Annotation of OpenXM/src/asir-doc/parts/groebner.texi, Revision 1.23
1.23 ! noro 1: @comment $OpenXM: OpenXM/src/asir-doc/parts/groebner.texi,v 1.22 2019/03/29 04:54:25 noro Exp $
1.2 noro 2: \BJP
1.1 noro 3: @node $B%0%l%V%J4pDl$N7W;;(B,,, Top
4: @chapter $B%0%l%V%J4pDl$N7W;;(B
1.2 noro 5: \E
6: \BEG
7: @node Groebner basis computation,,, Top
8: @chapter Groebner basis computation
9: \E
1.1 noro 10:
11: @menu
1.2 noro 12: \BJP
1.1 noro 13: * $BJ,;6I=8=B?9`<0(B::
14: * $B%U%!%$%k$NFI$_9~$_(B::
15: * $B4pK\E*$JH!?t(B::
16: * $B7W;;$*$h$SI=<($N@)8f(B::
17: * $B9`=g=x$N@_Dj(B::
1.13 noro 18: * Weight::
1.1 noro 19: * $BM-M}<0$r78?t$H$9$k%0%l%V%J4pDl7W;;(B::
20: * $B4pDlJQ49(B::
1.5 noro 21: * Weyl $BBe?t(B::
1.23 ! noro 22: * $BB?9`<04D>e$N2C72(B::
1.1 noro 23: * $B%0%l%V%J4pDl$K4X$9$kH!?t(B::
1.2 noro 24: \E
25: \BEG
26: * Distributed polynomial::
27: * Reading files::
28: * Fundamental functions::
29: * Controlling Groebner basis computations::
30: * Setting term orderings::
1.13 noro 31: * Weight::
1.2 noro 32: * Groebner basis computation with rational function coefficients::
33: * Change of ordering::
1.5 noro 34: * Weyl algebra::
1.23 ! noro 35: * Module over a polynomial ring::
1.2 noro 36: * Functions for Groebner basis computation::
37: \E
1.1 noro 38: @end menu
39:
1.2 noro 40: \BJP
1.1 noro 41: @node $BJ,;6I=8=B?9`<0(B,,, $B%0%l%V%J4pDl$N7W;;(B
42: @section $BJ,;6I=8=B?9`<0(B
1.2 noro 43: \E
44: \BEG
45: @node Distributed polynomial,,, Groebner basis computation
46: @section Distributed polynomial
47: \E
1.1 noro 48:
49: @noindent
1.2 noro 50: \BJP
1.1 noro 51: $BJ,;6I=8=B?9`<0$H$O(B, $BB?9`<0$NFbIt7A<0$N0l$D$G$"$k(B. $BDL>o$NB?9`<0(B
52: (@code{type} $B$,(B 2) $B$O(B, $B:F5"I=8=$H8F$P$l$k7A<0$GI=8=$5$l$F$$$k(B. $B$9$J$o(B
53: $B$A(B, $BFCDj$NJQ?t$r<gJQ?t$H$9$k(B 1 $BJQ?tB?9`<0$G(B, $B$=$NB>$NJQ?t$O(B, $B$=$N(B 1 $BJQ(B
54: $B?tB?9`<0$N78?t$K(B, $B<gJQ?t$r4^$^$J$$B?9`<0$H$7$F8=$l$k(B. $B$3$N78?t$,(B, $B$^$?(B,
55: $B$"$kJQ?t$r<gJQ?t$H$9$kB?9`<0$H$J$C$F$$$k$3$H$+$i:F5"I=8=$H8F$P$l$k(B.
1.2 noro 56: \E
57: \BEG
58: A distributed polynomial is a polynomial with a special internal
59: representation different from the ordinary one.
60:
61: An ordinary polynomial (having @code{type} 2) is internally represented
62: in a format, called recursive representation.
63: In fact, it is represented as an uni-variate polynomial with respect to
64: a fixed variable, called main variable of that polynomial,
65: where the other variables appear in the coefficients which may again
66: polynomials in such variables other than the previous main variable.
67: A polynomial in the coefficients is again represented as
68: an uni-variate polynomial in a certain fixed variable,
69: the main variable. Thus, by this recursive structure of polynomial
70: representation, it is called the `recursive representation.'
71: \E
1.1 noro 72:
73: @iftex
74: @tex
1.2 noro 75: \JP $(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 ))$
76: \EG $(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 ))$
1.1 noro 77: @end tex
78: @end iftex
79: @ifinfo
80: @example
81: (x+y+z)^2 = 1 x^2 + (2 y + (2 z)) x + ((2 z) y + (1 z^2 ))
82: @end example
83: @end ifinfo
84:
85: @noindent
1.2 noro 86: \BJP
1.1 noro 87: $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
88: $BI=8=$H8F$V(B.
1.2 noro 89: \E
90: \BEG
91: On the other hand,
92: we call a representation the distributed representation of a polynomial,
93: if a polynomial is represented, according to its original meaning,
94: as a sum of monomials,
95: where a monomial is the product of power product of variables
96: and a coefficient. We call a polynomial, represented in such an
97: internal format, a distributed polynomial. (This naming may sounds
98: something strange.)
99: \E
1.1 noro 100:
101: @iftex
102: @tex
1.2 noro 103: \JP $(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$
104: \EG $(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$
1.1 noro 105: @end tex
106: @end iftex
107: @ifinfo
108: @example
109: (x+y+z)^2 = 1 x^2 + 2 xy + 2 xz + 1 y^2 + 2 yz +1 z^2$
110: @end example
111: @end ifinfo
112:
113: @noindent
1.2 noro 114: \BJP
1.1 noro 115: $B%0%l%V%J4pDl7W;;$K$*$$$F$O(B, $BC19`<0$KCmL\$7$FA`:n$r9T$&$?$aB?9`<0$,J,;6I=8=(B
116: $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,
117: $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$$$k(B.
118: $B$3$3$G(B, $B8e$N@bL@$N$?$a$K(B, $B$$$/$D$+$N8@MU$rDj5A$7$F$*$/(B.
1.2 noro 119: \E
120: \BEG
121: For computation of Groebner basis, efficient operation is expected if
122: polynomials are represented in a distributed representation,
123: because major operations for Groebner basis are performed with respect
124: to monomials.
125: From this view point, we provide the object type distributed polynomial
126: with its object identification number 9, and objects having such a type
127: are available by @b{Asir} language.
128:
129: Here, we provide several definitions for the later description.
130: \E
1.1 noro 131:
132: @table @b
1.2 noro 133: \BJP
1.1 noro 134: @item $B9`(B (term)
135: $BJQ?t$NQQ@Q(B. $B$9$J$o$A(B, $B78?t(B 1 $B$NC19`<0$N$3$H(B. @b{Asir} $B$K$*$$$F$O(B,
1.2 noro 136: \E
137: \BEG
138: @item term
139: The power product of variables, i.e., a monomial with coefficient 1.
140: In an @b{Asir} session, it is displayed in the form like
141: \E
1.1 noro 142:
143: @example
144: <<0,1,2,3,4>>
145: @end example
146:
1.2 noro 147: \BJP
1.1 noro 148: $B$H$$$&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
149: $B$r<($9(B. $B3FJQ?t$r(B @code{a}, @code{b}, @code{c}, @code{d}, @code{e} $B$H$9$k$H(B
150: $B$3$N9`$O(B @code{b*c^2*d^3*e^4} $B$rI=$9(B.
1.2 noro 151: \E
152: \BEG
153: and also can be input in such a form.
154: This example shows a term in 5 variables. If we assume the 5 variables
155: as @code{a}, @code{b}, @code{c}, @code{d}, and @code{e},
156: the term represents @code{b*c^2*d^3*e^4} in the ordinary expression.
157: \E
1.1 noro 158:
1.2 noro 159: \BJP
1.1 noro 160: @item $B9`=g=x(B (term order)
161: $BJ,;6I=8=B?9`<0$K$*$1$k9`$O(B, $B<!$N@-<A$rK~$?$9A4=g=x$K$h$j@0Ns$5$l$k(B.
1.2 noro 162: \E
163: \BEG
164: @item term order
165: Terms are ordered according to a total order with the following properties.
166: \E
1.1 noro 167:
168: @enumerate
169: @item
1.2 noro 170: \JP $BG$0U$N9`(B @code{t} $B$KBP$7(B @code{t} > 1
171: \EG For all @code{t} @code{t} > 1.
1.1 noro 172:
173: @item
1.2 noro 174: \JP @code{t}, @code{s}, @code{u} $B$r9`$H$9$k;~(B, @code{t} > @code{s} $B$J$i$P(B @code{tu} > @code{su}
175: \EG For all @code{t}, @code{s}, @code{u} @code{t} > @code{s} implies @code{tu} > @code{su}.
1.1 noro 176: @end enumerate
177:
1.2 noro 178: \BJP
1.1 noro 179: $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)
180: $B$H9`=g=x7?(B ($B?t(B, $B%j%9%H$^$?$O9TNs(B) $B$K$h$j;XDj$5$l$k(B.
1.2 noro 181: \E
182: \BEG
183: Such a total order is called a term ordering. A term ordering is specified
184: by a variable ordering (a list of variables) and a type of term ordering
185: (an integer, a list or a matrix).
186: \E
1.1 noro 187:
1.2 noro 188: \BJP
1.1 noro 189: @item $BC19`<0(B (monomial)
190: $B9`$H78?t$N@Q(B.
1.2 noro 191: \E
192: \BEG
193: @item monomial
194: The product of a term and a coefficient.
195: In an @b{Asir} session, it is displayed in the form like
196: \E
1.1 noro 197:
198: @example
199: 2*<<0,1,2,3,4>>
200: @end example
201:
1.2 noro 202: \JP $B$H$$$&7A$GI=<($5$l(B, $B$^$?(B, $B$3$N7A$GF~NO2DG=$G$"$k(B.
203: \EG and also can be input in such a form.
1.1 noro 204:
1.2 noro 205: \BJP
1.19 noro 206: @item $BF,9`(B (head term)
1.1 noro 207: @itemx $BF,C19`<0(B (head monomial)
208: @itemx $BF,78?t(B (head coefficient)
209: $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
210: $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
211: $B$H8F$V(B.
1.2 noro 212: \E
213: \BEG
1.19 noro 214: @item head term
1.2 noro 215: @itemx head monomial
216: @itemx head coefficient
217:
218: Monomials in a distributed polynomial is sorted by a total order.
219: In such representation, we call the monomial that is maximum
220: with respect to the order the head monomial, and its term and coefficient
221: the head term and the head coefficient respectively.
222: \E
1.1 noro 223: @end table
224:
1.20 takayama 225: @noindent
226: ChangeLog
227: @itemize @bullet
228: \BJP
229: @item $BJ,;6I=8=B?9`<0$OG$0U$N%*%V%8%'%/%H$r78?t$K$b$F$k$h$&$K$J$C$?(B.
230: $B$^$?2C72$N(Bk$B@.J,$NMWAG$r<!$N7A<0(B <<d0,d1,...:k>> $B$GI=8=$9$k$h$&$K$J$C$?(B (2017-08-31).
231: \E
232: \BEG
233: @item Distributed polynomials accept objects as coefficients.
234: The k-th element of a free module is expressed as <<d0,d1,...:k>> (2017-08-31).
235: \E
236: @item
237: 1.15 algnum.c,
238: 1.53 ctrl.c,
239: 1.66 dp-supp.c,
240: 1.105 dp.c,
241: 1.73 gr.c,
242: 1.4 reduct.c,
243: 1.16 _distm.c,
244: 1.17 dalg.c,
245: 1.52 dist.c,
246: 1.20 distm.c,
247: 1.8 gmpq.c,
248: 1.238 engine/nd.c,
249: 1.102 ca.h,
250: 1.411 version.h,
251: 1.28 cpexpr.c,
252: 1.42 pexpr.c,
253: 1.20 pexpr_body.c,
254: 1.40 spexpr.c,
255: 1.27 arith.c,
256: 1.77 eval.c,
257: 1.56 parse.h,
258: 1.37 parse.y,
259: 1.8 stdio.c,
260: 1.31 plotf.c
261: @end itemize
262:
1.2 noro 263: \BJP
1.1 noro 264: @node $B%U%!%$%k$NFI$_9~$_(B,,, $B%0%l%V%J4pDl$N7W;;(B
265: @section $B%U%!%$%k$NFI$_9~$_(B
1.2 noro 266: \E
267: \BEG
268: @node Reading files,,, Groebner basis computation
269: @section Reading files
270: \E
1.1 noro 271:
272: @noindent
1.2 noro 273: \BJP
1.1 noro 274: $B%0%l%V%J4pDl$r7W;;$9$k$?$a$N4pK\E*$JH!?t$O(B @code{dp_gr_main()} $B$*$h$S(B
1.5 noro 275: @code{dp_gr_mod_main()}, @code{dp_gr_f_main()}
276: $B$J$k(B 3 $B$D$NAH$_9~$_H!?t$G$"$k$,(B, $BDL>o$O(B, $B%Q%i%a%?(B
1.1 noro 277: $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.
278: $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
279: $B$_9~$`$3$H$K$h$j;HMQ2DG=$H$J$k(B. @samp{gr} $B$O(B, @b{Asir} $B$NI8=`(B
1.5 noro 280: $B%i%$%V%i%j%G%#%l%/%H%j$KCV$+$l$F$$$k(B.
1.2 noro 281: \E
282: \BEG
1.5 noro 283: Facilities for computing Groebner bases are
284: @code{dp_gr_main()}, @code{dp_gr_mod_main()}and @code{dp_gr_f_main()}.
285: To call these functions,
286: it is necessary to set several parameters correctly and it is convenient
287: to use a set of interface functions provided in the library file
288: @samp{gr}.
1.2 noro 289: The facilities will be ready to use after you load the package by
290: @code{load()}. The package @samp{gr} is placed in the standard library
1.5 noro 291: directory of @b{Asir}.
1.2 noro 292: \E
1.1 noro 293:
294: @example
295: [0] load("gr")$
296: @end example
297:
1.2 noro 298: \BJP
1.1 noro 299: @node $B4pK\E*$JH!?t(B,,, $B%0%l%V%J4pDl$N7W;;(B
300: @section $B4pK\E*$JH!?t(B
1.2 noro 301: \E
302: \BEG
303: @node Fundamental functions,,, Groebner basis computation
304: @section Fundamental functions
305: \E
1.1 noro 306:
307: @noindent
1.2 noro 308: \BJP
1.1 noro 309: @samp{gr} $B$G$O?tB?$/$NH!?t$,Dj5A$5$l$F$$$k$,(B, $BD>@\(B
310: $B%0%l%V%J4pDl$r7W;;$9$k$?$a$N%H%C%W%l%Y%k$O<!$N(B 3 $B$D$G$"$k(B.
311: $B0J2<$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,
312: @var{order} $B$OJQ?t=g=x7?(B, @var{p} $B$O(B @code{2^27} $BL$K~$NAG?t$G$"$k(B.
1.2 noro 313: \E
314: \BEG
315: There are many functions and options defined in the package @samp{gr}.
316: Usually not so many of them are used. Top level functions for Groebner
317: basis computation are the following three functions.
318:
319: In the following description, @var{plist}, @var{vlist}, @var{order}
320: and @var{p} stand for a list of polynomials, a list of variables
321: (indeterminates), a type of term ordering and a prime less than
322: @code{2^27} respectively.
323: \E
1.1 noro 324:
325: @table @code
326: @item gr(@var{plist},@var{vlist},@var{order})
327:
1.2 noro 328: \BJP
1.1 noro 329: Gebauer-Moeller $B$K$h$k(B useless pair elimination criteria, sugar
330: strategy $B$*$h$S(B Traverso $B$K$h$k(B trace-lifting $B$rMQ$$$?(B Buchberger $B%"%k(B
331: $B%4%j%:%`$K$h$kM-M}?t78?t%0%l%V%J4pDl7W;;H!?t(B. $B0lHL$K$O$3$NH!?t$rMQ$$$k(B.
1.2 noro 332: \E
333: \BEG
334: Function that computes Groebner bases over the rationals. The
335: algorithm is Buchberger algorithm with useless pair elimination
336: criteria by Gebauer-Moeller, sugar strategy and trace-lifting by
337: Traverso. For ordinary computation, this function is used.
338: \E
1.1 noro 339:
340: @item hgr(@var{plist},@var{vlist},@var{order})
341:
1.2 noro 342: \BJP
1.1 noro 343: $BF~NOB?9`<0$r@F<!2=$7$?8e(B @code{gr()} $B$N%0%l%V%J4pDl8uJd@8@.It$K$h$j8u(B
344: $BJd@8@.$7(B, $BHs@F<!2=(B, interreduce $B$7$?$b$N$r(B @code{gr()} $B$N%0%l%V%J4pDl(B
345: $B%A%'%C%/It$G%A%'%C%/$9$k(B. 0 $B<!85%7%9%F%`(B ($B2r$N8D?t$,M-8B8D$NJ}Dx<07O(B)
346: $B$N>l9g(B, sugar strategy $B$,78?tKDD%$r0z$-5/$3$9>l9g$,$"$k(B. $B$3$N$h$&$J>l(B
347: $B9g(B, strategy $B$r@F<!2=$K$h$k(B strategy $B$KCV$-49$($k$3$H$K$h$j78?tKDD%$r(B
348: $BM^@)$9$k$3$H$,$G$-$k>l9g$,B?$$(B.
1.2 noro 349: \E
350: \BEG
351: After homogenizing the input polynomials a candidate of the \gr basis
352: is computed by trace-lifting. Then the candidate is dehomogenized and
353: checked whether it is indeed a Groebner basis of the input. Sugar
354: strategy often causes intermediate coefficient swells. It is
355: empirically known that the combination of homogenization and supresses
356: the swells for such cases.
