Annotation of OpenXM/src/asir-doc/parts/groebner.texi, Revision 1.1
1.1 ! noro 1: @node $B%0%l%V%J4pDl$N7W;;(B,,, Top
! 2: @chapter $B%0%l%V%J4pDl$N7W;;(B
! 3:
! 4: @menu
! 5: * $BJ,;6I=8=B?9`<0(B::
! 6: * $B%U%!%$%k$NFI$_9~$_(B::
! 7: * $B4pK\E*$JH!?t(B::
! 8: * $B7W;;$*$h$SI=<($N@)8f(B::
! 9: * $B9`=g=x$N@_Dj(B::
! 10: * $BM-M}<0$r78?t$H$9$k%0%l%V%J4pDl7W;;(B::
! 11: * $B4pDlJQ49(B::
! 12: * $B%0%l%V%J4pDl$K4X$9$kH!?t(B::
! 13: @end menu
! 14:
! 15: @node $BJ,;6I=8=B?9`<0(B,,, $B%0%l%V%J4pDl$N7W;;(B
! 16: @section $BJ,;6I=8=B?9`<0(B
! 17:
! 18: @noindent
! 19: $BJ,;6I=8=B?9`<0$H$O(B, $BB?9`<0$NFbIt7A<0$N0l$D$G$"$k(B. $BDL>o$NB?9`<0(B
! 20: (@code{type} $B$,(B 2) $B$O(B, $B:F5"I=8=$H8F$P$l$k7A<0$GI=8=$5$l$F$$$k(B. $B$9$J$o(B
! 21: $B$A(B, $BFCDj$NJQ?t$r<gJQ?t$H$9$k(B 1 $BJQ?tB?9`<0$G(B, $B$=$NB>$NJQ?t$O(B, $B$=$N(B 1 $BJQ(B
! 22: $B?tB?9`<0$N78?t$K(B, $B<gJQ?t$r4^$^$J$$B?9`<0$H$7$F8=$l$k(B. $B$3$N78?t$,(B, $B$^$?(B,
! 23: $B$"$kJQ?t$r<gJQ?t$H$9$kB?9`<0$H$J$C$F$$$k$3$H$+$i:F5"I=8=$H8F$P$l$k(B.
! 24:
! 25:
! 26: @iftex
! 27: @tex
! 28: $(x+y+z)^2 = 1 \cdot x^2 + (2 \cdot y + (2 \cdot z)) \cdot x + ((2 \cdot z) \cdot y + (1 \cdot z^2 ))$
! 29: @end tex
! 30: @end iftex
! 31: @ifinfo
! 32: @example
! 33: (x+y+z)^2 = 1 x^2 + (2 y + (2 z)) x + ((2 z) y + (1 z^2 ))
! 34: @end example
! 35: @end ifinfo
! 36:
! 37: @noindent
! 38: $B$3$l$KBP$7(B, $BB?9`<0$r(B, $BJQ?t$NQQ@Q$H78?t$N@Q$NOB$H$7$FI=8=$7$?$b$N$rJ,;6(B
! 39: $BI=8=$H8F$V(B.
! 40:
! 41: @iftex
! 42: @tex
! 43: $(x+y+z)^2 = 1 \cdot x^2 + 2 \cdot xy + 2 \cdot xz + 1 \cdot y^2 + 2 \cdot yz +1 \cdot z^2$
! 44: @end tex
! 45: @end iftex
! 46: @ifinfo
! 47: @example
! 48: (x+y+z)^2 = 1 x^2 + 2 xy + 2 xz + 1 y^2 + 2 yz +1 z^2$
! 49: @end example
! 50: @end ifinfo
! 51:
! 52: @noindent
! 53: $B%0%l%V%J4pDl7W;;$K$*$$$F$O(B, $BC19`<0$KCmL\$7$FA`:n$r9T$&$?$aB?9`<0$,J,;6I=8=(B
! 54: $B$5$l$F$$$kJ}$,$h$j8zN($N$h$$1i;;$,2DG=$K$J$k(B. $B$3$N$?$a(B, $BJ,;6I=8=B?9`<0$,(B,
! 55: $B<1JL;R(B 9 $B$N7?$H$7$F(B @b{Asir} $B$N%H%C%W%l%Y%k$+$iMxMQ2DG=$H$J$C$F$$$k(B.
! 56: $B$3$3$G(B, $B8e$N@bL@$N$?$a$K(B, $B$$$/$D$+$N8@MU$rDj5A$7$F$*$/(B.
! 57:
! 58: @table @b
! 59: @item $B9`(B (term)
! 60: $BJQ?t$NQQ@Q(B. $B$9$J$o$A(B, $B78?t(B 1 $B$NC19`<0$N$3$H(B. @b{Asir} $B$K$*$$$F$O(B,
! 61:
! 62: @example
! 63: <<0,1,2,3,4>>
! 64: @end example
! 65:
! 66: $B$H$$$&7A$GI=<($5$l(B, $B$^$?(B, $B$3$N7A$GF~NO2DG=$G$"$k(B. $B$3$NNc$O(B, 5 $BJQ?t$N9`(B
! 67: $B$r<($9(B. $B3FJQ?t$r(B @code{a}, @code{b}, @code{c}, @code{d}, @code{e} $B$H$9$k$H(B
! 68: $B$3$N9`$O(B @code{b*c^2*d^3*e^4} $B$rI=$9(B.
! 69:
! 70: @item $B9`=g=x(B (term order)
! 71: $BJ,;6I=8=B?9`<0$K$*$1$k9`$O(B, $B<!$N@-<A$rK~$?$9A4=g=x$K$h$j@0Ns$5$l$k(B.
! 72:
! 73: @enumerate
! 74: @item
! 75: $BG$0U$N9`(B @code{t} $B$KBP$7(B @code{t} > 1
! 76:
! 77: @item
! 78: @code{t}, @code{s}, @code{u} $B$r9`$H$9$k;~(B, @code{t} > @code{s} $B$J$i$P(B
! 79: @code{tu} > @code{su}
! 80: @end enumerate
! 81:
! 82: $B$3$N@-<A$rK~$?$9A4=g=x$r9`=g=x$H8F$V(B. $B$3$N=g=x$OJQ?t=g=x(B ($BJQ?t$N%j%9%H(B)
! 83: $B$H9`=g=x7?(B ($B?t(B, $B%j%9%H$^$?$O9TNs(B) $B$K$h$j;XDj$5$l$k(B.
! 84:
! 85: @item $BC19`<0(B (monomial)
! 86: $B9`$H78?t$N@Q(B.
! 87:
! 88: @example
! 89: 2*<<0,1,2,3,4>>
! 90: @end example
! 91:
! 92: $B$H$$$&7A$GI=<($5$l(B, $B$^$?(B, $B$3$N7A$GF~NO2DG=$G$"$k(B.
! 93:
! 94: @itemx $BF,C19`<0(B (head monomial)
! 95: @item $BF,9`(B (head term)
! 96: @itemx $BF,78?t(B (head coefficient)
! 97: $BJ,;6I=8=B?9`<0$K$*$1$k3FC19`<0$O(B, $B9`=g=x$K$h$j@0Ns$5$l$k(B. $B$3$N;~=g(B
! 98: $B=x:GBg$NC19`<0$rF,C19`<0(B, $B$=$l$K8=$l$k9`(B, $B78?t$r$=$l$>$lF,9`(B, $BF,78?t(B
! 99: $B$H8F$V(B.
! 100: @end table
! 101:
! 102: @node $B%U%!%$%k$NFI$_9~$_(B,,, $B%0%l%V%J4pDl$N7W;;(B
! 103: @section $B%U%!%$%k$NFI$_9~$_(B
! 104:
! 105: @noindent
! 106: $B%0%l%V%J4pDl$r7W;;$9$k$?$a$N4pK\E*$JH!?t$O(B @code{dp_gr_main()} $B$*$h$S(B
! 107: @code{dp_gr_mod_main()} $B$J$k(B 2 $B$D$NAH$_9~$_H!?t$G$"$k$,(B, $BDL>o$O(B, $B%Q%i%a%?(B
! 108: $B@_Dj$J$I$r9T$C$?$N$A$3$l$i$r8F$S=P$9%f!<%6H!?t$rMQ$$$k$N$,JXMx$G$"$k(B.
! 109: $B$3$l$i$N%f!<%6H!?t$O(B, $B%U%!%$%k(B @samp{gr} $B$r(B @code{load()} $B$K$h$jFI(B
! 110: $B$_9~$`$3$H$K$h$j;HMQ2DG=$H$J$k(B. @samp{gr} $B$O(B, @b{Asir} $B$NI8=`(B
! 111: $B%i%$%V%i%j%G%#%l%/%H%j$KCV$+$l$F$$$k(B. $B$h$C$F(B, $B4D6-JQ?t(B @code{ASIR_LIBDIR}
! 112: $B$rFC$K0[$J$k%Q%9$K@_Dj$7$J$$8B$j(B, $B%U%!%$%kL>$N$_$GFI$_9~$`$3$H$,$G$-$k(B.
! 113:
! 114: @example
! 115: [0] load("gr")$
! 116: @end example
! 117:
! 118: @node $B4pK\E*$JH!?t(B,,, $B%0%l%V%J4pDl$N7W;;(B
! 119: @section $B4pK\E*$JH!?t(B
! 120:
! 121: @noindent
! 122: @samp{gr} $B$G$O?tB?$/$NH!?t$,Dj5A$5$l$F$$$k$,(B, $BD>@\(B
! 123: $B%0%l%V%J4pDl$r7W;;$9$k$?$a$N%H%C%W%l%Y%k$O<!$N(B 3 $B$D$G$"$k(B.
! 124: $B0J2<$G(B, @var{plist} $B$OB?9`<0$N%j%9%H(B, @var{vlist} $B$OJQ?t(B ($BITDj85(B) $B$N%j%9%H(B,
! 125: @var{order} $B$OJQ?t=g=x7?(B, @var{p} $B$O(B @code{2^27} $BL$K~$NAG?t$G$"$k(B.
! 126:
! 127: @table @code
! 128: @item gr(@var{plist},@var{vlist},@var{order})
! 129:
! 130: Gebauer-Moeller $B$K$h$k(B useless pair elimination criteria, sugar
! 131: strategy $B$*$h$S(B Traverso $B$K$h$k(B trace-lifting $B$rMQ$$$?(B Buchberger $B%"%k(B
! 132: $B%4%j%:%`$K$h$kM-M}?t78?t%0%l%V%J4pDl7W;;H!?t(B. $B0lHL$K$O$3$NH!?t$rMQ$$$k(B.
! 133:
! 134: @item hgr(@var{plist},@var{vlist},@var{order})
! 135:
! 136: $BF~NOB?9`<0$r@F<!2=$7$?8e(B @code{gr()} $B$N%0%l%V%J4pDl8uJd@8@.It$K$h$j8u(B
! 137: $BJd@8@.$7(B, $BHs@F<!2=(B, interreduce $B$7$?$b$N$r(B @code{gr()} $B$N%0%l%V%J4pDl(B
! 138: $B%A%'%C%/It$G%A%'%C%/$9$k(B. 0 $B<!85%7%9%F%`(B ($B2r$N8D?t$,M-8B8D$NJ}Dx<07O(B)
! 139: $B$N>l9g(B, sugar strategy $B$,78?tKDD%$r0z$-5/$3$9>l9g$,$"$k(B. $B$3$N$h$&$J>l(B
! 140: $B9g(B, strategy $B$r@F<!2=$K$h$k(B strategy $B$KCV$-49$($k$3$H$K$h$j78?tKDD%$r(B
! 141: $BM^@)$9$k$3$H$,$G$-$k>l9g$,B?$$(B.
! 142:
! 143: @item gr_mod(@var{plist},@var{vlist},@var{order},@var{p})
! 144:
! 145: Gebauer-Moeller $B$K$h$k(B useless pair elimination criteria, sugar
! 146: strategy $B$*$h$S(B Buchberger $B%"%k%4%j%:%`$K$h$k(B GF(p) $B78?t%0%l%V%J4pDl7W(B
! 147: $B;;H!?t(B.
! 148:
! 149: @end table
! 150:
! 151: @node $B7W;;$*$h$SI=<($N@)8f(B,,, $B%0%l%V%J4pDl$N7W;;(B
! 152: @section $B7W;;$*$h$SI=<($N@)8f(B
! 153:
! 154: @noindent
! 155: $B%0%l%V%J4pDl$N7W;;$K$*$$$F(B, $B$5$^$6$^$J%Q%i%a%?@_Dj$r9T$&$3$H$K$h$j7W;;(B,
! 156: $BI=<($r@)8f$9$k$3$H$,$G$-$k(B. $B$3$l$i$O(B, $BAH$_9~$_H!?t(B @code{dp_gr_flags()}
! 157: $B$K$h$j@_Dj;2>H$9$k$3$H$,$G$-$k(B. $BL50z?t$G(B @code{dp_gr_flags()} $B$r<B9T$9$k(B
! 158: $B$H(B, $B8=:_@_Dj$5$l$F$$$k%Q%i%a%?$,(B, $BL>A0$HCM$N%j%9%H$GJV$5$l$k(B.
! 159:
! 160: @example
! 161: [100] dp_gr_flags();
! 162: [Demand,0,NoSugar,0,NoCriB,0,NoGC,0,NoMC,0,NoRA,0,NoGCD,0,Top,0,ShowMag,1,
! 163: Print,1,Stat,0,Reverse,0,InterReduce,0,Multiple,0]
! 164: [101]
! 165: @end example
! 166:
! 167: $B0J2<$G(B, $B3F%Q%i%a%?$N0UL#$r@bL@$9$k(B. on $B$N>l9g$H$O(B, $B%Q%i%a%?$,(B 0 $B$G$J$$>l9g$r(B
! 168: $B$$$&(B. $B$3$l$i$N%Q%i%a%?$N=i4|CM$OA4$F(B 0 (off) $B$G$"$k(B.
! 169:
! 170:
! 171: @table @code
! 172: @item NoSugar
! 173: on $B$N>l9g(B, sugar strategy $B$NBe$o$j$K(B Buchberger$B$N(B normal strategy $B$,MQ(B
! 174: $B$$$i$l$k(B.
! 175:
! 176: @item NoCriB
! 177: on $B$N>l9g(B, $BITI,MWBP8!=P5,=`$N$&$A(B, $B5,=`(B B $B$rE,MQ$7$J$$(B.
! 178:
! 179: @item NoGC
! 180: on $B$N>l9g(B, $B7k2L$,%0%l%V%J4pDl$K$J$C$F$$$k$+$I$&$+$N%A%'%C%/$r9T$o$J$$(B.
! 181:
! 182: @item NoMC
! 183: on $B$N>l9g(B, $B7k2L$,F~NO%$%G%"%k$HF1Ey$N%$%G%"%k$G$"$k$+$I$&$+$N%A%'%C%/(B
! 184: $B$r9T$o$J$$(B.
! 185:
! 186: @item NoRA
! 187: on $B$N>l9g(B, $B7k2L$r(B reduced $B%0%l%V%J4pDl$K$9$k$?$a$N(B
! 188: interreduce $B$r9T$o$J$$(B.
! 189:
! 190: @item NoGCD
! 191: on $B$N>l9g(B, $BM-M}<078?t$N%0%l%V%J4pDl7W;;$K$*$$$F(B, $B@8@.$5$l$?B?9`<0$N(B,
! 192: $B78?t$N(B content $B$r$H$i$J$$(B.
! 193:
! 194: @item Top
! 195: on $B$N>l9g(B, normal form $B7W;;$K$*$$$FF,9`>C5n$N$_$r9T$&(B.
! 196:
! 197: @item Interreduce
! 198: on $B$N>l9g(B, $BB?9`<0$r@8@.$9$kKh$K(B, $B$=$l$^$G@8@.$5$l$?4pDl$r$=$NB?9`<0$K(B
! 199: $B$h$k(B normal form $B$GCV$-49$($k(B.
! 200:
! 201: @item Reverse
! 202: on $B$N>l9g(B, normal form $B7W;;$N:]$N(B reducer $B$r(B, $B?7$7$/@8@.$5$l$?$b$N$rM%(B
! 203: $B@h$7$FA*$V(B.
! 204:
! 205: @item Print
! 206: on $B$N>l9g(B, $B%0%l%V%J4pDl7W;;$NESCf$K$*$1$k$5$^$6$^$J>pJs$rI=<($9$k(B.
! 207:
! 208: @item Stat
! 209: on $B$G(B @code{Print} $B$,(B off $B$J$i$P(B, @code{Print} $B$,(B on $B$N$H$-I=<($5(B
! 210: $B$l$k%G!<%?$NFb(B, $B=87W%G!<%?$N$_$,I=<($5$l$k(B.
! 211:
! 212: @item ShowMag
! 213: on $B$G(B @code{Print} $B$,(B on $B$J$i$P(B, $B@8@.$,@8@.$5$l$kKh$K(B, $B$=$NB?9`<0$N(B
! 214: $B78?t$N%S%C%HD9$NOB$rI=<($7(B, $B:G8e$K(B, $B$=$l$i$NOB$N:GBgCM$rI=<($9$k(B.
! 215:
! 216: @item Multiple
! 217: 0 $B$G$J$$@0?t$N;~(B, $BM-M}?t>e$N@55,7A7W;;$K$*$$$F(B, $B78?t$N%S%C%HD9$NOB$,(B
! 218: @code{Multiple} $BG\$K$J$k$4$H$K78?tA4BN$N(B GCD $B$,7W;;$5$l(B, $B$=$N(B GCD $B$G(B
! 219: $B3d$C$?B?9`<0$r4JLs$9$k(B. @code{Multiple} $B$,(B 1 $B$J$i$P(B, $B4JLs$9$k$4$H$K(B
! 220: GCD $B7W;;$,9T$o$l0lHL$K$O8zN($,0-$/$J$k$,(B, @code{Multiple} $B$r(B 2 $BDxEY(B
! 221: $B$H$9$k$H(B, $B5pBg$J@0?t$,78?t$K8=$l$k>l9g(B, $B8zN($,NI$/$J$k>l9g$,$"$k(B.
