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version 1.1, 2000/03/01 02:25:51 version 1.3, 2000/03/28 02:02:29
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   %$OpenXM$
 \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
Line 410  $<_h$ $B$O(B $BNc(B \ref{horder} $B$G=R$Y$?(B or
 \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
Line 536  return $Red$
Line 537  return $Red$
 $B$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.
   
 $$F' = \{h=poly(B_i) \mid h\neq 0, HT(h)\notin \{HT(r)\mid r\in Red\}\}$$  $$F' = \{h=poly(B_i) \mid h\neq 0, HT(h)\notin \{HT(r)\mid r\in Red\}\}$$
 $$Red' = \{h=poly(B_i) \mid h\neq 0, HT(h)\in \{HT(r)\mid r\in Red\}\}$$  $$Red' = \{h=poly(B_i) \mid h\neq 0, HT(h)\in \{HT(r)\mid r\in Red\}\}$$
Line 578  S-$BB?9`<0$N(B sugar $B$N:G>.$N$b$N$rA4$FA*$V(B.
Line 579  S-$BB?9`<0$N(B sugar $B$N:G>.$N$b$N$rA4$FA*$V(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  $B$"$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$*$$$F$O(B $B$ 
Line 625  Faug\`ere $B$K$h$l$P(B, $F_4$ $B$K$*$$$F$O(B $B$ 
 $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.
   
Line 637  $det(A'')$ $B$KHf$Y$F2r$N78?t$,>.$5$$>l9g$G$"$k(B. 
Line 638  $det(A'')$ $B$KHf$Y$F2r$N78?t$,>.$5$$>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 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. $B$?$@$7(B, $B$3$N>l9g$K$O(B, $B9TNs$N3F9T$N(B 0 $B4JLs%A%'%C%/(B
 $B$,I,?\$G$"$k(B.  $B$,I,?\$G$"$k(B.
Line 731  $F_4$ $B$OK\<AE*$K$O(B Buchberger $B%"%k%4%j%:%`$N2
Line 734  $F_4$ $B$OK\<AE*$K$O(B Buchberger $B%"%k%4%j%:%`$N2
   
 \centerline{{\tt http://posso.lib6.fr/\til jcf/bench.html}}  \centerline{{\tt http://posso.lib6.fr/\til jcf/bench.html}}
 \noi  \noi
 $B$K$"$k%G!<%?$K5Z$V$Y$/$b$J$$(B.  $B$K$"$k%G!<%?$K5Z$V$Y$/$b$J$$(B. $B$?$H$($P(B, odd order $BJ}Dx<0$G(B, P6-200MHz
 $B$?$H$($P(B, McKay $B$G(B, P6-200MHz PC $B>e$G(B 54 $BIC$HJs9p$5$l$F$$$k(B. $B$7$+$7(B,  PC $B>e$G(B 54 $BIC$HJs9p$5$l$F$$$k(B. $B$7$+$7(B, Faug\`ere\cite{F} $B$K(B,
 Faug\`ere\cite{F} $B$K(B,  
 Moreover, since big integer computations could be done by means of p-adic  
 or multi modular arithmetics it means that the cost of an integer computation is roughly  
   
   Moreover, since big integer computations could be done by means of
   p-adic or multi modular arithmetics it means that the cost of an
   integer computation is roughly
   
 \centerline{time of modular computation * size of the output coeffs}  \centerline{time of modular computation * size of the output coeffs}
 \noi  \noi
 $B$H=q$+$l$F$$$k$3$H$+$i(B, $BCf4V4pDl$KBP$9$k@5Ev@-%A%'%C%/$I$3$m$+(B, $BC1$J$k(B  $B$H=q$+$l$F$$$k$3$H$+$i(B, $BCf4V4pDl$KBP$9$k@5Ev@-%A%'%C%/$I$3$m$+(B, $BC1$J$k(B
 modular \gr $B4pDl7W;;$r(B, $BCf9q>jM>DjM}$K$h$k7k2L$,(B stable $B$K$J$k$^$G7+$jJV$7$F(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
 $B$$$k$@$1(B, $B$H$$$&2DG=@-$b<N$F@Z$l$J$$(B. $BH`$,$b$&>/$7<BAu$rL@$i$+$K$7$F$/$l$k(B  $B$$$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$$$k(B.  $B$3$H$rK>$s$G$$$k(B.
 Faug\`ere $B$O(B,  Faug\`ere $B$O(B,
Line 751  $F_4$ $B$G$O(B ``make no choice''
Line 755  $F_4$ $B$G$O(B ``make no choice''
 $B$G$O$J$$(B.  $B$G$O$J$$(B.
 \end{enumerate}  \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,  $B$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  $B8zN($KBg$-$/:9$,=P$k$3$H$O(B, odd order $BJ}Dx<0$NNc$+$iL@$i$+$G$"$k(B. $B$^$?(B,
 $B$7$F$O(B, $B%"%k%4%j%:%`$N4pK\9=B$$OL@$i$+$K(B Buchberger$B%"%k%4%j%:%`$G$"$j(B  2. $B$K4X$7$F$O(B, $B%"%k%4%j%:%`$N4pK\9=B$$OL@$i$+$K(B Buchberger$B%"%k%4%j%:%`(B
 $B5?Ld$G$"$k(B. $B$H$O$$$((B, $B=>Mh$N(B Buchberger$B%"%k%4%j%:%`$h$j8zN($h$/7W;;$G(B  $B$G$"$j5?Ld$G$"$k(B. $B$H$O$$$((B, $B=>Mh$N(B Buchberger$B%"%k%4%j%:%`$h$j8zN($h$/(B
 $B$-$k>l9g$,$"$k$3$H$O3N$+$G$"$j(B, $B3FIt$N2~NI$r4^$a$F(B, $B$h$j8zN($h$$<BAu$r(B  $B7W;;$G$-$k>l9g$,$"$k$3$H$O3N$+$G$"$j(B, $B3FIt$N2~NI$r4^$a$F(B, $B$h$j8zN($h$$(B
 $BL\;X$9$3$H$,<BMQ>e=EMW$G$"$k$H9M$($i$l$k(B.  $B<BAu$rL\;X$9$3$H$,<BMQ>e=EMW$G$"$k$H9M$($i$l$k(B.
   
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