $B$N(B

$BM%0L@-$,<gD%$G$-$J$/$J$C$F$-$?!#(B

$B0lJ}$G!"(BPC $B$KEk:\$G$-$k%a%b%jNL$b?t(B GB $B$KC#$7!"(BCPU $B$b$I$s$I$s9bB.2=$7!"(B

$B%0%l%V%J!<4pDl7W;;$N1~MQHO0O$O$I$s$I$sBg$-$/$J$C$F$$$k!#$=$3$G!"(B

$B$3$l$^$G$N$5$^$6$^$J7P83$*$h$S!"<BAu$K4X$9$k:G6a$NCN8+$r$b$H$K!"(B

$B$G$-$k8B$j9bB.$JJ,;6I=8=B?9`<07W;;$*$h$S%0%l%V%J!<4pDl7W;;$r<B8=$9$k(B

$B%Q%C%1!<%8$r?75,$K=q$/$3$H$K$7$?!#(B

$B%Q%C%1!<%8(B {\bf nd} (New Distributed polynomial package) $B$r?75,$K=q$/$3$H$K$7$?!#(B

\section{$B9bB.2=$N9)IW(B}

\section{$B8zN(2=$N9)IW(B}

Buchberger $B%"%k%4%j%:%`$K4X$7$F$O!"(B

Gebauer-Moeller $B$N(B useless pair detection$B!"(Bsugar strategy $B$J$I$K(B

$B$h$j!"%"%k%4%j%:%`E*$K$O$"$kDxEY8G$^$C$?$,!":G6a$K$J$C$F$$$/$D$+(B

$B<BAu$K4X$9$kDs0F$,$J$5$l$?!#:#2s$N<BAu$K:N$jF~$l$?$b$N$K$D$$$F(B

$B@bL@$9$k!#(B

$B@bL@$9$k(B.

\begin{itemize}

\begin{enumerate}

\item geobucket

$B$3$l$O!"B?9`<0$N2C;;$r8zN(2=$9$k$?$a$NJ}K!$G$"$k!#(B

$B$3$l$O!"B?9`<0$N2C;;$r8zN(2=$9$k$?$a$NJ}K!$G$"$j(B,

\cite{Geo} $B$GDs0F$5$l(B, Macaulay2, Singular $B$J$IB?$/$N%7%9%F%`$G(B

$B:NMQ$5$l(B, $B<B:]$K8z2L$,$"$k$3$H$,<B>Z$5$l$F$$$k(B.

$B@55,2=7W;;$G$O(B, $B?tB?$/$NB?9`<0$N2C;;$,9T$o$l$k$,(B, $BHs>o$K(B

$B9`?t$NB?$$B?9`<0$K(B, $B9`?t$NHf3SE*>/$J$$B?9`<0$r7+$jJV$7B-$9$h$&$J>l9g(B,

$B9`$I$&$7$NHf3S1i;;$N%3%9%H$,BgJQBg$-$/$J$k(B. geobucket $B$H$O(B, $BB?9`<0$r(B

Line 85 $b^i$ $B$h$jBg$-$1$l$P(B $g[i+1]$ $B$K2C$($k(B, $

Line 92 $b^i$ $B$h$jBg$-$1$l$P(B $g[i+1]$ $B$K2C$($k(B, $

$B>r7o$,K~$?$5$l$k$^$GB3$1$k(B. $B$3$l$K$h$j(B, $BOB$K8=$o$l$kB?9`<0$N9`$NAm?t$r(B

$N$ $B$H$9$k$H$-(B, $O(N\log N)$ $B$N%3%9%H$GB?9`<0$NOB$,7W;;$G$-$k(B.

\item $B2DJQD9;X?t%Y%/%H%k(B

{\tt oDL} $B$N%a%s%P!<$G$O(B, $BC19`<0$NJQ?t$N3F;X?t$r(B 32 bit $B8GDj$GI=8=$7$F(B

$B$$$?$,(B, $BB?$/$N>l9g$3$l$O2aJ,$G$"$j(B, $B7k2L$H$7$F(B, $BB?9`<0$N$b$D>pJsNL(B

$B$$$?$,(B, $BB?$/$N>l9g$3$l$O2aJ,$G$"$k(B.