! 222:
! 223: @item Demand
! 224: $B@5Ev$J%G%#%l%/%H%jL>(B ($BJ8;zNs(B) $B$rCM$K;}$D$H$-(B, $B@8@.$5$l$?B?9`<0$O%a%b%j(B
! 225: $BCf$K$*$+$l$:(B, $B$=$N%G%#%l%/%H%jCf$K%P%$%J%j%G!<%?$H$7$FCV$+$l(B, $B$=$NB?9`(B
! 226: $B<0$rMQ$$$k(B normal form $B7W;;$N:](B, $B<+F0E*$K%a%b%jCf$K%m!<%I$5$l$k(B. $B3FB?(B
! 227: $B9`<0$O(B, $BFbIt$G$N%$%s%G%C%/%9$r%U%!%$%kL>$K;}$D%U%!%$%k$K3JG<$5$l$k(B.
! 228: $B$3$3$G;XDj$5$l$?%G%#%l%/%H%j$K=q$+$l$?%U%!%$%k$O<+F0E*$K$O>C5n$5$l$J$$(B
! 229: $B$?$a(B, $B%f!<%6$,@UG$$r;}$C$F>C5n$9$kI,MW$,$"$k(B.
! 230: @end table
! 231:
! 232: @noindent
! 233: @code{Print} $B$,(B 0 $B$G$J$$>l9g<!$N$h$&$J%G!<%?$,I=<($5$l$k(B.
! 234:
! 235: @example
! 236: [93] gr(cyclic(4),[c0,c1,c2,c3],0)$
! 237: mod= 99999989, eval = []
! 238: (0)(0)<<0,2,0,0>>(2,3),nb=2,nab=5,rp=2,sugar=2,mag=4
! 239: (0)(0)<<0,1,2,0>>(1,2),nb=3,nab=6,rp=2,sugar=3,mag=4
! 240: (0)(0)<<0,1,1,2>>(0,1),nb=4,nab=7,rp=3,sugar=4,mag=6
! 241: .
! 242: (0)(0)<<0,0,3,2>>(5,6),nb=5,nab=8,rp=2,sugar=5,mag=4
! 243: (0)(0)<<0,1,0,4>>(4,6),nb=6,nab=9,rp=3,sugar=5,mag=4
! 244: (0)(0)<<0,0,2,4>>(6,8),nb=7,nab=10,rp=4,sugar=6,mag=6
! 245: ....gb done
! 246: reduceall
! 247: .......
! 248: membercheck
! 249: (0,0)(0,0)(0,0)(0,0)
! 250: gbcheck total 8 pairs
! 251: ........
! 252: UP=(0,0)SP=(0,0)SPM=(0,0)NF=(0,0)NFM=(0.010002,0)ZNFM=(0.010002,0)PZ=(0,0)
! 253: NP=(0,0)MP=(0,0)RA=(0,0)MC=(0,0)GC=(0,0)T=40,B=0 M=8 F=6 D=12 ZR=5 NZR=6
! 254: Max_mag=6
! 255: [94]
! 256: @end example
! 257:
! 258: @noindent
! 259: $B:G=i$KI=<($5$l$k(B @code{mod}, @code{eval} $B$O(B, trace-lifting $B$GMQ$$$i$l$kK!(B
! 260: $B$G$"$k(B. @code{mod} $B$OAG?t(B, @code{eval} $B$OM-M}<078?t$N>l9g$KMQ$$$i$l$k(B
! 261: $B?t$N%j%9%H$G$"$k(B.
! 262:
! 263: @noindent
! 264: $B7W;;ESCf$GB?9`<0$,@8@.$5$l$kKh$K<!$N7A$N%G!<%?$,I=<($5$l$k(B.
! 265:
! 266: @example
! 267: (TNF)(TCONT)HT(INDEX),nb=NB,nab=NAB,rp=RP,sugar=S,mag=M
! 268: @end example
! 269:
! 270: @noindent
! 271: $B$=$l$i$N0UL#$O<!$NDL$j(B.
! 272:
! 273: @table @code
! 274: @item TNF
! 275: normal form $B7W;;;~4V(B ($BIC(B)
! 276:
! 277: @item TCONT
! 278: content $B7W;;;~4V(B ($BIC(B)
! 279:
! 280: @item HT
! 281: $B@8@.$5$l$?B?9`<0$NF,9`(B
! 282:
! 283: @item INDEX
! 284: S-$BB?9`<0$r9=@.$9$kB?9`<0$N%$%s%G%C%/%9$N%Z%"(B
! 285:
! 286: @item NB
! 287: $B8=:_$N(B, $B>iD9@-$r=|$$$?4pDl$N?t(B
! 288:
! 289: @item NAB
! 290: $B8=:_$^$G$K@8@.$5$l$?4pDl$N?t(B
! 291:
! 292: @item RP
! 293: $B;D$j$N%Z%"$N?t(B
! 294:
! 295: @item S
! 296: $B@8@.$5$l$?B?9`<0$N(B sugar $B$NCM(B
! 297:
! 298: @item M
! 299: $B@8@.$5$l$?B?9`<0$N78?t$N%S%C%HD9$NOB(B (@code{ShowMag} $B$,(B on $B$N;~$KI=<($5$l$k(B. )
! 300: @end table
! 301:
! 302: @noindent
! 303: $B:G8e$K(B, $B=87W%G!<%?$,I=<($5$l$k(B. $B0UL#$O<!$NDL$j(B.
! 304: ($B;~4V$NI=<($K$*$$$F(B, $B?t;z$,(B 2 $B$D$"$k$b$N$O(B, $B7W;;;~4V$H(B GC $B;~4V$N%Z%"$G$"$k(B.)
! 305:
! 306:
! 307: @table @code
! 308: @item UP
! 309: $B%Z%"$N%j%9%H$NA`:n$K$+$+$C$?;~4V(B
! 310:
! 311: @item SP
! 312: $BM-M}?t>e$N(B S-$BB?9`<07W;;;~4V(B
! 313:
! 314: @item SPM
! 315: $BM-8BBN>e$N(B S-$BB?9`<07W;;;~4V(B
! 316:
! 317: @item NF
! 318: $BM-M}?t>e$N(B normal form $B7W;;;~4V(B
! 319:
! 320: @item NFM
! 321: $BM-8BBN>e$N(B normal form $B7W;;;~4V(B
! 322:
! 323: @item ZNFM
! 324: @code{NFM} $B$NFb(B, 0 $B$X$N(B reduction $B$K$+$+$C$?;~4V(B
! 325:
! 326: @item PZ
! 327: content $B7W;;;~4V(B
! 328:
! 329: @item NP
! 330: $BM-M}?t78?tB?9`<0$N78?t$KBP$9$k>jM>1i;;$N7W;;;~4V(B
! 331:
! 332: @item MP
! 333: S-$BB?9`<0$r@8@.$9$k%Z%"$NA*Br$K$+$+$C$?;~4V(B
! 334:
! 335: @item RA
! 336: interreduce $B7W;;;~4V(B
! 337:
! 338: @item MC
! 339: trace-lifting $B$K$*$1$k(B, $BF~NOB?9`<0$N%a%s%P%7%C%W7W;;;~4V(B
! 340:
! 341: @item GC
! 342: $B7k2L$N%0%l%V%J4pDl8uJd$N%0%l%V%J4pDl%A%'%C%/;~4V(B
! 343:
! 344: @item T
! 345: $B@8@.$5$l$?%Z%"$N?t(B
! 346:
! 347: @item B, M, F, D
! 348: $B3F(B criterion $B$K$h$j=|$+$l$?%Z%"$N?t(B
! 349:
! 350: @item ZR
! 351: 0 $B$K(B reduce $B$5$l$?%Z%"$N?t(B
! 352:
! 353: @item NZR
! 354: 0 $B$G$J$$B?9`<0$K(B reduce $B$5$l$?%Z%"$N?t(B
! 355:
! 356: @item Max_mag
! 357: $B@8@.$5$l$?B?9`<0$N(B, $B78?t$N%S%C%HD9$NOB$N:GBgCM(B
! 358:
! 359: @end table
! 360:
! 361: @node $B9`=g=x$N@_Dj(B,,, $B%0%l%V%J4pDl$N7W;;(B
! 362: @section $B9`=g=x$N@_Dj(B
! 363:
! 364: @noindent
! 365: $B9`$OFbIt$G$O(B, $B3FJQ?t$K4X$9$k;X?t$r@.J,$H$9$k@0?t%Y%/%H%k$H$7$FI=8=$5$l(B
! 366: $B$k(B. $BB?9`<0$rJ,;6I=8=B?9`<0$KJQ49$9$k:](B, $B3FJQ?t$,$I$N@.J,$KBP1~$9$k$+$r(B
! 367: $B;XDj$9$k$N$,(B, $BJQ?t%j%9%H$G$"$k(B. $B$5$i$K(B, $B$=$l$i@0?t%Y%/%H%k$NA4=g=x$r(B
! 368: $B;XDj$9$k$N$,9`=g=x$N7?$G$"$k(B. $B9`=g=x7?$O(B, $B?t(B, $B?t$N%j%9%H$"$k$$$O(B
! 369: $B9TNs$GI=8=$5$l$k(B.
! 370:
! 371: @noindent
! 372: $B4pK\E*$J9`=g=x7?$H$7$F<!$N(B 3 $B$D$,$"$k(B.
! 373:
! 374: @table @code
! 375: @item 0 (DegRevLex; @b{$BA4<!?t5U<-=q<0=g=x(B})
! 376:
! 377: $B0lHL$K(B, $B$3$N=g=x$K$h$k%0%l%V%J4pDl7W;;$,:G$b9bB.$G$"$k(B. $B$?$@$7(B,
! 378: $BJ}Dx<0$r2r$/$H$$$&L\E*$KMQ$$$k$3$H$O(B, $B0lHL$K$O$G$-$J$$(B. $B$3$N(B
! 379: $B=g=x$K$h$k%0%l%V%J4pDl$O(B, $B2r$N8D?t$N7W;;(B, $B%$%G%"%k$N%a%s%P%7%C%W$d(B,
! 380: $BB>$NJQ?t=g=x$X$N4pDlJQ49$N$?$a$N%=!<%9$H$7$FMQ$$$i$l$k(B.
! 381:
! 382: @item 1 (DegLex; @b{$BA4<!?t<-=q<0=g=x(B})
! 383:
! 384: $B$3$N=g=x$b(B, $B<-=q<0=g=x$KHf$Y$F9bB.$K%0%l%V%J4pDl$r5a$a$k$3$H$,$G$-$k$,(B,
! 385: @code{DegRevLex} $B$HF1MMD>@\$=$N7k2L$rMQ$$$k$3$H$O:$Fq$G$"$k(B. $B$7$+$7(B,
! 386: $B<-=q<0=g=x$N%0%l%V%J4pDl$r5a$a$k:]$K(B, $B@F<!2=8e$K$3$N=g=x$G%0%l%V%J4pDl(B
! 387: $B$r5a$a$F$$$k(B.
! 388:
! 389: @item 2 (Lex; @b{$B<-=q<0=g=x(B})
! 390:
! 391: $B$3$N=g=x$K$h$k%0%l%V%J4pDl$O(B, $BJ}Dx<0$r2r$/>l9g$K:GE,$N7A$N4pDl$rM?$($k$,(B
! 392: $B7W;;;~4V$,$+$+$j2a$.$k$N$,FqE@$G$"$k(B. $BFC$K(B, $B2r$,M-8B8D$N>l9g(B, $B7k2L$N(B
! 393: $B78?t$,6K$a$FD9Bg$JB?G\D9?t$K$J$k>l9g$,B?$$(B. $B$3$N>l9g(B, @code{gr()},
! 394: @code{hgr()} $B$K$h$k7W;;$,6K$a$FM-8z$K$J$k>l9g$,B?$$(B.
! 395: @end table
! 396:
! 397: @noindent
! 398: $B$3$l$i$rAH$_9g$o$;$F%j%9%H$G;XDj$9$k$3$H$K$h$j(B, $BMM!9$J>C5n=g=x$,;XDj$G$-$k(B.
! 399: $B$3$l$O(B,
! 400:
! 401: @code{[[O1,L1],[O2,L2],...]}
! 402:
! 403: @noindent
! 404: $B$G;XDj$5$l$k(B. @code{Oi} $B$O(B 0, 1, 2 $B$N$$$:$l$+$G(B, @code{Li} $B$OJQ?t$N8D(B
! 405: $B?t$rI=$9(B. $B$3$N;XDj$O(B, $BJQ?t$r@hF,$+$i(B @code{L1}, @code{L2} , ...$B8D(B
! 406: $B$:$D$NAH$KJ,$1(B, $B$=$l$>$l$NJQ?t$K4X$7(B, $B=g$K(B @code{O1}, @code{O2},
! 407: ...$B$N9`=g=x7?$GBg>.$,7hDj$9$k$^$GHf3S$9$k$3$H$r0UL#$9$k(B. $B$3$N7?$N(B
! 408: $B=g=x$O0lHL$K>C5n=g=x$H8F$P$l$k(B.
! 409:
! 410: @noindent
! 411: $B$5$i$K(B, $B9TNs$K$h$j9`=g=x$r;XDj$9$k$3$H$,$G$-$k(B. $B0lHL$K(B, @code{n} $B9T(B
! 412: @code{m} $BNs$N<B?t9TNs(B @code{M} $B$,<!$N@-<A$r;}$D$H$9$k(B.
! 413:
! 414: @enumerate
! 415: @item
! 416: $BD9$5(B @code{m} $B$N@0?t%Y%/%H%k(B @code{v} $B$KBP$7(B @code{Mv=0} $B$H(B @code{v=0} $B$OF1CM(B.
! 417:
! 418: @item
! 419: $BHsIi@.J,$r;}$DD9$5(B @code{m} $B$N(B 0 $B$G$J$$@0?t%Y%/%H%k(B @code{v} $B$KBP$7(B,
! 420: @code{Mv} $B$N(B 0 $B$G$J$$:G=i$N@.J,$OHsIi(B.
! 421: @end enumerate
! 422:
! 423: @noindent
! 424: $B$3$N;~(B, 2 $B$D$N%Y%/%H%k(B @code{t}, @code{s} $B$KBP$7(B,
! 425: @code{t>s} $B$r(B, @code{M(t-s)} $B$N(B 0 $B$G$J$$:G=i$N@.J,$,HsIi(B,
! 426: $B$GDj5A$9$k$3$H$K$h$j9`=g=x$,Dj5A$G$-$k(B.
! 427:
! 428: @noindent
! 429: $B9`=g=x7?$O(B, @code{gr()} $B$J$I$N0z?t$H$7$F;XDj$5$l$kB>(B, $BAH$_9~$_H!?t(B
! 430: @code{dp_ord()} $B$G;XDj$5$l(B, $B$5$^$6$^$JH!?t$N<B9T$N:]$K;2>H$5$l$k(B.
! 431:
! 432: @noindent
! 433: $B$3$l$i$N=g=x$N6qBNE*$JDj5A$*$h$S%0%l%V%J4pDl$K4X$9$k99$K>\$7$$2r@b$O(B
! 434: @code{[Becker,Weispfenning]} $B$J$I$r;2>H$N$3$H(B.
! 435:
! 436: @noindent
! 437: $B9`=g=x7?$N@_Dj$NB>$K(B, $BJQ?t$N=g=x<+BN$b7W;;;~4V$KBg$-$J1F6A$rM?$($k(B.
! 438:
! 439: @example
! 440: [90] B=[x^10-t,x^8-z,x^31-x^6-x-y]$
! 441: [91] gr(B,[x,y,z,t],2);
! 442: [x^2-2*y^7+(-41*t^2-13*t-1)*y^2+(2*t^17-12*t^14+42*t^12+30*t^11-168*t^9
! 443: -40*t^8+70*t^7+252*t^6+30*t^5-140*t^4-168*t^3+2*t^2-12*t+16)*z^2*y
! 444: +(-12*t^16+72*t^13-28*t^11-180*t^10+112*t^8+240*t^7+28*t^6-127*t^5
! 445: -167*t^4-55*t^3+30*t^2+58*t-15)*z^4,
! 446: (y+t^2*z^2)*x+y^7+(20*t^2+6*t+1)*y^2+(-t^17+6*t^14-21*t^12-15*t^11+84*t^9
! 447: +20*t^8-35*t^7-126*t^6-15*t^5+70*t^4+84*t^3-t^2+5*t-9)*z^2*y+(6*t^16-36*t^13
! 448: +14*t^11+90*t^10-56*t^8-120*t^7-14*t^6+64*t^5+84*t^4+27*t^3-16*t^2-30*t+7)*z^4,
! 449: (t^3-1)*x-y^6+(-6*t^13+24*t^10-20*t^8-36*t^7+40*t^5+24*t^4-6*t^3-20*t^2-6*t-1)*y
! 450: +(t^17-6*t^14+9*t^12+15*t^11-36*t^9-20*t^8-5*t^7+54*t^6+15*t^5+10*t^4-36*t^3
! 451: -11*t^2-5*t+9)*z^2,
! 452: -y^8-8*t*y^3+16*z^2*y^2+(-8*t^16+48*t^13-56*t^11-120*t^10+224*t^8+160*t^7
! 453: -56*t^6-336*t^5-112*t^4+112*t^3+224*t^2+24*t-56)*z^4*y+(t^24-8*t^21+20*t^19
! 454: +28*t^18-120*t^16-56*t^15+14*t^14+300*t^13+70*t^12-56*t^11-400*t^10-84*t^9
! 455: +84*t^8+268*t^7+84*t^6-56*t^5-63*t^4-36*t^3+46*t^2-12*t+1)*z,
! 456: 2*t*y^5+z*y^2+(-2*t^11+8*t^8-20*t^6-12*t^5+40*t^3+8*t^2-10*t-20)*z^3*y+8*t^14
! 457: -32*t^11+48*t^8-t^7-32*t^5-6*t^4+9*t^2-t,
! 458: -z*y^3+(t^7-2*t^4+3*t^2+t)*y+(-2*t^6+4*t^3+2*t-2)*z^2,
! 459: 2*t^2*y^3+z^2*y^2+(-2*t^5+4*t^2-6)*z^4*y+(4*t^8-t^7-8*t^5+2*t^4-4*t^3+5*t^2-t)*z,
! 460: z^3*y^2+2*t^3*y+(-t^7+2*t^4+t^2-t)*z^2,
! 461: -t*z*y^2-2*z^3*y+t^8-2*t^5-t^3+t^2,
! 462: -t^3*y^2-2*t^2*z^2*y+(t^6-2*t^3-t+1)*z^4,
! 463: z^5-t^4]
! 464: [93] gr(B,[t,z,y,x],2);
! 465: [x^10-t,x^8-z,x^31-x^6-x-y]
! 466: @end example
! 467:
! 468: @noindent
! 469: $BJQ?t=g=x(B @code{[x,y,z,t]} $B$K$*$1$k%0%l%V%J4pDl$O(B, $B4pDl$N?t$bB?$/(B, $B$=$l$>$l$N(B
! 470: $B<0$bBg$-$$(B. $B$7$+$7(B, $B=g=x(B @code{[t,z,y,x]} $B$K$b$H$G$O(B, @code{B} $B$,$9$G$K(B
! 471: $B%0%l%V%J4pDl$H$J$C$F$$$k(B. $BBg;(GD$K$$$($P(B, $B<-=q<0=g=x$G%0%l%V%J4pDl$r5a$a$k(B
! 472: $B$3$H$O(B, $B:8B&$N(B ($B=g=x$N9b$$(B) $BJQ?t$r(B, $B1&B&$N(B ($B=g=x$NDc$$(B) $BJQ?t$G=q$-I=$9(B
! 473: $B$3$H$G$"$j(B, $B$3$NNc$N>l9g$O(B, @code{t}, @code{z}, @code{y} $B$,4{$K(B
! 474: @code{x} $B$GI=$5$l$F$$$k$3$H$+$i$3$N$h$&$J6KC<$J7k2L$H$J$C$?$o$1$G$"$k(B.