357: \E
1.1 noro 358:
359: @item gr_mod(@var{plist},@var{vlist},@var{order},@var{p})
360:
1.2 noro 361: \BJP
1.1 noro 362: Gebauer-Moeller $B$K$h$k(B useless pair elimination criteria, sugar
363: strategy $B$*$h$S(B Buchberger $B%"%k%4%j%:%`$K$h$k(B GF(p) $B78?t%0%l%V%J4pDl7W(B
364: $B;;H!?t(B.
1.2 noro 365: \E
366: \BEG
367: Function that computes Groebner bases over GF(@var{p}). The same
368: algorithm as @code{gr()} is used.
369: \E
1.1 noro 370:
371: @end table
372:
1.2 noro 373: \BJP
1.1 noro 374: @node $B7W;;$*$h$SI=<($N@)8f(B,,, $B%0%l%V%J4pDl$N7W;;(B
375: @section $B7W;;$*$h$SI=<($N@)8f(B
1.2 noro 376: \E
377: \BEG
378: @node Controlling Groebner basis computations,,, Groebner basis computation
379: @section Controlling Groebner basis computations
380: \E
1.1 noro 381:
382: @noindent
1.2 noro 383: \BJP
1.1 noro 384: $B%0%l%V%J4pDl$N7W;;$K$*$$$F(B, $B$5$^$6$^$J%Q%i%a%?@_Dj$r9T$&$3$H$K$h$j7W;;(B,
385: $BI=<($r@)8f$9$k$3$H$,$G$-$k(B. $B$3$l$i$O(B, $BAH$_9~$_H!?t(B @code{dp_gr_flags()}
386: $B$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
387: $B$H(B, $B8=:_@_Dj$5$l$F$$$k%Q%i%a%?$,(B, $BL>A0$HCM$N%j%9%H$GJV$5$l$k(B.
1.2 noro 388: \E
389: \BEG
390: One can cotrol a Groebner basis computation by setting various parameters.
391: These parameters can be set and examined by a built-in function
392: @code{dp_gr_flags()}. Without argument it returns the current settings.
393: \E
1.1 noro 394:
395: @example
396: [100] dp_gr_flags();
1.5 noro 397: [Demand,0,NoSugar,0,NoCriB,0,NoGC,0,NoMC,0,NoRA,0,NoGCD,0,Top,0,
398: ShowMag,1,Print,1,Stat,0,Reverse,0,InterReduce,0,Multiple,0]
1.1 noro 399: [101]
400: @end example
401:
1.2 noro 402: \BJP
1.1 noro 403: $B0J2<$G(B, $B3F%Q%i%a%?$N0UL#$r@bL@$9$k(B. on $B$N>l9g$H$O(B, $B%Q%i%a%?$,(B 0 $B$G$J$$>l9g$r(B
404: $B$$$&(B. $B$3$l$i$N%Q%i%a%?$N=i4|CM$OA4$F(B 0 (off) $B$G$"$k(B.
1.2 noro 405: \E
406: \BEG
407: The return value is a list which contains the names of parameters and their
408: values. The meaning of the parameters are as follows. `on' means that the
409: parameter is not zero.
410: \E
1.1 noro 411:
412: @table @code
413: @item NoSugar
1.2 noro 414: \BJP
1.1 noro 415: on $B$N>l9g(B, sugar strategy $B$NBe$o$j$K(B Buchberger$B$N(B normal strategy $B$,MQ(B
416: $B$$$i$l$k(B.
1.2 noro 417: \E
418: \BEG
419: If `on', Buchberger's normal strategy is used instead of sugar strategy.
420: \E
1.1 noro 421:
422: @item NoCriB
1.2 noro 423: \JP on $B$N>l9g(B, $BITI,MWBP8!=P5,=`$N$&$A(B, $B5,=`(B B $B$rE,MQ$7$J$$(B.
424: \EG If `on', criterion B among the Gebauer-Moeller's criteria is not applied.
1.1 noro 425:
426: @item NoGC
1.2 noro 427: \JP on $B$N>l9g(B, $B7k2L$,%0%l%V%J4pDl$K$J$C$F$$$k$+$I$&$+$N%A%'%C%/$r9T$o$J$$(B.
428: \BEG
429: If `on', the check that a Groebner basis candidate is indeed a Groebner basis,
430: is not executed.
431: \E
1.1 noro 432:
433: @item NoMC
1.2 noro 434: \BJP
1.1 noro 435: on $B$N>l9g(B, $B7k2L$,F~NO%$%G%"%k$HF1Ey$N%$%G%"%k$G$"$k$+$I$&$+$N%A%'%C%/(B
436: $B$r9T$o$J$$(B.
1.2 noro 437: \E
438: \BEG
439: If `on', the check that the resulting polynomials generates the same ideal as
440: the ideal generated by the input, is not executed.
441: \E
1.1 noro 442:
443: @item NoRA
1.2 noro 444: \BJP
1.1 noro 445: on $B$N>l9g(B, $B7k2L$r(B reduced $B%0%l%V%J4pDl$K$9$k$?$a$N(B
446: interreduce $B$r9T$o$J$$(B.
1.2 noro 447: \E
448: \BEG
449: If `on', the interreduction, which makes the Groebner basis reduced, is not
450: executed.
451: \E
1.1 noro 452:
453: @item NoGCD
1.2 noro 454: \BJP
1.1 noro 455: on $B$N>l9g(B, $BM-M}<078?t$N%0%l%V%J4pDl7W;;$K$*$$$F(B, $B@8@.$5$l$?B?9`<0$N(B,
456: $B78?t$N(B content $B$r$H$i$J$$(B.
1.2 noro 457: \E
458: \BEG
459: If `on', content removals are not executed during a Groebner basis computation
460: over a rational function field.
461: \E
1.1 noro 462:
463: @item Top
1.2 noro 464: \JP on $B$N>l9g(B, normal form $B7W;;$K$*$$$FF,9`>C5n$N$_$r9T$&(B.
465: \EG If `on', Only the head term of the polynomial being reduced is reduced.
1.1 noro 466:
1.2 noro 467: @comment @item Interreduce
468: @comment \BJP
469: @comment 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
470: @comment $B$h$k(B normal form $B$GCV$-49$($k(B.
471: @comment \E
472: @comment \BEG
473: @comment If `on', intermediate basis elements are reduced by using a newly generated
474: @comment basis element.
475: @comment \E
1.1 noro 476:
477: @item Reverse
1.2 noro 478: \BJP
1.1 noro 479: 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
480: $B@h$7$FA*$V(B.
1.2 noro 481: \E
482: \BEG
483: If `on', the selection strategy of reducer in a normal form computation
484: is such that a newer reducer is used first.
485: \E
1.1 noro 486:
487: @item Print
1.2 noro 488: \JP on $B$N>l9g(B, $B%0%l%V%J4pDl7W;;$NESCf$K$*$1$k$5$^$6$^$J>pJs$rI=<($9$k(B.
489: \BEG
490: If `on', various informations during a Groebner basis computation is
491: displayed.
492: \E
1.1 noro 493:
1.7 noro 494: @item PrintShort
495: \JP on $B$G!"(BPrint $B$,(B off $B$N>l9g(B, $B%0%l%V%J4pDl7W;;$NESCf$N>pJs$rC;=L7A$GI=<($9$k(B.
496: \BEG
497: If `on' and Print is `off', short information during a Groebner basis computation is
498: displayed.
499: \E
500:
1.1 noro 501: @item Stat
1.2 noro 502: \BJP
1.1 noro 503: 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
504: $B$l$k%G!<%?$NFb(B, $B=87W%G!<%?$N$_$,I=<($5$l$k(B.
1.2 noro 505: \E
506: \BEG
507: If `on', a summary of informations is shown after a Groebner basis
508: computation. Note that the summary is always shown if @code{Print} is `on'.
509: \E
1.1 noro 510:
511: @item ShowMag
1.2 noro 512: \BJP
1.1 noro 513: 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
514: $B78?t$N%S%C%HD9$NOB$rI=<($7(B, $B:G8e$K(B, $B$=$l$i$NOB$N:GBgCM$rI=<($9$k(B.
1.2 noro 515: \E
516: \BEG
517: If `on' and @code{Print} is `on', the sum of bit length of
518: coefficients of a generated basis element, which we call @var{magnitude},
519: is shown after every normal computation. After comleting the
520: computation the maximal value among the sums is shown.
521: \E
1.1 noro 522:
1.7 noro 523: @item Content
524: @itemx Multiple
1.2 noro 525: \BJP
1.7 noro 526: 0 $B$G$J$$M-M}?t$N;~(B, $BM-M}?t>e$N@55,7A7W;;$K$*$$$F(B, $B78?t$N%S%C%HD9$NOB$,(B
527: @code{Content} $BG\$K$J$k$4$H$K78?tA4BN$N(B GCD $B$,7W;;$5$l(B, $B$=$N(B GCD $B$G(B
528: $B3d$C$?B?9`<0$r4JLs$9$k(B. @code{Content} $B$,(B 1 $B$J$i$P(B, $B4JLs$9$k$4$H$K(B
529: GCD $B7W;;$,9T$o$l0lHL$K$O8zN($,0-$/$J$k$,(B, @code{Content} $B$r(B 2 $BDxEY(B
1.1 noro 530: $B$H$9$k$H(B, $B5pBg$J@0?t$,78?t$K8=$l$k>l9g(B, $B8zN($,NI$/$J$k>l9g$,$"$k(B.
1.7 noro 531: backward compatibility $B$N$?$a!"(B@code{Multiple} $B$G@0?tCM$r;XDj$G$-$k(B.
1.2 noro 532: \E
533: \BEG
1.7 noro 534: If a non-zero rational number, in a normal form computation
1.2 noro 535: over the rationals, the integer content of the polynomial being
1.7 noro 536: reduced is removed when its magnitude becomes @code{Content} times
1.2 noro 537: larger than a registered value, which is set to the magnitude of the
538: input polynomial. After each content removal the registered value is
1.7 noro 539: set to the magnitude of the resulting polynomial. @code{Content} is
1.2 noro 540: equal to 1, the simiplification is done after every normal form computation.
1.7 noro 541: It is empirically known that it is often efficient to set @code{Content} to 2
1.2 noro 542: for the case where large integers appear during the computation.
1.7 noro 543: An integer value can be set by the keyword @code{Multiple} for
544: backward compatibility.
1.2 noro 545: \E
1.1 noro 546:
547: @item Demand
1.2 noro 548:
549: \BJP
1.1 noro 550: $B@5Ev$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
551: $BCf$K$*$+$l$:(B, $B$=$N%G%#%l%/%H%jCf$K%P%$%J%j%G!<%?$H$7$FCV$+$l(B, $B$=$NB?9`(B
552: $B<0$rMQ$$$k(B normal form $B7W;;$N:](B, $B<+F0E*$K%a%b%jCf$K%m!<%I$5$l$k(B. $B3FB?(B
553: $B9`<0$O(B, $BFbIt$G$N%$%s%G%C%/%9$r%U%!%$%kL>$K;}$D%U%!%$%k$K3JG<$5$l$k(B.
554: $B$3$3$G;XDj$5$l$?%G%#%l%/%H%j$K=q$+$l$?%U%!%$%k$O<+F0E*$K$O>C5n$5$l$J$$(B
555: $B$?$a(B, $B%f!<%6$,@UG$$r;}$C$F>C5n$9$kI,MW$,$"$k(B.
1.2 noro 556: \E
557: \BEG
558: If the value (a character string) is a valid directory name, then
559: generated basis elements are put in the directory and are loaded on
560: demand during normal form computations. Each elements is saved in the
561: binary form and its name coincides with the index internally used in
562: the computation. These binary files are not removed automatically
563: and one should remove them by hand.
564: \E
1.1 noro 565: @end table
566:
567: @noindent
1.2 noro 568: \JP @code{Print} $B$,(B 0 $B$G$J$$>l9g<!$N$h$&$J%G!<%?$,I=<($5$l$k(B.
569: \EG If @code{Print} is `on', the following informations are shown.
1.1 noro 570:
571: @example
572: [93] gr(cyclic(4),[c0,c1,c2,c3],0)$
573: mod= 99999989, eval = []
574: (0)(0)<<0,2,0,0>>(2,3),nb=2,nab=5,rp=2,sugar=2,mag=4
575: (0)(0)<<0,1,2,0>>(1,2),nb=3,nab=6,rp=2,sugar=3,mag=4
576: (0)(0)<<0,1,1,2>>(0,1),nb=4,nab=7,rp=3,sugar=4,mag=6
577: .
578: (0)(0)<<0,0,3,2>>(5,6),nb=5,nab=8,rp=2,sugar=5,mag=4
579: (0)(0)<<0,1,0,4>>(4,6),nb=6,nab=9,rp=3,sugar=5,mag=4
580: (0)(0)<<0,0,2,4>>(6,8),nb=7,nab=10,rp=4,sugar=6,mag=6
581: ....gb done
582: reduceall
583: .......
584: membercheck
585: (0,0)(0,0)(0,0)(0,0)
586: gbcheck total 8 pairs
587: ........
1.5 noro 588: UP=(0,0)SP=(0,0)SPM=(0,0)NF=(0,0)NFM=(0.010002,0)ZNFM=(0.010002,0)
589: PZ=(0,0)NP=(0,0)MP=(0,0)RA=(0,0)MC=(0,0)GC=(0,0)T=40,B=0 M=8 F=6
590: D=12 ZR=5 NZR=6 Max_mag=6
1.1 noro 591: [94]
592: @end example
593:
594: @noindent
1.2 noro 595: \BJP
1.1 noro 596: $B:G=i$KI=<($5$l$k(B @code{mod}, @code{eval} $B$O(B, trace-lifting $B$GMQ$$$i$l$kK!(B
597: $B$G$"$k(B. @code{mod} $B$OAG?t(B, @code{eval} $B$OM-M}<078?t$N>l9g$KMQ$$$i$l$k(B
598: $B?t$N%j%9%H$G$"$k(B.
1.2 noro 599: \E
600: \BEG
601: In this example @code{mod} and @code{eval} indicate moduli used in
602: trace-lifting. @code{mod} is a prime and @code{eval} is a list of integers
603: used for evaluation when the ground field is a field of rational functions.
604: \E
1.1 noro 605:
606: @noindent
1.2 noro 607: \JP $B7W;;ESCf$GB?9`<0$,@8@.$5$l$kKh$K<!$N7A$N%G!<%?$,I=<($5$l$k(B.
608: \EG The following information is shown after every normal form computation.
1.1 noro 609:
610: @example
611: (TNF)(TCONT)HT(INDEX),nb=NB,nab=NAB,rp=RP,sugar=S,mag=M
612: @end example
613:
614: @noindent
1.2 noro 615: \JP $B$=$l$i$N0UL#$O<!$NDL$j(B.
616: \EG Meaning of each component is as follows.
1.1 noro 617:
618: @table @code
1.2 noro 619: \BJP
1.1 noro 620: @item TNF
1.2 noro 621:
1.1 noro 622: normal form $B7W;;;~4V(B ($BIC(B)
623:
624: @item TCONT
1.2 noro 625:
1.1 noro 626: content $B7W;;;~4V(B ($BIC(B)
627:
628: @item HT
1.2 noro 629:
1.1 noro 630: $B@8@.$5$l$?B?9`<0$NF,9`(B
631:
632: @item INDEX
1.2 noro 633:
1.1 noro 634: S-$BB?9`<0$r9=@.$9$kB?9`<0$N%$%s%G%C%/%9$N%Z%"(B
635:
636: @item NB
1.2 noro 637:
1.1 noro 638: $B8=:_$N(B, $B>iD9@-$r=|$$$?4pDl$N?t(B
639:
640: @item NAB
1.2 noro 641:
1.1 noro 642: $B8=:_$^$G$K@8@.$5$l$?4pDl$N?t(B
643:
644: @item RP
1.2 noro 645:
1.1 noro 646: $B;D$j$N%Z%"$N?t(B
647:
648: @item S
1.2 noro 649:
1.1 noro 650: $B@8@.$5$l$?B?9`<0$N(B sugar $B$NCM(B
651:
652: @item M
1.2 noro 653:
1.1 noro 654: $B@8@.$5$l$?B?9`<0$N78?t$N%S%C%HD9$NOB(B (@code{ShowMag} $B$,(B on $B$N;~$KI=<($5$l$k(B. )
1.2 noro 655: \E
656: \BEG
657: @item TNF
658:
659: CPU time for normal form computation (second)
660:
661: @item TCONT
662:
663: CPU time for content removal(second)
664:
665: @item HT
666:
667: Head term of the generated basis element
668:
669: @item INDEX
670:
671: Pair of indices which corresponds to the reduced S-polynomial
672:
673: @item NB
674:
675: Number of basis elements after removing redundancy
676:
677: @item NAB
678:
679: Number of all the basis elements
680:
681: @item RP
682:
683: Number of remaining pairs
684:
685: @item S
686:
687: Sugar of the generated basis element
688:
689: @item M
690:
691: Magnitude of the genrated basis element (shown if @code{ShowMag} is `on'.)
692: \E
1.1 noro 693: @end table
694:
695: @noindent
1.2 noro 696: \BJP
1.1 noro 697: $B:G8e$K(B, $B=87W%G!<%?$,I=<($5$l$k(B. $B0UL#$O<!$NDL$j(B.
698: ($B;~4V$NI=<($K$*$$$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.)
1.2 noro 699: \E
700: \BEG
701: The summary of the informations shown after a Groebner basis
702: computation is as follows. If a component shows timings and it
703: contains two numbers, they are a pair of time for computation and time
704: for garbage colection.
