| version 1.1.1.1, 2000/03/01 02:25:51 |
version 1.4, 2001/02/27 08:07:24 |
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%$OpenXM: OpenXM/doc/compalg/effgr.tex,v 1.3 2000/03/28 02:02:29 noro Exp $ |
| \chapter{$B%0%l%V%J4pDl7W;;$N8zN(2=(B} |
\chapter{$B%0%l%V%J4pDl7W;;$N8zN(2=(B} |
| |
|
| $BBh(B \ref{chapgr} $B>O$G%0%l%V%J4pDl$r7W;;$9$k%"%k%4%j%:%`$G$"$k(B |
$BBh(B \ref{chapgr} $B>O$G%0%l%V%J4pDl$r7W;;$9$k%"%k%4%j%:%`$G$"$k(B |
| Line 409 $<_h$ $B$O(B $BNc(B \ref{horder} $B$G=R$Y$?(B or |
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| Line 411 $<_h$ $B$O(B $BNc(B \ref{horder} $B$G=R$Y$?(B or |
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| \section{$F_4$ $B%"%k%4%j%:%`(B} |
\section{$F_4$ $B%"%k%4%j%:%`(B} |
| |
|
| $BBe?tJ}Dx<05a2r$K8B$i$:(B, $BBe?t4v2?$K$*$1$kITJQNL$N7W;;$J$I$K$*$$$F$b(B |
$BBe?tJ}Dx<05a2r$K8B$i$:(B, $BBe?t4v2?$K$*$1$kITJQNL$N7W;;$J$I$K$*$$$F$b(B |
| $BG$0UF~NOB?9`<0=89g$+$i$N(B \gr $B4pDl$N7W;;$O(B, $B7W;;NLE*$K$_$F(B dominant |
$BG$0UF~NOB?9`<0=89g$+$i$N%0%l%V%J4pDl$N7W;;$O(B, $B7W;;NLE*$K$_$F(B dominant |
| step $B$H$J$k$3$H$,B?$$(B. $B$3$N$h$&$J>l9g$N7W;;K!$H$7$F$O(B Buchberger |
step $B$H$J$k$3$H$,B?$$(B. $B$3$N$h$&$J>l9g$N7W;;K!$H$7$F$O(B Buchberger |
| $B%"%k%4%j%:%`$,$[$\M#0l$NJ}K!$G$"$C$?$,(B, $B:G6a(B Faug\`ere $B$K$h$j(B |
$B%"%k%4%j%:%`$,$[$\M#0l$NJ}K!$G$"$C$?$,(B, $B:G6a(B Faug\`ere $B$K$h$j(B |
| $F_4$ ($B$"$k$$$O(B $F_5$) $B%"%k%4%j%:%`$,Ds0F$5$l(B, $B$=$N9bB.@-$,(B |
$F_4$ ($B$"$k$$$O(B $F_5$) $B%"%k%4%j%:%`$,Ds0F$5$l(B, $B$=$N9bB.@-$,(B |
|
|
| $B$r4^$^$J$$(B. |
$B$r4^$^$J$$(B. |
| \end{itemize} |
\end{itemize} |
| $B$3$l$O(B, $t_1 > t_2 > \cdots$ $B$r?7$?$JJQ?t$H$_$F(B, $B$3$N=g=x$G(B reduced $B$J(B |
$B$3$l$O(B, $t_1 > t_2 > \cdots$ $B$r?7$?$JJQ?t$H$_$F(B, $B$3$N=g=x$G(B reduced $B$J(B |
| \gr $B4pDl$r7W;;$7$?7k2L$KBP1~$9$k(B. |
$B%0%l%V%J4pDl$r7W;;$7$?7k2L$KBP1~$9$k(B. |
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|