! 475: $B<B:]$K8=$l$k7W;;$K$*$$$F$O(B, $B$3$N$h$&$KA*$V$Y$-JQ?t=g=x$,L@$i$+$G$"$k(B
! 476: $B$3$H$O>/$J$/(B, $B;n9T:x8m$,I,MW$J>l9g$b$"$k(B.
! 477:
! 478: @node $BM-M}<0$r78?t$H$9$k%0%l%V%J4pDl7W;;(B,,, $B%0%l%V%J4pDl$N7W;;(B
! 479: @section $BM-M}<0$r78?t$H$9$k%0%l%V%J4pDl7W;;(B
! 480:
! 481: @noindent
! 482: @code{gr()} $B$J$I$N%H%C%W%l%Y%kH!?t$O(B, $B$$$:$l$b(B, $BF~NOB?9`<0%j%9%H$K(B
! 483: $B8=$l$kJQ?t(B ($BITDj85(B) $B$H(B, $BJQ?t%j%9%H$K8=$l$kJQ?t$rHf3S$7$F(B, $BJQ?t%j%9%H$K(B
! 484: $B$J$$JQ?t$,F~NOB?9`<0$K8=$l$F$$$k>l9g$K$O(B, $B<+F0E*$K(B, $B$=$NJQ?t$r(B, $B78?t(B
! 485: $BBN$N85$H$7$F07$&(B.
! 486:
! 487: @example
! 488: [64] gr([a*x+b*y-c,d*x+e*y-f],[x,y],2);
! 489: [(-e*a+d*b)*x-f*b+e*c,(-e*a+d*b)*y+f*a-d*c]
! 490: @end example
! 491:
! 492: @noindent
! 493: $B$3$NNc$G$O(B, @code{a}, @code{b}, @code{c}, @code{d} $B$,78?tBN$N85$H$7$F(B
! 494: $B07$o$l$k(B. $B$9$J$o$A(B, $BM-M}H!?tBN(B
! 495: @b{F} = @b{Q}(@code{a},@code{b},@code{c},@code{d}) $B>e$N(B 2 $BJQ?tB?9`<04D(B
! 496: @b{F}[@code{x},@code{y}] $B$K$*$1$k%0%l%V%J4pDl$r5a$a$k$3$H$K$J$k(B.
! 497: $BCm0U$9$Y$-$3$H$O(B,
! 498: $B78?t$,BN$H$7$F07$o$l$F$$$k$3$H$G$"$k(B. $B$9$J$o$A(B, $B78?t$N4V$KB?9`<0(B
! 499: $B$H$7$F$N6&DL0x;R$,$"$C$?>l9g$K$O(B, $B7k2L$+$i$=$N0x;R$O=|$+$l$F$$$k(B
! 500: $B$?$a(B, $BM-M}?tBN>e$NB?9`<04D>e$NLdBj$H$7$F9M$($?>l9g$N7k2L$H$O0lHL(B
! 501: $B$K$O0[$J$k(B. $B$^$?(B, $B<g$H$7$F7W;;8zN(>e$NLdBj$N$?$a(B, $BJ,;6I=8=B?9`<0(B
! 502: $B$N78?t$H$7$F<B:]$K5v$5$l$k$N$OB?9`<0$^$G$G$"$k(B. $B$9$J$o$A(B, $BJ,Jl$r(B
! 503: $B;}$DM-M}<0$OJ,;6I=8=B?9`<0$N78?t$H$7$F$O5v$5$l$J$$(B.
! 504:
! 505: @node $B4pDlJQ49(B,,, $B%0%l%V%J4pDl$N7W;;(B
! 506: @section $B4pDlJQ49(B
! 507:
! 508: @noindent
! 509: $B<-=q<0=g=x$N%0%l%V%J4pDl$r5a$a$k>l9g(B, $BD>@\(B @code{gr()} $B$J$I$r5/F0$9$k(B
! 510: $B$h$j(B, $B0lC6B>$N=g=x(B ($BNc$($PA4<!?t5U<-=q<0=g=x(B) $B$N%0%l%V%J4pDl$r7W;;$7$F(B,
! 511: $B$=$l$rF~NO$H$7$F<-=q<0=g=x$N%0%l%V%J4pDl$r7W;;$9$kJ}$,8zN($,$h$$>l9g(B
! 512: $B$,$"$k(B. $B$^$?(B, $BF~NO$,2?$i$+$N=g=x$G$N%0%l%V%J4pDl$K$J$C$F$$$k>l9g(B, $B4pDl(B
! 513: $BJQ49$H8F$P$l$kJ}K!$K$h$j(B, Buchberger $B%"%k%4%j%:%`$K$h$i$:$K8zN(NI$/(B
! 514: $B<-=q<0=g=x$N%0%l%V%J4pDl$,7W;;$G$-$k>l9g$,$"$k(B. $B$3$N$h$&$JL\E*$N$?$a$N(B
! 515: $BH!?t$,(B, $B%f!<%6Dj5AH!?t$H$7$F(B @samp{gr} $B$K$$$/$D$+Dj5A$5$l$F$$$k(B.
! 516: $B0J2<$N(B 2 $B$D$NH!?t$O(B, $BJQ?t=g=x(B @var{vlist1}, $B9`=g=x7?(B @var{order} $B$G(B
! 517: $B4{$K%0%l%V%J4pDl$H$J$C$F$$$kB?9`<0%j%9%H(B @var{gbase} $B$r(B, $BJQ?t=g=x(B
! 518: @var{vlist2} $B$K$*$1$k<-=q<0=g=x$N%0%l%V%J4pDl$KJQ49$9$kH!?t$G$"$k(B.
! 519:
! 520: @table @code
! 521: @item tolex(@var{gbase},@var{vlist1},@var{order},@var{vlist2})
! 522:
! 523: $B$3$NH!?t$O(B, @var{gbase} $B$,M-M}?tBN>e$N%7%9%F%`$N>l9g$K$N$_;HMQ2DG=$G$"$k(B.
! 524: $B$3$NH!?t$O(B, $B<-=q<0=g=x$N%0%l%V%J4pDl$r(B, $BM-8BBN>e$G7W;;$5$l$?%0%l%V%J4pDl(B
! 525: $B$r?w7?$H$7$F(B, $BL$Dj78?tK!$*$h$S(B Hensel $B9=@.$K$h$j5a$a$k$b$N$G$"$k(B.
! 526:
! 527: @item tolex_tl(@var{gbase},@var{vlist1},@var{order},@var{vlist2},@var{homo})
! 528:
! 529: $B$3$NH!?t$O(B, $B<-=q<0=g=x$N%0%l%V%J4pDl$r(B Buchberger $B%"%k%4%j%:%`$K$h$j5a(B
! 530: $B$a$k$b$N$G$"$k$,(B, $BF~NO$,$"$k=g=x$K$*$1$k%0%l%V%J4pDl$G$"$k>l9g$N(B
! 531: trace-lifting$B$K$*$1$k%0%l%V%J4pDl8uJd$NF,9`(B, $BF,78?t$N@-<A$rMxMQ$7$F(B,
! 532: $B:G=*E*$J%0%l%V%J4pDl%A%'%C%/(B, $B%$%G%"%k%a%s%P%7%C%W%A%'%C%/$r>JN,$7$F$$(B
! 533: $B$k$?$a(B, $BC1$K(BBuchberger $B%"%k%4%j%:%`$r7+$jJV$9$h$j8zN($h$/7W;;$G$-$k(B.
! 534: $B99$K(B, $BF~NO$,(B 0 $B<!85%7%9%F%`$N>l9g(B, $B<+F0E*$K$b$&(B 1 $B$D$NCf4VE*$J9`=g=x$r(B
! 535: $B7PM3$7$F<-=q<0=g=x$N%0%l%V%J4pDl$r7W;;$9$k(B. $BB?$/$N>l9g(B, $B$3$NJ}K!$O(B,
! 536: $BD>@\<-=q<0=g=x$N7W;;$r9T$&$h$j8zN($,$h$$(B. ($B$b$A$m$sNc30$"$j(B. )
! 537: $B0z?t(B @var{homo} $B$,(B 0 $B$G$J$$;~(B, @code{hgr()} $B$HF1MM$K@F<!2=$r7PM3$7$F(B
! 538: $B7W;;$r9T$&(B.
! 539:
! 540: @end table
! 541:
! 542: @noindent
! 543: $B$=$NB>(B, 0 $B<!85%7%9%F%`$KBP$7(B, $BM?$($i$l$?B?9`<0$N:G>.B?9`<0$r5a$a$k(B
! 544: $BH!?t(B, 0 $B<!85%7%9%F%`$N2r$r(B, $B$h$j%3%s%Q%/%H$KI=8=$9$k$?$a$NH!?t$J$I$,(B
! 545: @samp{gr} $B$GDj5A$5$l$F$$$k(B. $B$3$l$i$K$D$$$F$O8D!9$NH!?t$N@bL@$r;2>H$N$3$H(B.
! 546:
! 547: @node $B%0%l%V%J4pDl$K4X$9$kH!?t(B,,, $B%0%l%V%J4pDl$N7W;;(B
! 548: @section $B%0%l%V%J4pDl$K4X$9$kH!?t(B
! 549:
! 550: @menu
! 551: * gr hgr gr_mod::
! 552: * lex_hensel lex_tl tolex tolex_d tolex_tl::
! 553: * lex_hensel_gsl tolex_gsl tolex_gsl_d::
! 554: * gr_minipoly minipoly::
! 555: * tolexm minipolym::
! 556: * dp_gr_main dp_gr_mod_main::
! 557: * dp_f4_main dp_f4_mod_main::
! 558: * dp_gr_flags dp_gr_print::
! 559: * dp_ord::
! 560: * dp_ptod::
! 561: * dp_dtop::
! 562: * dp_mod dp_rat::
! 563: * dp_homo dp_dehomo::
! 564: * dp_ptozp dp_prim::
! 565: * dp_nf dp_nf_mod dp_true_nf dp_true_nf_mod::
! 566: * dp_hm dp_ht dp_hc dp_rest::
! 567: * dp_td dp_sugar::
! 568: * dp_lcm::
! 569: * dp_redble::
! 570: * dp_subd::
! 571: * dp_mbase::
! 572: * dp_mag::
! 573: * dp_red dp_red_mod::
! 574: * dp_sp dp_sp_mod::
! 575: * p_nf p_nf_mod p_true_nf p_true_nf_mod ::
! 576: * p_terms::
! 577: * gb_comp::
! 578: * katsura hkatsura cyclic hcyclic::
! 579: * dp_vtoe dp_etov::
! 580: * lex_hensel_gsl tolex_gsl tolex_gsl_d::
! 581: @end menu
! 582:
! 583: @node gr hgr gr_mod,,, $B%0%l%V%J4pDl$K4X$9$kH!?t(B
! 584: @subsection @code{gr}, @code{hgr}, @code{gr_mod}, @code{dgr}
! 585: @findex gr
! 586: @findex hgr
! 587: @findex gr_mod
! 588: @findex dgr
! 589:
! 590: @table @t
! 591: @item gr(@var{plist},@var{vlist},@var{order})
! 592: @itemx hgr(@var{plist},@var{vlist},@var{order})
! 593: @itemx gr_mod(@var{plist},@var{vlist},@var{order},@var{p})
! 594: @itemx dgr(@var{plist},@var{vlist},@var{order},@var{procs})
! 595: :: $B%0%l%V%J4pDl$N7W;;(B
! 596: @end table
! 597:
! 598: @table @var
! 599: @item return
! 600: $B%j%9%H(B
! 601: @item plist, vlist, procs
! 602: $B%j%9%H(B
! 603: @item order
! 604: $B?t(B, $B%j%9%H$^$?$O9TNs(B
! 605: @item p
! 606: 2^27 $BL$K~$NAG?t(B
! 607: @end table
! 608:
! 609: @itemize @bullet
! 610: @item
! 611: $BI8=`%i%$%V%i%j$N(B @samp{gr} $B$GDj5A$5$l$F$$$k(B.
! 612: @item
! 613: $B$$$:$l$b(B, $BB?9`<0%j%9%H(B @var{plist} $B$N(B, $BJQ?t=g=x(B @var{vlist}, $B9`=g=x7?(B
! 614: @var{order} $B$K4X$9$k%0%l%V%J4pDl$r5a$a$k(B. @code{gr()}, @code{hgr()}
! 615: $B$O(B $BM-M}?t78?t(B, @code{gr_mod()} $B$O(B GF(@var{p}) $B78?t$H$7$F7W;;$9$k(B.
! 616: @item
! 617: @var{vlist} $B$OITDj85$N%j%9%H(B. @var{vlist} $B$K8=$l$J$$ITDj85$O(B,
! 618: $B78?tBN$KB0$9$k$H8+$J$5$l$k(B.
! 619: @item
! 620: @code{gr()}, trace-lifting ($B%b%8%e%i1i;;$rMQ$$$?9bB.2=(B) $B$*$h$S(B sugar
! 621: strategy $B$K$h$k7W;;(B, @code{hgr()} $B$O(B trace-lifting $B$*$h$S(B
! 622: $B@F<!2=$K$h$k(B $B6:@5$5$l$?(B sugar strategy $B$K$h$k7W;;$r9T$&(B.
! 623: @item
! 624: @code{dgr()} $B$O(B, @code{gr()}, @code{dgr()} $B$r(B
! 625: $B;R%W%m%;%9%j%9%H(B @var{procs} $B$N(B 2 $B$D$N%W%m%;%9$K$h$jF1;~$K7W;;$5$;(B,
! 626: $B@h$K7k2L$rJV$7$?J}$N7k2L$rJV$9(B. $B7k2L$OF10l$G$"$k$,(B, $B$I$A$i$NJ}K!$,(B
! 627: $B9bB.$+0lHL$K$OITL@$N$?$a(B, $B<B:]$N7P2a;~4V$rC;=L$9$k$N$KM-8z$G$"$k(B.
! 628: @item
! 629: @code{dgr()} $B$GI=<($5$l$k;~4V$O(B, $B$3$NH!?t$,<B9T$5$l$F$$$k%W%m%;%9$G$N(B
! 630: CPU $B;~4V$G$"$j(B, $B$3$NH!?t$N>l9g$O$[$H$s$IDL?.$N$?$a$N;~4V$G$"$k(B.
! 631: @end itemize
! 632:
! 633: @example
! 634: [0] load("gr")$
! 635: [64] load("cyclic")$
! 636: [74] G=gr(cyclic(5),[c0,c1,c2,c3,c4],2);
! 637: [c4^15+122*c4^10-122*c4^5-1,...]
! 638: [75] GM=gr_mod(cyclic(5),[c0,c1,c2,c3,c4],2,31991)$
! 639: 24628*c4^15+29453*c4^10+2538*c4^5+7363
! 640: [76] (G[0]*24628-GM[0])%31991;
! 641: 0
! 642: @end example
! 643:
! 644: @table @t
! 645: @item $B;2>H(B
! 646: @comment @fref{dp_gr_main dp_gr_mod_main},
! 647: @fref{dp_gr_main dp_gr_mod_main},
! 648: @fref{dp_ord}.
! 649: @end table
! 650:
! 651: @node lex_hensel lex_tl tolex tolex_d tolex_tl,,, $B%0%l%V%J4pDl$K4X$9$kH!?t(B
! 652: @subsection @code{lex_hensel}, @code{lex_tl}, @code{tolex}, @code{tolex_d}, @code{tolex_tl}
! 653: @findex lex_hensel
! 654: @findex lex_tl
! 655: @findex tolex
! 656: @findex tolex_d
! 657: @findex tolex_tl
! 658:
! 659: @table @t
! 660: @item lex_hensel(@var{plist},@var{vlist1},@var{order},@var{vlist2},@var{homo})
! 661: @itemx lex_tl(@var{plist},@var{vlist1},@var{order},@var{vlist2},@var{homo})
! 662: :: $B4pDlJQ49$K$h$k<-=q<0=g=x%0%l%V%J4pDl$N7W;;(B
! 663: @item tolex(@var{plist},@var{vlist1},@var{order},@var{vlist2})
! 664: @itemx tolex_d(@var{plist},@var{vlist1},@var{order},@var{vlist2},@var{procs})
! 665: @itemx tolex_tl(@var{plist},@var{vlist1},@var{order},@var{vlist2},@var{homo})
! 666: :: $B%0%l%V%J4pDl$rF~NO$H$9$k(B, $B4pDlJQ49$K$h$k<-=q<0=g=x%0%l%V%J4pDl$N7W;;(B
! 667: @end table
! 668:
! 669: @table @var
! 670: @item return
! 671: $B%j%9%H(B
! 672: @item plist, vlist1, vlist2, procs
! 673: $B%j%9%H(B
! 674: @item order
! 675: $B?t(B, $B%j%9%H$^$?$O9TNs(B
! 676: @item homo
! 677: $B%U%i%0(B
! 678: @end table
! 679:
! 680: @itemize @bullet
! 681: @item
! 682: $BI8=`%i%$%V%i%j$N(B @samp{gr} $B$GDj5A$5$l$F$$$k(B.