705: \E
1.1 noro 706:
707: @table @code
1.2 noro 708: \BJP
1.1 noro 709: @item UP
1.2 noro 710:
1.1 noro 711: $B%Z%"$N%j%9%H$NA`:n$K$+$+$C$?;~4V(B
712:
713: @item SP
1.2 noro 714:
1.1 noro 715: $BM-M}?t>e$N(B S-$BB?9`<07W;;;~4V(B
716:
717: @item SPM
1.2 noro 718:
1.1 noro 719: $BM-8BBN>e$N(B S-$BB?9`<07W;;;~4V(B
720:
721: @item NF
1.2 noro 722:
1.1 noro 723: $BM-M}?t>e$N(B normal form $B7W;;;~4V(B
724:
725: @item NFM
1.2 noro 726:
1.1 noro 727: $BM-8BBN>e$N(B normal form $B7W;;;~4V(B
728:
729: @item ZNFM
1.2 noro 730:
1.1 noro 731: @code{NFM} $B$NFb(B, 0 $B$X$N(B reduction $B$K$+$+$C$?;~4V(B
732:
733: @item PZ
1.2 noro 734:
1.1 noro 735: content $B7W;;;~4V(B
736:
737: @item NP
1.2 noro 738:
1.1 noro 739: $BM-M}?t78?tB?9`<0$N78?t$KBP$9$k>jM>1i;;$N7W;;;~4V(B
740:
741: @item MP
1.2 noro 742:
1.1 noro 743: S-$BB?9`<0$r@8@.$9$k%Z%"$NA*Br$K$+$+$C$?;~4V(B
744:
745: @item RA
1.2 noro 746:
1.1 noro 747: interreduce $B7W;;;~4V(B
748:
749: @item MC
1.2 noro 750:
1.1 noro 751: trace-lifting $B$K$*$1$k(B, $BF~NOB?9`<0$N%a%s%P%7%C%W7W;;;~4V(B
752:
753: @item GC
1.2 noro 754:
1.1 noro 755: $B7k2L$N%0%l%V%J4pDl8uJd$N%0%l%V%J4pDl%A%'%C%/;~4V(B
756:
757: @item T
1.2 noro 758:
1.1 noro 759: $B@8@.$5$l$?%Z%"$N?t(B
760:
761: @item B, M, F, D
1.2 noro 762:
1.1 noro 763: $B3F(B criterion $B$K$h$j=|$+$l$?%Z%"$N?t(B
764:
765: @item ZR
1.2 noro 766:
1.1 noro 767: 0 $B$K(B reduce $B$5$l$?%Z%"$N?t(B
768:
769: @item NZR
1.2 noro 770:
1.1 noro 771: 0 $B$G$J$$B?9`<0$K(B reduce $B$5$l$?%Z%"$N?t(B
772:
773: @item Max_mag
1.2 noro 774:
1.1 noro 775: $B@8@.$5$l$?B?9`<0$N(B, $B78?t$N%S%C%HD9$NOB$N:GBgCM(B
1.2 noro 776: \E
777: \BEG
778: @item UP
779:
780: Time to manipulate the list of critical pairs
781:
782: @item SP
783:
784: Time to compute S-polynomials over the rationals
785:
786: @item SPM
787:
788: Time to compute S-polynomials over a finite field
789:
790: @item NF
791:
792: Time to compute normal forms over the rationals
793:
794: @item NFM
795:
796: Time to compute normal forms over a finite field
797:
798: @item ZNFM
799:
800: Time for zero reductions in @code{NFM}
801:
802: @item PZ
803:
804: Time to remove integer contets
805:
806: @item NP
807:
808: Time to compute remainders for coefficients of polynomials with coeffieints
809: in the rationals
810:
811: @item MP
812:
813: Time to select pairs from which S-polynomials are computed
814:
815: @item RA
816:
817: Time to interreduce the Groebner basis candidate
818:
819: @item MC
1.1 noro 820:
1.2 noro 821: Time to check that each input polynomial is a member of the ideal
822: generated by the Groebner basis candidate.
823:
824: @item GC
825:
826: Time to check that the Groebner basis candidate is a Groebner basis
827:
828: @item T
829:
830: Number of critical pairs generated
831:
832: @item B, M, F, D
833:
834: Number of critical pairs removed by using each criterion
835:
836: @item ZR
837:
838: Number of S-polynomials reduced to 0
839:
840: @item NZR
841:
842: Number of S-polynomials reduced to non-zero results
843:
844: @item Max_mag
845:
846: Maximal magnitude among all the generated polynomials
847: \E
1.1 noro 848: @end table
849:
1.2 noro 850: \BJP
1.1 noro 851: @node $B9`=g=x$N@_Dj(B,,, $B%0%l%V%J4pDl$N7W;;(B
852: @section $B9`=g=x$N@_Dj(B
1.2 noro 853: \E
854: \BEG
855: @node Setting term orderings,,, Groebner basis computation
856: @section Setting term orderings
857: \E
1.1 noro 858:
859: @noindent
1.2 noro 860: \BJP
1.1 noro 861: $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
862: $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
863: $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
864: $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
865: $B9TNs$GI=8=$5$l$k(B.
1.2 noro 866: \E
867: \BEG
868: A term is internally represented as an integer vector whose components
869: are exponents with respect to variables. A variable list specifies the
870: correspondences between variables and components. A type of term ordering
871: specifies a total order for integer vectors. A type of term ordering is
872: represented by an integer, a list of integer or matrices.
873: \E
1.1 noro 874:
875: @noindent
1.2 noro 876: \JP $B4pK\E*$J9`=g=x7?$H$7$F<!$N(B 3 $B$D$,$"$k(B.
877: \EG There are following three fundamental types.
1.1 noro 878:
879: @table @code
1.2 noro 880: \JP @item 0 (DegRevLex; @b{$BA4<!?t5U<-=q<0=g=x(B})
881: \EG @item 0 (DegRevLex; @b{total degree reverse lexicographic ordering})
1.1 noro 882:
1.2 noro 883: \BJP
1.1 noro 884: $B0lHL$K(B, $B$3$N=g=x$K$h$k%0%l%V%J4pDl7W;;$,:G$b9bB.$G$"$k(B. $B$?$@$7(B,
885: $BJ}Dx<0$r2r$/$H$$$&L\E*$KMQ$$$k$3$H$O(B, $B0lHL$K$O$G$-$J$$(B. $B$3$N(B
886: $B=g=x$K$h$k%0%l%V%J4pDl$O(B, $B2r$N8D?t$N7W;;(B, $B%$%G%"%k$N%a%s%P%7%C%W$d(B,
887: $BB>$NJQ?t=g=x$X$N4pDlJQ49$N$?$a$N%=!<%9$H$7$FMQ$$$i$l$k(B.
1.2 noro 888: \E
889: \BEG
890: In general, computation by this ordering shows the fastest speed
891: in most Groebner basis computations.
892: However, for the purpose to solve polynomial equations, this type
893: of ordering is, in general, not so suitable.
894: The Groebner bases obtained by this ordering is used for computing
895: the number of solutions, solving ideal membership problem and seeds
896: for conversion to other Groebner bases under different ordering.
897: \E
1.1 noro 898:
1.2 noro 899: \JP @item 1 (DegLex; @b{$BA4<!?t<-=q<0=g=x(B})
900: \EG @item 1 (DegLex; @b{total degree lexicographic ordering})
1.1 noro 901:
1.2 noro 902: \BJP
1.1 noro 903: $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,
904: @code{DegRevLex} $B$HF1MMD>@\$=$N7k2L$rMQ$$$k$3$H$O:$Fq$G$"$k(B. $B$7$+$7(B,
905: $B<-=q<0=g=x$N%0%l%V%J4pDl$r5a$a$k:]$K(B, $B@F<!2=8e$K$3$N=g=x$G%0%l%V%J4pDl(B
906: $B$r5a$a$F$$$k(B.
1.2 noro 907: \E
908: \BEG
909: By this type term ordering, Groebner bases are obtained fairly faster
910: than Lex (lexicographic) ordering, too.
911: Alike the @code{DegRevLex} ordering, the result, in general, cannot directly
912: be used for solving polynomial equations.
913: It is used, however, in such a way
914: that a Groebner basis is computed in this ordering after homogenization
915: to obtain the final lexicographic Groebner basis.
916: \E
1.1 noro 917:
1.2 noro 918: \JP @item 2 (Lex; @b{$B<-=q<0=g=x(B})
919: \EG @item 2 (Lex; @b{lexicographic ordering})
1.1 noro 920:
1.2 noro 921: \BJP
1.1 noro 922: $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
923: $B7W;;;~4V$,$+$+$j2a$.$k$N$,FqE@$G$"$k(B. $BFC$K(B, $B2r$,M-8B8D$N>l9g(B, $B7k2L$N(B
924: $B78?t$,6K$a$FD9Bg$JB?G\D9?t$K$J$k>l9g$,B?$$(B. $B$3$N>l9g(B, @code{gr()},
925: @code{hgr()} $B$K$h$k7W;;$,6K$a$FM-8z$K$J$k>l9g$,B?$$(B.
1.2 noro 926: \E
927: \BEG
928: Groebner bases computed by this ordering give the most convenient
929: Groebner bases for solving the polynomial equations.
930: The only and serious shortcoming is the enormously long computation
931: time.
932: It is often observed that the number coefficients of the result becomes
933: very very long integers, especially if the ideal is 0-dimensional.
934: For such a case, it is empirically true for many cases
935: that i.e., computation by
936: @code{gr()} and/or @code{hgr()} may be quite effective.
937: \E
1.1 noro 938: @end table
939:
940: @noindent
1.2 noro 941: \BJP
1.1 noro 942: $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.
943: $B$3$l$O(B,
1.2 noro 944: \E
945: \BEG
946: By combining these fundamental orderingl into a list, one can make
947: various term ordering called elimination orderings.
948: \E
1.1 noro 949:
950: @code{[[O1,L1],[O2,L2],...]}
951:
952: @noindent
1.2 noro 953: \BJP
1.1 noro 954: $B$G;XDj$5$l$k(B. @code{Oi} $B$O(B 0, 1, 2 $B$N$$$:$l$+$G(B, @code{Li} $B$OJQ?t$N8D(B
955: $B?t$rI=$9(B. $B$3$N;XDj$O(B, $BJQ?t$r@hF,$+$i(B @code{L1}, @code{L2} , ...$B8D(B
956: $B$:$D$NAH$KJ,$1(B, $B$=$l$>$l$NJQ?t$K4X$7(B, $B=g$K(B @code{O1}, @code{O2},
957: ...$B$N9`=g=x7?$GBg>.$,7hDj$9$k$^$GHf3S$9$k$3$H$r0UL#$9$k(B. $B$3$N7?$N(B
958: $B=g=x$O0lHL$K>C5n=g=x$H8F$P$l$k(B.
1.2 noro 959: \E
960: \BEG
961: In this example @code{Oi} indicates 0, 1 or 2 and @code{Li} indicates
962: the number of variables subject to the correspoinding orderings.
963: This specification means the following.
964:
965: The variable list is separated into sub lists from left to right where
966: the @code{i}-th list contains @code{Li} members and it corresponds to
967: the ordering of type @code{Oi}. The result of a comparison is equal
968: to that for the leftmost different sub components. This type of ordering
969: is called an elimination ordering.
970: \E
1.1 noro 971:
972: @noindent
1.2 noro 973: \BJP
1.1 noro 974: $B$5$i$K(B, $B9TNs$K$h$j9`=g=x$r;XDj$9$k$3$H$,$G$-$k(B. $B0lHL$K(B, @code{n} $B9T(B
975: @code{m} $BNs$N<B?t9TNs(B @code{M} $B$,<!$N@-<A$r;}$D$H$9$k(B.
1.2 noro 976: \E
977: \BEG
978: Furthermore one can specify a term ordering by a matix.
979: Suppose that a real @code{n}, @code{m} matrix @code{M} has the
980: following properties.
981: \E
1.1 noro 982:
983: @enumerate
984: @item
1.2 noro 985: \JP $BD9$5(B @code{m} $B$N@0?t%Y%/%H%k(B @code{v} $B$KBP$7(B @code{Mv=0} $B$H(B @code{v=0} $B$OF1CM(B.
986: \BEG
987: For all integer verctors @code{v} of length @code{m} @code{Mv=0} is equivalent
988: to @code{v=0}.
989: \E
1.1 noro 990:
991: @item
1.2 noro 992: \BJP
1.1 noro 993: $BHsIi@.J,$r;}$DD9$5(B @code{m} $B$N(B 0 $B$G$J$$@0?t%Y%/%H%k(B @code{v} $B$KBP$7(B,
994: @code{Mv} $B$N(B 0 $B$G$J$$:G=i$N@.J,$OHsIi(B.
1.2 noro 995: \E
996: \BEG
997: For all non-negative integer vectors @code{v} the first non-zero component
998: of @code{Mv} is non-negative.
999: \E
1.1 noro 1000: @end enumerate
1001:
1002: @noindent
1.2 noro 1003: \BJP
1.1 noro 1004: $B$3$N;~(B, 2 $B$D$N%Y%/%H%k(B @code{t}, @code{s} $B$KBP$7(B,
1005: @code{t>s} $B$r(B, @code{M(t-s)} $B$N(B 0 $B$G$J$$:G=i$N@.J,$,HsIi(B,
1006: $B$GDj5A$9$k$3$H$K$h$j9`=g=x$,Dj5A$G$-$k(B.
1.2 noro 1007: \E
1008: \BEG
1009: Then we can define a term ordering such that, for two vectors
1010: @code{t}, @code{s}, @code{t>s} means that the first non-zero component
1011: of @code{M(t-s)} is non-negative.
1012: \E
1.1 noro 1013:
1014: @noindent
1.2 noro 1015: \BJP
1.1 noro 1016: $B9`=g=x7?$O(B, @code{gr()} $B$J$I$N0z?t$H$7$F;XDj$5$l$kB>(B, $BAH$_9~$_H!?t(B
1017: @code{dp_ord()} $B$G;XDj$5$l(B, $B$5$^$6$^$JH!?t$N<B9T$N:]$K;2>H$5$l$k(B.
1.2 noro 1018: \E
1019: \BEG
1020: Types of term orderings are used as arguments of functions such as
1021: @code{gr()}. It is also set internally by @code{dp_ord()} and is used
1022: during executions of various functions.
1023: \E
1.1 noro 1024:
1025: @noindent
1.2 noro 1026: \BJP
1.1 noro 1027: $B$3$l$i$N=g=x$N6qBNE*$JDj5A$*$h$S%0%l%V%J4pDl$K4X$9$k99$K>\$7$$2r@b$O(B
1028: @code{[Becker,Weispfenning]} $B$J$I$r;2>H$N$3$H(B.
1.2 noro 1029: \E
1030: \BEG
1031: For concrete definitions of term ordering and more information
1032: about Groebner basis, refer to, for example, the book
1033: @code{[Becker,Weispfenning]}.
1034: \E
1.1 noro 1035:
1036: @noindent
1.2 noro 1037: \JP $B9`=g=x7?$N@_Dj$NB>$K(B, $BJQ?t$N=g=x<+BN$b7W;;;~4V$KBg$-$J1F6A$rM?$($k(B.
1038: \BEG
1039: Note that the variable ordering have strong effects on the computation
1040: time as well as the choice of types of term orderings.
1041: \E
1.1 noro 1042:
1043: @example
1044: [90] B=[x^10-t,x^8-z,x^31-x^6-x-y]$
1045: [91] gr(B,[x,y,z,t],2);
1046: [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
1047: -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
1048: +(-12*t^16+72*t^13-28*t^11-180*t^10+112*t^8+240*t^7+28*t^6-127*t^5
1049: -167*t^4-55*t^3+30*t^2+58*t-15)*z^4,
1.5 noro 1050: (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
1051: +84*t^9+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
1052: +(6*t^16-36*t^13+14*t^11+90*t^10-56*t^8-120*t^7-14*t^6+64*t^5+84*t^4
1053: +27*t^3-16*t^2-30*t+7)*z^4,
1054: (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
1055: -6*t-1)*y+(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
1056: +10*t^4-36*t^3-11*t^2-5*t+9)*z^2,
1.1 noro 1057: -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
1.5 noro 1058: -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
1059: +20*t^19+28*t^18-120*t^16-56*t^15+14*t^14+300*t^13+70*t^12-56*t^11
1060: -400*t^10-84*t^9+84*t^8+268*t^7+84*t^6-56*t^5-63*t^4-36*t^3+46*t^2
1061: -12*t+1)*z,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
1062: -10*t-20)*z^3*y+8*t^14-32*t^11+48*t^8-t^7-32*t^5-6*t^4+9*t^2-t,
1.1 noro 1063: -z*y^3+(t^7-2*t^4+3*t^2+t)*y+(-2*t^6+4*t^3+2*t-2)*z^2,
1.5 noro 1064: 2*t^2*y^3+z^2*y^2+(-2*t^5+4*t^2-6)*z^4*y
1065: +(4*t^8-t^7-8*t^5+2*t^4-4*t^3+5*t^2-t)*z,
1.1 noro 1066: z^3*y^2+2*t^3*y+(-t^7+2*t^4+t^2-t)*z^2,
1067: -t*z*y^2-2*z^3*y+t^8-2*t^5-t^3+t^2,
1.5 noro 1068: -t^3*y^2-2*t^2*z^2*y+(t^6-2*t^3-t+1)*z^4,z^5-t^4]
1.1 noro 1069: [93] gr(B,[t,z,y,x],2);
1070: [x^10-t,x^8-z,x^31-x^6-x-y]
1071: @end example
1072:
1073: @noindent
1.2 noro 1074: \BJP
1.1 noro 1075: $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
1076: $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
1077: $B%0%l%V%J4pDl$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
1078: $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
1079: $B$3$H$G$"$j(B, $B$3$NNc$N>l9g$O(B, @code{t}, @code{z}, @code{y} $B$,4{$K(B
1080: @code{x} $B$GI=$5$l$F$$$k$3$H$+$i$3$N$h$&$J6KC<$J7k2L$H$J$C$?$o$1$G$"$k(B.
1081: $B<B:]$K8=$l$k7W;;$K$*$$$F$O(B, $B$3$N$h$&$KA*$V$Y$-JQ?t=g=x$,L@$i$+$G$"$k(B
1082: $B$3$H$O>/$J$/(B, $B;n9T:x8m$,I,MW$J>l9g$b$"$k(B.