| $$F' = \{h=poly(B_i) \mid h\neq 0, HT(h)\notin \{HT(r)\mid r\in Red\}\}$$ |
$$F' = \{h=poly(B_i) \mid h\neq 0, HT(h)\notin \{HT(r)\mid r\in Red\}\}$$ |
| $$Red' = \{h=poly(B_i) \mid h\neq 0, HT(h)\in \{HT(r)\mid r\in Red\}\}$$ |
$$Red' = \{h=poly(B_i) \mid h\neq 0, HT(h)\in \{HT(r)\mid r\in Red\}\}$$ |
| Line 578 S-$BB?9`<0$N(B sugar $B$N:G>.$N$b$N$rA4$FA*$V(B. |
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| Line 580 S-$BB?9`<0$N(B sugar $B$N:G>.$N$b$N$rA4$FA*$V(B. |
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| $B$"$k(B sugar $B$N(B S-$BB?9`<0$r=8$a$F7W;;$7$?>l9g(B, $B@8@.$5$l$?4pDl$N(B sugar $B$b(B |
$B$"$k(B sugar $B$N(B S-$BB?9`<0$r=8$a$F7W;;$7$?>l9g(B, $B@8@.$5$l$?4pDl$N(B sugar $B$b(B |
| $B$=$NCM$G$"$k$H$7$F(B, $B:F5"E*$K(B sugar $B$NCM$rDj$a$k$3$H$H$9$k(B. |
$B$=$NCM$G$"$k$H$7$F(B, $B:F5"E*$K(B sugar $B$NCM$rDj$a$k$3$H$H$9$k(B. |
| $BFC$K(B, $BF~NOB?9`<0=89g$,(B homogeneous $B$N>l9g(B, $B$3$N(B strategy $B$K$h$j(B, |
$BFC$K(B, $BF~NOB?9`<0=89g$,(B homogeneous $B$N>l9g(B, $B$3$N(B strategy $B$K$h$j(B, |
| $B3F%9%F%C%W$G7W;;$5$l$k4pDl$O(B, $B<!?t$,(B $d$ $B$N>l9g(B, {\bf reduced} $B$J(B \gr |
$B3F%9%F%C%W$G7W;;$5$l$k4pDl$O(B, $B<!?t$,(B $d$ $B$N>l9g(B, {\bf reduced} $B$J%0%l%V%J(B |
| $B4pDl$N$&$A$N(B, $d$ $B<!$N85A4$F$H$J$k(B. $B$3$l$O(B, homogeneous ideal $B$N(B |
$B4pDl$N$&$A$N(B, $d$ $B<!$N85A4$F$H$J$k(B. $B$3$l$O(B, homogeneous ideal $B$N(B |
| $B>l9g(B, |
$B>l9g<!$,@.$jN)$D$3$H$+$iJ,$+$k(B. |
| |
|
| \begin{enumerate} |
\begin{enumerate} |
| \item $d$ $B<!$N4pDl$r@8@.$9$k$?$a$N(B S-$BB?9`<0$OA4$F(B $d-1$ $B<!0J2<$N(B |
\item $d$ $B<!$N4pDl$r@8@.$9$k$?$a$N(B S-$BB?9`<0$OA4$F(B $d-1$ $B<!0J2<$N(B |
| Line 624 Faug\`ere $B$K$h$l$P(B, $F_4$ $B$K$*$$$F$O(B $B$ |
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| Line 626 Faug\`ere $B$K$h$l$P(B, $F_4$ $B$K$*$$$F$O(B $B$ |
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| $B$$$:$l$K$;$h(B, modular $B7W;;$K$h$kJ}K!$,8zN($r>e$2$k>l9g$H$$$&$N$O(B, |