! 683: @item
! 684: @code{lex_hensel()}, @code{lex_tl()} $B$O(B,
! 685: $BB?9`<0%j%9%H(B @var{plist} $B$N(B, $BJQ?t=g=x(B @var{vlist1}, $B9`=g=x7?(B
! 686: @var{order} $B$K4X$9$k%0%l%V%J4pDl$r5a$a(B, $B$=$l$r(B, $BJQ?t=g=x(B @var{vlist2}
! 687: $B$N<-=q<0=g=x%0%l%V%J4pDl$KJQ49$9$k(B.
! 688: @item
! 689: @code{tolex()}, @code{tolex_tl()} $B$O(B,
! 690: $BJQ?t=g=x(B @var{vlist1}, $B9`=g=x7?(B @var{order} $B$K4X$9$k%0%l%V%J4pDl$G$"$k(B
! 691: $BB?9`<0%j%9%H(B @var{plist} $B$rJQ?t=g=x(B @var{vlist2} $B$N<-=q<0=g=x%0%l%V%J(B
! 692: $B4pDl$KJQ49$9$k(B.
! 693: @code{tolex_d()} $B$O(B, @code{tolex()} $B$K$*$1$k(B, $B3F4pDl$N7W;;$r(B, $B;R%W%m%;%9(B
! 694: $B%j%9%H(B @var{procs} $B$N3F%W%m%;%9$KJ,;67W;;$5$;$k(B.
! 695: @item
! 696: @code{lex_hensel()}, @code{lex_tl()} $B$K$*$$$F$O(B, $B<-=q<0=g=x%0%l%V%J4pDl$N(B
! 697: $B7W;;$O<!$N$h$&$K9T$o$l$k(B. (@code{[Noro,Yokoyama]} $B;2>H(B.)
! 698:
! 699: @enumerate
! 700: @item
! 701: @var{vlist1}, @var{order} $B$K4X$9$k%0%l%V%J4pDl(B @var{G0} $B$r7W;;$9$k(B.
! 702: (@code{lex_hensel()} $B$N$_(B. )
! 703: @item
! 704: @var{G0} $B$N3F85$N(B @var{vlist2} $B$K4X$9$k<-=q<0=g=x$K$*$1$kF,78?t$r3d$i$J$$(B
! 705: $B$h$&$JAG?t(B @var{p} $B$rA*$S(B, GF(@var{p}) $B>e$G$N<-=q<0=g=x%0%l%V%J4pDl(B
! 706: @var{Gp} $B$r7W;;$9$k(B.
! 707: @item
! 708: @var{Gp} $B$K8=$l$k$9$Y$F$N9`$N(B, @var{G0} $B$K4X$9$k@55,7A(B @var{NF} $B$r7W;;$9$k(B.
! 709: @item
! 710: @var{Gp} $B$N3F85(B @var{f} $B$K$D$-(B, @var{f} $B$N78?t$rL$Dj78?t$G(B,
! 711: @var{f} $B$N3F9`$rBP1~$9$k(B @var{NF} $B$N85$GCV$-49$((B, $B3F9`$N78?t$r(B 0 $B$HCV$$$?(B,
! 712: $BL$Dj78?t$K4X$9$k@~7AJ}Dx<07O(B @var{Lf} $B$r:n$k(B.
! 713: @item
! 714: @var{Lf} $B$,(B, $BK!(B @var{p} $B$G0l0U2r$r;}$D$3$H$rMQ$$$F(B @var{Lf} $B$N2r$r(B
! 715: $BK!(B @var{p}$B$N2r$+$i(B Hensel $B9=@.$K$h$j5a$a$k(B.
! 716: @item
! 717: $B$9$Y$F$N(B @var{Gp} $B$N85$K$D$-@~7AJ}Dx<0$,2r$1$?$i$=$N2rA4BN$,5a$a$k(B
! 718: $B<-=q<0=g=x$G$N%0%l%V%J4pDl(B. $B$b$7$I$l$+$N@~7AJ}Dx<0$N5a2r$K<:GT$7$?$i(B,
! 719: @var{p} $B$r$H$jD>$7$F$d$jD>$9(B.
! 720: @end enumerate
! 721:
! 722: @item
! 723: @code{lex_tl()}, @code{tolex_tl()} $B$K$*$$$F$O(B, $B<-=q<0=g=x%0%l%V%J4pDl$N(B
! 724: $B7W;;$O<!$N$h$&$K9T$o$l$k(B.
! 725:
! 726: @enumerate
! 727: @item
! 728: @var{vlist1}, @var{order} $B$K4X$9$k%0%l%V%J4pDl(B @var{G0} $B$r7W;;$9$k(B.
! 729: (@code{lex_hensel()} $B$N$_(B. )
! 730: @item
! 731: @var{G0} $B$,(B 0 $B<!85%7%9%F%`$G$J$$$H$-(B, @var{G0} $B$rF~NO$H$7$F(B,
! 732: @var{G0} $B$N3F85$N(B @var{vlist2} $B$K4X$9$k<-=q<0=g=x$K$*$1$kF,78?t$r3d$i$J$$(B
! 733: $B$h$&$JAG?t(B @var{p} $B$rA*$S(B, @var{p} $B$rMQ$$$?(B trace-lifting $B$K$h$j<-=q<0(B
! 734: $B=g=x$N%0%l%V%J4pDl8uJd$r5a$a(B, $B$b$75a$^$C$?$J$i%A%'%C%/$J$7$K$=$l$,5a$a$k(B
! 735: $B%0%l%V%J4pDl$H$J$k(B. $B$b$7<:GT$7$?$i(B, @var{p} $B$r$H$jD>$7$F$d$jD>$9(B.
! 736: @item
! 737: @var{G0} $B$,(B 0 $B<!85%7%9%F%`$N$H$-(B, @var{G0} $B$rF~NO$H$7$F(B,
! 738: $B$^$:(B, @var{vlist2} $B$N:G8e$NJQ?t0J30$r>C5n$9$k>C5n=g=x$K$h$j(B
! 739: $B%0%l%V%J4pDl(B @var{G1} $B$r7W;;$7(B, $B$=$l$+$i<-=q<0=g=x$N%0%l%V%J4pDl$r(B
! 740: $B7W;;$9$k(B. $B$=$N:](B, $B3F%9%F%C%W$G$O(B, $BF~NO$N3F85$N(B, $B5a$a$k=g=x$K$*$1$k(B
! 741: $BF,78?t$r3d$i$J$$AG?t$rMQ$$$?(B trace-lifting $B$G%0%l%V%J4pDl8uJd$r5a$a(B,
! 742: $B$b$75a$^$C$?$i%A%'%C%/$J$7$K$=$l$,$=$N=g=x$G$N%0%l%V%J4pDl$H$J$k(B.
! 743: @end enumerate
! 744:
! 745: @item
! 746: $BM-M}<078?t$N7W;;$O(B, @code{lex_tl()}, @code{tolex_tl()} $B$N$_<u$1IU$1$k(B.
! 747: @item
! 748: @code{homo} $B$,(B 0 $B$G$J$$>l9g(B, $BFbIt$G5/F0$5$l$k(B Buchberger $B%"%k%4%j%:%`$K(B
! 749: $B$*$$$F(B, $B@F<!2=$,9T$o$l$k(B.
! 750: @item
! 751: @code{tolex_d()} $B$GI=<($5$l$k;~4V$O(B, $B$3$NH!?t$,<B9T$5$l$F$$$k%W%m%;%9$K(B
! 752: $B$*$$$F9T$o$l$?7W;;$KBP1~$7$F$$$F(B, $B;R%W%m%;%9$K$*$1$k;~4V$O4^$^$l$J$$(B.
! 753: @end itemize
! 754:
! 755: @example
! 756: [78] K=katsura(5)$
! 757: 30msec + gc : 20msec
! 758: [79] V=[u5,u4,u3,u2,u1,u0]$
! 759: 0msec
! 760: [80] G0=hgr(K,V,2)$
! 761: 91.558sec + gc : 15.583sec
! 762: [81] G1=lex_hensel(K,V,0,V,0)$
! 763: 49.049sec + gc : 9.961sec
! 764: [82] G2=lex_tl(K,V,0,V,1)$
! 765: 31.186sec + gc : 3.500sec
! 766: [83] gb_comp(G0,G1);
! 767: 1
! 768: 10msec
! 769: [84] gb_comp(G0,G2);
! 770: 1
! 771: @end example
! 772:
! 773: @table @t
! 774: @item $B;2>H(B
! 775: @fref{dp_gr_main dp_gr_mod_main},
! 776: @fref{dp_ord}, @fref{$BJ,;67W;;(B}
! 777: @end table
! 778:
! 779: @node lex_hensel_gsl tolex_gsl tolex_gsl_d,,, $B%0%l%V%J4pDl$K4X$9$kH!?t(B
! 780: @subsection @code{lex_hensel_gsl}, @code{tolex_gsl}, @code{tolex_gsl_d}
! 781: @findex lex_hensel_gsl
! 782: @findex tolex_gsl
! 783: @findex tolex_gsl_d
! 784:
! 785: @table @t
! 786: @item lex_hensel_gsl(@var{plist},@var{vlist1},@var{order},@var{vlist2},@var{homo})
! 787: :: GSL $B7A<0$N%$%G%"%k4pDl$N7W;;(B
! 788: @item tolex_gsl(@var{plist},@var{vlist1},@var{order},@var{vlist2},@var{homo})
! 789: @itemx tolex_gsl_d(@var{plist},@var{vlist1},@var{order},@var{vlist2},@var{homo},@var{procs})
! 790: :: $B%0%l%V%J4pDl$rF~NO$H$9$k(B, GSL $B7A<0$N%$%G%"%k4pDl$N7W;;(B
! 791: @end table
! 792:
! 793: @table @var
! 794: @item return
! 795: $B%j%9%H(B
! 796: @item plist, vlist1, vlist2, procs
! 797: $B%j%9%H(B
! 798: @item order
! 799: $B?t(B, $B%j%9%H$^$?$O9TNs(B
! 800: @item homo
! 801: $B%U%i%0(B
! 802: @end table
! 803:
! 804: @itemize @bullet
! 805: @item
! 806: @code{lex_hensel_gsl()} $B$O(B @code{lex_hensel()} $B$N(B, @code{tolex_gsl()} $B$O(B
! 807: @code{tolex()} $B$NJQ<o$G(B, $B7k2L$N$_$,0[$J$k(B.
! 808: @code{tolex_gsl_d()} $B$O(B, $B4pDl7W;;$r(B, @code{procs} $B$G;XDj$5$l$k;R%W%m%;%9$K(B
! 809: $BJ,;67W;;$5$;$k(B.
! 810: @item
! 811: $BF~NO$,(B 0 $B<!85%7%9%F%`$G(B, $B$=$N<-=q<0=g=x%0%l%V%J4pDl$,(B
! 812: @code{[f0,x1-f1,...,xn-fn]} (@code{f0},...,@code{fn} $B$O(B
! 813: @code{x0} $B$N(B 1 $BJQ?tB?9`<0(B) $B$J$k7A(B ($B$3$l$r(B SL $B7A<0$H8F$V(B) $B$r;}$D>l9g(B,
! 814: @code{[[x1,g1,d1],...,[xn,gn,dn],[x0,f0,f0']]} $B$J$k%j%9%H(B ($B$3$l$r(B GSL $B7A<0$H8F$V(B)
! 815: $B$rJV$9(B.
! 816: $B$3$3$G(B, @code{gi} $B$O(B, @code{f0'fi-gi} $B$,(B @code{f0} $B$G3d$j@Z$l$k$h$&$J(B
! 817: @code{x0} $B$N(B1 $BJQ?tB?9`<0$G(B,
! 818: $B2r$O(B @code{f0(x0)=0} $B$J$k(B @code{x0} $B$KBP$7(B, @code{[x1=g1/(d1*f0'),...,xn=gn/(dn*f0')]}
! 819: $B$H$J$k(B. $B<-=q<0=g=x%0%l%V%J4pDl$,>e$N$h$&$J7A$G$J$$>l9g(B, @code{tolex()} $B$K(B
! 820: $B$h$kDL>o$N%0%l%V%J4pDl$rJV$9(B.
! 821: @item
! 822: GSL $B7A<0$K$h$jI=$5$l$k4pDl$O%0%l%V%J4pDl$G$O$J$$$,(B, $B0lHL$K78?t$,(B SL $B7A<0(B
! 823: $B$N%0%l%V%J4pDl$h$jHs>o$K>.$5$$$?$a7W;;$bB.$/(B, $B2r$b5a$a$d$9$$(B.
! 824: @code{tolex_gsl_d()} $B$GI=<($5$l$k;~4V$O(B, $B$3$NH!?t$,<B9T$5$l$F$$$k%W%m%;%9$K(B
! 825: $B$*$$$F9T$o$l$?7W;;$KBP1~$7$F$$$F(B, $B;R%W%m%;%9$K$*$1$k;~4V$O4^$^$l$J$$(B.
! 826: @end itemize
! 827:
! 828: @example
! 829: [103] K=katsura(5)$
! 830: [104] V=[u5,u4,u3,u2,u1,u0]$
! 831: [105] G0=gr(K,V,0)$
! 832: [106] GSL=tolex_gsl(G0,V,0,V)$
! 833: [107] GSL[0];
! 834: [u1,8635837421130477667200000000*u0^31-...]
! 835: [108] GSL[1];
! 836: [u2,10352277157007342793600000000*u0^31-...]
! 837: [109] GSL[5];
! 838: [u0,11771021876193064124640000000*u0^32-...,376672700038178051988480000000*u0^31-...]
! 839: @end example
! 840:
! 841: @table @t
! 842: @item $B;2>H(B
! 843: @fref{lex_hensel lex_tl tolex tolex_d tolex_tl},
! 844: @fref{$BJ,;67W;;(B}
! 845: @end table
! 846:
! 847: @node gr_minipoly minipoly,,, $B%0%l%V%J4pDl$K4X$9$kH!?t(B
! 848: @subsection @code{gr_minipoly}, @code{minipoly}
! 849: @findex gr_minipoly
! 850: @findex minipoly
! 851:
! 852: @table @t
! 853: @item gr_minipoly(@var{plist},@var{vlist},@var{order},@var{poly},@var{v},@var{homo})
! 854: :: $BB?9`<0$N(B, $B%$%G%"%k$rK!$H$7$?:G>.B?9`<0$N7W;;(B
! 855: @item minipoly(@var{plist},@var{vlist},@var{order},@var{poly},@var{v})
! 856: :: $B%0%l%V%J4pDl$rF~NO$H$9$k(B, $BB?9`<0$N:G>.B?9`<0$N7W;;(B
! 857: @end table
! 858:
! 859: @table @var
! 860: @item return
! 861: $BB?9`<0(B
! 862: @item plist, vlist
! 863: $B%j%9%H(B
! 864: @item order
! 865: $B?t(B, $B%j%9%H$^$?$O9TNs(B
! 866: @item poly
! 867: $BB?9`<0(B
! 868: @item v
! 869: $BITDj85(B
! 870: @item homo
! 871: $B%U%i%0(B
! 872: @end table
! 873:
! 874: @itemize @bullet
! 875: @item
! 876: @code{gr_minipoly()} $B$O%0%l%V%J4pDl$N7W;;$+$i9T$$(B, @code{minipoly()} $B$O(B
! 877: $BF~NO$r%0%l%V%J4pDl$H$_$J$9(B.
! 878: @item
! 879: $B%$%G%"%k(B I $B$,BN(B K $B>e$NB?9`<04D(B K[X] $B$N(B 0 $B<!85%$%G%"%k$N;~(B,
! 880: K[@var{v}] $B$N85(B f(@var{v}) $B$K(B f(@var{p}) mod I $B$rBP1~$5$;$k(B
! 881: $B4D=`F17?$N3K$O(B 0 $B$G$J$$B?9`<0$K$h$j@8@.$5$l$k(B. $B$3$N@8@.85$r(B @var{p}
! 882: $B$N(B, $BK!(B @var{I} $B$G$N:G>.B?9`<0$H8F$V(B.
! 883: @item
! 884: @code{gr_minipoly()}, @code{minipoly()} $B$O(B, $BB?9`<0(B @var{p} $B$N:G>.B?9`<0(B
! 885: $B$r5a$a(B, @var{v} $B$rJQ?t$H$9$kB?9`<0$H$7$FJV$9(B.
! 886: @item
! 887: $B:G>.B?9`<0$O(B, $B%0%l%V%J4pDl$N(B 1 $B$D$N85$H$7$F7W;;$9$k$3$H$b$G$-$k$,(B,
! 888: $B:G>.B?9`<0$N$_$r5a$a$?$$>l9g(B, @code{minipoly()}, @code{gr_minipoly()} $B$O(B
! 889: $B%0%l%V%J4pDl$rMQ$$$kJ}K!$KHf$Y$F8zN($,$h$$(B.
! 890: @item
! 891: @code{gr_minipoly()} $B$K;XDj$9$k9`=g=x$H$7$F$O(B, $BDL>oA4<!?t5U<-=q<0=g=x$r(B
! 892: $BMQ$$$k(B.
! 893: @end itemize
! 894:
! 895: @example
! 896: [117] G=tolex(G0,V,0,V)$
! 897: 43.818sec + gc : 11.202sec
! 898: [118] GSL=tolex_gsl(G0,V,0,V)$
! 899: 17.123sec + gc : 2.590sec
! 900: [119] MP=minipoly(G0,V,0,u0,z)$
! 901: 4.370sec + gc : 780msec
! 902: @end example
! 903:
! 904: @table @t
! 905: @item $B;2>H(B
! 906: @fref{lex_hensel lex_tl tolex tolex_d tolex_tl}.