1.2 noro 1083: \E
1084: \BEG
1085: As you see in the above example, the Groebner base under variable
1086: ordering @code{[x,y,z,t]} has a lot of bases and each base itself is
1087: large. Under variable ordering @code{[t,z,y,x]}, however, @code{B} itself
1088: is already the Groebner basis.
1089: Roughly speaking, to obtain a Groebner base under the lexicographic
1090: ordering is to express the variables on the left (having higher order)
1091: in terms of variables on the right (having lower order).
1092: In the example, variables @code{t}, @code{z}, and @code{y} are already
1093: expressed by variable @code{x}, and the above explanation justifies
1094: such a drastic experimental results.
1095: In practice, however, optimum ordering for variables may not known
1096: beforehand, and some heuristic trial may be inevitable.
1.13 noro 1097: \E
1098:
1099: \BJP
1100: @node Weight ,,, $B%0%l%V%J4pDl$N7W;;(B
1101: @section Weight
1102: \E
1103: \BEG
1104: @node Weight,,, Groebner basis computation
1105: @section Weight
1106: \E
1107: \BJP
1108: $BA0@a$G>R2p$7$?9`=g=x$O(B, $B3FJQ?t$K(B weight ($B=E$_(B) $B$r@_Dj$9$k$3$H$G(B
1109: $B$h$j0lHLE*$J$b$N$H$J$k(B.
1110: \E
1111: \BEG
1.14 noro 1112: Term orderings introduced in the previous section can be generalized
1.13 noro 1113: by setting a weight for each variable.
1114: \E
1115: @example
1116: [0] dp_td(<<1,1,1>>);
1117: 3
1118: [1] dp_set_weight([1,2,3])$
1119: [2] dp_td(<<1,1,1>>);
1120: 6
1121: @end example
1122: \BJP
1123: $BC19`<0$NA4<!?t$r7W;;$9$k:](B, $B%G%U%)%k%H$G$O(B
1124: $B3FJQ?t$N;X?t$NOB$rA4<!?t$H$9$k(B. $B$3$l$O3FJQ?t$N(B weight $B$r(B 1 $B$H(B
1125: $B9M$($F$$$k$3$H$KAjEv$9$k(B. $B$3$NNc$G$O(B, $BBh0l(B, $BBhFs(B, $BBh;0JQ?t$N(B
1126: weight $B$r$=$l$>$l(B 1,2,3 $B$H;XDj$7$F$$$k(B. $B$3$N$?$a(B, @code{<<1,1,1>>}
1127: $B$NA4<!?t(B ($B0J2<$G$O$3$l$rC19`<0$N(B weight $B$H8F$V(B) $B$,(B @code{1*1+1*2+1*3=6} $B$H$J$k(B.
1128: weight $B$r@_Dj$9$k$3$H$G(B, $BF1$89`=g=x7?$N$b$H$G0[$J$k9`=g=x$,Dj5A$G$-$k(B.
1129: $BNc$($P(B, weight $B$r$&$^$/@_Dj$9$k$3$H$G(B, $BB?9`<0$r(B weighted homogeneous
1130: $B$K$9$k$3$H$,$G$-$k>l9g$,$"$k(B.
1131: \E
1132: \BEG
1133: By default, the total degree of a monomial is equal to
1134: the sum of all exponents. This means that the weight for each variable
1135: is set to 1.
1136: In this example, the weights for the first, the second and the third
1137: variable are set to 1, 2 and 3 respectively.
1138: Therefore the total degree of @code{<<1,1,1>>} under this weight,
1139: which is called the weight of the monomial, is @code{1*1+1*2+1*3=6}.
1.14 noro 1140: By setting weights, different term orderings can be set under a type of
1141: term ordeing. In some case a polynomial can
1142: be made weighted homogeneous by setting an appropriate weight.
1.13 noro 1143: \E
1144:
1145: \BJP
1146: $B3FJQ?t$KBP$9$k(B weight $B$r$^$H$a$?$b$N$r(B weight vector $B$H8F$V(B.
1147: $B$9$Y$F$N@.J,$,@5$G$"$j(B, $B%0%l%V%J4pDl7W;;$K$*$$$F(B, $BA4<!?t$N(B
1148: $BBe$o$j$KMQ$$$i$l$k$b$N$rFC$K(B sugar weight $B$H8F$V$3$H$K$9$k(B.
1149: sugar strategy $B$K$*$$$F(B, $BA4<!?t$NBe$o$j$K;H$o$l$k$+$i$G$"$k(B.
1150: $B0lJ}$G(B, $B3F@.J,$,I,$:$7$b@5$H$O8B$i$J$$(B weight vector $B$O(B,
1151: sugar weight $B$H$7$F@_Dj$9$k$3$H$O$G$-$J$$$,(B, $B9`=g=x$N0lHL2=$K$O(B
1152: $BM-MQ$G$"$k(B. $B$3$l$i$O(B, $B9TNs$K$h$k9`=g=x$N@_Dj$K$9$G$K8=$l$F(B
1153: $B$$$k(B. $B$9$J$o$A(B, $B9`=g=x$rDj5A$9$k9TNs$N3F9T$,(B, $B0l$D$N(B weight vector
1154: $B$H8+$J$5$l$k(B. $B$^$?(B, $B%V%m%C%/=g=x$O(B, $B3F%V%m%C%/$N(B
1155: $BJQ?t$KBP1~$9$k@.J,$N$_(B 1 $B$GB>$O(B 0 $B$N(B weight vector $B$K$h$kHf3S$r(B
1156: $B:G=i$K9T$C$F$+$i(B, $B3F%V%m%C%/Kh$N(B tie breaking $B$r9T$&$3$H$KAjEv$9$k(B.
1157: \E
1158:
1159: \BEG
1160: A list of weights for all variables is called a weight vector.
1161: A weight vector is called a sugar weight vector if
1162: its elements are all positive and it is used for computing
1163: a weighted total degree of a monomial, because such a weight
1164: is used instead of total degree in sugar strategy.
1165: On the other hand, a weight vector whose elements are not necessarily
1166: positive cannot be set as a sugar weight, but it is useful for
1167: generalizing term order. In fact, such a weight vector already
1168: appeared in a matrix order. That is, each row of a matrix defining
1169: a term order is regarded as a weight vector. A block order
1170: is also considered as a refinement of comparison by weight vectors.
1171: It compares two terms by using a weight vector whose elements
1172: corresponding to variables in a block is 1 and 0 otherwise,
1173: then it applies a tie breaker.
1.14 noro 1174: \E
1175:
1176: \BJP
1177: weight vector $B$N@_Dj$O(B @code{dp_set_weight()} $B$G9T$&$3$H$,$G$-$k(B
1178: $B$,(B, $B9`=g=x$r;XDj$9$k:]$NB>$N%Q%i%a%?(B ($B9`=g=x7?(B, $BJQ?t=g=x(B) $B$H(B
1179: $B$^$H$a$F@_Dj$G$-$k$3$H$,K>$^$7$$(B. $B$3$N$?$a(B, $B<!$N$h$&$J7A$G$b(B
1180: $B9`=g=x$,;XDj$G$-$k(B.
1181: \E
1182: \BEG
1183: A weight vector can be set by using @code{dp_set_weight()}.
1184: However it is more preferable if a weight vector can be set
1185: together with other parapmeters such as a type of term ordering
1186: and a variable order. This is realized as follows.
1187: \E
1.13 noro 1188:
1.14 noro 1189: @example
1190: [64] B=[x+y+z-6,x*y+y*z+z*x-11,x*y*z-6]$
1191: [65] dp_gr_main(B|v=[x,y,z],sugarweight=[3,2,1],order=0);
1192: [z^3-6*z^2+11*z-6,x+y+z-6,-y^2+(-z+6)*y-z^2+6*z-11]
1193: [66] dp_gr_main(B|v=[y,z,x],order=[[1,1,0],[0,1,0],[0,0,1]]);
1194: [x^3-6*x^2+11*x-6,x+y+z-6,-x^2+(-y+6)*x-y^2+6*y-11]
1195: [67] dp_gr_main(B|v=[y,z,x],order=[[x,1,y,2,z,3]]);
1196: [x+y+z-6,x^3-6*x^2+11*x-6,-x^2+(-y+6)*x-y^2+6*y-11]
1197: @end example
1198:
1199: \BJP
1200: $B$$$:$l$NNc$K$*$$$F$b(B, $B9`=g=x$O(B option $B$H$7$F;XDj$5$l$F$$$k(B.
1201: $B:G=i$NNc$G$O(B @code{v} $B$K$h$jJQ?t=g=x$r(B, @code{sugarweight} $B$K$h$j(B
1202: sugar weight vector $B$r(B, @code{order}$B$K$h$j9`=g=x7?$r;XDj$7$F$$$k(B.
1203: $BFs$DL\$NNc$K$*$1$k(B @code{order} $B$N;XDj$O(B matrix order $B$HF1MM$G$"$k(B.
1204: $B$9$J$o$A(B, $B;XDj$5$l$?(B weight vector $B$r:8$+$i=g$K;H$C$F(B weight $B$NHf3S(B
1205: $B$r9T$&(B. $B;0$DL\$NNc$bF1MM$G$"$k$,(B, $B$3$3$G$O(B weight vector $B$NMWAG$r(B
1206: $BJQ?tKh$K;XDj$7$F$$$k(B. $B;XDj$,$J$$$b$N$O(B 0 $B$H$J$k(B. $B;0$DL\$NNc$G$O(B,
1207: @code{order} $B$K$h$k;XDj$G$O9`=g=x$,7hDj$7$J$$(B. $B$3$N>l9g$K$O(B,
1208: tie breaker $B$H$7$FA4<!?t5U<-=q<0=g=x$,<+F0E*$K@_Dj$5$l$k(B.
1209: $B$3$N;XDjJ}K!$O(B, @code{dp_gr_main}, @code{dp_gr_mod_main} $B$J$I(B
1210: $B$NAH$_9~$_4X?t$G$N$_2DG=$G$"$j(B, @code{gr} $B$J$I$N%f!<%6Dj5A4X?t(B
1211: $B$G$OL$BP1~$G$"$k(B.
1212: \E
1213: \BEG
1214: In each example, a term ordering is specified as options.
1215: In the first example, a variable order, a sugar weight vector
1216: and a type of term ordering are specified by options @code{v},
1217: @code{sugarweight} and @code{order} respectively.
1218: In the second example, an option @code{order} is used
1219: to set a matrix ordering. That is, the specified weight vectors
1220: are used from left to right for comparing terms.
1221: The third example shows a variant of specifying a weight vector,
1222: where each component of a weight vector is specified variable by variable,
1223: and unspecified components are set to zero. In this example,
1224: a term order is not determined only by the specified weight vector.
1225: In such a case a tie breaker by the graded reverse lexicographic ordering
1226: is set automatically.
1227: This type of a term ordering specification can be applied only to builtin
1228: functions such as @code{dp_gr_main()}, @code{dp_gr_mod_main()}, not to
1229: user defined functions such as @code{gr()}.
1.2 noro 1230: \E
1.1 noro 1231:
1.2 noro 1232: \BJP
1.1 noro 1233: @node $BM-M}<0$r78?t$H$9$k%0%l%V%J4pDl7W;;(B,,, $B%0%l%V%J4pDl$N7W;;(B
1234: @section $BM-M}<0$r78?t$H$9$k%0%l%V%J4pDl7W;;(B
1.2 noro 1235: \E
1236: \BEG
1237: @node Groebner basis computation with rational function coefficients,,, Groebner basis computation
1238: @section Groebner basis computation with rational function coefficients
1239: \E
1.1 noro 1240:
1241: @noindent
1.2 noro 1242: \BJP
1.1 noro 1243: @code{gr()} $B$J$I$N%H%C%W%l%Y%kH!?t$O(B, $B$$$:$l$b(B, $BF~NOB?9`<0%j%9%H$K(B
1244: $B8=$l$kJQ?t(B ($BITDj85(B) $B$H(B, $BJQ?t%j%9%H$K8=$l$kJQ?t$rHf3S$7$F(B, $BJQ?t%j%9%H$K(B
1245: $B$J$$JQ?t$,F~NOB?9`<0$K8=$l$F$$$k>l9g$K$O(B, $B<+F0E*$K(B, $B$=$NJQ?t$r(B, $B78?t(B
1246: $BBN$N85$H$7$F07$&(B.
1.2 noro 1247: \E
1248: \BEG
1249: Such variables that appear within the input polynomials but
1250: not appearing in the input variable list are automatically treated
1251: as elements in the coefficient field
1252: by top level functions, such as @code{gr()}.
1253: \E
1.1 noro 1254:
1255: @example
1256: [64] gr([a*x+b*y-c,d*x+e*y-f],[x,y],2);
1257: [(-e*a+d*b)*x-f*b+e*c,(-e*a+d*b)*y+f*a-d*c]
1258: @end example
1259:
1260: @noindent
1.2 noro 1261: \BJP
1.1 noro 1262: $B$3$NNc$G$O(B, @code{a}, @code{b}, @code{c}, @code{d} $B$,78?tBN$N85$H$7$F(B
1263: $B07$o$l$k(B. $B$9$J$o$A(B, $BM-M}H!?tBN(B
1264: @b{F} = @b{Q}(@code{a},@code{b},@code{c},@code{d}) $B>e$N(B 2 $BJQ?tB?9`<04D(B
1265: @b{F}[@code{x},@code{y}] $B$K$*$1$k%0%l%V%J4pDl$r5a$a$k$3$H$K$J$k(B.
1266: $BCm0U$9$Y$-$3$H$O(B,
1267: $B78?t$,BN$H$7$F07$o$l$F$$$k$3$H$G$"$k(B. $B$9$J$o$A(B, $B78?t$N4V$KB?9`<0(B
1268: $B$H$7$F$N6&DL0x;R$,$"$C$?>l9g$K$O(B, $B7k2L$+$i$=$N0x;R$O=|$+$l$F$$$k(B
1269: $B$?$a(B, $BM-M}?tBN>e$NB?9`<04D>e$NLdBj$H$7$F9M$($?>l9g$N7k2L$H$O0lHL(B
1270: $B$K$O0[$J$k(B. $B$^$?(B, $B<g$H$7$F7W;;8zN(>e$NLdBj$N$?$a(B, $BJ,;6I=8=B?9`<0(B
1271: $B$N78?t$H$7$F<B:]$K5v$5$l$k$N$OB?9`<0$^$G$G$"$k(B. $B$9$J$o$A(B, $BJ,Jl$r(B
1272: $B;}$DM-M}<0$OJ,;6I=8=B?9`<0$N78?t$H$7$F$O5v$5$l$J$$(B.
1.2 noro 1273: \E
1274: \BEG
1275: In this example, variables @code{a}, @code{b}, @code{c}, and @code{d}
1276: are treated as elements in the coefficient field.
1277: In this case, a Groebner basis is computed
1278: on a bi-variate polynomial ring
1279: @b{F}[@code{x},@code{y}]
1280: over rational function field
1281: @b{F} = @b{Q}(@code{a},@code{b},@code{c},@code{d}).
1282: Notice that coefficients are considered as a member in a field.
1283: As a consequence, polynomial factors common to the coefficients
1284: are removed so that the result, in general, is different from
1285: the result that would be obtained when the problem is considered
1286: as a computation of Groebner basis over a polynomial ring
1287: with rational function coefficients.
1288: And note that coefficients of a distributed polynomial are limited
1289: to numbers and polynomials because of efficiency.
1290: \E
1.1 noro 1291:
1.2 noro 1292: \BJP
1.1 noro 1293: @node $B4pDlJQ49(B,,, $B%0%l%V%J4pDl$N7W;;(B
1294: @section $B4pDlJQ49(B
1.2 noro 1295: \E
1296: \BEG
1297: @node Change of ordering,,, Groebner basis computation
1298: @section Change of orderng
1299: \E
1.1 noro 1300:
1301: @noindent
1.2 noro 1302: \BJP
1.1 noro 1303: $B<-=q<0=g=x$N%0%l%V%J4pDl$r5a$a$k>l9g(B, $BD>@\(B @code{gr()} $B$J$I$r5/F0$9$k(B
1304: $B$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,
1305: $B$=$l$rF~NO$H$7$F<-=q<0=g=x$N%0%l%V%J4pDl$r7W;;$9$kJ}$,8zN($,$h$$>l9g(B
1306: $B$,$"$k(B. $B$^$?(B, $BF~NO$,2?$i$+$N=g=x$G$N%0%l%V%J4pDl$K$J$C$F$$$k>l9g(B, $B4pDl(B
1307: $BJQ49$H8F$P$l$kJ}K!$K$h$j(B, Buchberger $B%"%k%4%j%:%`$K$h$i$:$K8zN(NI$/(B
1308: $B<-=q<0=g=x$N%0%l%V%J4pDl$,7W;;$G$-$k>l9g$,$"$k(B. $B$3$N$h$&$JL\E*$N$?$a$N(B
1309: $BH!?t$,(B, $B%f!<%6Dj5AH!?t$H$7$F(B @samp{gr} $B$K$$$/$D$+Dj5A$5$l$F$$$k(B.
1310: $B0J2<$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
1311: $B4{$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
1312: @var{vlist2} $B$K$*$1$k<-=q<0=g=x$N%0%l%V%J4pDl$KJQ49$9$kH!?t$G$"$k(B.
1.2 noro 1313: \E
1314: \BEG
1315: When we compute a lex order Groebner basis, it is often efficient to
1316: compute it via Groebner basis with respect to another order such as
1317: degree reverse lex order, rather than to compute it directory by
1318: @code{gr()} etc. If we know that an input is a Groebner basis with
1319: respect to an order, we can apply special methods called change of
1320: ordering for a Groebner basis computation with respect to another
1321: order, without using Buchberger algorithm. The following two functions
1322: are ones for change of ordering such that they convert a Groebner
1323: basis @var{gbase} with respect to the variable order @var{vlist1} and
1324: the order type @var{order} into a lex Groebner basis with respect
1325: to the variable order @var{vlist2}.
