Annotation of OpenXM/src/asir-doc/parts/groebner.texi, Revision 1.1.1.1
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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