! 907: @end table
! 908:
! 909: @node tolexm minipolym,,, $B%0%l%V%J4pDl$K4X$9$kH!?t(B
! 910: @subsection @code{tolexm}, @code{minipolym}
! 911: @findex tolexm
! 912: @findex minipolym
! 913:
! 914: @table @t
! 915: @item tolexm(@var{plist},@var{vlist1},@var{order},@var{vlist2},@var{mod})
! 916: :: $BK!(B @var{mod} $B$G$N4pDlJQ49$K$h$k%0%l%V%J4pDl7W;;(B
! 917: @item minipolym(@var{plist},@var{vlist1},@var{order},@var{poly},@var{v},@var{mod})
! 918: :: $BK!(B @var{mod} $B$G$N%0%l%V%J4pDl$K$h$kB?9`<0$N:G>.B?9`<0$N7W;;(B
! 919: @end table
! 920:
! 921: @table @var
! 922: @item return
! 923: @code{tolexm()} : $B%j%9%H(B, @code{minipolym()} : $BB?9`<0(B
! 924: @item plist, vlist1, vlist2
! 925: $B%j%9%H(B
! 926: @item order
! 927: $B?t(B, $B%j%9%H$^$?$O9TNs(B
! 928: @item mod
! 929: $BAG?t(B
! 930: @end table
! 931:
! 932: @itemize @bullet
! 933: @item
! 934: $BF~NO(B @var{plist} $B$O$$$:$l$b(B $BJQ?t=g=x(B @var{vlist1}, $B9`=g=x7?(B @var{order},
! 935: $BK!(B @var{mod} $B$K$*$1$k%0%l%V%J4pDl$G$J$1$l$P$J$i$J$$(B.
! 936: @item
! 937: @code{minipolym()} $B$O(B @code{minipoly} $B$KBP1~$9$k7W;;$rK!(B @var{mod}$B$G9T$&(B.
! 938: @item
! 939: @code{tolexm()} $B$O(B FGLM $BK!$K$h$k4pDlJQ49$K$h$j(B @var{vlist2},
! 940: $B<-=q<0=g=x$K$h$k%0%l%V%J4pDl$r7W;;$9$k(B.
! 941: @end itemize
! 942:
! 943: @example
! 944: [197] tolexm(G0,V,0,V,31991);
! 945: [8271*u0^31+10435*u0^30+816*u0^29+26809*u0^28+...,...]
! 946: [198] minipolym(G0,V,0,u0,z,31991);
! 947: z^32+11405*z^31+20868*z^30+21602*z^29+...
! 948: @end example
! 949:
! 950: @table @t
! 951: @item $B;2>H(B
! 952: @fref{lex_hensel lex_tl tolex tolex_d tolex_tl},
! 953: @fref{gr_minipoly minipoly}.
! 954: @end table
! 955:
! 956: @node dp_gr_main dp_gr_mod_main,,, $B%0%l%V%J4pDl$K4X$9$kH!?t(B
! 957: @subsection @code{dp_gr_main}, @code{dp_gr_mod_main}
! 958: @findex dp_gr_main
! 959: @findex dp_gr_mod_main
! 960:
! 961: @table @t
! 962: @item dp_gr_main(@var{plist},@var{vlist},@var{homo},@var{modular},@var{order})
! 963: @itemx dp_gr_mod_main(@var{plist},@var{vlist},@var{homo},@var{modular},@var{order})
! 964: :: $B%0%l%V%J4pDl$N7W;;(B ($BAH$_9~$_H!?t(B)
! 965: @end table
! 966:
! 967: @table @var
! 968: @item return
! 969: $B%j%9%H(B
! 970: @item plist, vlist
! 971: $B%j%9%H(B
! 972: @item order
! 973: $B?t(B, $B%j%9%H$^$?$O9TNs(B
! 974: @item homo
! 975: $B%U%i%0(B
! 976: @item modular
! 977: $B%U%i%0$^$?$OAG?t(B
! 978: @end table
! 979:
! 980: @itemize @bullet
! 981: @item
! 982: $B$3$l$i$NH!?t$O(B, $B%0%l%V%J4pDl7W;;$N4pK\E*AH$_9~$_H!?t$G$"$j(B, @code{gr()},
! 983: @code{hgr()}, @code{gr_mod()} $B$J$I$O$9$Y$F$3$l$i$NH!?t$r8F$S=P$7$F7W;;(B
! 984: $B$r9T$C$F$$$k(B.
! 985: @item
! 986: $B%U%i%0(B @var{homo} $B$,(B 0 $B$G$J$$;~(B, $BF~NO$r@F<!2=$7$F$+$i(B Buchberger $B%"%k%4%j%:%`(B
! 987: $B$r<B9T$9$k(B.
! 988: @item
! 989: @code{dp_gr_mod_main()} $B$KBP$7$F$O(B, @var{modular} $B$O(B, GF(@var{modular}) $B>e(B
! 990: $B$G$N7W;;$r0UL#$9$k(B.
! 991: @code{dp_gr_main()} $B$KBP$7$F$O(B, @var{modular} $B$O<!$N$h$&$J0UL#$r;}$D(B.
! 992: @enumerate
! 993: @item
! 994: @var{modular} $B$,(B 1 $B$N;~(B, trace-lifting $B$K$h$k7W;;$r9T$&(B. $BAG?t$O(B
! 995: @code{lprime(0)} $B$+$i=g$K@.8y$9$k$^$G(B @code{lprime()} $B$r8F$S=P$7$F@8@.$9$k(B.
! 996: @item
! 997: @var{modular} $B$,(B 2 $B0J>e$N<+A3?t$N;~(B, $B$=$NCM$rAG?t$H$_$J$7$F(B trace-lifting
! 998: $B$r9T$&(B. $B$=$NAG?t$G<:GT$7$?>l9g(B, 0 $B$rJV$9(B.
! 999: @item
! 1000: @var{modular} $B$,Ii$N>l9g(B,
! 1001: @var{-modular} $B$KBP$7$F>e=R$N5,B'$,E,MQ$5$l$k$,(B, trace-lifting $B$N:G=*(B
! 1002: $BCJ3,$N%0%l%V%J4pDl%A%'%C%/$H%$%G%"%k%a%s%P%7%C%W%A%'%C%/$,>JN,$5$l$k(B.
! 1003: @end enumerate
! 1004:
! 1005: @item
! 1006: @code{gr(P,V,O)} $B$O(B @code{dp_gr_main(P,V,0,1,O)}, @code{hgr(P,V,O)} $B$O(B
! 1007: @code{dp_gr_main(P,V,1,1,O)}, @code{gr_mod(P,V,O,M)} $B$O(B
! 1008: @code{dp_gr_mod_main(P,V,0,M,O)} $B$r$=$l$>$l<B9T$9$k(B.
! 1009: @item
! 1010: @var{homo}, @var{modular} $B$NB>$K(B, @code{dp_gr_flags()} $B$G@_Dj$5$l$k(B
! 1011: $B$5$^$6$^$J%U%i%0$K$h$j7W;;$,@)8f$5$l$k(B.
! 1012: @end itemize
! 1013:
! 1014: @table @t
! 1015: @item $B;2>H(B
! 1016: @fref{dp_ord},
! 1017: @fref{dp_gr_flags dp_gr_print},
! 1018: @fref{gr hgr gr_mod},
! 1019: @fref{$B7W;;$*$h$SI=<($N@)8f(B}.
! 1020: @end table
! 1021:
! 1022: @node dp_f4_main dp_f4_mod_main,,, $B%0%l%V%J4pDl$K4X$9$kH!?t(B
! 1023: @subsection @code{dp_f4_main}, @code{dp_f4_mod_main}
! 1024: @findex dp_f4_main
! 1025: @findex dp_f4_mod_main
! 1026:
! 1027: @table @t
! 1028: @item dp_f4_main(@var{plist},@var{vlist},@var{order})
! 1029: @itemx dp_f4_mod_main(@var{plist},@var{vlist},@var{order})
! 1030: :: F4 $B%"%k%4%j%:%`$K$h$k%0%l%V%J4pDl$N7W;;(B ($BAH$_9~$_H!?t(B)
! 1031: @end table
! 1032:
! 1033: @table @var
! 1034: @item return
! 1035: $B%j%9%H(B
! 1036: @item plist, vlist
! 1037: $B%j%9%H(B
! 1038: @item order
! 1039: $B?t(B, $B%j%9%H$^$?$O9TNs(B
! 1040: @end table
! 1041:
! 1042: @itemize @bullet
! 1043: @item
! 1044: F4 $B%"%k%4%j%:%`$K$h$j%0%l%V%J4pDl$N7W;;$r9T$&(B.
! 1045: @item
! 1046: F4 $B%"%k%4%j%:%`$O(B, J.C. Faugere $B$K$h$jDs>'$5$l$??7@$Be%0%l%V%J4pDl(B
! 1047: $B;;K!$G$"$j(B, $BK\<BAu$O(B, $BCf9q>jM>DjM}$K$h$k@~7AJ}Dx<05a2r$rMQ$$$?(B
! 1048: $B;n83E*$J<BAu$G$"$k(B.
! 1049: @item
! 1050: $B0z?t$*$h$SF0:n$O$=$l$>$l(B @code{dp_gr_main()}, @code{dp_gr_mod_main()}
! 1051: $B$HF1MM$G$"$k(B.
! 1052: @end itemize
! 1053:
! 1054: @table @t
! 1055: @item $B;2>H(B
! 1056: @fref{dp_ord},
! 1057: @fref{dp_gr_flags dp_gr_print},
! 1058: @fref{gr hgr gr_mod},
! 1059: @fref{$B7W;;$*$h$SI=<($N@)8f(B}.
! 1060: @end table
! 1061:
! 1062: @node dp_gr_flags dp_gr_print,,, $B%0%l%V%J4pDl$K4X$9$kH!?t(B
! 1063: @subsection @code{dp_gr_flags}, @code{dp_gr_print}
! 1064: @findex dp_gr_flags
! 1065: @findex dp_gr_print
! 1066:
! 1067: @table @t
! 1068: @item dp_gr_flags([@var{list}])
! 1069: @itemx dp_gr_print([@var{0|1}])
! 1070: :: $B7W;;$*$h$SI=<(MQ%Q%i%a%?$N@_Dj(B, $B;2>H(B
! 1071: @end table
! 1072:
! 1073: @table @var
! 1074: @item return
! 1075: $B@_DjCM(B
! 1076: @item list
! 1077: $B%j%9%H(B
! 1078: @end table
! 1079:
! 1080: @itemize @bullet
! 1081: @item
! 1082: @code{dp_gr_main()}, @code{dp_gr_mod_main()} $B<B9T;~$K$*$1$k$5$^$6$^(B
! 1083: $B$J%Q%i%a%?$r@_Dj(B, $B;2>H$9$k(B.
! 1084: @item
! 1085: $B0z?t$,$J$$>l9g(B, $B8=:_$N@_Dj$,JV$5$l$k(B.
! 1086: @item
! 1087: $B0z?t$O(B, @code{["Print",1,"NoSugar",1,...]} $B$J$k7A$N%j%9%H$G(B, $B:8$+$i=g$K(B
! 1088: $B@_Dj$5$l$k(B. $B%Q%i%a%?L>$OJ8;zNs$GM?$($kI,MW$,$"$k(B.
! 1089: @item
! 1090: @code{dp_gr_print()} $B$O(B, $BFC$K%Q%i%a%?(B @code{Print} $B$NCM$rD>@\@_Dj(B, $B;2>H(B
! 1091: $B$G$-$k(B. $B$3$l$O(B, @code{dp_gr_main()} $B$J$I$r%5%V%k!<%A%s$H$7$FMQ$$$k%f!<%6(B
! 1092: $BH!?t$K$*$$$F(B, @code{Print} $B$NCM$r8+$F(B, $B$=$N%5%V%k!<%A%s$,Cf4V>pJs$NI=<((B
! 1093: $B$r9T$&:]$K(B, $B?WB.$K%U%i%0$r8+$k$3$H$,$G$-$k$h$&$KMQ0U$5$l$F$$$k(B.
! 1094: @end itemize
! 1095:
! 1096: @table @t
! 1097: @item $B;2>H(B
! 1098: @fref{$B7W;;$*$h$SI=<($N@)8f(B}
! 1099: @end table
! 1100:
! 1101: @node dp_ord,,, $B%0%l%V%J4pDl$K4X$9$kH!?t(B
! 1102: @subsection @code{dp_ord}
! 1103: @findex dp_ord
! 1104:
! 1105: @table @t
! 1106: @item dp_ord([@var{order}])
! 1107: :: $BJQ?t=g=x7?$N@_Dj(B, $B;2>H(B
! 1108: @end table
! 1109:
! 1110: @table @var
! 1111: @item return
! 1112: $BJQ?t=g=x7?(B ($B?t(B, $B%j%9%H$^$?$O9TNs(B)
! 1113: @item order
! 1114: $B?t(B, $B%j%9%H$^$?$O9TNs(B
! 1115: @end table
! 1116:
! 1117: @itemize @bullet
! 1118: @item
! 1119: $B0z?t$,$"$k;~(B, $BJQ?t=g=x7?$r(B @var{order} $B$K@_Dj$9$k(B. $B0z?t$,$J$$;~(B,
! 1120: $B8=:_@_Dj$5$l$F$$$kJQ?t=g=x7?$rJV$9(B.
! 1121:
! 1122: @item
! 1123: $BJ,;6I=8=B?9`<0$K4X$9$kH!?t(B, $B1i;;$O0z?t$H$7$FJQ?t=g=x7?$r$H$k$b$N$H$H$i$J$$$b$N(B
! 1124: $B$,$"$j(B, $B$H$i$J$$$b$N$K4X$7$F$O(B, $B$=$N;~E@$G@_Dj$5$l$F$$$kCM$rMQ$$$F7W;;$,(B
! 1125: $B9T$o$l$k(B.
! 1126:
! 1127: @item
! 1128: @code{gr()} $B$J$I(B, $B0z?t$H$7$FJQ?t=g=x7?$r$H$k$b$N$O(B, $BFbIt$G(B @code{dp_ord()}
! 1129: $B$r8F$S=P$7(B, $BJQ?t=g=x7?$r@_Dj$9$k(B. $B$3$N@_Dj$O(B, $B7W;;=*N;8e$b@8$-;D$k(B.
! 1130:
! 1131: @item
! 1132: $BJ,;6I=8=B?9`<0$N;MB'1i;;$b(B, $B@_Dj$5$l$F$$$kCM$rMQ$$$F7W;;$5$l$k(B. $B=>$C$F(B,
! 1133: $B$=$NB?9`<0$,@8@.$5$l$?;~E@$K$*$1$kJQ?t=g=x7?$,(B, $B;MB'1i;;;~$K@5$7$/@_Dj(B
! 1134: $B$5$l$F$$$J$1$l$P$J$i$J$$(B. $B$^$?(B, $B1i;;BP>]$H$J$kB?9`<0$O(B, $BF10l$NJQ?t=g=x(B
! 1135: $B7?$K4p$E$$$F@8@.$5$l$?$b$N$G$J$1$l$P$J$i$J$$(B.
! 1136:
! 1137: @item
! 1138: $B%H%C%W%l%Y%kH!?t0J30$NH!?t$rD>@\8F$S=P$9>l9g$K$O(B, $B$3$NH!?t$K$h$j(B
! 1139: $BJQ?t=g=x7?$r@5$7$/@_Dj$7$J$1$l$P$J$i$J$$(B.
! 1140: @end itemize
! 1141:
! 1142: @example
! 1143: [19] dp_ord(0)$
! 1144: [20] <<1,2,3>>+<<3,1,1>>;
! 1145: (1)*<<1,2,3>>+(1)*<<3,1,1>>
! 1146: [21] dp_ord(2)$
! 1147: [22] <<1,2,3>>+<<3,1,1>>;
! 1148: (1)*<<3,1,1>>+(1)*<<1,2,3>>
! 1149: @end example
! 1150:
! 1151: @table @t
! 1152: @item $B;2>H(B
! 1153: @fref{$B9`=g=x$N@_Dj(B}
! 1154: @end table
! 1155:
! 1156: @node dp_ptod,,, $B%0%l%V%J4pDl$K4X$9$kH!?t(B
! 1157: @subsection @code{dp_ptod}
! 1158: @findex dp_ptod
! 1159:
! 1160: @table @t
! 1161: @item dp_ptod(@var{poly},@var{vlist})
! 1162: :: $BB?9`<0$rJ,;6I=8=B?9`<0$KJQ49$9$k(B.
! 1163: @end table
! 1164:
! 1165: @table @var
! 1166: @item return
! 1167: $BJ,;6I=8=B?9`<0(B
! 1168: @item poly
! 1169: $BB?9`<0(B
! 1170: @item vlist
! 1171: $B%j%9%H(B
! 1172: @end table
! 1173:
! 1174: @itemize @bullet
! 1175: @item
! 1176: $BJQ?t=g=x(B @var{vlist} $B$*$h$S8=:_$NJQ?t=g=x7?$K=>$C$FJ,;6I=8=B?9`<0$KJQ49$9$k(B.
! 1177: @item
! 1178: @var{vlist} $B$K4^$^$l$J$$ITDj85$O(B, $B78?tBN$KB0$9$k$H$7$FJQ49$5$l$k(B.
! 1179: @end itemize
! 1180:
! 1181: @example
! 1182: [50] dp_ord(0);
! 1183: 1
! 1184: [51] dp_ptod((x+y+z)^2,[x,y,z]);
! 1185: (1)*<<2,0,0>>+(2)*<<1,1,0>>+(1)*<<0,2,0>>+(2)*<<1,0,1>>+(2)*<<0,1,1>>
! 1186: +(1)*<<0,0,2>>
! 1187: [52] dp_ptod((x+y+z)^2,[x,y]);
! 1188: (1)*<<2,0>>+(2)*<<1,1>>+(1)*<<0,2>>+(2*z)*<<1,0>>+(2*z)*<<0,1>>+(z^2)*<<0,0>>
! 1189: @end example
! 1190:
! 1191: @table @t
! 1192: @item $B;2>H(B
! 1193: @fref{dp_dtop},
! 1194: @fref{dp_ord}.