1326: \E
1.1 noro 1327:
1328: @table @code
1329: @item tolex(@var{gbase},@var{vlist1},@var{order},@var{vlist2})
1330:
1.2 noro 1331: \BJP
1.1 noro 1332: $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.
1333: $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
1334: $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.
1.2 noro 1335: \E
1336: \BEG
1337: This function can be used only when @var{gbase} is an ideal over the
1338: rationals. The input @var{gbase} must be a Groebner basis with respect
1339: to the variable order @var{vlist1} and the order type @var{order}. Moreover
1340: the ideal generated by @var{gbase} must be zero-dimensional.
1341: This computes the lex Groebner basis of @var{gbase}
1342: by using the modular change of ordering algorithm. The algorithm first
1343: computes the lex Groebner basis over a finite field. Then each element
1344: in the lex Groebner basis over the rationals is computed with undetermined
1345: coefficient method and linear equation solving by Hensel lifting.
1346: \E
1.1 noro 1347:
1348: @item tolex_tl(@var{gbase},@var{vlist1},@var{order},@var{vlist2},@var{homo})
1349:
1.2 noro 1350: \BJP
1.1 noro 1351: $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
1352: $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
1353: trace-lifting$B$K$*$1$k%0%l%V%J4pDl8uJd$NF,9`(B, $BF,78?t$N@-<A$rMxMQ$7$F(B,
1354: $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
1355: $B$k$?$a(B, $BC1$K(BBuchberger $B%"%k%4%j%:%`$r7+$jJV$9$h$j8zN($h$/7W;;$G$-$k(B.
1356: $B99$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
1357: $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,
1358: $BD>@\<-=q<0=g=x$N7W;;$r9T$&$h$j8zN($,$h$$(B. ($B$b$A$m$sNc30$"$j(B. )
1359: $B0z?t(B @var{homo} $B$,(B 0 $B$G$J$$;~(B, @code{hgr()} $B$HF1MM$K@F<!2=$r7PM3$7$F(B
1360: $B7W;;$r9T$&(B.
1.2 noro 1361: \E
1362: \BEG
1363: This function computes the lex Groebner basis of @var{gbase}. The
1364: input @var{gbase} must be a Groebner basis with respect to the
1365: variable order @var{vlist1} and the order type @var{order}.
1366: Buchberger algorithm with trace lifting is used to compute the lex
1367: Groebner basis, however the Groebner basis check and the ideal
1368: membership check can be omitted by using several properties derived
1369: from the fact that the input is a Groebner basis. So it is more
1370: efficient than simple repetition of Buchberger algorithm. If the input
1371: is zero-dimensional, this function inserts automatically a computation
1372: of Groebner basis with respect to an elimination order, which makes
1373: the whole computation more efficient for many cases. If @var{homo} is
1374: not equal to 0, homogenization is used in each step.
1375: \E
1.1 noro 1376: @end table
1377:
1378: @noindent
1.2 noro 1379: \BJP
1.1 noro 1380: $B$=$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
1381: $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
1382: @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.
1.2 noro 1383: \E
1384: \BEG
1385: For zero-dimensional systems, there are several fuctions to
1386: compute the minimal polynomial of a polynomial and or a more compact
1387: representation for zeros of the system. They are all defined in @samp{gr}.
1388: Refer to the sections for each functions.
1389: \E
1.1 noro 1390:
1.2 noro 1391: \BJP
1.6 noro 1392: @node Weyl $BBe?t(B,,, $B%0%l%V%J4pDl$N7W;;(B
1393: @section Weyl $BBe?t(B
1394: \E
1395: \BEG
1396: @node Weyl algebra,,, Groebner basis computation
1397: @section Weyl algebra
1398: \E
1399:
1400: @noindent
1401:
1402: \BJP
1403: $B$3$l$^$G$O(B, $BDL>o$N2D49$JB?9`<04D$K$*$1$k%0%l%V%J4pDl7W;;$K$D$$$F(B
1404: $B=R$Y$F$-$?$,(B, $B%0%l%V%J4pDl$NM}O@$O(B, $B$"$k>r7o$rK~$?$9Hs2D49$J(B
1405: $B4D$K$b3HD%$G$-$k(B. $B$3$N$h$&$J4D$NCf$G(B, $B1~MQ>e$b=EMW$J(B,
1406: Weyl $BBe?t(B, $B$9$J$o$AB?9`<04D>e$NHyJ,:nMQAG4D$N1i;;$*$h$S(B
1407: $B%0%l%V%J4pDl7W;;$,(B Risa/Asir $B$K<BAu$5$l$F$$$k(B.
1408:
1409: $BBN(B @code{K} $B>e$N(B @code{n} $B<!85(B Weyl $BBe?t(B
1410: @code{D=K<x1,@dots{},xn,D1,@dots{},Dn>} $B$O(B
1411: \E
1412:
1413: \BEG
1414: So far we have explained Groebner basis computation in
1415: commutative polynomial rings. However Groebner basis can be
1416: considered in more general non-commutative rings.
1417: Weyl algebra is one of such rings and
1418: Risa/Asir implements fundamental operations
1419: in Weyl algebra and Groebner basis computation in Weyl algebra.
1420:
1421: The @code{n} dimensional Weyl algebra over a field @code{K},
1422: @code{D=K<x1,@dots{},xn,D1,@dots{},Dn>} is a non-commutative
1423: algebra which has the following fundamental relations:
1424: \E
1425:
1426: @code{xi*xj-xj*xi=0}, @code{Di*Dj-Dj*Di=0}, @code{Di*xj-xj*Di=0} (@code{i!=j}),
1427: @code{Di*xi-xi*Di=1}
1428:
1429: \BJP
1430: $B$H$$$&4pK\4X78$r;}$D4D$G$"$k(B. @code{D} $B$O(B $BB?9`<04D(B @code{K[x1,@dots{},xn]} $B$r78?t(B
1431: $B$H$9$kHyJ,:nMQAG4D$G(B, @code{Di} $B$O(B @code{xi} $B$K$h$kHyJ,$rI=$9(B. $B8r494X78$K$h$j(B,
1432: @code{D} $B$N85$O(B, @code{x1^i1*@dots{}*xn^in*D1^j1*@dots{}*Dn^jn} $B$J$kC19`(B
1433: $B<0$N(B @code{K} $B@~7A7k9g$H$7$F=q$-I=$9$3$H$,$G$-$k(B.
1434: Risa/Asir $B$K$*$$$F$O(B, $B$3$NC19`<0$r(B, $B2D49$JB?9`<0$HF1MM$K(B
1435: @code{<<i1,@dots{},in,j1,@dots{},jn>>} $B$GI=$9(B. $B$9$J$o$A(B, @code{D} $B$N85$b(B
1436: $BJ,;6I=8=B?9`<0$H$7$FI=$5$l$k(B. $B2C8:;;$O(B, $B2D49$N>l9g$HF1MM$K(B, @code{+}, @code{-}
1437: $B$K$h$j(B
1438: $B<B9T$G$-$k$,(B, $B>h;;$O(B, $BHs2D49@-$r9MN8$7$F(B @code{dp_weyl_mul()} $B$H$$$&4X?t(B
1439: $B$K$h$j<B9T$9$k(B.
1440: \E
1441:
1442: \BEG
1443: @code{D} is the ring of differential operators whose coefficients
1444: are polynomials in @code{K[x1,@dots{},xn]} and
1445: @code{Di} denotes the differentiation with respect to @code{xi}.
1446: According to the commutation relation,
1447: elements of @code{D} can be represented as a @code{K}-linear combination
1448: of monomials @code{x1^i1*@dots{}*xn^in*D1^j1*@dots{}*Dn^jn}.
1449: In Risa/Asir, this type of monomial is represented
1450: by @code{<<i1,@dots{},in,j1,@dots{},jn>>} as in the case of commutative
1451: polynomial.
1452: That is, elements of @code{D} are represented by distributed polynomials.
1453: Addition and subtraction can be done by @code{+}, @code{-},
1454: but multiplication is done by calling @code{dp_weyl_mul()} because of
1455: the non-commutativity of @code{D}.
1456: \E
1457:
1458: @example
1459: [0] A=<<1,2,2,1>>;
1460: (1)*<<1,2,2,1>>
1461: [1] B=<<2,1,1,2>>;
1462: (1)*<<2,1,1,2>>
1463: [2] A*B;
1464: (1)*<<3,3,3,3>>
1465: [3] dp_weyl_mul(A,B);
1466: (1)*<<3,3,3,3>>+(1)*<<3,2,3,2>>+(4)*<<2,3,2,3>>+(4)*<<2,2,2,2>>
1467: +(2)*<<1,3,1,3>>+(2)*<<1,2,1,2>>
1468: @end example
1469:
1470: \BJP
1471: $B%0%l%V%J4pDl7W;;$K$D$$$F$b(B, Weyl $BBe?t@lMQ$N4X?t$H$7$F(B,
1472: $B<!$N4X?t$,MQ0U$7$F$"$k(B.
1473: \E
1474: \BEG
1475: The following functions are avilable for Groebner basis computation
1476: in Weyl algebra:
1477: \E
1478: @code{dp_weyl_gr_main()},
1479: @code{dp_weyl_gr_mod_main()},
1480: @code{dp_weyl_gr_f_main()},
1481: @code{dp_weyl_f4_main()},
1482: @code{dp_weyl_f4_mod_main()}.
1483: \BJP
1484: $B$^$?(B, $B1~MQ$H$7$F(B, global b $B4X?t$N7W;;$,<BAu$5$l$F$$$k(B.
1485: \E
1486: \BEG
1487: Computation of the global b function is implemented as an application.
1488: \E
1489:
1490: \BJP
1.23 ! noro 1491: @node $BB?9`<04D>e$N2C72(B,,, $B%0%l%V%J4pDl$N7W;;(B
! 1492: @section $BB?9`<04D>e$N2C72(B
! 1493: \E
! 1494: \BEG
! 1495: @node Module over a polynomial ring,,, Groebner basis computation
! 1496: @section Module over a polynomial ring
! 1497: \E
! 1498:
! 1499: @noindent
! 1500:
! 1501: \BJP
! 1502: $BB?9`<04D>e$N<+M32C72$N85$O(B, $B2C72C19`<0(B te_i $B$N@~7?OB$H$7$FFbItI=8=$5$l$k(B.
! 1503: $B$3$3$G(B t $B$OB?9`<04D$NC19`<0(B, e_i $B$O<+M32C72$NI8=`4pDl$G$"$k(B. $B2C72C19`<0$O(B, $BB?9`<04D$NC19`<0(B
! 1504: $B$K0LCV(B i $B$rDI2C$7$?(B @code{<<a,b,...,c:i>>} $B$GI=$9(B. $B2C72B?9`<0(B, $B$9$J$o$A2C72C19`<0$N@~7?OB$O(B,
! 1505: $B@_Dj$5$l$F$$$k2C729`=g=x$K$7$?$,$C$F9_=g$K@0Ns$5$l$k(B. $B2C729`=g=x$K$O0J2<$N(B3$B<oN`$,$"$k(B.
! 1506:
! 1507: @table @code
! 1508: @item TOP $B=g=x(B
! 1509:
! 1510: $B$3$l$O(B, te_i > se_j $B$H$J$k$N$O(B t>s $B$^$?$O(B (t=s $B$+$D(B i<j) $B$H$J$k$h$&$J9`=g=x$G$"$k(B. $B$3$3$G(B,
! 1511: t, s $B$NHf3S$OB?9`<04D$K@_Dj$5$l$F$$$k=g=x$G9T$&(B.
! 1512: $B$3$N7?$N=g=x$O(B, @code{dp_ord([0,Ord])} $B$K(B
! 1513: $B$h$j@_Dj$9$k(B. $B$3$3$G(B, @code{Ord} $B$OB?9`<04D$N=g=x7?$G$"$k(B.
! 1514:
! 1515: @item POT $B=g=x(B
! 1516:
! 1517: $B$3$l$O(B, te_i > se_j $B$H$J$k$N$O(B i<j $B$^$?$O(B (i=j $B$+$D(B t>s) $B$H$J$k$h$&$J9`=g=x$G$"$k(B. $B$3$3$G(B,
! 1518: t, s $B$NHf3S$OB?9`<04D$K@_Dj$5$l$F$$$k=g=x$G9T$&(B.
! 1519: $B$3$N7?$N=g=x$O(B, @code{dp_ord([1,Ord])} $B$K(B
! 1520: $B$h$j@_Dj$9$k(B. $B$3$3$G(B, @code{Ord} $B$OB?9`<04D$N=g=x7?$G$"$k(B.
! 1521:
! 1522: @item Schreyer $B7?=g=x(B
! 1523:
! 1524: $B3FI8=`4pDl(B e_i $B$KBP$7(B, $BJL$N<+M32C72$N2C72C19`<0(B T_i $B$,M?$($i$l$F$$$F(B, te_i > se_j $B$H$J$k$N$O(B
! 1525: tT_i > sT_j $B$^$?$O(B (tT_i=sT_j $B$+$D(B i<j) $B$H$J$k$h$&$J9`=g=x$G$"$k(B. $B$3$3$G(B tT_i, sT_j $B$N(B
! 1526: $BHf3S$O(B, $B$3$l$i$,=jB0$9$k<+M32C72$K@_Dj$5$l$F$$$k=g=x$G9T$&(B.
! 1527: $B$3$N7?$N=g=x$O(B, $BDL>o:F5"E*$K@_Dj$5$l$k(B. $B$9$J$o$A(B, T_i $B$,=jB0$9$k<+M32C72$N=g=x$b(B Schreyer $B7?(B
! 1528: $B$G$"$k$+(B, $B$^$?$O%\%H%`$H$J$k(B TOP, POT $B$J$I$N9`=g=x$H$J$k(B.
! 1529: $B$3$N7?$N=g=x$O(B @code{dpm_set_schreyer([H_1,H_2,...])} $B$K$h$j;XDj$9$k(B. $B$3$3$G(B,
! 1530: @code{H_i=[T_1,T_2,...]} $B$O2C72C19`<0$N%j%9%H$G(B, @code{[H_2,...]} $B$GDj5A$5$l$k(B Schreyer $B7?9`=g=x$r(B
! 1531: @code{tT_i} $B$i$KE,MQ$9$k$H$$$&0UL#$G$"$k(B.
! 1532: @end table
! 1533:
! 1534: $B2C72B?9`<0$rF~NO$9$kJ}K!$H$7$F$O(B, @code{<<a,b,...:i>>} $B$J$k7A<0$GD>@\F~NO$9$kB>$K(B,
! 1535: $BB?9`<0%j%9%H$r:n$j(B, @code{dpm_ltod()} $B$K$h$jJQ49$9$kJ}K!$b$"$k(B.
! 1536: \E
! 1537: \BEG
! 1538: not yet
! 1539: \E
! 1540:
! 1541: \BJP
1.1 noro 1542: @node $B%0%l%V%J4pDl$K4X$9$kH!?t(B,,, $B%0%l%V%J4pDl$N7W;;(B
1543: @section $B%0%l%V%J4pDl$K4X$9$kH!?t(B
1.2 noro 1544: \E
1545: \BEG
1546: @node Functions for Groebner basis computation,,, Groebner basis computation
1547: @section Functions for Groebner basis computation
1548: \E
1.1 noro 1549:
1550: @menu
1551: * gr hgr gr_mod::
1552: * lex_hensel lex_tl tolex tolex_d tolex_tl::
1553: * lex_hensel_gsl tolex_gsl tolex_gsl_d::
1554: * gr_minipoly minipoly::
1555: * tolexm minipolym::
1.6 noro 1556: * dp_gr_main dp_gr_mod_main dp_gr_f_main dp_weyl_gr_main dp_weyl_gr_mod_main dp_weyl_gr_f_main::
1557: * dp_f4_main dp_f4_mod_main dp_weyl_f4_main dp_weyl_f4_mod_main::
1.17 noro 1558: * nd_gr nd_gr_trace nd_f4 nd_f4_trace nd_weyl_gr nd_weyl_gr_trace::
1.22 noro 1559: * nd_gr_postproc nd_weyl_gr_postproc::
1.1 noro 1560: * dp_gr_flags dp_gr_print::
1561: * dp_ord::
1.18 noro 1562: * dp_set_weight dp_set_top_weight dp_weyl_set_weight::
1.1 noro 1563: * dp_ptod::
1564: * dp_dtop::
1565: * dp_mod dp_rat::
1566: * dp_homo dp_dehomo::
1567: * dp_ptozp dp_prim::
1.18 noro 1568: * dp_nf dp_nf_mod dp_true_nf dp_true_nf_mod dp_weyl_nf dp_weyl_nf_mod::
1.1 noro 1569: * dp_hm dp_ht dp_hc dp_rest::
1.23 ! noro 1570: * dpm_hm dpm_ht dpm_hc dpm_hp dpm_rest::
1.1 noro 1571: * dp_td dp_sugar::
1572: * dp_lcm::
1573: * dp_redble::
1574: * dp_subd::
1575: * dp_mbase::
1576: * dp_mag::
1577: * dp_red dp_red_mod::
1578: * dp_sp dp_sp_mod::
1579: * p_nf p_nf_mod p_true_nf p_true_nf_mod ::
1580: * p_terms::
1581: * gb_comp::
1582: * katsura hkatsura cyclic hcyclic::
1583: * dp_vtoe dp_etov::
1584: * lex_hensel_gsl tolex_gsl tolex_gsl_d::
1.3 noro 1585: * primadec primedec::
1.5 noro 1586: * primedec_mod::
1.10 noro 1587: * bfunction bfct generic_bfct ann ann0::
1.1 noro 1588: @end menu
1589:
1.2 noro 1590: \JP @node gr hgr gr_mod,,, $B%0%l%V%J4pDl$K4X$9$kH!?t(B
1591: \EG @node gr hgr gr_mod,,, Functions for Groebner basis computation
1.1 noro 1592: @subsection @code{gr}, @code{hgr}, @code{gr_mod}, @code{dgr}
1593: @findex gr
1594: @findex hgr
1595: @findex gr_mod
1596: @findex dgr
1597:
1598: @table @t
1599: @item gr(@var{plist},@var{vlist},@var{order})
1600: @itemx hgr(@var{plist},@var{vlist},@var{order})
1601: @itemx gr_mod(@var{plist},@var{vlist},@var{order},@var{p})
1602: @itemx dgr(@var{plist},@var{vlist},@var{order},@var{procs})
1.2 noro 1603: \JP :: $B%0%l%V%J4pDl$N7W;;(B
1604: \EG :: Groebner basis computation
1.1 noro 1605: @end table
1606:
1607: @table @var
1608: @item return
1.2 noro 1609: \JP $B%j%9%H(B
1610: \EG list
1.4 noro 1611: @item plist vlist procs
1.2 noro 1612: \JP $B%j%9%H(B
1613: \EG list
1.1 noro 1614: @item order
1.2 noro 1615: \JP $B?t(B, $B%j%9%H$^$?$O9TNs(B
1616: \EG number, list or matrix
1.1 noro 1617: @item p
1.2 noro 1618: \JP 2^27 $BL$K~$NAG?t(B
1619: \EG prime less than 2^27
1.1 noro 1620: @end table
1621:
1622: @itemize @bullet
1.2 noro 1623: \BJP
1.1 noro 1624: @item
1625: $BI8=`%i%$%V%i%j$N(B @samp{gr} $B$GDj5A$5$l$F$$$k(B.