$B$h$j$bM>J,$K%a%b%j$,I,MW$H$J$C$F$$$?(B. $B$3$l$KBP$7$F(B, $BI,MW:G>.8B$N(B

$BI,MW:G>.8B$N(B bit $BD9$r;X?t$K3d$jEv$F$k$3$H$K$9$l$P(B, 32 bit $BCf$KJ#?t$N(B

bit $BD9$r;X?t$K3d$jEv$F$F$*$-(B, $B$"$U$l$,@8$8$k:]$K(B, $B%5%$%:$rJQ99$7$F(B

$B;X?t$rJ];}$G$-(B, $BHf3S(B, $BOB$J$I$r0lEY$KJ#?t8D<B9T$G$-$k(B. $B$^$?(B, $B;X?t$N(B

$BB?9`<0$r:n$j$J$*$9$H$$$&$N$,$3$NJ}K!$G$"$k(B.

$BJ];}$KI,MW$J%a%b%jNL$b8:$k(B. $B$3$&$9$k$H(B, $BOB$G$"$U$l$,@8$8$k>l9g$,$"$k$,(B,

$B$3$N>l9g$K$O%5%$%:$rJQ99$7$F(B

$BB?9`<0$r:n$j$J$*$9(B. $B$3$l$O(B \cite{Singular}

$B$GDs0F$5$l$F$$$kJ}K!$G$"$k(B.

\item $BG[Ns$K$h$kB?9`<0$NJ];}(B

Line 124 $i$ $B$N>.$5$$=g$+$iC5$7$F(B, $t$ $B$r3d$j@Z$k:G=i$

Line 135 $i$ $B$N>.$5$$=g$+$iC5$7$F(B, $t$ $B$r3d$j@Z$k:G=i$

$B$h$$$H$5$l$k(B ($BNc30$b$"$k$,(B). $B$3$N$?$a(B, $t$ $B$N(B reducer $B$O$"$l$P0l0U(B

$B$K$-$^$k(B.

$t$ $B$N(B reducer $g_t$ $B$,8+$D$+$C$?$i(B,

$t$ $B$N%O%C%7%eCM(B $h_t$ $B$r7W;;$7$F(B, $B$"$k%F!<%V%k$N(B $h_t$ $B$N0LCV$K(B,

$t$ $B$N%O%C%7%eCM(B $h_t$ $B$r7W;;$7$F(B, $B%O%C%7%e%F!<%V%k$N(B $h_t$ $B$N0LCV$K(B,

$(t,g_t)$ $B$rEPO?$9$k(B. $t$ $B$N(B reducer $B$rC5$9:]$K$O(B, $h_t$ $B$N0LCV(B

$B$KEPO?$5$l$?%G!<%?$+$i(B, $t$ $B$N(B reducer $B$rC5$7$F(B, $B$b$7$"$l$P$=$l$r(B

$BMQ$$$l$P$h$$(B.

\item $B@F<!$N>l9g$N8zN(2=(B

$B0lHL$K$O(B, $B?7$?$K@8@.$5$l$?Cf4V4pDl$G(B, $B4{B8$NCf4V4pDl$N@55,2=$O9T$o$J$$$,(B,

$BF~NO$,@F<!$N>l9g$K$O(B, $B$"$k(B(weight $B$D$-(B)$BA4<!?t$N=hM}$,=*$C$?;~E@$G(B

$B$=$N<!?t$NCf4V4pDl$I$&$7$G(B inter reduction $B$r9T$&(B. $B$3$N>l9g(B, $BF,9`$O(B

$BJQ2=$7$J$$$N$G(B, criteria $B$X$N1F6A$O$J$/(B, $B$^$?(B, $BDc$$A4<!?t$+$i=g$K(B

$BCf4V4pDl$r@8@.$7$F$$$l$P(B, $B4{$K(B, $B8=<!?t$^$G$N4JLs%0%l%V%J!<4pDl$N(B

$B$9$Y$F$NMWAG$,F@$i$l$F$$$k$N$G(B, $B$3$l$^$G$K(B 0 $B$K4JLs$5$l$?(B S-poly $B$O(B

$B$d$O$j?7$7$$4pDl$G$b(B 0 $B$K4JLs$5$l$k(B. $B$3$N=hM}$r9T$&$3$H$K$h$j(B,

$B0J9_$N7W;;$,4JLs4pDl$K$h$j@55,2=$5$l$k$3$H$K$J$j(B, $B@55,2=$,8zN(2=(B

$B$5$l$k$3$H$,4|BT$G$-$k(B.