$B$$$:$l$K$;$h(B, modular $B7W;;$K$h$kJ}K!$,8zN($r>e$2$k>l9g$H$$$&$N$O(B, |
| $det(A'')$ $B$KHf$Y$F2r$N78?t$,>.$5$$>l9g$G$"$k(B. $B$3$l$O0lHL$K4|BT(B |
$det(A'')$ $B$KHf$Y$F2r$N78?t$,>.$5$$>l9g$G$"$k(B. $B$3$l$O0lHL$K4|BT(B |
| $B$G$-$k$3$H$G$O$J$$$,(B, $BA0@a$G=R$Y$?$h$&$K(B, homogeneous $B$N>l9g$K(B $F_4$ |
$B$G$-$k$3$H$G$O$J$$$,(B, $BA0@a$G=R$Y$?$h$&$K(B, homogeneous $B$N>l9g$K(B $F_4$ |
| $B$,(B reduced $B$J(B \gr $B4pDl$N0lIt$r@8@.$9$k(B, $B$H$$$&$3$H$+$i(B, reduced $B$K(B |
$B$,(B reduced $B$J%0%l%V%J4pDl$N0lIt$r@8@.$9$k(B, $B$H$$$&$3$H$+$i(B, reduced $B$K(B |
| $B$7$?>l9g$K78?t$,>.$5$/$J$k$h$&$JLdBj$G$O(B modular $B7W;;$,8zN($r8~>e(B |
$B$7$?>l9g$K78?t$,>.$5$/$J$k$h$&$JLdBj$G$O(B modular $B7W;;$,8zN($r8~>e(B |
| $B$5$;$k$3$H$,4|BT$G$-$k(B. |
$B$5$;$k$3$H$,4|BT$G$-$k(B. |
| |
|
| \subsection{$B<B83(B} |
\subsection{$BNc(B} |
| |
|
| \cite{REP} $B$G$O(B, 4 $BJQ?t$NBe?tJ}Dx<07O(B (odd order $BJ}Dx<0(B) $B$N%0%l%V%J4pDl7W;;(B |
\cite{REP} $B$G$O(B, 4 $BJQ?t$NBe?tJ}Dx<07O(B (odd order $BJ}Dx<0(B) $B$N%0%l%V%J4pDl7W;;(B |
| $B$r(B Buchberger $B%"%k%4%j%:%`$K$h$j9T$C$?(B. $B$3$N7W;;$K8=$l$kCf4V4pDl$K$*$$(B |
$B$r(B Buchberger $B%"%k%4%j%:%`$K$h$j9T$C$?(B. $B$3$N7W;;$K8=$l$kCf4V4pDl$K$*$$(B |
| Line 637 $det(A'')$ $B$KHf$Y$F2r$N78?t$,>.$5$$>l9g$G$"$k(B. |
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| Line 639 $det(A'')$ $B$KHf$Y$F2r$N78?t$,>.$5$$>l9g$G$"$k(B. |
|
| order $BJ}Dx<0$r7W;;$9$k>l9g(B, $B:G8e$K78?t$,BgJQ>.$5$$4pDl$,2?K\$+=P$F=*N;(B |
order $BJ}Dx<0$r7W;;$9$k>l9g(B, $B:G8e$K78?t$,BgJQ>.$5$$4pDl$,2?K\$+=P$F=*N;(B |
| $B$9$k(B. $BFC$K(B, $B:G=i$K8=$l$k(B, $B78?t$N>.$5$$4pDl$,(B, 16 $B<!$N4pDl$N$&$A:G8e$N(B |
$B$9$k(B. $BFC$K(B, $B:G=i$K8=$l$k(B, $B78?t$N>.$5$$4pDl$,(B, 16 $B<!$N4pDl$N$&$A:G8e$N(B |
| $B$b$N$G$"$C$?$?$a(B, $B$=$N4pDl$rMQ$$$F0l$DA0$N4pDl$r4JLs$7$?$H$3$m(B, $B78?t$,(B |
$B$b$N$G$"$C$?$?$a(B, $B$=$N4pDl$rMQ$$$F0l$DA0$N4pDl$r4JLs$7$?$H$3$m(B, $B78?t$,(B |