1626: @item
1.21 takayama 1627: gr $B$rL>A0$K4^$`4X?t$O8=:_%a%s%F$5$l$F$$$J$$(B. @code{nd_gr}$B7O$N4X?t$rBe$o$j$KMxMQ$9$Y$-$G$"$k(B(@fref{nd_gr nd_gr_trace nd_f4 nd_f4_trace nd_weyl_gr nd_weyl_gr_trace}).
1628: @item
1.1 noro 1629: $B$$$:$l$b(B, $BB?9`<0%j%9%H(B @var{plist} $B$N(B, $BJQ?t=g=x(B @var{vlist}, $B9`=g=x7?(B
1630: @var{order} $B$K4X$9$k%0%l%V%J4pDl$r5a$a$k(B. @code{gr()}, @code{hgr()}
1631: $B$O(B $BM-M}?t78?t(B, @code{gr_mod()} $B$O(B GF(@var{p}) $B78?t$H$7$F7W;;$9$k(B.
1632: @item
1633: @var{vlist} $B$OITDj85$N%j%9%H(B. @var{vlist} $B$K8=$l$J$$ITDj85$O(B,
1634: $B78?tBN$KB0$9$k$H8+$J$5$l$k(B.
1635: @item
1636: @code{gr()}, trace-lifting ($B%b%8%e%i1i;;$rMQ$$$?9bB.2=(B) $B$*$h$S(B sugar
1637: strategy $B$K$h$k7W;;(B, @code{hgr()} $B$O(B trace-lifting $B$*$h$S(B
1638: $B@F<!2=$K$h$k(B $B6:@5$5$l$?(B sugar strategy $B$K$h$k7W;;$r9T$&(B.
1639: @item
1.16 fujiwara 1640: @code{dgr()} $B$O(B, @code{gr()}, @code{hgr()} $B$r(B
1.1 noro 1641: $B;R%W%m%;%9%j%9%H(B @var{procs} $B$N(B 2 $B$D$N%W%m%;%9$K$h$jF1;~$K7W;;$5$;(B,
1642: $B@h$K7k2L$rJV$7$?J}$N7k2L$rJV$9(B. $B7k2L$OF10l$G$"$k$,(B, $B$I$A$i$NJ}K!$,(B
1643: $B9bB.$+0lHL$K$OITL@$N$?$a(B, $B<B:]$N7P2a;~4V$rC;=L$9$k$N$KM-8z$G$"$k(B.
1644: @item
1645: @code{dgr()} $B$GI=<($5$l$k;~4V$O(B, $B$3$NH!?t$,<B9T$5$l$F$$$k%W%m%;%9$G$N(B
1646: CPU $B;~4V$G$"$j(B, $B$3$NH!?t$N>l9g$O$[$H$s$IDL?.$N$?$a$N;~4V$G$"$k(B.
1.12 takayama 1647: @item
1648: $BB?9`<0%j%9%H(B @var{plist} $B$NMWAG$,J,;6I=8=B?9`<0$N>l9g$O(B
1649: $B7k2L$bJ,;6I=8=B?9`<0$N%j%9%H$G$"$k(B.
1650: $B$3$N>l9g(B, $B0z?t$NJ,;6B?9`<0$OM?$($i$l$?=g=x$K=>$$(B @code{dp_sort} $B$G(B
1651: $B%=!<%H$5$l$F$+$i7W;;$5$l$k(B.
1652: $BB?9`<0%j%9%H$NMWAG$,J,;6I=8=B?9`<0$N>l9g$b(B
1653: $BJQ?t$N?tJ,$NITDj85$N%j%9%H$r(B @var{vlist} $B0z?t$H$7$FM?$($J$$$H$$$1$J$$(B
1654: ($B%@%_!<(B).
1.2 noro 1655: \E
1656: \BEG
1657: @item
1658: These functions are defined in @samp{gr} in the standard library
1659: directory.
1.21 takayama 1660: @item
1661: Functions of which names contains gr are obsolted.
1662: Functions of @code{nd_gr} families should be used (@fref{nd_gr nd_gr_trace nd_f4 nd_f4_trace nd_weyl_gr nd_weyl_gr_trace}).
1.2 noro 1663: @item
1664: They compute a Groebner basis of a polynomial list @var{plist} with
1665: respect to the variable order @var{vlist} and the order type @var{order}.
1666: @code{gr()} and @code{hgr()} compute a Groebner basis over the rationals
1667: and @code{gr_mod} computes over GF(@var{p}).
1668: @item
1669: Variables not included in @var{vlist} are regarded as
1670: included in the ground field.
1671: @item
1672: @code{gr()} uses trace-lifting (an improvement by modular computation)
1673: and sugar strategy.
1674: @code{hgr()} uses trace-lifting and a cured sugar strategy
1675: by using homogenization.
1676: @item
1677: @code{dgr()} executes @code{gr()}, @code{dgr()} simultaneously on
1678: two process in a child process list @var{procs} and returns
1679: the result obtained first. The results returned from both the process
1680: should be equal, but it is not known in advance which method is faster.
1681: Therefore this function is useful to reduce the actual elapsed time.
1682: @item
1683: The CPU time shown after an exection of @code{dgr()} indicates
1684: that of the master process, and most of the time corresponds to the time
1685: for communication.
1.12 takayama 1686: @item
1687: When the elements of @var{plist} are distributed polynomials,
1688: the result is also a list of distributed polynomials.
1689: In this case, firstly the elements of @var{plist} is sorted by @code{dp_sort}
1690: and the Grobner basis computation is started.
1691: Variables must be given in @var{vlist} even in this case
1692: (these variables are dummy).
1.2 noro 1693: \E
1.1 noro 1694: @end itemize
1695:
1696: @example
1697: [0] load("gr")$
1698: [64] load("cyclic")$
1699: [74] G=gr(cyclic(5),[c0,c1,c2,c3,c4],2);
1700: [c4^15+122*c4^10-122*c4^5-1,...]
1701: [75] GM=gr_mod(cyclic(5),[c0,c1,c2,c3,c4],2,31991)$
1702: 24628*c4^15+29453*c4^10+2538*c4^5+7363
1703: [76] (G[0]*24628-GM[0])%31991;
1704: 0
1705: @end example
1706:
1707: @table @t
1.2 noro 1708: \JP @item $B;2>H(B
1709: \EG @item References
1.6 noro 1710: @fref{dp_gr_main dp_gr_mod_main dp_gr_f_main dp_weyl_gr_main dp_weyl_gr_mod_main dp_weyl_gr_f_main},
1.1 noro 1711: @fref{dp_ord}.
1712: @end table
1713:
1.2 noro 1714: \JP @node lex_hensel lex_tl tolex tolex_d tolex_tl,,, $B%0%l%V%J4pDl$K4X$9$kH!?t(B
1715: \EG @node lex_hensel lex_tl tolex tolex_d tolex_tl,,, Functions for Groebner basis computation
1.1 noro 1716: @subsection @code{lex_hensel}, @code{lex_tl}, @code{tolex}, @code{tolex_d}, @code{tolex_tl}
1717: @findex lex_hensel
1718: @findex lex_tl
1719: @findex tolex
1720: @findex tolex_d
1721: @findex tolex_tl
1722:
1723: @table @t
1724: @item lex_hensel(@var{plist},@var{vlist1},@var{order},@var{vlist2},@var{homo})
1725: @itemx lex_tl(@var{plist},@var{vlist1},@var{order},@var{vlist2},@var{homo})
1.2 noro 1726: \JP :: $B4pDlJQ49$K$h$k<-=q<0=g=x%0%l%V%J4pDl$N7W;;(B
1727: \EG:: Groebner basis computation with respect to a lex order by change of ordering
1.1 noro 1728: @item tolex(@var{plist},@var{vlist1},@var{order},@var{vlist2})
1729: @itemx tolex_d(@var{plist},@var{vlist1},@var{order},@var{vlist2},@var{procs})
1730: @itemx tolex_tl(@var{plist},@var{vlist1},@var{order},@var{vlist2},@var{homo})
1.2 noro 1731: \JP :: $B%0%l%V%J4pDl$rF~NO$H$9$k(B, $B4pDlJQ49$K$h$k<-=q<0=g=x%0%l%V%J4pDl$N7W;;(B
1732: \EG :: Groebner basis computation with respect to a lex order by change of ordering, starting from a Groebner basis
1.1 noro 1733: @end table
1734:
1735: @table @var
1736: @item return
1.2 noro 1737: \JP $B%j%9%H(B
1738: \EG list
1.4 noro 1739: @item plist vlist1 vlist2 procs
1.2 noro 1740: \JP $B%j%9%H(B
1741: \EG list
1.1 noro 1742: @item order
1.2 noro 1743: \JP $B?t(B, $B%j%9%H$^$?$O9TNs(B
1744: \EG number, list or matrix
1.1 noro 1745: @item homo
1.2 noro 1746: \JP $B%U%i%0(B
1747: \EG flag
1.1 noro 1748: @end table
1749:
1750: @itemize @bullet
1.2 noro 1751: \BJP
1.1 noro 1752: @item
1753: $BI8=`%i%$%V%i%j$N(B @samp{gr} $B$GDj5A$5$l$F$$$k(B.
1754: @item
1755: @code{lex_hensel()}, @code{lex_tl()} $B$O(B,
1756: $BB?9`<0%j%9%H(B @var{plist} $B$N(B, $BJQ?t=g=x(B @var{vlist1}, $B9`=g=x7?(B
1757: @var{order} $B$K4X$9$k%0%l%V%J4pDl$r5a$a(B, $B$=$l$r(B, $BJQ?t=g=x(B @var{vlist2}
1758: $B$N<-=q<0=g=x%0%l%V%J4pDl$KJQ49$9$k(B.
1759: @item
1760: @code{tolex()}, @code{tolex_tl()} $B$O(B,
1761: $BJQ?t=g=x(B @var{vlist1}, $B9`=g=x7?(B @var{order} $B$K4X$9$k%0%l%V%J4pDl$G$"$k(B
1762: $BB?9`<0%j%9%H(B @var{plist} $B$rJQ?t=g=x(B @var{vlist2} $B$N<-=q<0=g=x%0%l%V%J(B
1763: $B4pDl$KJQ49$9$k(B.
1764: @code{tolex_d()} $B$O(B, @code{tolex()} $B$K$*$1$k(B, $B3F4pDl$N7W;;$r(B, $B;R%W%m%;%9(B
1765: $B%j%9%H(B @var{procs} $B$N3F%W%m%;%9$KJ,;67W;;$5$;$k(B.
1766: @item
1767: @code{lex_hensel()}, @code{lex_tl()} $B$K$*$$$F$O(B, $B<-=q<0=g=x%0%l%V%J4pDl$N(B
1768: $B7W;;$O<!$N$h$&$K9T$o$l$k(B. (@code{[Noro,Yokoyama]} $B;2>H(B.)
1769: @enumerate
1770: @item
1771: @var{vlist1}, @var{order} $B$K4X$9$k%0%l%V%J4pDl(B @var{G0} $B$r7W;;$9$k(B.
1772: (@code{lex_hensel()} $B$N$_(B. )
1773: @item
1774: @var{G0} $B$N3F85$N(B @var{vlist2} $B$K4X$9$k<-=q<0=g=x$K$*$1$kF,78?t$r3d$i$J$$(B
1775: $B$h$&$JAG?t(B @var{p} $B$rA*$S(B, GF(@var{p}) $B>e$G$N<-=q<0=g=x%0%l%V%J4pDl(B
1776: @var{Gp} $B$r7W;;$9$k(B.
1777: @item
1778: @var{Gp} $B$K8=$l$k$9$Y$F$N9`$N(B, @var{G0} $B$K4X$9$k@55,7A(B @var{NF} $B$r7W;;$9$k(B.
1779: @item
1780: @var{Gp} $B$N3F85(B @var{f} $B$K$D$-(B, @var{f} $B$N78?t$rL$Dj78?t$G(B,
1781: @var{f} $B$N3F9`$rBP1~$9$k(B @var{NF} $B$N85$GCV$-49$((B, $B3F9`$N78?t$r(B 0 $B$HCV$$$?(B,
1782: $BL$Dj78?t$K4X$9$k@~7AJ}Dx<07O(B @var{Lf} $B$r:n$k(B.
1783: @item
1784: @var{Lf} $B$,(B, $BK!(B @var{p} $B$G0l0U2r$r;}$D$3$H$rMQ$$$F(B @var{Lf} $B$N2r$r(B
1785: $BK!(B @var{p}$B$N2r$+$i(B Hensel $B9=@.$K$h$j5a$a$k(B.
1786: @item
1787: $B$9$Y$F$N(B @var{Gp} $B$N85$K$D$-@~7AJ}Dx<0$,2r$1$?$i$=$N2rA4BN$,5a$a$k(B
1788: $B<-=q<0=g=x$G$N%0%l%V%J4pDl(B. $B$b$7$I$l$+$N@~7AJ}Dx<0$N5a2r$K<:GT$7$?$i(B,
1789: @var{p} $B$r$H$jD>$7$F$d$jD>$9(B.
1790: @end enumerate
1791:
1792: @item
1793: @code{lex_tl()}, @code{tolex_tl()} $B$K$*$$$F$O(B, $B<-=q<0=g=x%0%l%V%J4pDl$N(B
1794: $B7W;;$O<!$N$h$&$K9T$o$l$k(B.
1795:
1796: @enumerate
1797: @item
1798: @var{vlist1}, @var{order} $B$K4X$9$k%0%l%V%J4pDl(B @var{G0} $B$r7W;;$9$k(B.
1799: (@code{lex_hensel()} $B$N$_(B. )
1800: @item
1801: @var{G0} $B$,(B 0 $B<!85%7%9%F%`$G$J$$$H$-(B, @var{G0} $B$rF~NO$H$7$F(B,
1802: @var{G0} $B$N3F85$N(B @var{vlist2} $B$K4X$9$k<-=q<0=g=x$K$*$1$kF,78?t$r3d$i$J$$(B
1803: $B$h$&$JAG?t(B @var{p} $B$rA*$S(B, @var{p} $B$rMQ$$$?(B trace-lifting $B$K$h$j<-=q<0(B
1804: $B=g=x$N%0%l%V%J4pDl8uJd$r5a$a(B, $B$b$75a$^$C$?$J$i%A%'%C%/$J$7$K$=$l$,5a$a$k(B
1805: $B%0%l%V%J4pDl$H$J$k(B. $B$b$7<:GT$7$?$i(B, @var{p} $B$r$H$jD>$7$F$d$jD>$9(B.
1806: @item
1807: @var{G0} $B$,(B 0 $B<!85%7%9%F%`$N$H$-(B, @var{G0} $B$rF~NO$H$7$F(B,
1808: $B$^$:(B, @var{vlist2} $B$N:G8e$NJQ?t0J30$r>C5n$9$k>C5n=g=x$K$h$j(B
1809: $B%0%l%V%J4pDl(B @var{G1} $B$r7W;;$7(B, $B$=$l$+$i<-=q<0=g=x$N%0%l%V%J4pDl$r(B
1810: $B7W;;$9$k(B. $B$=$N:](B, $B3F%9%F%C%W$G$O(B, $BF~NO$N3F85$N(B, $B5a$a$k=g=x$K$*$1$k(B
1811: $BF,78?t$r3d$i$J$$AG?t$rMQ$$$?(B trace-lifting $B$G%0%l%V%J4pDl8uJd$r5a$a(B,
1812: $B$b$75a$^$C$?$i%A%'%C%/$J$7$K$=$l$,$=$N=g=x$G$N%0%l%V%J4pDl$H$J$k(B.
1813: @end enumerate
1814:
1815: @item
1816: $BM-M}<078?t$N7W;;$O(B, @code{lex_tl()}, @code{tolex_tl()} $B$N$_<u$1IU$1$k(B.
1817: @item
1818: @code{homo} $B$,(B 0 $B$G$J$$>l9g(B, $BFbIt$G5/F0$5$l$k(B Buchberger $B%"%k%4%j%:%`$K(B
1819: $B$*$$$F(B, $B@F<!2=$,9T$o$l$k(B.