! 1195: @end table
! 1196:
! 1197: @node dp_dtop,,, $B%0%l%V%J4pDl$K4X$9$kH!?t(B
! 1198: @subsection @code{dp_dtop}
! 1199: @findex dp_dtop
! 1200:
! 1201: @table @t
! 1202: @item dp_dtop(@var{dpoly},@var{vlist})
! 1203: :: $BJ,;6I=8=B?9`<0$rB?9`<0$KJQ49$9$k(B.
! 1204: @end table
! 1205:
! 1206: @table @var
! 1207: @item return
! 1208: $BB?9`<0(B
! 1209: @item dpoly
! 1210: $BJ,;6I=8=B?9`<0(B
! 1211: @item vlist
! 1212: $B%j%9%H(B
! 1213: @end table
! 1214:
! 1215: @itemize @bullet
! 1216: @item
! 1217: $BJ,;6I=8=B?9`<0$r(B, $BM?$($i$l$?ITDj85%j%9%H$rMQ$$$FB?9`<0$KJQ49$9$k(B.
! 1218: @item
! 1219: $BITDj85%j%9%H$O(B, $BD9$5J,;6I=8=B?9`<0$NJQ?t$N8D?t$H0lCW$7$F$$$l$P2?$G$b$h$$(B.
! 1220: @end itemize
! 1221:
! 1222: @example
! 1223: [53] T=dp_ptod((x+y+z)^2,[x,y]);
! 1224: (1)*<<2,0>>+(2)*<<1,1>>+(1)*<<0,2>>+(2*z)*<<1,0>>+(2*z)*<<0,1>>+(z^2)*<<0,0>>
! 1225: [54] P=dp_dtop(T,[a,b]);
! 1226: z^2+(2*a+2*b)*z+a^2+2*b*a+b^2
! 1227: @end example
! 1228:
! 1229: @node dp_mod dp_rat,,, $B%0%l%V%J4pDl$K4X$9$kH!?t(B
! 1230: @subsection @code{dp_mod}, @code{dp_rat}
! 1231: @findex dp_mod
! 1232: @findex dp_rat
! 1233:
! 1234: @table @t
! 1235: @item dp_mod(@var{p},@var{mod},@var{subst})
! 1236: :: $BM-M}?t78?tJ,;6I=8=B?9`<0$NM-8BBN78?t$X$NJQ49(B
! 1237: @item dp_rat(@var{p})
! 1238: :: $BM-8BBN78?tJ,;6I=8=B?9`<0$NM-M}?t78?t$X$NJQ49(B
! 1239: @end table
! 1240:
! 1241: @table @var
! 1242: @item return
! 1243: $BJ,;6I=8=B?9`<0(B
! 1244: @item p
! 1245: $BJ,;6I=8=B?9`<0(B
! 1246: @item mod
! 1247: $BAG?t(B
! 1248: @item subst
! 1249: $B%j%9%H(B
! 1250: @end table
! 1251:
! 1252: @itemize @bullet
! 1253: @item
! 1254: @code{dp_nf_mod()}, @code{dp_true_nf_mod()} $B$O(B, $BF~NO$H$7$FM-8BBN78?t$N(B
! 1255: $BJ,;6I=8=B?9`<0$rI,MW$H$9$k(B. $B$3$N$h$&$J>l9g(B, @code{dp_mod()} $B$K$h$j(B
! 1256: $BM-M}?t78?tJ,;6I=8=B?9`<0$rJQ49$7$FMQ$$$k$3$H$,$G$-$k(B. $B$^$?(B, $BF@$i$l$?(B
! 1257: $B7k2L$O(B, $BM-8BBN78?tB?9`<0$H$O1i;;$G$-$k$,(B, $BM-M}?t78?tB?9`<0$H$O1i;;$G$-$J$$(B
! 1258: $B$?$a(B, @code{dp_rat()} $B$K$h$jJQ49$9$kI,MW$,$"$k(B.
! 1259: @item
! 1260: $BM-8BBN78?t$N1i;;$K$*$$$F$O(B, $B$"$i$+$8$a(B @code{setmod()} $B$K$h$jM-8BBN$N85$N(B
! 1261: $B8D?t$r;XDj$7$F$*$/I,MW$,$"$k(B.
! 1262: @item
! 1263: @var{subst} $B$O(B, $B78?t$,M-M}<0$N>l9g(B, $B$=$NM-M}<0$NJQ?t$K$"$i$+$8$a?t$rBeF~(B
! 1264: $B$7$?8eM-8BBN78?t$KJQ49$9$k$H$$$&A`:n$r9T$&:]$N(B, $BBeF~CM$r;XDj$9$k$b$N$G(B,
! 1265: @code{[[@var{var},@var{value}],...]} $B$N7A$N%j%9%H$G$"$k(B.
! 1266: @end itemize
! 1267:
! 1268: @example
! 1269: @end example
! 1270:
! 1271: @table @t
! 1272: @item $B;2>H(B
! 1273: @fref{dp_nf dp_nf_mod dp_true_nf dp_true_nf_mod},
! 1274: @fref{subst psubst},
! 1275: @fref{setmod}.
! 1276: @end table
! 1277:
! 1278: @node dp_homo dp_dehomo,,, $B%0%l%V%J4pDl$K4X$9$kH!?t(B
! 1279: @subsection @code{dp_homo}, @code{dp_dehomo}
! 1280: @findex dp_homo
! 1281: @findex dp_dehomo
! 1282:
! 1283: @table @t
! 1284: @item dp_homo(@var{dpoly})
! 1285: :: $BJ,;6I=8=B?9`<0$N@F<!2=(B
! 1286: @item dp_dehomo(@var{dpoly})
! 1287: :: $B@F<!J,;6I=8=B?9`<0$NHs@F<!2=(B
! 1288: @end table
! 1289:
! 1290: @table @var
! 1291: @item return
! 1292: $BJ,;6I=8=B?9`<0(B
! 1293: @item dpoly
! 1294: $BJ,;6I=8=B?9`<0(B
! 1295: @end table
! 1296:
! 1297: @itemize @bullet
! 1298: @item
! 1299: @code{dp_homo()} $B$O(B, @var{dpoly} $B$N(B $B3F9`(B @var{t} $B$K$D$$$F(B, $B;X?t%Y%/%H%k$ND9$5$r(B
! 1300: 1 $B?-$P$7(B, $B:G8e$N@.J,$NCM$r(B @var{d}-@code{deg(@var{t})}
! 1301: (@var{d} $B$O(B @var{dpoly} $B$NA4<!?t(B) $B$H$7$?J,;6I=8=B?9`<0$rJV$9(B.
! 1302: @item
! 1303: @code{dp_dehomo()} $B$O(B, @var{dpoly} $B$N3F9`$K$D$$$F(B, $B;X?t%Y%/%H%k$N:G8e$N@.J,(B
! 1304: $B$r<h$j=|$$$?J,;6B?9`<0$rJV$9(B.
! 1305: @item
! 1306: $B$$$:$l$b(B, $B@8@.$5$l$?B?9`<0$rMQ$$$?1i;;$r9T$&>l9g(B, $B$=$l$i$KE,9g$9$k9`=g=x$r(B
! 1307: $B@5$7$/@_Dj$9$kI,MW$,$"$k(B.
! 1308: @item
! 1309: @code{hgr()} $B$J$I$K$*$$$F(B, $BFbItE*$KMQ$$$i$l$F$$$k(B.
! 1310: @end itemize
! 1311:
! 1312: @example
! 1313: [202] X=<<1,2,3>>+3*<<1,2,1>>;
! 1314: (1)*<<1,2,3>>+(3)*<<1,2,1>>
! 1315: [203] dp_homo(X);
! 1316: (1)*<<1,2,3,0>>+(3)*<<1,2,1,2>>
! 1317: [204] dp_dehomo(@@);
! 1318: (1)*<<1,2,3>>+(3)*<<1,2,1>>
! 1319: @end example
! 1320:
! 1321: @table @t
! 1322: @item $B;2>H(B
! 1323: @fref{gr hgr gr_mod}.
! 1324: @end table
! 1325:
! 1326: @node dp_ptozp dp_prim,,, $B%0%l%V%J4pDl$K4X$9$kH!?t(B
! 1327: @subsection @code{dp_ptozp}, @code{dp_prim}
! 1328: @findex dp_ptozp
! 1329: @findex dp_prim
! 1330:
! 1331: @table @t
! 1332: @item dp_ptozp(@var{dpoly})
! 1333: :: $BDj?tG\$7$F78?t$r@0?t78?t$+$D78?t$N@0?t(B GCD $B$r(B 1 $B$K$9$k(B.
! 1334: @itemx dp_prim(@var{dpoly})
! 1335: :: $BM-M}<0G\$7$F78?t$r@0?t78?tB?9`<078?t$+$D78?t$NB?9`<0(B GCD $B$r(B 1 $B$K$9$k(B.
! 1336: @end table
! 1337:
! 1338: @table @var
! 1339: @item return
! 1340: $BJ,;6I=8=B?9`<0(B
! 1341: @item dpoly
! 1342: $BJ,;6I=8=B?9`<0(B
! 1343: @end table
! 1344:
! 1345: @itemize @bullet
! 1346: @item
! 1347: @code{dp_ptozp()} $B$O(B, @code{ptozp()} $B$KAjEv$9$kA`:n$rJ,;6I=8=B?9`<0$K(B
! 1348: $BBP$7$F9T$&(B. $B78?t$,B?9`<0$r4^$`>l9g(B, $B78?t$K4^$^$l$kB?9`<06&DL0x;R$O(B
! 1349: $B<h$j=|$+$J$$(B.
! 1350: @item
! 1351: @code{dp_prim()} $B$O(B, $B78?t$,B?9`<0$r4^$`>l9g(B, $B78?t$K4^$^$l$kB?9`<06&DL0x;R(B
! 1352: $B$r<h$j=|$/(B.
! 1353: @end itemize
! 1354:
! 1355: @example
! 1356: [208] X=dp_ptod(3*(x-y)*(y-z)*(z-x),[x]);
! 1357: (-3*y+3*z)*<<2>>+(3*y^2-3*z^2)*<<1>>+(-3*z*y^2+3*z^2*y)*<<0>>
! 1358: [209] dp_ptozp(X);
! 1359: (-y+z)*<<2>>+(y^2-z^2)*<<1>>+(-z*y^2+z^2*y)*<<0>>
! 1360: [210] dp_prim(X);
! 1361: (1)*<<2>>+(-y-z)*<<1>>+(z*y)*<<0>>
! 1362: @end example
! 1363:
! 1364: @table @t
! 1365: @item $B;2>H(B
! 1366: @fref{ptozp}.
! 1367: @end table
! 1368:
! 1369: @node dp_nf dp_nf_mod dp_true_nf dp_true_nf_mod,,, $B%0%l%V%J4pDl$K4X$9$kH!?t(B
! 1370: @subsection @code{dp_nf}, @code{dp_nf_mod}, @code{dp_true_nf}, @code{dp_true_nf_mod}
! 1371: @findex dp_nf
! 1372: @findex dp_true_nf
! 1373: @findex dp_nf_mod
! 1374: @findex dp_true_nf_mod
! 1375:
! 1376: @table @t
! 1377: @item dp_nf(@var{indexlist},@var{dpoly},@var{dpolyarray},@var{fullreduce})
! 1378: @item dp_nf_mod(@var{indexlist},@var{dpoly},@var{dpolyarray},@var{fullreduce},@var{mod})
! 1379: :: $BJ,;6I=8=B?9`<0$N@55,7A$r5a$a$k(B. ($B7k2L$ODj?tG\$5$l$F$$$k2DG=@-$"$j(B)
! 1380:
! 1381: @item dp_true_nf(@var{indexlist},@var{dpoly},@var{dpolyarray},@var{fullreduce})
! 1382: @item dp_true_nf_mod(@var{indexlist},@var{dpoly},@var{dpolyarray},@var{fullreduce},@var{mod})
! 1383: :: $BJ,;6I=8=B?9`<0$N@55,7A$r5a$a$k(B. ($B??$N7k2L$r(B @code{[$BJ,;R(B, $BJ,Jl(B]} $B$N7A$GJV$9(B)
! 1384: @end table
! 1385:
! 1386: @table @var
! 1387: @item return
! 1388: @code{dp_nf()} : $BJ,;6I=8=B?9`<0(B, @code{dp_true_nf()} : $B%j%9%H(B
! 1389: @item indexlist
! 1390: $B%j%9%H(B
! 1391: @item dpoly
! 1392: $BJ,;6I=8=B?9`<0(B
! 1393: @item dpolyarray
! 1394: $BG[Ns(B
! 1395: @item fullreduce
! 1396: $B%U%i%0(B
! 1397: @item mod
! 1398: $BAG?t(B
! 1399: @end table
! 1400:
! 1401: @itemize @bullet
! 1402: @item
! 1403: $BJ,;6I=8=B?9`<0(B @var{dpoly} $B$N@55,7A$r5a$a$k(B.
! 1404: @item
! 1405: @code{dp_nf_mod()}, @code{dp_true_nf_mod()} $B$NF~NO$O(B, @code{dp_mod()} $B$J$I(B
! 1406: $B$K$h$j(B, $BM-8BBN>e$NJ,;6I=8=B?9`<0$K$J$C$F$$$J$1$l$P$J$i$J$$(B.
! 1407: @item
! 1408: $B7k2L$KM-M}?t(B, $BM-M}<0$,4^$^$l$k$N$rHr$1$k$?$a(B, @code{dp_nf()} $B$O(B
! 1409: $B??$NCM$NDj?tG\$NCM$rJV$9(B. $BM-M}<078?t$N>l9g$N(B @code{dp_nf_mod()} $B$bF1MM(B
! 1410: $B$G$"$k$,(B, $B78?tBN$,M-8BBN$N>l9g(B @code{dp_nf_mod()} $B$O??$NCM$rJV$9(B.
! 1411: @item
! 1412: @code{dp_true_nf()}, @code{dp_true_nf_mod()} $B$O(B,
! 1413: @code{[@var{nm},@var{dn}]} $B$J$k7A$N%j%9%H$rJV$9(B.
! 1414: $B$?$@$7(B, @var{nm} $B$O78?t$KJ,?t(B, $BM-M}<0$r4^$^$J$$J,;6I=8=B?9`<0(B, @var{dn} $B$O(B
! 1415: $B?t$^$?$OB?9`<0$G(B @var{nm}/@var{dn} $B$,??$NCM$H$J$k(B.
! 1416: @item
! 1417: @var{dpolyarray} $B$OJ,;6I=8=B?9`<0$rMWAG$H$9$k%Y%/%H%k(B,
! 1418: @var{indexlist} $B$O@55,2=7W;;$KMQ$$$k(B @var{dpolyarray} $B$NMWAG$N%$%s%G%C%/%9(B
! 1419: $B$N%j%9%H(B.
! 1420: @item
! 1421: @var{fullreduce} $B$,(B 0 $B$G$J$$$H$-A4$F$N9`$KBP$7$F4JLs$r9T$&(B. @var{fullreduce}
! 1422: $B$,(B 0 $B$N$H$-F,9`$N$_$KBP$7$F4JLs$r9T$&(B.
! 1423: @item
! 1424: @var{indexlist} $B$G;XDj$5$l$?B?9`<0$O(B, $BA0$NJ}$N$b$N$,M%@hE*$K;H$o$l$k(B.
! 1425: @item
! 1426: $B0lHL$K$O(B @var{indexlist} $B$NM?$(J}$K$h$jH!?t$NCM$O0[$J$k2DG=@-$,$"$k$,(B,
! 1427: $B%0%l%V%J4pDl$KBP$7$F$O0l0UE*$KDj$^$k(B.
! 1428: @item
! 1429: $BJ,;6I=8=$G$J$$8GDj$5$l$?B?9`<0=89g$K$h$k@55,7A$rB??t5a$a$kI,MW$,$"$k>l9g(B
! 1430: $B$KJXMx$G$"$k(B. $BC10l$N1i;;$K4X$7$F$O(B, @code{p_nf}, @code{p_true_nf} $B$r(B
! 1431: $BMQ$$$k$H$h$$(B.
! 1432:
! 1433: @end itemize
! 1434:
! 1435: @example
! 1436: [0] load("gr")$
! 1437: [64] load("katsura")$
! 1438: [69] K=katsura(4)$
! 1439: [70] dp_ord(2)$
! 1440: [71] V=[u0,u1,u2,u3,u4]$
! 1441: [72] DP1=newvect(length(K),map(dp_ptod,K,V))$
! 1442: [73] G=gr(K,V,2)$
! 1443: [74] DP2=newvect(length(G),map(dp_ptod,G,V))$
! 1444: [75] T=dp_ptod((u0-u1+u2-u3+u4)^2,V)$
! 1445: [76] dp_dtop(dp_nf([0,1,2,3,4],T,DP1,1),V);
! 1446: u4^2+(6*u3+2*u2+6*u1-2)*u4+9*u3^2+(6*u2+18*u1-6)*u3+u2^2+(6*u1-2)*u2+9*u1^2-6*u1+1
! 1447: [77] dp_dtop(dp_nf([4,3,2,1,0],T,DP1,1),V);
! 1448: -5*u4^2+(-4*u3-4*u2-4*u1)*u4-u3^2-3*u3-u2^2+(2*u1-1)*u2-2*u1^2-3*u1+1
! 1449: [78] dp_dtop(dp_nf([0,1,2,3,4],T,DP2,1),V);
! 1450: -1138087976845165778088612297273078520347097001020471455633353049221045677593
! 1451: 0005716505560062087150928400876150217079820311439477560587583488*u4^15+...
! 1452: [79] dp_dtop(dp_nf([4,3,2,1,0],T,DP2,1),V);
! 1453: -1138087976845165778088612297273078520347097001020471455633353049221045677593
! 1454: 0005716505560062087150928400876150217079820311439477560587583488*u4^15+...
! 1455: [80] @@78==@@79;
! 1456: 1
! 1457: @end example
! 1458:
! 1459: @table @t
! 1460: @item $B;2>H(B
! 1461: @fref{dp_dtop},
! 1462: @fref{dp_ord},
! 1463: @fref{dp_mod dp_rat},
! 1464: @fref{p_nf p_nf_mod p_true_nf p_true_nf_mod}.