| $B>.$5$/$J$C$?$?$a(B, $B$=$NA`:n$r4pDlA4$F$KE,MQ$7$?$H$3$m(B, 16 $B<!$N4pDlA4$F(B |
$B>.$5$/$J$C$?(B. $B$=$NA`:n$r4pDlA4$F$KE,MQ$7$?$H$3$m(B, 16 $B<!$N4pDlA4$F$N78(B |
| $B$N78?t$,(B, inter reduction $B$K$h$j>.$5$/$G$-$k$3$H$,J,$+$C$?$N$G$"$k(B. |
$B?t$,(B, inter reduction $B$K$h$j>.$5$/$G$-$k$3$H$,J,$+$C$?$N$G$"$k(B. $B4{$K(B |
| $B4{$K=R$Y$?$h$&$K(B, $B$3$N$h$&$J>l9g$K$O(B, modular $B7W;;$K$h$k2r8uJd$N7W;;$,(B |
$B=R$Y$?$h$&$K(B, $B$3$N$h$&$J>l9g$K$O(B, modular $B7W;;$K$h$k2r8uJd$N7W;;$,M-8z(B |
| $BM-8z$H$J$k(B. $B$7$+$7(B, 15 $B<!$N4pDl$r(B inter reduction $B$7$?$H$3$m(B, $B78?t$NBg(B |
$B$H$J$k(B. $B$7$+$7(B, 15 $B<!$N4pDl$r(B inter reduction $B$7$?$H$3$m(B, $B78?t$NBg$-$5(B |
| $B$-$5$O$[$H$s$IJQ$o$i$:(B, $BBg$-$$$^$^$G$"$C$?(B. $B0J>e$N$h$&$JGX7J$N$b$H$K(B, |
$B$O$[$H$s$IJQ$o$i$:(B, $BBg$-$$$^$^$G$"$C$?(B. $B0J>e$N$h$&$JGX7J$N$b$H$K(B, $B8=:_(B |
| $B8=:_$N<BAu$G$N(B McKay $BLdBj$KBP$9$k(B Buchberger$B%"%k%4%j%:%`7W;;(B |
$B$N<BAu$G$N(B odd order $BJ}Dx<0$KBP$9$k(B Buchberger$B%"%k%4%j%:%`7W;;(B |
| (homogenization+trace lifting) $B$*$h$S(B $F_4$ $B$NHf3S$r<($9(B. |
(homogenization+trace lifting) $B$*$h$S(B $F_4$ $B$NHf3S$r<($9(B. $BI=$G(B, |
| |
GaussElim, ChRem, IntRat, Check $B$O$=$l$>$l(B Gauss $B>C5n(B, $BCf9q>jM>DjM}(B, |
| |
$B@0?t(B-$BM-M}?tJQ49(B, $B7k2L$N%A%'%C%/$K$+$+$C$?;~4V$r<($9(B. |
| |
|
| \begin{table}[hbtp] |
\begin{table}[hbtp] |
| \caption{$B7W;;;~4V$NHf3S(B} |
\caption{$B7W;;;~4V$NHf3S(B} |
| Line 713 $F_4$ non homo & 937 & 902 & --- & --- & --- & --- & 3 |
|
| Line 717 $F_4$ non homo & 937 & 902 & --- & --- & --- & --- & 3 |
|
| $B$J$k$3$H$KBP1~$7$F$$$k(B. $B$=$l0J>e$N<!?t$K$D$$$F$O(B, $B>.$5$$78?t$r;}$D(B |
$B$J$k$3$H$KBP1~$7$F$$$k(B. $B$=$l0J>e$N<!?t$K$D$$$F$O(B, $B>.$5$$78?t$r;}$D(B |
| $BB?9`<0$K$h$k4JLs2=$G:Q$`$3$H$,9bB.2=$NM}M3$G$"$k(B. |
$BB?9`<0$K$h$k4JLs2=$G:Q$`$3$H$,9bB.2=$NM}M3$G$"$k(B. |
| |
|
| \subsection{$B9M;!(B} |
\subsection{$B$^$H$a(B} |
| |
|
| $F_4$ $B$OK\<AE*$K$O(B Buchberger $B%"%k%4%j%:%`$N2~NI$H9M$($i$l$k$,(B, $B<!$N$h$&$J(B |