1820: @item
1821: @code{tolex_d()} $B$GI=<($5$l$k;~4V$O(B, $B$3$NH!?t$,<B9T$5$l$F$$$k%W%m%;%9$K(B
1822: $B$*$$$F9T$o$l$?7W;;$KBP1~$7$F$$$F(B, $B;R%W%m%;%9$K$*$1$k;~4V$O4^$^$l$J$$(B.
1.2 noro 1823: \E
1824: \BEG
1825: @item
1826: These functions are defined in @samp{gr} in the standard library
1827: directory.
1828: @item
1829: @code{lex_hensel()} and @code{lex_tl()} first compute a Groebner basis
1830: with respect to the variable order @var{vlist1} and the order type @var{order}.
1831: Then the Groebner basis is converted into a lex order Groebner basis
1832: with respect to the varable order @var{vlist2}.
1833: @item
1834: @code{tolex()} and @code{tolex_tl()} convert a Groebner basis @var{plist}
1835: with respect to the variable order @var{vlist1} and the order type @var{order}
1836: into a lex order Groebner basis
1837: with respect to the varable order @var{vlist2}.
1838: @code{tolex_d()} does computations of basis elements in @code{tolex()}
1839: in parallel on the processes in a child process list @var{procs}.
1840: @item
1841: In @code{lex_hensel()} and @code{tolex_hensel()} a lex order Groebner basis
1842: is computed as follows.(Refer to @code{[Noro,Yokoyama]}.)
1843: @enumerate
1844: @item
1845: Compute a Groebner basis @var{G0} with respect to @var{vlist1} and @var{order}.
1846: (Only in @code{lex_hensel()}. )
1847: @item
1848: Choose a prime which does not divide head coefficients of elements in @var{G0}
1849: with respect to @var{vlist1} and @var{order}. Then compute a lex order
1850: Groebner basis @var{Gp} over GF(@var{p}) with respect to @var{vlist2}.
1851: @item
1852: Compute @var{NF}, the set of all the normal forms with respect to
1853: @var{G0} of terms appearing in @var{Gp}.
1854: @item
1855: For each element @var{f} in @var{Gp}, replace coefficients and terms in @var{f}
1856: with undetermined coefficients and the corresponding polynomials in @var{NF}
1857: respectively, and generate a system of liear equation @var{Lf} by equating
1858: the coefficients of terms in the replaced polynomial with 0.
1859: @item
1860: Solve @var{Lf} by Hensel lifting, starting from the unique mod @var{p}
1861: solution.
1862: @item
1863: If all the linear equations generated from the elements in @var{Gp}
1864: could be solved, then the set of solutions corresponds to a lex order
1865: Groebner basis. Otherwise redo the whole process with another @var{p}.
1866: @end enumerate
1867:
1868: @item
1869: In @code{lex_tl()} and @code{tolex_tl()} a lex order Groebner basis
1870: is computed as follows.(Refer to @code{[Noro,Yokoyama]}.)
1871:
1872: @enumerate
1873: @item
1874: Compute a Groebner basis @var{G0} with respect to @var{vlist1} and @var{order}.
1875: (Only in @code{lex_tl()}. )
1876: @item
1877: If @var{G0} is not zero-dimensional, choose a prime which does not divide
1878: head coefficients of elements in @var{G0} with respect to @var{vlist1} and
1879: @var{order}. Then compute a candidate of a lex order Groebner basis
1880: via trace lifting with @var{p}. If it succeeds the candidate is indeed
1881: a lex order Groebner basis without any check. Otherwise redo the whole
1882: process with another @var{p}.
1883: @item
1884:
1885: If @var{G0} is zero-dimensional, starting from @var{G0},
1886: compute a Groebner basis @var{G1} with respect to an elimination order
1887: to eliminate variables other than the last varibale in @var{vlist2}.
1888: Then compute a lex order Groebner basis stating from @var{G1}. These
1889: computations are done by trace lifting and the selection of a mudulus
1890: @var{p} is the same as in non zero-dimensional cases.
1891: @end enumerate
1892:
1893: @item
1894: Computations with rational function coefficients can be done only by
1895: @code{lex_tl()} and @code{tolex_tl()}.
1896: @item
1897: If @code{homo} is not equal to 0, homogenization is used in Buchberger
1898: algorithm.
1899: @item
1900: The CPU time shown after an execution of @code{tolex_d()} indicates
1901: that of the master process, and it does not include the time in child
1902: processes.
1903: \E
1.1 noro 1904: @end itemize
1905:
1906: @example
1907: [78] K=katsura(5)$
1908: 30msec + gc : 20msec
1909: [79] V=[u5,u4,u3,u2,u1,u0]$
1910: 0msec
1911: [80] G0=hgr(K,V,2)$
1912: 91.558sec + gc : 15.583sec
1913: [81] G1=lex_hensel(K,V,0,V,0)$
1914: 49.049sec + gc : 9.961sec
1915: [82] G2=lex_tl(K,V,0,V,1)$
1916: 31.186sec + gc : 3.500sec
1917: [83] gb_comp(G0,G1);
1918: 1
1919: 10msec
1920: [84] gb_comp(G0,G2);
1921: 1
1922: @end example
1923:
1924: @table @t
1.2 noro 1925: \JP @item $B;2>H(B
1926: \EG @item References
1.6 noro 1927: @fref{dp_gr_main dp_gr_mod_main dp_gr_f_main dp_weyl_gr_main dp_weyl_gr_mod_main dp_weyl_gr_f_main},
1.2 noro 1928: \JP @fref{dp_ord}, @fref{$BJ,;67W;;(B}
1929: \EG @fref{dp_ord}, @fref{Distributed computation}
1.1 noro 1930: @end table
1931:
1.2 noro 1932: \JP @node lex_hensel_gsl tolex_gsl tolex_gsl_d,,, $B%0%l%V%J4pDl$K4X$9$kH!?t(B
1933: \EG @node lex_hensel_gsl tolex_gsl tolex_gsl_d,,, Functions for Groebner basis computation
1.1 noro 1934: @subsection @code{lex_hensel_gsl}, @code{tolex_gsl}, @code{tolex_gsl_d}
1935: @findex lex_hensel_gsl
1936: @findex tolex_gsl
1937: @findex tolex_gsl_d
1938:
1939: @table @t
1940: @item lex_hensel_gsl(@var{plist},@var{vlist1},@var{order},@var{vlist2},@var{homo})
1.2 noro 1941: \JP :: GSL $B7A<0$N%$%G%"%k4pDl$N7W;;(B
1942: \EG ::Computation of an GSL form ideal basis
1.8 noro 1943: @item tolex_gsl(@var{plist},@var{vlist1},@var{order},@var{vlist2})
1944: @itemx tolex_gsl_d(@var{plist},@var{vlist1},@var{order},@var{vlist2},@var{procs})
1.2 noro 1945: \JP :: $B%0%l%V%J4pDl$rF~NO$H$9$k(B, GSL $B7A<0$N%$%G%"%k4pDl$N7W;;(B
1946: \EG :: Computation of an GSL form ideal basis stating from a Groebner basis
1.1 noro 1947: @end table
1948:
1949: @table @var
1950: @item return
1.2 noro 1951: \JP $B%j%9%H(B
1952: \EG list
1.4 noro 1953: @item plist vlist1 vlist2 procs
1.2 noro 1954: \JP $B%j%9%H(B
1955: \EG list
1.1 noro 1956: @item order
1.2 noro 1957: \JP $B?t(B, $B%j%9%H$^$?$O9TNs(B
1958: \EG number, list or matrix
1.1 noro 1959: @item homo
1.2 noro 1960: \JP $B%U%i%0(B
1961: \EG flag
1.1 noro 1962: @end table
1963:
1964: @itemize @bullet
1.2 noro 1965: \BJP
1.1 noro 1966: @item
1967: @code{lex_hensel_gsl()} $B$O(B @code{lex_hensel()} $B$N(B, @code{tolex_gsl()} $B$O(B
1968: @code{tolex()} $B$NJQ<o$G(B, $B7k2L$N$_$,0[$J$k(B.
1969: @code{tolex_gsl_d()} $B$O(B, $B4pDl7W;;$r(B, @code{procs} $B$G;XDj$5$l$k;R%W%m%;%9$K(B
1970: $BJ,;67W;;$5$;$k(B.
1971: @item
1972: $BF~NO$,(B 0 $B<!85%7%9%F%`$G(B, $B$=$N<-=q<0=g=x%0%l%V%J4pDl$,(B
1973: @code{[f0,x1-f1,...,xn-fn]} (@code{f0},...,@code{fn} $B$O(B
1974: @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,
1975: @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)
1976: $B$rJV$9(B.
1.2 noro 1977: $B$3$3$G(B, @code{gi} $B$O(B, @code{di*f0'*fi-gi} $B$,(B @code{f0} $B$G3d$j@Z$l$k$h$&$J(B
1.1 noro 1978: @code{x0} $B$N(B1 $BJQ?tB?9`<0$G(B,
1979: $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')]}
1980: $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
1981: $B$h$kDL>o$N%0%l%V%J4pDl$rJV$9(B.
1982: @item
1983: GSL $B7A<0$K$h$jI=$5$l$k4pDl$O%0%l%V%J4pDl$G$O$J$$$,(B, $B0lHL$K78?t$,(B SL $B7A<0(B
1984: $B$N%0%l%V%J4pDl$h$jHs>o$K>.$5$$$?$a7W;;$bB.$/(B, $B2r$b5a$a$d$9$$(B.
1985: @code{tolex_gsl_d()} $B$GI=<($5$l$k;~4V$O(B, $B$3$NH!?t$,<B9T$5$l$F$$$k%W%m%;%9$K(B
1986: $B$*$$$F9T$o$l$?7W;;$KBP1~$7$F$$$F(B, $B;R%W%m%;%9$K$*$1$k;~4V$O4^$^$l$J$$(B.
1.2 noro 1987: \E
1988: \BEG
1989: @item
1990: @code{lex_hensel_gsl()} and @code{lex_hensel()} are variants of
1991: @code{tolex_gsl()} and @code{tolex()} respectively. The results are
1992: Groebner basis or a kind of ideal basis, called GSL form.
1993: @code{tolex_gsl_d()} does basis computations in parallel on child
1994: processes specified in @code{procs}.
1995:
1996: @item
1997: If the input is zero-dimensional and a lex order Groebner basis has
1998: the form @code{[f0,x1-f1,...,xn-fn]} (@code{f0},...,@code{fn} are
1999: univariate polynomials of @code{x0}; SL form), then this these
2000: functions return a list such as
2001: @code{[[x1,g1,d1],...,[xn,gn,dn],[x0,f0,f0']]} (GSL form). In this list
2002: @code{gi} is a univariate polynomial of @code{x0} such that
2003: @code{di*f0'*fi-gi} divides @code{f0} and the roots of the input ideal is
2004: @code{[x1=g1/(d1*f0'),...,xn=gn/(dn*f0')]} for @code{x0}
2005: such that @code{f0(x0)=0}.
2006: If the lex order Groebner basis does not have the above form,
2007: these functions return
2008: a lex order Groebner basis computed by @code{tolex()}.
2009: @item
2010: Though an ideal basis represented as GSL form is not a Groebner basis
2011: we can expect that the coefficients are much smaller than those in a Groebner
2012: basis and that the computation is efficient.
2013: The CPU time shown after an execution of @code{tolex_gsl_d()} indicates
2014: that of the master process, and it does not include the time in child
2015: processes.
2016: \E
1.1 noro 2017: @end itemize
2018:
2019: @example
2020: [103] K=katsura(5)$
2021: [104] V=[u5,u4,u3,u2,u1,u0]$
2022: [105] G0=gr(K,V,0)$
2023: [106] GSL=tolex_gsl(G0,V,0,V)$
2024: [107] GSL[0];
2025: [u1,8635837421130477667200000000*u0^31-...]
2026: [108] GSL[1];
2027: [u2,10352277157007342793600000000*u0^31-...]
2028: [109] GSL[5];
1.5 noro 2029: [u0,11771021876193064124640000000*u0^32-...,
2030: 376672700038178051988480000000*u0^31-...]
1.1 noro 2031: @end example
2032:
2033: @table @t
1.2 noro 2034: \JP @item $B;2>H(B
2035: \EG @item References
1.1 noro 2036: @fref{lex_hensel lex_tl tolex tolex_d tolex_tl},
1.2 noro 2037: \JP @fref{$BJ,;67W;;(B}
2038: \EG @fref{Distributed computation}
1.1 noro 2039: @end table
2040:
1.2 noro 2041: \JP @node gr_minipoly minipoly,,, $B%0%l%V%J4pDl$K4X$9$kH!?t(B
2042: \EG @node gr_minipoly minipoly,,, Functions for Groebner basis computation
1.1 noro 2043: @subsection @code{gr_minipoly}, @code{minipoly}
2044: @findex gr_minipoly
2045: @findex minipoly
2046:
2047: @table @t
2048: @item gr_minipoly(@var{plist},@var{vlist},@var{order},@var{poly},@var{v},@var{homo})
1.2 noro 2049: \JP :: $BB?9`<0$N(B, $B%$%G%"%k$rK!$H$7$?:G>.B?9`<0$N7W;;(B
2050: \EG :: Computation of the minimal polynomial of a polynomial modulo an ideal
1.1 noro 2051: @item minipoly(@var{plist},@var{vlist},@var{order},@var{poly},@var{v})
1.2 noro 2052: \JP :: $B%0%l%V%J4pDl$rF~NO$H$9$k(B, $BB?9`<0$N:G>.B?9`<0$N7W;;(B
2053: \EG :: Computation of the minimal polynomial of a polynomial modulo an ideal
1.1 noro 2054: @end table
2055:
2056: @table @var
2057: @item return
1.2 noro 2058: \JP $BB?9`<0(B
2059: \EG polynomial
1.4 noro 2060: @item plist vlist
1.2 noro 2061: \JP $B%j%9%H(B
2062: \EG list
1.1 noro 2063: @item order
1.2 noro 2064: \JP $B?t(B, $B%j%9%H$^$?$O9TNs(B
2065: \EG number, list or matrix
1.1 noro 2066: @item poly
1.2 noro 2067: \JP $BB?9`<0(B
2068: \EG polynomial
1.1 noro 2069: @item v
1.2 noro 2070: \JP $BITDj85(B
2071: \EG indeterminate
1.1 noro 2072: @item homo
1.2 noro 2073: \JP $B%U%i%0(B
2074: \EG flag
1.1 noro 2075: @end table
2076:
2077: @itemize @bullet
1.2 noro 2078: \BJP
1.1 noro 2079: @item
2080: @code{gr_minipoly()} $B$O%0%l%V%J4pDl$N7W;;$+$i9T$$(B, @code{minipoly()} $B$O(B
2081: $BF~NO$r%0%l%V%J4pDl$H$_$J$9(B.
2082: @item
2083: $B%$%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,
2084: K[@var{v}] $B$N85(B f(@var{v}) $B$K(B f(@var{p}) mod I $B$rBP1~$5$;$k(B
2085: $B4D=`F17?$N3K$O(B 0 $B$G$J$$B?9`<0$K$h$j@8@.$5$l$k(B. $B$3$N@8@.85$r(B @var{p}
2086: $B$N(B, $BK!(B @var{I} $B$G$N:G>.B?9`<0$H8F$V(B.
2087: @item
2088: @code{gr_minipoly()}, @code{minipoly()} $B$O(B, $BB?9`<0(B @var{p} $B$N:G>.B?9`<0(B
2089: $B$r5a$a(B, @var{v} $B$rJQ?t$H$9$kB?9`<0$H$7$FJV$9(B.
2090: @item
2091: $B:G>.B?9`<0$O(B, $B%0%l%V%J4pDl$N(B 1 $B$D$N85$H$7$F7W;;$9$k$3$H$b$G$-$k$,(B,
2092: $B:G>.B?9`<0$N$_$r5a$a$?$$>l9g(B, @code{minipoly()}, @code{gr_minipoly()} $B$O(B
2093: $B%0%l%V%J4pDl$rMQ$$$kJ}K!$KHf$Y$F8zN($,$h$$(B.
2094: @item
2095: @code{gr_minipoly()} $B$K;XDj$9$k9`=g=x$H$7$F$O(B, $BDL>oA4<!?t5U<-=q<0=g=x$r(B
2096: $BMQ$$$k(B.
1.2 noro 2097: \E
2098: \BEG
2099: @item
2100: @code{gr_minipoly()} begins by computing a Groebner basis.
2101: @code{minipoly()} regards an input as a Groebner basis with respect to
2102: the variable order @var{vlist} and the order type @var{order}.
2103: @item
2104: Let K be a field. If an ideal @var{I} in K[X] is zero-dimensional, then, for
2105: a polynomial @var{p} in K[X], the kernel of a homomorphism from
2106: K[@var{v}] to K[X]/@var{I} which maps f(@var{v}) to f(@var{p}) mod @var{I}
2107: is generated by a polynomial. The generator is called the minimal polynomial
2108: of @var{p} modulo @var{I}.
2109: @item
2110: @code{gr_minipoly()} and @code{minipoly()} computes the minimal polynomial
2111: of a polynomial @var{p} and returns it as a polynomial of @var{v}.
2112: @item
2113: The minimal polynomial can be computed as an element of a Groebner basis.
2114: But if we are only interested in the minimal polynomial,
2115: @code{minipoly()} and @code{gr_minipoly()} can compute it more efficiently
2116: than methods using Groebner basis computation.
2117: @item
2118: It is recommended to use a degree reverse lex order as a term order
2119: for @code{gr_minipoly()}.
2120: \E
1.1 noro 2121: @end itemize
2122:
2123: @example
2124: [117] G=tolex(G0,V,0,V)$
2125: 43.818sec + gc : 11.202sec
2126: [118] GSL=tolex_gsl(G0,V,0,V)$
2127: 17.123sec + gc : 2.590sec
2128: [119] MP=minipoly(G0,V,0,u0,z)$
2129: 4.370sec + gc : 780msec
2130: @end example
2131:
2132: @table @t
1.2 noro 2133: \JP @item $B;2>H(B
2134: \EG @item References
1.1 noro 2135: @fref{lex_hensel lex_tl tolex tolex_d tolex_tl}.