! 1465: @end table
! 1466:
! 1467: @node dp_hm dp_ht dp_hc dp_rest,,, $B%0%l%V%J4pDl$K4X$9$kH!?t(B
! 1468: @subsection @code{dp_hm}, @code{dp_ht}, @code{dp_hc}, @code{dp_rest}
! 1469: @findex dp_hm
! 1470: @findex dp_ht
! 1471: @findex dp_hc
! 1472: @findex dp_rest
! 1473:
! 1474: @table @t
! 1475: @item dp_hm(@var{dpoly})
! 1476: :: $BF,C19`<0$r<h$j=P$9(B.
! 1477: @item dp_ht(@var{dpoly})
! 1478: :: $BF,9`$r<h$j=P$9(B.
! 1479: @item dp_hc(@var{dpoly})
! 1480: :: $BF,78?t$r<h$j=P$9(B.
! 1481: @item dp_rest(@var{dpoly})
! 1482: :: $BF,C19`<0$r<h$j=|$$$?;D$j$rJV$9(B.
! 1483: @end table
! 1484:
! 1485: @table @var
! 1486: @item return
! 1487: @code{dp_hm()}, @code{dp_ht()}, @code{dp_rest()} : $BJ,;6I=8=B?9`<0(B,
! 1488: @code{dp_hc()} : $B?t$^$?$OB?9`<0(B
! 1489: @item dpoly
! 1490: $BJ,;6I=8=B?9`<0(B
! 1491: @end table
! 1492:
! 1493: @itemize @bullet
! 1494: @item
! 1495: $B$3$l$i$O(B, $BJ,;6I=8=B?9`<0$N3FItJ,$r<h$j=P$9$?$a$NH!?t$G$"$k(B.
! 1496: @item
! 1497: $BJ,;6I=8=B?9`<0(B @var{p} $B$KBP$7<!$,@.$jN)$D(B.
! 1498: @table @code
! 1499: @item @var{p} = dp_hm(@var{p}) + dp_rest(@var{p})
! 1500: @item dp_hm(@var{p}) = dp_hc(@var{p}) dp_ht(@var{p})
! 1501: @end table
! 1502: @end itemize
! 1503:
! 1504: @example
! 1505: [87] dp_ord(0)$
! 1506: [88] X=ptozp((a46^2+7/10*a46+7/48)*u3^4-50/27*a46^2-35/27*a46-49/216)$
! 1507: [89] T=dp_ptod(X,[u3,u4,a46])$
! 1508: [90] dp_hm(T);
! 1509: (2160)*<<4,0,2>>
! 1510: [91] dp_ht(T);
! 1511: (1)*<<4,0,2>>
! 1512: [92] dp_hc(T);
! 1513: 2160
! 1514: [93] dp_rest(T);
! 1515: (1512)*<<4,0,1>>+(315)*<<4,0,0>>+(-4000)*<<0,0,2>>+(-2800)*<<0,0,1>>
! 1516: +(-490)*<<0,0,0>>
! 1517: @end example
! 1518:
! 1519: @node dp_td dp_sugar,,, $B%0%l%V%J4pDl$K4X$9$kH!?t(B
! 1520: @subsection @code{dp_td}, @code{dp_sugar}
! 1521: @findex dp_td
! 1522: @findex dp_sugar
! 1523:
! 1524: @table @t
! 1525: @item dp_td(@var{dpoly})
! 1526: :: $BF,9`$NA4<!?t$rJV$9(B.
! 1527: @item dp_sugar(@var{dpoly})
! 1528: :: $BB?9`<0$N(B @code{sugar} $B$rJV$9(B.
! 1529: @end table
! 1530:
! 1531: @table @var
! 1532: @item return
! 1533: $B<+A3?t(B
! 1534: @item dpoly
! 1535: $BJ,;6I=8=B?9`<0(B
! 1536: @item onoff
! 1537: $B%U%i%0(B
! 1538: @end table
! 1539:
! 1540: @itemize @bullet
! 1541: @item
! 1542: @code{dp_td()} $B$O(B, $BF,9`$NA4<!?t(B, $B$9$J$o$A3FJQ?t$N;X?t$NOB$rJV$9(B.
! 1543: @item
! 1544: $BJ,;6I=8=B?9`<0$,@8@.$5$l$k$H(B, @code{sugar} $B$H8F$P$l$k$"$k@0?t$,IUM?(B
! 1545: $B$5$l$k(B. $B$3$NCM$O(B $B2>A[E*$K@F<!2=$7$F7W;;$7$?>l9g$K7k2L$,;}$DA4<!?t$NCM$H$J$k(B.
! 1546: @item
! 1547: @code{sugar} $B$O(B, $B%0%l%V%J4pDl7W;;$K$*$1$k@55,2=BP$NA*Br$N%9%H%i%F%8$r(B
! 1548: $B7hDj$9$k$?$a$N=EMW$J;X?K$H$J$k(B.
! 1549: @end itemize
! 1550:
! 1551: @example
! 1552: [74] dp_ord(0)$
! 1553: [75] X=<<1,2>>+<<0,1>>$
! 1554: [76] Y=<<1,2>>+<<1,0>>$
! 1555: [77] Z=X-Y;
! 1556: (-1)*<<1,0>>+(1)*<<0,1>>
! 1557: [78] dp_sugar(T);
! 1558: 3
! 1559: @end example
! 1560:
! 1561: @node dp_lcm,,, $B%0%l%V%J4pDl$K4X$9$kH!?t(B
! 1562: @subsection @code{dp_lcm}
! 1563: @findex dp_lcm
! 1564:
! 1565: @table @t
! 1566: @item dp_lcm(@var{dpoly1},@var{dpoly2})
! 1567: :: $B:G>.8xG\9`$rJV$9(B.
! 1568: @end table
! 1569:
! 1570: @table @var
! 1571: @item return
! 1572: $BJ,;6I=8=B?9`<0(B
! 1573: @item dpoly1, dpoly2
! 1574: $BJ,;6I=8=B?9`<0(B
! 1575: @end table
! 1576:
! 1577: @itemize @bullet
! 1578: @item
! 1579: $B$=$l$>$l$N0z?t$NF,9`$N:G>.8xG\9`$rJV$9(B. $B78?t$O(B 1 $B$G$"$k(B.
! 1580: @end itemize
! 1581:
! 1582: @example
! 1583: [100] dp_lcm(<<1,2,3,4,5>>,<<5,4,3,2,1>>);
! 1584: (1)*<<5,4,3,4,5>>
! 1585: @end example
! 1586:
! 1587: @table @t
! 1588: @item $B;2>H(B
! 1589: @fref{p_nf p_nf_mod p_true_nf p_true_nf_mod}.
! 1590: @end table
! 1591:
! 1592: @node dp_redble,,, $B%0%l%V%J4pDl$K4X$9$kH!?t(B
! 1593: @subsection @code{dp_redble}
! 1594: @findex dp_redble
! 1595:
! 1596: @table @t
! 1597: @item dp_redble(@var{dpoly1},@var{dpoly2})
! 1598: :: $BF,9`$I$&$7$,@0=|2DG=$+$I$&$+D4$Y$k(B.
! 1599: @end table
! 1600:
! 1601: @table @var
! 1602: @item return
! 1603: $B@0?t(B
! 1604: @item dpoly1, dpoly2
! 1605: $BJ,;6I=8=B?9`<0(B
! 1606: @end table
! 1607:
! 1608: @itemize @bullet
! 1609: @item
! 1610: @var{dpoly1} $B$NF,9`$,(B @var{dpoly2} $B$NF,9`$G3d$j@Z$l$l$P(B 1, $B3d$j@Z$l$J$1$l$P(B
! 1611: 0 $B$rJV$9(B.
! 1612: @item
! 1613: $BB?9`<0$N4JLs$r9T$&:](B, $B$I$N9`$r4JLs$G$-$k$+$rC5$9$N$KMQ$$$k(B.
! 1614: @end itemize
! 1615:
! 1616: @example
! 1617: [148] C;
! 1618: (1)*<<1,1,1,0,0>>+(1)*<<0,1,1,1,0>>+(1)*<<1,1,0,0,1>>+(1)*<<1,0,0,1,1>>
! 1619: [149] T;
! 1620: (3)*<<2,1,0,0,0>>+(3)*<<1,2,0,0,0>>+(1)*<<0,3,0,0,0>>+(6)*<<1,1,1,0,0>>
! 1621: [150] for ( ; T; T = dp_rest(T)) print(dp_redble(T,C));
! 1622: 0
! 1623: 0
! 1624: 0
! 1625: 1
! 1626: @end example
! 1627:
! 1628: @table @t
! 1629: @item $B;2>H(B
! 1630: @fref{dp_red dp_red_mod}.
! 1631: @end table
! 1632:
! 1633: @node dp_subd,,, $B%0%l%V%J4pDl$K4X$9$kH!?t(B
! 1634: @subsection @code{dp_subd}
! 1635: @findex dp_subd
! 1636:
! 1637: @table @t
! 1638: @item dp_subd(@var{dpoly1},@var{dpoly2})
! 1639: :: $BF,9`$N>&C19`<0$rJV$9(B.
! 1640: @end table
! 1641:
! 1642: @table @var
! 1643: @item return
! 1644: $BJ,;6I=8=B?9`<0(B
! 1645: @item dpoly1, dpoly2
! 1646: $BJ,;6I=8=B?9`<0(B
! 1647: @end table
! 1648:
! 1649: @itemize @bullet
! 1650: @item
! 1651: @code{dp_ht(@var{dpoly1})/dp_ht(@var{dpoly2})} $B$r5a$a$k(B. $B7k2L$N78?t$O(B 1
! 1652: $B$G$"$k(B.
! 1653: @item
! 1654: $B3d$j@Z$l$k$3$H$,$"$i$+$8$a$o$+$C$F$$$kI,MW$,$"$k(B.
! 1655: @end itemize
! 1656:
! 1657: @example
! 1658: [162] dp_subd(<<1,2,3,4,5>>,<<1,1,2,3,4>>);
! 1659: (1)*<<0,1,1,1,1>>
! 1660: @end example
! 1661:
! 1662: @table @t
! 1663: @item $B;2>H(B
! 1664: @fref{dp_red dp_red_mod}.
! 1665: @end table
! 1666:
! 1667: @node dp_vtoe dp_etov,,, $B%0%l%V%J4pDl$K4X$9$kH!?t(B
! 1668: @subsection @code{dp_vtoe}, @code{dp_etov}
! 1669: @findex dp_vtoe
! 1670: @findex dp_etov
! 1671:
! 1672: @table @t
! 1673: @item dp_vtoe(@var{vect})
! 1674: :: $B;X?t%Y%/%H%k$r9`$KJQ49(B
! 1675: @item dp_etov(@var{dpoly})
! 1676: :: $BF,9`$r;X?t%Y%/%H%k$KJQ49(B
! 1677: @end table
! 1678:
! 1679: @table @var
! 1680: @item return
! 1681: @code{dp_vtoe} : $BJ,;6I=8=B?9`<0(B, @code{dp_etov} : $B%Y%/%H%k(B
! 1682: @item vect
! 1683: $B%Y%/%H%k(B
! 1684: @item dpoly
! 1685: $BJ,;6I=8=B?9`<0(B
! 1686: @end table
! 1687:
! 1688: @itemize @bullet
! 1689: @item
! 1690: @code{dp_vtoe()} $B$O(B, $B%Y%/%H%k(B @var{vect} $B$r;X?t%Y%/%H%k$H$9$k9`$r@8@.$9$k(B.
! 1691: @item
! 1692: @code{dp_etov()} $B$O(B, $BJ,;6I=8=B?9`<0(B @code{dpoly} $B$NF,9`$N;X?t%Y%/%H%k$r(B
! 1693: $B%Y%/%H%k$KJQ49$9$k(B.
! 1694: @end itemize
! 1695:
! 1696: @example
! 1697: [211] X=<<1,2,3>>;
! 1698: (1)*<<1,2,3>>
! 1699: [212] V=dp_etov(X);
! 1700: [ 1 2 3 ]
! 1701: [213] V[2]++$
! 1702: [214] Y=dp_vtoe(V);
! 1703: (1)*<<1,2,4>>
! 1704: @end example
! 1705:
! 1706: @node dp_mbase,,, $B%0%l%V%J4pDl$K4X$9$kH!?t(B
! 1707: @subsection @code{dp_mbase}
! 1708: @findex dp_mbase
! 1709:
! 1710: @table @t
! 1711: @item dp_mbase(@var{dplist})
! 1712: :: monomial $B4pDl$N7W;;(B
! 1713: @end table
! 1714:
! 1715: @table @var
! 1716: @item return
! 1717: $BJ,;6I=8=B?9`<0$N%j%9%H(B
! 1718: @item dplist
! 1719: $BJ,;6I=8=B?9`<0$N%j%9%H(B
! 1720: @end table
! 1721:
! 1722: @itemize @bullet
! 1723: @item
! 1724: $B$"$k=g=x$G%0%l%V%J4pDl$H$J$C$F$$$kB?9`<0=89g$N(B, $B$=$N=g=x$K4X$9$kJ,;6I=8=(B
! 1725: $B$G$"$k(B @var{dplist} $B$K$D$$$F(B,
! 1726: @var{dplist} $B$,(B K[X] $BCf$G@8@.$9$k%$%G%"%k(B I $B$,(B 0 $B<!85$N;~(B,
! 1727: K $B>eM-8B<!85@~7A6u4V$G$"$k(B K[X]/I $B$N(B monomial $B$K$h$k4pDl$r5a$a$k(B.
! 1728: @item
! 1729: $BF@$i$l$?4pDl$N8D?t$,(B, K[X]/I $B$N(B K-$B@~7A6u4V$H$7$F$N<!85$KEy$7$$(B.
! 1730: @end itemize
! 1731:
! 1732: @example
! 1733: [215] K=katsura(5)$
! 1734: [216] V=[u5,u4,u3,u2,u1,u0]$
! 1735: [217] G0=gr(K,V,0)$
! 1736: [218] H=map(dp_ptod,G0,V)$
! 1737: [219] map(dp_ptod,dp_mbase(H),V)$
! 1738: [u0^5,u4*u0^3,u3*u0^3,u2*u0^3,u1*u0^3,u0^4,u3^2*u0,u2*u3*u0,u1*u3*u0,
! 1739: u1*u2*u0,u1^2*u0,u4*u0^2,u3*u0^2,u2*u0^2,u1*u0^2,u0^3,u3^2,u2*u3,u1*u3,
! 1740: u1*u2,u1^2,u4*u0,u3*u0,u2*u0,u1*u0,u0^2,u4,u3,u2,u1,u0,1]
! 1741: @end example
! 1742:
! 1743: @table @t
! 1744: @item $B;2>H(B
! 1745: @fref{gr hgr gr_mod}.
! 1746: @end table
! 1747:
! 1748: @node dp_mag,,, $B%0%l%V%J4pDl$K4X$9$kH!?t(B
! 1749: @subsection @code{dp_mag}
! 1750: @findex dp_mag
! 1751:
! 1752: @table @t
! 1753: @item dp_mag(@var{p})
! 1754: :: $B78?t$N%S%C%HD9$NOB$rJV$9(B
! 1755: @end table
! 1756:
! 1757: @table @var
! 1758: @item return
! 1759: $B?t(B
! 1760: @item p
! 1761: $BJ,;6I=8=B?9`<0(B
! 1762: @end table
! 1763:
! 1764: @itemize @bullet
! 1765: @item
! 1766: $BJ,;6I=8=B?9`<0$N78?t$K8=$l$kM-M}?t$K$D$-(B, $B$=$NJ,JlJ,;R(B ($B@0?t$N>l9g$OJ,;R(B)
! 1767: $B$N%S%C%HD9$NAmOB$rJV$9(B.
! 1768: @item
! 1769: $BBP>]$H$J$kB?9`<0$NBg$-$5$NL\0B$H$7$FM-8z$G$"$k(B. $BFC$K(B, 0 $B<!85%7%9%F%`$K$*$$$F$O(B
! 1770: $B78?tKDD%$,LdBj$H$J$j(B, $BESCf@8@.$5$l$kB?9`<0$,78?tKDD%$r5/$3$7$F$$$k$+$I$&$+(B
! 1771: $B$NH=Dj$KLrN)$D(B.
! 1772: @item
! 1773: @code{dp_gr_flags()} $B$G(B, @code{ShowMag}, @code{Print} $B$r(B on $B$K$9$k$3$H$K$h$j(B
! 1774: $BESCf@8@.$5$l$kB?9`<0$K$?$$$9$k(B @code{dp_mag()} $B$NCM$r8+$k$3$H$,$G$-$k(B.
! 1775: @end itemize
! 1776:
! 1777: @example
! 1778: [221] X=dp_ptod((x+2*y)^10,[x,y])$
! 1779: [222] dp_mag(X);
! 1780: 115
! 1781: @end example
! 1782:
! 1783: @table @t
! 1784: @item $B;2>H(B
! 1785: @fref{dp_gr_flags dp_gr_print}.
! 1786: @end table
! 1787:
! 1788: @node dp_red dp_red_mod,,, $B%0%l%V%J4pDl$K4X$9$kH!?t(B
! 1789: @subsection @code{dp_red}, @code{dp_red_mod}
! 1790: @findex dp_red
! 1791: @findex dp_red_mod
! 1792:
! 1793: @table @t
! 1794: @item dp_red(@var{dpoly1},@var{dpoly2},@var{dpoly3})
! 1795: @item dp_red_mod(@var{dpoly1},@var{dpoly2},@var{dpoly3},@var{mod})
! 1796: :: $B0l2s$N4JLsA`:n(B
! 1797: @end table
! 1798:
! 1799: @table @var
! 1800: @item return
! 1801: $B%j%9%H(B
! 1802: @item dpoly1, dpoly2, dpoly3
! 1803: $BJ,;6I=8=B?9`<0(B
! 1804: @item vlist
! 1805: $B%j%9%H(B
! 1806: @item mod
! 1807: $BAG?t(B
! 1808: @end table
! 1809:
! 1810: @itemize @bullet
! 1811: @item
! 1812: @var{dpoly1} + @var{dpoly2} $B$J$kJ,;6I=8=B?9`<0$r(B @var{dpoly3} $B$G(B
! 1813: 1 $B2s4JLs$9$k(B.