\item $B%a%b%j4IM}(B

$B7W;;ESCf(B, $B$5$^$6$^$JBg$-$5$NNN0h$,7+$jJV$7I,MW$H$J$k(B. $BFC$KB?$/I,MW$H$5(B

Line 137 $(t,g_t)$ $B$rEPO?$9$k(B. $t$ $B$N(B reducer $B$r

Line 160 $(t,g_t)$ $B$rEPO?$9$k(B. $t$ $B$N(B reducer $B$r

$B$,0lDj$N%3%9%H$rH<$&$?$a$G$"$k(B. $B$3$N4IM}$O(B nd $B%Q%C%1!<%8Fb$GJD$8$F$*(B

$B$j(B, $B$+$D%U%j!<%j%9%H$N(B root $B$r(B 0 $B$K$7$F$*$1$P(B, $B$$$:$l(B GC $B$K$h$j2s<}$5(B

$B$l$k(B.

\end{itemize}

\end{enumerate}

\section{$B4pK\%G!<%?9=B$(B}

\vskip 5mm

%\vskip 5mm

\begin{tabular}{cc}

%\begin{tabular}{cc}

\begin{minipage}{.5\hsize}

%\begin{minipage}{.5\hsize}

\begin{verbatim}

%\begin{verbatim}

typedef struct oND {

%typedef struct oND {

struct oNM *body;

% struct oNM *body;

int nv;

% int nv,len,sugar;

int len;

%} *ND;

int sugar;

%\end{verbatim}

} *ND;

%\end{minipage}

\end{verbatim}

\end{minipage}

%\begin{minipage}{.5\hsize}

%\begin{verbatim}

\begin{minipage}{.5\hsize}

%typedef struct oNDV {

\begin{verbatim}

% struct oNMV *body;

typedef struct oNDV {

% int nv,len,sugar;

struct oNMV *body;

%} *NDV;

int nv;

%\end{verbatim}

int len;

%\end{minipage}

int sugar;

%\end{tabular}

} *NDV;

%\vskip 5mm

\end{verbatim}

\end{minipage}

\end{tabular}

\vskip 5mm

$B$3$l$i$O(B, $B$$$:$l$bJ,;6I=8=B?9`<0$rJ];}$9$k$?$a$N9=B$BN$G$"$k(B.

$BJ,;6I=8=B?9`<0$rJ];}$9$k$?$a$N9=B$BN$,Fs$DDj5A$5$l$F$$$k(B.

{\tt ND} $B$O(B linked list $B7A<0$N(B, {\tt NDV} $B$OG[Ns7A<0$NB?9`<0(B

$B$rI=$9(B.

$B$rI=$9(B. $BA0<T$O<!$G=R$Y$k(B {\tt oNM} $B$X$N(B, $B8e<T$O(B {\tt oNMV} $B$X$N(B

$B%]%$%s%?$r;}$C$F$$$k(B.

\vskip 5mm

\begin{tabular}{cc}

Line 195 typedef struct oNMV {

Line 215 typedef struct oNMV {

\vskip 5mm

$B$3$l$i$O(B, $BC19`<0$rI=$9$?$a$N9=B$BN$G$"$k(B. {\tt dl} $B$OC19`<0$N;X?t%Y%/(B

$B%H%k$rI=$7$F$*$j!"<B:]$K$OJQ?t$N8D?tJ,$ND9$5$NG[Ns$,%;%C%H$5$l$k(B.

$B%H%k$rI=$7$F$*$j!"<B:]$K$O(B, $B9=B$BN:n@.;~E@$G$N;X?t$N(Bbit $BD9$HJQ?t$N(B

$B8D?t$K1~$8$?D9$5$NG[Ns$NBg$-$5J,$NNN0h$,3NJ]$5$l$k(B.

