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version 1.3, 2000/03/28 02:02:29 version 1.4, 2001/02/27 08:07:24
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 %$OpenXM$  
   %$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 629  $det(A'')$ $B$KHf$Y$F2r$N78?t$,>.$5$$>l9g$G$"$k(B. 
Line 630  $det(A'')$ $B$KHf$Y$F2r$N78?t$,>.$5$$>l9g$G$"$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 716  $F_4$ 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 727  $F_4$ $B$OK\<AE*$K$O(B Buchberger $B%"%k%4%j%:%`$N2
Line 727  $F_4$ $B$OK\<AE*$K$O(B Buchberger $B%"%k%4%j%:%`$N2
 $BA]$-=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$5$l$?9TNs$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$$(B. $B$?$H$($P(B, odd order $BJ}Dx<0$G(B, P6-200MHz  $B5pBg$J%a%b%j$rI,MW$H$9$k>l9g$,$7$P$7$P@8$:$k(B.
 PC $B>e$G(B 54 $BIC$HJs9p$5$l$F$$$k(B. $B$7$+$7(B, Faug\`ere\cite{F} $B$K(B,  \item  $B0lHL$K9TNs$OAB$H$J$k$?$a(B, $BBg5,LOAB9TNs$r8zN($h$/J];}$7(B, $BA]$-=P$9I,MW(B
   $B$,$"$k(B.
 Moreover, since big integer computations could be done by means of  \item modular $B7W;;$r9T$&>l9g(B, $B9TNs$N3F9T$N(B 0 $B4JLs%A%'%C%/(B
 p-adic or multi modular arithmetics it means that the cost of an  $B$K$h$k%*!<%P!<%X%C%I$,LdBj$H$J$k(B.
 integer computation is roughly  \end{itemize}
   $B$J$I$N:$Fq$K$h$j(B, $B<BMQE*$J<BAu$r9T$&$K$O(B Buchberger $B%"%k%4%j%:%`$HF1MM(B
 \centerline{time of modular computation * size of the output coeffs}  $B$5$^$6$^$J9)IW$,I,MW$H$J$k(B. $B$H$O$$$((B, $BA0@a$NNc$G8+$?$h$&$K(B, $B<B83E*$J<B(B
 \noi  $BAu$G$b=>Mh$N(B Buchberger $B%"%k%4%j%:%`$h$j8zN($h$/7W;;$G$-$k>l9g$,$"$k$3(B
 $B$H=q$+$l$F$$$k$3$H$+$i(B, $BCf4V4pDl$KBP$9$k@5Ev@-%A%'%C%/$I$3$m$+(B, $BC1$J$k(B  $B$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
 modular $B%0%l%V%J4pDl7W;;$r(B, $BCf9q>jM>DjM}$K$h$k7k2L$,(B stable $B$K$J$k$^$G7+$jJV$7$F(B  $B4|BT$5$l$k(B.
 $B$$$k$@$1(B, $B$H$$$&2DG=@-$b<N$F@Z$l$J$$(B. $BH`$,$b$&>/$7<BAu$rL@$i$+$K$7$F$/$l$k(B  
 $B$3$H$rK>$s$G$$$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$$$k$,(B, 1. $B$K4X$7$F$O(B, critcal pair $B$N(Bsubset $B$r$I$&A*$V$+$G(B,  
 $B8zN($KBg$-$/:9$,=P$k$3$H$O(B, odd order $BJ}Dx<0$NNc$+$iL@$i$+$G$"$k(B. $B$^$?(B,  
 2. $B$K4X$7$F$O(B, $B%"%k%4%j%:%`$N4pK\9=B$$OL@$i$+$K(B Buchberger$B%"%k%4%j%:%`(B  
 $B$G$"$j5?Ld$G$"$k(B. $B$H$O$$$((B, $B=>Mh$N(B Buchberger$B%"%k%4%j%:%`$h$j8zN($h$/(B  
 $B7W;;$G$-$k>l9g$,$"$k$3$H$O3N$+$G$"$j(B, $B3FIt$N2~NI$r4^$a$F(B, $B$h$j8zN($h$$(B  
 $B<BAu$rL\;X$9$3$H$,<BMQ>e=EMW$G$"$k$H9M$($i$l$k(B.  
   
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