! 1814: @item
! 1815: @code{dp_red_mod()} $B$NF~NO$O(B, $BA4$FM-8BBN78?t$KJQ49$5$l$F$$$kI,MW$,$"$k(B.
! 1816: @item
! 1817: $B4JLs$5$l$k9`$O(B @var{dpoly2} $B$NF,9`$G$"$k(B. $B=>$C$F(B, @var{dpoly2} $B$N(B
! 1818: $BF,9`$,(B @var{dpoly3} $B$NF,9`$G3d$j@Z$l$k$3$H$,$"$i$+$8$a$o$+$C$F$$$J$1$l$P(B
! 1819: $B$J$i$J$$(B.
! 1820: @item
! 1821: $B0z?t$,@0?t78?t$N;~(B, $B4JLs$O(B, $BJ,?t$,8=$l$J$$$h$&(B, $B@0?t(B @var{a}, @var{b},
! 1822: $B9`(B @var{t} $B$K$h$j(B @var{a(dpoly1 + dpoly2)-bt dpoly3} $B$H$7$F7W;;$5$l$k(B.
! 1823: @item
! 1824: $B7k2L$O(B, @code{[@var{a dpoly1},@var{a dpoly2 - bt dpoly3}]} $B$J$k%j%9%H$G$"$k(B.
! 1825: @end itemize
! 1826:
! 1827: @example
! 1828: [157] D=(3)*<<2,1,0,0,0>>+(3)*<<1,2,0,0,0>>+(1)*<<0,3,0,0,0>>;
! 1829: (3)*<<2,1,0,0,0>>+(3)*<<1,2,0,0,0>>+(1)*<<0,3,0,0,0>>
! 1830: [158] R=(6)*<<1,1,1,0,0>>;
! 1831: (6)*<<1,1,1,0,0>>
! 1832: [159] C=12*<<1,1,1,0,0>>+(1)*<<0,1,1,1,0>>+(1)*<<1,1,0,0,1>>;
! 1833: (12)*<<1,1,1,0,0>>+(1)*<<0,1,1,1,0>>+(1)*<<1,1,0,0,1>>
! 1834: [160] dp_red(D,R,C);
! 1835: [(6)*<<2,1,0,0,0>>+(6)*<<1,2,0,0,0>>+(2)*<<0,3,0,0,0>>,(-1)*<<0,1,1,1,0>>
! 1836: +(-1)*<<1,1,0,0,1>>]
! 1837: @end example
! 1838:
! 1839: @table @t
! 1840: @item $B;2>H(B
! 1841: @fref{dp_mod dp_rat}.
! 1842: @end table
! 1843:
! 1844: @node dp_sp dp_sp_mod,,, $B%0%l%V%J4pDl$K4X$9$kH!?t(B
! 1845: @subsection @code{dp_sp}, @code{dp_sp_mod}
! 1846: @findex dp_sp
! 1847: @findex dp_sp_mod
! 1848:
! 1849: @table @t
! 1850: @item dp_sp(@var{dpoly1},@var{dpoly2})
! 1851: @item dp_sp_mod(@var{dpoly1},@var{dpoly2},@var{mod})
! 1852: :: S-$BB?9`<0$N7W;;(B
! 1853: @end table
! 1854:
! 1855: @table @var
! 1856: @item return
! 1857: $BJ,;6I=8=B?9`<0(B
! 1858: @item dpoly1, dpoly2
! 1859: $BJ,;6I=8=B?9`<0(B
! 1860: @item mod
! 1861: $BAG?t(B
! 1862: @end table
! 1863:
! 1864: @itemize @bullet
! 1865: @item
! 1866: @var{dpoly1}, @var{dpoly2} $B$N(B S-$BB?9`<0$r7W;;$9$k(B.
! 1867: @item
! 1868: @code{dp_sp_mod()} $B$NF~NO$O(B, $BA4$FM-8BBN78?t$KJQ49$5$l$F$$$kI,MW$,$"$k(B.
! 1869: @item
! 1870: $B7k2L$KM-M}?t(B, $BM-M}<0$,F~$k$N$rHr$1$k$?$a(B, $B7k2L$,Dj?tG\(B, $B$"$k$$$OB?9`<0(B
! 1871: $BG\$5$l$F$$$k2DG=@-$,$"$k(B.
! 1872: @end itemize
! 1873:
! 1874: @example
! 1875: [227] X=dp_ptod(x^2*y+x*y,[x,y]);
! 1876: (1)*<<2,1>>+(1)*<<1,1>>
! 1877: [228] Y=dp_ptod(x*y^2+x*y,[x,y]);
! 1878: (1)*<<1,2>>+(1)*<<1,1>>
! 1879: [229] dp_sp(X,Y);
! 1880: (-1)*<<2,1>>+(1)*<<1,2>>
! 1881: @end example
! 1882:
! 1883: @table @t
! 1884: @item $B;2>H(B
! 1885: @fref{dp_mod dp_rat}.
! 1886: @end table
! 1887: @node p_nf p_nf_mod p_true_nf p_true_nf_mod,,, $B%0%l%V%J4pDl$K4X$9$kH!?t(B
! 1888: @subsection @code{p_nf}, @code{p_nf_mod}, @code{p_true_nf}, @code{p_true_nf_mod}
! 1889: @findex p_nf
! 1890: @findex p_nf_mod
! 1891: @findex p_true_nf
! 1892: @findex p_true_nf_mod
! 1893:
! 1894: @table @t
! 1895: @item p_nf(@var{poly},@var{plist},@var{vlist},@var{order})
! 1896: @itemx p_nf_mod(@var{poly},@var{plist},@var{vlist},@var{order},@var{mod})
! 1897: :: $BI=8=B?9`<0$N@55,7A$r5a$a$k(B. ($B7k2L$ODj?tG\$5$l$F$$$k2DG=@-$"$j(B)
! 1898: @item p_true_nf(@var{poly},@var{plist},@var{vlist},@var{order})
! 1899: @itemx p_true_nf_mod(@var{poly},@var{plist},@var{vlist},@var{order},@var{mod})
! 1900: :: $BI=8=B?9`<0$N@55,7A$r5a$a$k(B. ($B??$N7k2L$r(B @code{[$BJ,;R(B, $BJ,Jl(B]} $B$N7A$GJV$9(B)
! 1901: @end table
! 1902:
! 1903: @table @var
! 1904: @item return
! 1905: @code{p_nf} : $BB?9`<0(B, @code{p_true_nf} : $B%j%9%H(B
! 1906: @item poly
! 1907: $BB?9`<0(B
! 1908: @item plist,vlist
! 1909: $B%j%9%H(B
! 1910: @item order
! 1911: $B?t(B, $B%j%9%H$^$?$O9TNs(B
! 1912: @item mod
! 1913: $BAG?t(B
! 1914: @end table
! 1915:
! 1916: @itemize @bullet
! 1917: @item
! 1918: @samp{gr} $B$GDj5A$5$l$F$$$k(B.
! 1919: @item
! 1920: $BB?9`<0$N(B, $BB?9`<0%j%9%H$K$h$k@55,7A$r5a$a$k(B.
! 1921: @item
! 1922: @code{dp_nf()}, @code{dp_true_nf()}, @code{dp_nf_mod()}, @code{dp_true_nf_mod}
! 1923: $B$KBP$9$k%$%s%?%U%'!<%9$G$"$k(B.
! 1924: @item
! 1925: @var{poly} $B$*$h$S(B @var{plist} $B$O(B, $BJQ?t=g=x(B @var{vlist} $B$*$h$S(B
! 1926: $BJQ?t=g=x7?(B @var{otype} $B$K=>$C$FJ,;6I=8=B?9`<0$KJQ49$5$l(B,
! 1927: @code{dp_nf()}, @code{dp_true_nf()}, @code{dp_nf_mod()},
! 1928: @code{dp_true_nf_mod()} $B$KEO$5$l$k(B.
! 1929: @item
! 1930: @code{dp_nf()}, @code{dp_true_nf()}, @code{dp_nf_mod()},
! 1931: @code{dp_true_nf_mod()} $B$O(B @var{fullreduce} $B$,(B 1 $B$G8F$S=P$5$l$k(B.
! 1932: @item
! 1933: $B7k2L$OB?9`<0$KJQ49$5$l$F=PNO$5$l$k(B.
! 1934: @item
! 1935: @code{p_true_nf()}, @code{p_true_nf_mod()} $B$N=PNO$K4X$7$F$O(B,
! 1936: @code{dp_true_nf()}, @code{dp_true_nf_mod()} $B$N9`$r;2>H(B.
! 1937: @end itemize
! 1938:
! 1939: @example
! 1940: [79] K = katsura(5)$
! 1941: [80] V = [u5,u4,u3,u2,u1,u0]$
! 1942: [81] G = hgr(K,V,2)$
! 1943: [82] p_nf(K[1],G,V,2);
! 1944: 0
! 1945: [83] L = p_true_nf(K[1]+1,G,V,2);
! 1946: [-1503...,-1503...]
! 1947: [84] L[0]/L[1];
! 1948: 1
! 1949: @end example
! 1950:
! 1951: @table @t
! 1952: @item $B;2>H(B
! 1953: @fref{dp_ptod},
! 1954: @fref{dp_dtop},
! 1955: @fref{dp_ord},
! 1956: @fref{dp_nf dp_nf_mod dp_true_nf dp_true_nf_mod}.
! 1957: @end table
! 1958:
! 1959: @node p_terms,,, $B%0%l%V%J4pDl$K4X$9$kH!?t(B
! 1960: @subsection @code{p_terms}
! 1961: @findex p_terms
! 1962:
! 1963: @table @t
! 1964: @item p_terms(@var{poly},@var{vlist},@var{order})
! 1965: :: $BB?9`<0$K$"$i$o$l$kC19`$r%j%9%H$K$9$k(B.
! 1966: @end table
! 1967:
! 1968: @table @var
! 1969: @item return
! 1970: $B%j%9%H(B
! 1971: @item poly
! 1972: $BB?9`<0(B
! 1973: @item vlist
! 1974: $B%j%9%H(B
! 1975: @item order
! 1976: $B?t(B, $B%j%9%H$^$?$O9TNs(B
! 1977: @end table
! 1978:
! 1979: @itemize @bullet
! 1980: @item
! 1981: @samp{gr} $B$GDj5A$5$l$F$$$k(B.
! 1982: @item
! 1983: $BB?9`<0$rC19`$KE83+$7$?;~$K8=$l$k9`$r%j%9%H$K$7$FJV$9(B.
! 1984: @var{vlist} $B$*$h$S(B @var{order} $B$K$h$jDj$^$k9`=g=x$K$h$j(B, $B=g=x$N9b$$$b$N(B
! 1985: $B$,%j%9%H$N@hF,$KMh$k$h$&$K%=!<%H$5$l$k(B.
! 1986: @item
! 1987: $B%0%l%V%J4pDl$O$7$P$7$P78?t$,5pBg$K$J$k$?$a(B, $B<B:]$K$I$N9`$,8=$l$F(B
! 1988: $B$$$k$N$+$r8+$k$?$a$J$I$KMQ$$$k(B.
! 1989: @end itemize
! 1990:
! 1991: @example
! 1992: [233] G=gr(katsura(5),[u5,u4,u3,u2,u1,u0],2)$
! 1993: [234] p_terms(G[0],[u5,u4,u3,u2,u1,u0],2);
! 1994: [u5,u0^31,u0^30,u0^29,u0^28,u0^27,u0^26,u0^25,u0^24,u0^23,u0^22,u0^21,u0^20,
! 1995: u0^19,u0^18,u0^17,u0^16,u0^15,u0^14,u0^13,u0^12,u0^11,u0^10,u0^9,u0^8,u0^7,
! 1996: u0^6,u0^5,u0^4,u0^3,u0^2,u0,1]
! 1997: @end example
! 1998:
! 1999: @node gb_comp,,, $B%0%l%V%J4pDl$K4X$9$kH!?t(B
! 2000: @subsection @code{gb_comp}
! 2001: @findex gb_comp
! 2002:
! 2003: @table @t
! 2004: @item gb_comp(@var{plist1}, @var{plist2})
! 2005: :: $BB?9`<0%j%9%H$,(B, $BId9f$r=|$$$F=89g$H$7$FEy$7$$$+$I$&$+D4$Y$k(B.
! 2006: @end table
! 2007:
! 2008: @table @var
! 2009: @item return 0 $B$^$?$O(B 1
! 2010: @item plist1, plist2
! 2011: @end table
! 2012:
! 2013: @itemize @bullet
! 2014: @item
! 2015: @var{plist1}, @var{plist2} $B$K$D$$$F(B, $BId9f$r=|$$$F=89g$H$7$FEy$7$$$+$I$&$+(B
! 2016: $BD4$Y$k(B.
! 2017: @item
! 2018: $B0[$J$kJ}K!$G5a$a$?%0%l%V%J4pDl$O(B, $B4pDl$N=g=x(B, $BId9f$,0[$J$k>l9g$,$"$j(B,
! 2019: $B$=$l$i$,Ey$7$$$+$I$&$+$rD4$Y$k$?$a$KMQ$$$k(B.
! 2020: @end itemize
! 2021:
! 2022: @example
! 2023: [243] C=cyclic(6)$
! 2024: [244] V=[c0,c1,c2,c3,c4,c5]$
! 2025: [245] G0=gr(C,V,0)$
! 2026: [246] G=tolex(G0,V,0,V)$
! 2027: [247] GG=lex_tl(C,V,0,V,0)$
! 2028: [248] gb_comp(G,GG);
! 2029: 1
! 2030: @end example
! 2031:
! 2032: @node katsura hkatsura cyclic hcyclic,,, $B%0%l%V%J4pDl$K4X$9$kH!?t(B
! 2033: @subsection @code{katsura}, @code{hkatsura}, @code{cyclic}, @code{hcyclic}
! 2034: @findex katsura
! 2035: @findex hkatsura
! 2036: @findex cyclic
! 2037: @findex hcyclic
! 2038:
! 2039: @table @t
! 2040: @item katsura(@var{n})
! 2041: @item hkatsura(@var{n})
! 2042: @item cyclic(@var{n})
! 2043: @item hcyclic(@var{n})
! 2044: :: $BB?9`<0%j%9%H$N@8@.(B
! 2045: @end table
! 2046:
! 2047: @table @var
! 2048: @item return
! 2049: $B%j%9%H(B
! 2050: @item n
! 2051: $B@0?t(B
! 2052: @end table
! 2053:
! 2054: @itemize @bullet
! 2055: @item
! 2056: @code{katsura()} $B$O(B @samp{katsura}, @code{cyclic()} $B$O(B @samp{cyclic}
! 2057: $B$GDj5A$5$l$F$$$k(B.
! 2058: @item
! 2059: $B%0%l%V%J4pDl7W;;$G$7$P$7$P%F%9%H(B, $B%Y%s%A%^!<%/$KMQ$$$i$l$k(B @code{katsura},
! 2060: @code{cyclic} $B$*$h$S$=$N@F<!2=$r@8@.$9$k(B.
! 2061: @item
! 2062: @code{cyclic} $B$O(B @code{Arnborg}, @code{Lazard}, @code{Davenport} $B$J$I$N(B
! 2063: $BL>$G8F$P$l$k$3$H$b$"$k(B.
! 2064: @end itemize
! 2065:
! 2066: @example
! 2067: [74] load("katsura")$
! 2068: [79] load("cyclic")$
! 2069: [89] katsura(5);
! 2070: [u0+2*u4+2*u3+2*u2+2*u1+2*u5-1,2*u4*u0-u4+2*u1*u3+u2^2+2*u5*u1,
! 2071: 2*u3*u0+2*u1*u4-u3+(2*u1+2*u5)*u2,2*u2*u0+2*u2*u4+(2*u1+2*u5)*u3-u2+u1^2,
! 2072: 2*u1*u0+(2*u3+2*u5)*u4+2*u2*u3+2*u1*u2-u1,
! 2073: u0^2-u0+2*u4^2+2*u3^2+2*u2^2+2*u1^2+2*u5^2]
! 2074: [90] hkatsura(5);
! 2075: [-t+u0+2*u4+2*u3+2*u2+2*u1+2*u5,
! 2076: -u4*t+2*u4*u0+2*u1*u3+u2^2+2*u5*u1,-u3*t+2*u3*u0+2*u1*u4+(2*u1+2*u5)*u2,
! 2077: -u2*t+2*u2*u0+2*u2*u4+(2*u1+2*u5)*u3+u1^2,
! 2078: -u1*t+2*u1*u0+(2*u3+2*u5)*u4+2*u2*u3+2*u1*u2,
! 2079: -u0*t+u0^2+2*u4^2+2*u3^2+2*u2^2+2*u1^2+2*u5^2]
! 2080: [91] cyclic(6);
! 2081: [c5*c4*c3*c2*c1*c0-1,
! 2082: ((((c4+c5)*c3+c5*c4)*c2+c5*c4*c3)*c1+c5*c4*c3*c2)*c0+c5*c4*c3*c2*c1,
! 2083: (((c3+c5)*c2+c5*c4)*c1+c5*c4*c3)*c0+c4*c3*c2*c1+c5*c4*c3*c2,
! 2084: ((c2+c5)*c1+c5*c4)*c0+c3*c2*c1+c4*c3*c2+c5*c4*c3,
! 2085: (c1+c5)*c0+c2*c1+c3*c2+c4*c3+c5*c4,c0+c1+c2+c3+c4+c5]
! 2086: [92] hcyclic(6);
! 2087: [-c^6+c5*c4*c3*c2*c1*c0,
! 2088: ((((c4+c5)*c3+c5*c4)*c2+c5*c4*c3)*c1+c5*c4*c3*c2)*c0+c5*c4*c3*c2*c1,
! 2089: (((c3+c5)*c2+c5*c4)*c1+c5*c4*c3)*c0+c4*c3*c2*c1+c5*c4*c3*c2,
! 2090: ((c2+c5)*c1+c5*c4)*c0+c3*c2*c1+c4*c3*c2+c5*c4*c3,
! 2091: (c1+c5)*c0+c2*c1+c3*c2+c4*c3+c5*c4,c0+c1+c2+c3+c4+c5]
! 2092: @end example
! 2093:
! 2094: @table @t
! 2095: @item $B;2>H(B
! 2096: @fref{dp_dtop}.
! 2097: @end table
! 2098:
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