{\tt NM} $B$O(B linked list $B7A<0$N(B, {\tt NMV} $B$OG[Ns7A<0$NB?9`<0$K$*$1$k(B

$BC19`<0$rI=$9(B. {\tt NDV} $B$O(B, {oNMV} $B$9$J$o$A9=B$BN$=$N$b$N$N(B

$BC19`<0$rI=$9(B. {\tt NDV} $B$O(B, {\tt oNMV} $B$9$J$o$A9=B$BN$=$N$b$N$N(B

$BG[Ns$X$N%]%$%s%?$r;}$D(B. {\tt NDC} $B$O78?t$rJ];}$9$k$?$a$NHFMQ$N6&MQBN(B

$BG[Ns$X$N%]%$%s%?$r;}$D(B.

$B$G$"$k(B.

\vskip 5mm

\begin{tabular}{cc}

\begin{minipage}{.5\hsize}

\begin{verbatim}

typedef union oNDC {

int m;

Line 207 typedef union oNDC {

Line 231 typedef union oNDC {

P p;

} *NDC;

\end{verbatim}

{\tt m} $B$O(B, $B0L?t$,(B 1 $B%o!<%I$G<}$^$kM-8BBN$N85$rJ];}$9$k$?$a$N(B

\end{minipage}

$B%a%s%P!<$G$"$k(B.

\begin{tabular}{cc}

\begin{minipage}{.5\hsize}

\begin{verbatim}

typedef struct oRHist {

Line 221 typedef struct oRHist {

Line 243 typedef struct oRHist {

} *RHist;

\end{verbatim}

\end{minipage}

\begin{minipage}{.5\hsize}

\begin{verbatim}

typedef struct oND_pairs {

struct oND_pairs *next;

int i1,i2;

int sugar;

UINT lcm[1];

} *ND_pairs;

\end{verbatim}

\end{minipage}

\end{tabular}

\vskip 5mm

{\tt NDC} $B$O78?t$rJ];}$9$k$?$a$NHFMQ$N6&MQBN$G$"$k(B.

{\tt m} $B$O(B, $B0L?t$,(B 1 $B%o!<%I$G<}$^$kM-8BBN$N85$rJ];}$9$k$?$a$N(B

$B%a%s%P!<$G$"$k(B.

{\tt RHist} $B$O(B reducer $B$NMzNr$r%O%C%7%e%F!<%V%k$KEPO?$9$k$?$a$N9=B$BN$G$"$k(B.

$B3F%(%s%H%j$O(B, {\tt RHist} $B$N%j%9%H$H$7$FEPO?$5$l$k(B. $B$^$?(B {\tt ND\_pairs}

$B3F%(%s%H%j$O(B, {\tt RHist} $B$N%j%9%H$H$7$FEPO?$5$l$k(B.

$B$O(B S-pair $B$rJ];}$9$k$?$a$N9=B$BN$G$"$j(B, $B$d$O$j%j%9%H$G$"$k(B.

\section{$B3FIt$N>\:Y(B}

\subsection{$B%I%i%$%P(B}

Line 278 $F_4$ $B$N3K?4$G$"$k(B, $BJ#?t$N(B S-pair $B$+$i

Line 292 $F_4$ $B$N3K?4$G$"$k(B, $BJ#?t$N(B S-pair $B$+$i

$B3FCf4V4pDl$KBP$7(B, $B3FJQ?t$KBP$9$k;X?t$N:GBgCM$r5-O?$7$F$*$-(B,

$B$=$N%Y%/%H%k$H$NOB$,$"$U$l$r5/$3$9>l9g$K:n$j$J$*$7$r$7$F$$$k(B.

\subsection{$BCf4V4pDl$N(B demand loading}

\subsection{$B$=$NB>(B}

{\tt dp} $B7O$GDs6!$5$l$F$$$k$N$HF1MM$K(B,

nd $B$K$*$$$F$b(B, $BCf4V4pDl$r%G%#%9%/>e$N;XDj$5$l$?%G%#%l%/%H%j$K(B

$BCV$/$3$H$,$G$-$k(B. $B;XDjJ}K!$O(B {\tt dp} $B7O$HF1MM(B {\tt dp\_gr\_flags()}

$B$G;XDj$9$k(B. $B%U%!%$%k$O(B {\tt dp} $B7O$HF1MM$N7A<0$J$N$G(B, {\tt bload()}

$B$GFI$`$3$H$,$G$-$k(B.