$F_4$ $B$OK\<AE*$K$O(B Buchberger $B%"%k%4%j%:%`$N2~NI$H9M$($i$l$k$,(B, $B<!$N$h$&$J(B |
| $BD9=j$r;}$D(B. |
$BD9=j$r;}$D(B. |
| |
|
| Line 722 $F_4$ $B$OK\<AE*$K$O(B Buchberger $B%"%k%4%j%:%`$N2 |
|
| Line 725 $F_4$ $B$OK\<AE*$K$O(B Buchberger $B%"%k%4%j%:%`$N2 |
|
| \item $B9TNs$NA]$-=P$7Cf$O(B, $B9`$N=g=xHf3S$,C1$J$k(B index $B$NHf3S$K$J$k(B. |
\item $B9TNs$NA]$-=P$7Cf$O(B, $B9`$N=g=xHf3S$,C1$J$k(B index $B$NHf3S$K$J$k(B. |
| \item $BCf4V4pDl$r(B reduced $B$"$k$$$O$=$l$K6a$$7A$KJ]$D$?$a(B, $B$=$N8e$N(B |
\item $BCf4V4pDl$r(B reduced $B$"$k$$$O$=$l$K6a$$7A$KJ]$D$?$a(B, $B$=$N8e$N(B |
| $BA]$-=P$77W;;$,8zN(2=$9$k(B. ($B!VBP3Q2=!W$5$l$?9TNs$K$h$k4JLs2=(B vs. $B!V;03Q2=!W(B |
$BA]$-=P$77W;;$,8zN(2=$9$k(B. ($B!VBP3Q2=!W$5$l$?9TNs$K$h$k4JLs2=(B vs. $B!V;03Q2=!W(B |
| $B$5$i$?9TNs$K$h$k4JLs2=(B) |
$B$5$l$?9TNs$K$h$k4JLs2=(B) |
| \item reduced $B$J4pDl$N78?t$,>.$5$/$J$k>l9g$K$O(B, modular $B7W;;$K$h$k(B |
\item reduced $B$J4pDl$N78?t$,>.$5$/$J$k>l9g$K$O(B, modular $B7W;;$K$h$k(B |
| $B8zN(2=$,4|BT$G$-$k(B. $B$?$@$7(B, $B$3$N>l9g$K$O(B, $B9TNs$N3F9T$N(B 0 $B4JLs%A%'%C%/(B |
$B8zN(2=$,4|BT$G$-$k(B. |
| $B$,I,?\$G$"$k(B. |
|
| \end{itemize} |
\end{itemize} |
| $B8zN($K4X$7$F$$$($P(B, Risa/Asir $B$G$N8=>u$O(B, Faug\`ere $B$N(B home page |
$B0lJ}$G(B, |
| |
|
| \centerline{{\tt http://posso.lib6.fr/\til jcf/bench.html}} |
\begin{itemize} |
| \noi |
\item $BB??t$N(B S-$BB?9`<0$*$h$S(B reducer $B$r0lEY$K9TNs$H$7$F07$&$?$a$K(B, |
| $B$K$"$k%G!<%?$K5Z$V$Y$/$b$J$$(B. |
$B5pBg$J%a%b%j$rI,MW$H$9$k>l9g$,$7$P$7$P@8$:$k(B. |
| $B$?$H$($P(B, McKay $B$G(B, P6-200MHz PC $B>e$G(B 54 $BIC$HJs9p$5$l$F$$$k(B. $B$7$+$7(B, |
\item $B0lHL$K9TNs$OAB$H$J$k$?$a(B, $BBg5,LOAB9TNs$r8zN($h$/J];}$7(B, $BA]$-=P$9I,MW(B |
| Faug\`ere\cite{F} $B$K(B, |
$B$,$"$k(B. |
| Moreover, since big integer computations could be done by means of p-adic |
\item modular $B7W;;$r9T$&>l9g(B, $B9TNs$N3F9T$N(B 0 $B4JLs%A%'%C%/(B |
| or multi modular arithmetics it means that the cost of an integer computation is roughly |
$B$K$h$k%*!<%P!<%X%C%I$,LdBj$H$J$k(B. |
| |
\end{itemize} |
| \centerline{time of modular computation * size of the output coeffs} |
$B$J$I$N:$Fq$K$h$j(B, $B<BMQE*$J<BAu$r9T$&$K$O(B Buchberger $B%"%k%4%j%:%`$HF1MM(B |
| \noi |
$B$5$^$6$^$J9)IW$,I,MW$H$J$k(B. $B$H$O$$$((B, $BA0@a$NNc$G8+$?$h$&$K(B, $B<B83E*$J<B(B |
| $B$H=q$+$l$F$$$k$3$H$+$i(B, $BCf4V4pDl$KBP$9$k@5Ev@-%A%'%C%/$I$3$m$+(B, $BC1$J$k(B |
$BAu$G$b=>Mh$N(B Buchberger $B%"%k%4%j%:%`$h$j8zN($h$/7W;;$G$-$k>l9g$,$"$k$3(B |
| modular \gr $B4pDl7W;;$r(B, $BCf9q>jM>DjM}$K$h$k7k2L$,(B stable $B$K$J$k$^$G7+$jJV$7$F(B |
$B$H$O3N$+$G(B, Faug\`ere $B$K$h$k$b$N0J30$K$b$5$^$6$^$J<BAu<B83$,9T$o$l$k$3$H$,(B |
| $B$$$k$@$1(B, $B$H$$$&2DG=@-$b<N$F@Z$l$J$$(B. $BH`$,$b$&>/$7<BAu$rL@$i$+$K$7$F$/$l$k(B |
$B4|BT$5$l$k(B. |
| $B$3$H$rK>$s$G$$$k(B. |
|
| Faug\`ere $B$O(B, |
|
| \begin{enumerate} |
|
| \item Buchberger $B%"%k%4%j%:%`$K$*$1$k(B selection strategy $B$HHf$Y$F(B, |
|
| $F_4$ $B$G$O(B ``make no choice'' |
|
| \item non homogeneous case $B$N5sF0$r8+$l$P(B, $F_4$ $B$O(B Buchberger $B%"%k%4%j%:%`(B |
|
| $B$G$O$J$$(B. |
|
| \end{enumerate} |
|
| $B$H=q$$$F$$$k$,(B, 1. $B$K4X$7$F$O(B, critcal pair $B$N(Bsubset $B$r$I$&A*$V$+$G(B, |
|
| $B8zN($KBg$-$/:9$,=P$k$3$H$O(B, McKay $B$NNc$+$iL@$i$+$G$"$k(B. $B$^$?(B, 2. $B$K4X(B |
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| $B$7$F$O(B, $B%"%k%4%j%:%`$N4pK\9=B$$OL@$i$+$K(B Buchberger$B%"%k%4%j%:%`$G$"$j(B |
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| $B5?Ld$G$"$k(B. $B$H$O$$$((B, $B=>Mh$N(B Buchberger$B%"%k%4%j%:%`$h$j8zN($h$/7W;;$G(B |
|
| $B$-$k>l9g$,$"$k$3$H$O3N$+$G$"$j(B, $B3FIt$N2~NI$r4^$a$F(B, $B$h$j8zN($h$$<BAu$r(B |
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| $BL\;X$9$3$H$,<BMQ>e=EMW$G$"$k$H9M$($i$l$k(B. |
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| |
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