$B$GFI$`$3$H$,$G$-$k(B. $B$^$?(B, $BM-M}?tBN>e$N>l9g(B,

\subsection{$BM-M}?tBN>e$G$N(B content $B=|5n(B}

$B@55,2=7W;;ESCf$G$N(B content $B=|5n$O(B, $B>o$K9T$o$l$k(B. $B8=>u$G$O(B

$BF,78?t$,(B 2 $BG\(B ($B8GDj(B) $B$K$J$C$?$H$-$K=|5n$,9T$o$l$k!#(B

\section{$B@-G=(B}

$B0lHL$K(B, $BM-8BBN>e$N7W;;$N>l9g(B, {\tt nd\_gr} $B$O(B {\tt dp\_gr\_mod\_main}

$B$h$j?tG\$+$i==?tG\9bB.$G$"$k(B. $B$^$?(B, $BLdBj$K$b$h$k$,(B, {\tt nd\_f4} $B$O(B

$B$h$j?tG\9bB.$G$"$k(B. $B$^$?(B, $BLdBj$K$b$h$k$,(B, {\tt nd\_f4} $B$O(B

{\tt nd\_gr} $B$N?tG\DxEY9bB.$J>l9g$,$"$k(B. $B$*$J$8$_$N(B cyclic-$n$ $B$G(B

$BHf3S$9$k$HI=(B \ref{tab:cyclic}$B$N$h$&$J7k2L$rF@$k(B.

\begin{table}[hbtp]

\begin{center}

\begin{tabular}{c||c|c|c|c} \hline

\begin{tabular}{c||c|c|c|c}

$n$ & {\tt nd\_gr} & {\tt nd\_f4} & Singular & {\tt dp\_gr\_mod\_main} \\ \hline

$n$ & {\tt nd\_gr} & Singular & {\tt nd\_f4} & {\tt dp\_gr\_mod\_main} \\ \hline

7 & & & & \\ \hline

7 & 5.1 & 5.0 & 1.8 & 17 \\

8 & & & & \\ \hline

8 & 124 & 135 & 34 & 564 \\

9 & & & & \\ \hline

9 & 27810 & 29725 & 3951 & --- \\

\end{tabular}

\end{center}

\caption{$GF(31991)$ $B>e$G$N(B DRL $B=g=x%0%l%V%J!<4pDl7W;;(B}

\caption{$GF(31991)$ $B>e$G$N(B DRL $B=g=x%0%l%V%J!<4pDl7W;;(B (cyclic-$n$)}

\label{tab:cyclic}

\end{table}

$B$3$N$h$&$K(B, $B>/$J$/$H$b(B cyclic-$n$ $B$G$O(B, nd $B$N<BAu$N8z2L$,==J,$K8=$o$l$F$$$k(B.

$BI=(B \ref{tab:janet} $B$O(B, $B<o!9$N%Y%s%A%^!<%/LdBj$N7W;;;~4V$r<($9(B.

$BI=(B \ref{tab:janet} $B$O(B, $B<o!9$N%Y%s%A%^!<%/LdBj(B \cite{janet} $B$N7W;;;~4V$r<($9(B.

\begin{table}[hbtp]

\begin{center}

\begin{tabular}{cc}

\begin{minipage}{.5\hsize}

\begin{tabular}{c||c|c|c}

& {\tt nd\_gr} & Singular & {\tt nd\_f4} \\ \hline

dl & 5.9 & 4.9 &4.0 \\

eco10 & 7.1 & 10 &3.1 \\

eco11 & 63 & 106 &23 \\

eco12 & 507 & 1012 &198 \\

extcyc6 & 11 & 9.4 &4.1 \\

extcyc7 & 1813 & 1283 &447 \\

f855 & 3.6 & 3.4 &2.5 \\

filter9 & 0.28 & 0.80 &3.2 \\

hairer2 & 5.9 & 3.8 &4.5 \\

hairer3 & 11 & 35 &* \\

hcyclic7 & 6.5 & 4.8 &3.1 \\

hcyclic8 & 213 & 163 &82 \\

hf744 & 1.1 & 1.1 &1.6 \\

hf855 & 25 & 25 &17 \\

ilias13 & 11 & 8.4 &6.0\\

ilias\_k\_2 & 3.1 & 2.7 &1.1

\end{tabular}

\end{minipage}

\begin{minipage}{.5\hsize}

\begin{tabular}{c||c|c|c}

& {\tt nd\_gr} & Singular & {\tt nd\_f4} \\ \hline

ilias\_k\_3 & 4.4 & 2.9 &1.2 \\

katsura10 & 285 & 218 &80 \\

katsura8 & 4.1 & 3.3 &1.3 \\

katsura9 & 35 & 29 &11 \\

noon7 & 4.4 & 1.8 &13 \\

noon8 & 35 & 18 &220 \\

pinchon1 & 3.6 & 1.0 &7.6 \\

rbpl & 1.0 & 0.89 &1.2 \\

redcyc7 & 3.5 & 3.3 &1.2 \\

redeco10 & 2.8 & 2.3 &1.3 \\

redeco11 & 24 & 18 &12 \\

redeco12 & 177 & 134 &74 \\

reimer6 & 11 & 32 &10 \\

reimer7 & 4000 & 4108 & 956 \\

virasoro & 1.8 & 1.4 & 0.65

\end{tabular}

\end{minipage}

\end{tabular}

\end{center}

\caption{$GF(31991)$ $B>e$G$N(B DRL $B=g=x%0%l%V%J!<4pDl7W;;(B}

\label{tab:janet}

\end{table}

$BM-M}?tBN>e$N7W;;$N>l9g(B, $BB?9`<0$d(B, $B;X?t%Y%/%H%k$NI=8=J}K!0J30$K(B, $BESCf$"$i$o$l$k(B

$B78?t$NKDD%$NJ}$,(B, $B7W;;;~4V$KBg$-$/1F6A$rM?$($k>l9g$,B?$$(B. $B$3$NE@$G$O(B

{\tt nd\_gr\_trace} $B$H(B {\tt dp\_gr\_main} $B$H$G$OBg:9$J$$$N$G3d0&$9$k$,(B,

$B$h$j0-$/$J$k$3$H$O$J$$(B.

$B$h$j0-$/$J$k$3$H$O$J$$(B. $BFC$K(B, weight $B$rE,@Z$K@_Dj$9$k$3$H$K$h$j(B \cite{Kimura},

$B78?tKDD%$K4X$7$F$b$h$j5sF0$N$h$$7W;;$,2DG=$H$J$k$3$H$KCm0U$7$F$*$/(B.

\section{$B:#8e$NM=Dj(B}

{\tt dp} $B7O$K$"$C$F(B nd $B$K$J$$5!G=$H$7$F(B, $BM-M}4X?tBN78?t$N%0%l%V%J!<4pDl(B

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$B$7$?$$$H9M$($F$$$k(B.

$B$7$?$$$H9M$($F$$$k(B. $B$^$?(B,

tangent cone $B%"%k%4%j%:%`$rMQ$$$?(B local ring $B$G$NI8=`4pDl(B

$B7W;;$b(B, reducer $B$rC5$94X?t$r?7$?$KMQ0U$9$k$3$H$GBP1~2DG=$H9M$($F$$$k(B.

\begin{thebibliography}{99}

\bibitem{Geo}

Yan, T., The Geobucket Data Structure for Polynomials.

Journal of Symbolic Computation, {\bf 25}, 3 (1998), 285-293.

\bibitem{Singular}

Sch\"onemann, H., Singular in a Framework for Polynomial Computations.

Joswig, M. and Takayama, N. (eds.), Algebra, Geometry, and Software Systems,

Springer (2003), 163-176.

\bibitem{janet}

{\tt http://invo.jinr.ru/}. $B$^$?(B {\tt http://www.symbolicdata.org}

$B$K$O$5$i$KB?$/$N%Y%s%A%^!<%/LdBj$,$*$$$F$"$k(B.

\bibitem{Kimura}

$BLZB<(B, $BLnO$(B, $B%0%l%V%J!<4pDl7W;;$N$?$a$N(B weight $B@8@.%"%k%4%j%:%`(B.

$BK\8&5f=82q$K$*$1$kH/I=(B (2003).

\end{thebibliography}

\end{document}

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