2136: @end table
2137:
1.2 noro 2138: \JP @node tolexm minipolym,,, $B%0%l%V%J4pDl$K4X$9$kH!?t(B
2139: \EG @node tolexm minipolym,,, Functions for Groebner basis computation
1.1 noro 2140: @subsection @code{tolexm}, @code{minipolym}
2141: @findex tolexm
2142: @findex minipolym
2143:
2144: @table @t
2145: @item tolexm(@var{plist},@var{vlist1},@var{order},@var{vlist2},@var{mod})
1.2 noro 2146: \JP :: $BK!(B @var{mod} $B$G$N4pDlJQ49$K$h$k%0%l%V%J4pDl7W;;(B
2147: \EG :: Groebner basis computation modulo @var{mod} by change of ordering.
1.1 noro 2148: @item minipolym(@var{plist},@var{vlist1},@var{order},@var{poly},@var{v},@var{mod})
1.2 noro 2149: \JP :: $BK!(B @var{mod} $B$G$N%0%l%V%J4pDl$K$h$kB?9`<0$N:G>.B?9`<0$N7W;;(B
2150: \EG :: Minimal polynomial computation modulo @var{mod} the same method as
1.1 noro 2151: @end table
2152:
2153: @table @var
2154: @item return
1.2 noro 2155: \JP @code{tolexm()} : $B%j%9%H(B, @code{minipolym()} : $BB?9`<0(B
2156: \EG @code{tolexm()} : list, @code{minipolym()} : polynomial
1.4 noro 2157: @item plist vlist1 vlist2
1.2 noro 2158: \JP $B%j%9%H(B
2159: \EG list
1.1 noro 2160: @item order
1.2 noro 2161: \JP $B?t(B, $B%j%9%H$^$?$O9TNs(B
2162: \EG number, list or matrix
1.1 noro 2163: @item mod
1.2 noro 2164: \JP $BAG?t(B
2165: \EG prime
1.1 noro 2166: @end table
2167:
2168: @itemize @bullet
1.2 noro 2169: \BJP
1.1 noro 2170: @item
2171: $BF~NO(B @var{plist} $B$O$$$:$l$b(B $BJQ?t=g=x(B @var{vlist1}, $B9`=g=x7?(B @var{order},
2172: $BK!(B @var{mod} $B$K$*$1$k%0%l%V%J4pDl$G$J$1$l$P$J$i$J$$(B.
2173: @item
2174: @code{minipolym()} $B$O(B @code{minipoly} $B$KBP1~$9$k7W;;$rK!(B @var{mod}$B$G9T$&(B.
2175: @item
2176: @code{tolexm()} $B$O(B FGLM $BK!$K$h$k4pDlJQ49$K$h$j(B @var{vlist2},
2177: $B<-=q<0=g=x$K$h$k%0%l%V%J4pDl$r7W;;$9$k(B.
1.2 noro 2178: \E
2179: \BEG
2180: @item
2181: An input @var{plist} must be a Groebner basis modulo @var{mod}
2182: with respect to the variable order @var{vlist1} and the order type @var{order}.
2183: @item
2184: @code{minipolym()} executes the same computation as in @code{minipoly}.
2185: @item
2186: @code{tolexm()} computes a lex order Groebner basis modulo @var{mod}
2187: with respect to the variable order @var{vlist2}, by using FGLM algorithm.
2188: \E
1.1 noro 2189: @end itemize
2190:
2191: @example
2192: [197] tolexm(G0,V,0,V,31991);
2193: [8271*u0^31+10435*u0^30+816*u0^29+26809*u0^28+...,...]
2194: [198] minipolym(G0,V,0,u0,z,31991);
2195: z^32+11405*z^31+20868*z^30+21602*z^29+...
2196: @end example
2197:
2198: @table @t
1.2 noro 2199: \JP @item $B;2>H(B
2200: \EG @item References
1.1 noro 2201: @fref{lex_hensel lex_tl tolex tolex_d tolex_tl},
2202: @fref{gr_minipoly minipoly}.
2203: @end table
2204:
1.6 noro 2205: \JP @node dp_gr_main dp_gr_mod_main dp_gr_f_main dp_weyl_gr_main dp_weyl_gr_mod_main dp_weyl_gr_f_main,,, $B%0%l%V%J4pDl$K4X$9$kH!?t(B
2206: \EG @node dp_gr_main dp_gr_mod_main dp_gr_f_main dp_weyl_gr_main dp_weyl_gr_mod_main dp_weyl_gr_f_main,,, Functions for Groebner basis computation
2207: @subsection @code{dp_gr_main}, @code{dp_gr_mod_main}, @code{dp_gr_f_main}, @code{dp_weyl_gr_main}, @code{dp_weyl_gr_mod_main}, @code{dp_weyl_gr_f_main}
1.1 noro 2208: @findex dp_gr_main
2209: @findex dp_gr_mod_main
1.5 noro 2210: @findex dp_gr_f_main
1.6 noro 2211: @findex dp_weyl_gr_main
2212: @findex dp_weyl_gr_mod_main
2213: @findex dp_weyl_gr_f_main
1.1 noro 2214:
2215: @table @t
2216: @item dp_gr_main(@var{plist},@var{vlist},@var{homo},@var{modular},@var{order})
2217: @itemx dp_gr_mod_main(@var{plist},@var{vlist},@var{homo},@var{modular},@var{order})
1.5 noro 2218: @itemx dp_gr_f_main(@var{plist},@var{vlist},@var{homo},@var{order})
1.6 noro 2219: @itemx dp_weyl_gr_main(@var{plist},@var{vlist},@var{homo},@var{modular},@var{order})
2220: @itemx dp_weyl_gr_mod_main(@var{plist},@var{vlist},@var{homo},@var{modular},@var{order})
2221: @itemx dp_weyl_gr_f_main(@var{plist},@var{vlist},@var{homo},@var{order})
1.2 noro 2222: \JP :: $B%0%l%V%J4pDl$N7W;;(B ($BAH$_9~$_H!?t(B)
2223: \EG :: Groebner basis computation (built-in functions)
1.1 noro 2224: @end table
2225:
2226: @table @var
2227: @item return
1.2 noro 2228: \JP $B%j%9%H(B
2229: \EG list
1.4 noro 2230: @item plist vlist
1.2 noro 2231: \JP $B%j%9%H(B
2232: \EG list
1.1 noro 2233: @item order
1.2 noro 2234: \JP $B?t(B, $B%j%9%H$^$?$O9TNs(B
2235: \EG number, list or matrix
1.1 noro 2236: @item homo
1.2 noro 2237: \JP $B%U%i%0(B
2238: \EG flag
1.1 noro 2239: @item modular
1.2 noro 2240: \JP $B%U%i%0$^$?$OAG?t(B
2241: \EG flag or prime
1.1 noro 2242: @end table
2243:
2244: @itemize @bullet
1.2 noro 2245: \BJP
1.1 noro 2246: @item
2247: $B$3$l$i$NH!?t$O(B, $B%0%l%V%J4pDl7W;;$N4pK\E*AH$_9~$_H!?t$G$"$j(B, @code{gr()},
2248: @code{hgr()}, @code{gr_mod()} $B$J$I$O$9$Y$F$3$l$i$NH!?t$r8F$S=P$7$F7W;;(B
1.6 noro 2249: $B$r9T$C$F$$$k(B. $B4X?tL>$K(B weyl $B$,F~$C$F$$$k$b$N$O(B, Weyl $BBe?t>e$N7W;;(B
2250: $B$N$?$a$N4X?t$G$"$k(B.
1.1 noro 2251: @item
1.6 noro 2252: @code{dp_gr_f_main()}, @code{dp_weyl_f_main()} $B$O(B, $B<o!9$NM-8BBN>e$N%0%l%V%J4pDl$r7W;;$9$k(B
1.5 noro 2253: $B>l9g$KMQ$$$k(B. $BF~NO$O(B, $B$"$i$+$8$a(B, @code{simp_ff()} $B$J$I$G(B,
2254: $B9M$($kM-8BBN>e$K<M1F$5$l$F$$$kI,MW$,$"$k(B.
2255: @item
1.1 noro 2256: $B%U%i%0(B @var{homo} $B$,(B 0 $B$G$J$$;~(B, $BF~NO$r@F<!2=$7$F$+$i(B Buchberger $B%"%k%4%j%:%`(B
2257: $B$r<B9T$9$k(B.
2258: @item
2259: @code{dp_gr_mod_main()} $B$KBP$7$F$O(B, @var{modular} $B$O(B, GF(@var{modular}) $B>e(B
2260: $B$G$N7W;;$r0UL#$9$k(B.
2261: @code{dp_gr_main()} $B$KBP$7$F$O(B, @var{modular} $B$O<!$N$h$&$J0UL#$r;}$D(B.
2262: @enumerate
2263: @item
2264: @var{modular} $B$,(B 1 $B$N;~(B, trace-lifting $B$K$h$k7W;;$r9T$&(B. $BAG?t$O(B
2265: @code{lprime(0)} $B$+$i=g$K@.8y$9$k$^$G(B @code{lprime()} $B$r8F$S=P$7$F@8@.$9$k(B.
2266: @item
2267: @var{modular} $B$,(B 2 $B0J>e$N<+A3?t$N;~(B, $B$=$NCM$rAG?t$H$_$J$7$F(B trace-lifting
2268: $B$r9T$&(B. $B$=$NAG?t$G<:GT$7$?>l9g(B, 0 $B$rJV$9(B.
2269: @item
2270: @var{modular} $B$,Ii$N>l9g(B,
2271: @var{-modular} $B$KBP$7$F>e=R$N5,B'$,E,MQ$5$l$k$,(B, trace-lifting $B$N:G=*(B
2272: $BCJ3,$N%0%l%V%J4pDl%A%'%C%/$H%$%G%"%k%a%s%P%7%C%W%A%'%C%/$,>JN,$5$l$k(B.
2273: @end enumerate
2274:
2275: @item
2276: @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
2277: @code{dp_gr_main(P,V,1,1,O)}, @code{gr_mod(P,V,O,M)} $B$O(B
2278: @code{dp_gr_mod_main(P,V,0,M,O)} $B$r$=$l$>$l<B9T$9$k(B.
2279: @item
2280: @var{homo}, @var{modular} $B$NB>$K(B, @code{dp_gr_flags()} $B$G@_Dj$5$l$k(B
2281: $B$5$^$6$^$J%U%i%0$K$h$j7W;;$,@)8f$5$l$k(B.
1.2 noro 2282: \E
2283: \BEG
2284: @item
2285: These functions are fundamental built-in functions for Groebner basis
2286: computation and @code{gr()},@code{hgr()} and @code{gr_mod()}
1.6 noro 2287: are all interfaces to these functions. Functions whose names
2288: contain weyl are those for computation in Weyl algebra.
1.2 noro 2289: @item
1.6 noro 2290: @code{dp_gr_f_main()} and @code{dp_weyl_gr_f_main()}
2291: are functions for Groebner basis computation
1.5 noro 2292: over various finite fields. Coefficients of input polynomials
2293: must be converted to elements of a finite field
2294: currently specified by @code{setmod_ff()}.
2295: @item
1.2 noro 2296: If @var{homo} is not equal to 0, homogenization is applied before entering
2297: Buchberger algorithm
2298: @item
2299: For @code{dp_gr_mod_main()}, @var{modular} means a computation over
2300: GF(@var{modular}).
2301: For @code{dp_gr_main()}, @var{modular} has the following mean.
2302: @enumerate
2303: @item
2304: If @var{modular} is 1 , trace lifting is used. Primes for trace lifting
2305: are generated by @code{lprime()}, starting from @code{lprime(0)}, until
2306: the computation succeeds.
2307: @item
2308: If @var{modular} is an integer greater than 1, the integer is regarded as a
2309: prime and trace lifting is executed by using the prime. If the computation
2310: fails then 0 is returned.
2311: @item
2312: If @var{modular} is negative, the above rule is applied for @var{-modular}
2313: but the Groebner basis check and ideal-membership check are omitted in
2314: the last stage of trace lifting.
2315: @end enumerate
2316:
2317: @item
2318: @code{gr(P,V,O)}, @code{hgr(P,V,O)} and @code{gr_mod(P,V,O,M)} execute
2319: @code{dp_gr_main(P,V,0,1,O)}, @code{dp_gr_main(P,V,1,1,O)}
2320: and @code{dp_gr_mod_main(P,V,0,M,O)} respectively.
2321: @item
2322: Actual computation is controlled by various parameters set by
2323: @code{dp_gr_flags()}, other then by @var{homo} and @var{modular}.
2324: \E
1.1 noro 2325: @end itemize
2326:
2327: @table @t
1.2 noro 2328: \JP @item $B;2>H(B
2329: \EG @item References
1.1 noro 2330: @fref{dp_ord},
2331: @fref{dp_gr_flags dp_gr_print},
2332: @fref{gr hgr gr_mod},
1.5 noro 2333: @fref{setmod_ff},
1.2 noro 2334: \JP @fref{$B7W;;$*$h$SI=<($N@)8f(B}.
2335: \EG @fref{Controlling Groebner basis computations}
1.1 noro 2336: @end table
2337:
1.6 noro 2338: \JP @node dp_f4_main dp_f4_mod_main dp_weyl_f4_main dp_weyl_f4_mod_main,,, $B%0%l%V%J4pDl$K4X$9$kH!?t(B
2339: \EG @node dp_f4_main dp_f4_mod_main dp_weyl_f4_main dp_weyl_f4_mod_main,,, Functions for Groebner basis computation
2340: @subsection @code{dp_f4_main}, @code{dp_f4_mod_main}, @code{dp_weyl_f4_main}, @code{dp_weyl_f4_mod_main}
1.1 noro 2341: @findex dp_f4_main
2342: @findex dp_f4_mod_main
1.6 noro 2343: @findex dp_weyl_f4_main
2344: @findex dp_weyl_f4_mod_main
1.1 noro 2345:
2346: @table @t
2347: @item dp_f4_main(@var{plist},@var{vlist},@var{order})
2348: @itemx dp_f4_mod_main(@var{plist},@var{vlist},@var{order})
1.6 noro 2349: @itemx dp_weyl_f4_main(@var{plist},@var{vlist},@var{order})
2350: @itemx dp_weyl_f4_mod_main(@var{plist},@var{vlist},@var{order})
1.2 noro 2351: \JP :: F4 $B%"%k%4%j%:%`$K$h$k%0%l%V%J4pDl$N7W;;(B ($BAH$_9~$_H!?t(B)
2352: \EG :: Groebner basis computation by F4 algorithm (built-in functions)
1.1 noro 2353: @end table
2354:
2355: @table @var
2356: @item return
1.2 noro 2357: \JP $B%j%9%H(B
2358: \EG list
1.4 noro 2359: @item plist vlist
1.2 noro 2360: \JP $B%j%9%H(B
2361: \EG list
1.1 noro 2362: @item order
1.2 noro 2363: \JP $B?t(B, $B%j%9%H$^$?$O9TNs(B
2364: \EG number, list or matrix
1.1 noro 2365: @end table
2366:
2367: @itemize @bullet
1.2 noro 2368: \BJP
1.1 noro 2369: @item
2370: F4 $B%"%k%4%j%:%`$K$h$j%0%l%V%J4pDl$N7W;;$r9T$&(B.
2371: @item
2372: F4 $B%"%k%4%j%:%`$O(B, J.C. Faugere $B$K$h$jDs>'$5$l$??7@$Be%0%l%V%J4pDl(B
2373: $B;;K!$G$"$j(B, $BK\<BAu$O(B, $BCf9q>jM>DjM}$K$h$k@~7AJ}Dx<05a2r$rMQ$$$?(B
2374: $B;n83E*$J<BAu$G$"$k(B.
2375: @item
1.6 noro 2376: $B@F<!2=$N0z?t$,$J$$$3$H$r=|$1$P(B, $B0z?t$*$h$SF0:n$O$=$l$>$l(B
2377: @code{dp_gr_main()}, @code{dp_gr_mod_main()},
2378: @code{dp_weyl_gr_main()}, @code{dp_weyl_gr_mod_main()}
1.1 noro 2379: $B$HF1MM$G$"$k(B.
1.2 noro 2380: \E
2381: \BEG
2382: @item
2383: These functions compute Groebner bases by F4 algorithm.
2384: @item
2385: F4 is a new generation algorithm for Groebner basis computation
2386: invented by J.C. Faugere. The current implementation of @code{dp_f4_main()}
2387: uses Chinese Remainder theorem and not highly optimized.
2388: @item
2389: Arguments and actions are the same as those of
1.6 noro 2390: @code{dp_gr_main()}, @code{dp_gr_mod_main()},
2391: @code{dp_weyl_gr_main()}, @code{dp_weyl_gr_mod_main()},
2392: except for lack of the argument for controlling homogenization.
1.2 noro 2393: \E
1.1 noro 2394: @end itemize
2395:
2396: @table @t
1.2 noro 2397: \JP @item $B;2>H(B
2398: \EG @item References
1.1 noro 2399: @fref{dp_ord},
2400: @fref{dp_gr_flags dp_gr_print},
2401: @fref{gr hgr gr_mod},
1.15 noro 2402: \JP @fref{$B7W;;$*$h$SI=<($N@)8f(B}.
2403: \EG @fref{Controlling Groebner basis computations}
2404: @end table
2405:
1.17 noro 2406: \JP @node nd_gr nd_gr_trace nd_f4 nd_f4_trace nd_weyl_gr nd_weyl_gr_trace,,, $B%0%l%V%J4pDl$K4X$9$kH!?t(B
2407: \EG @node nd_gr nd_gr_trace nd_f4 nd_f4_trace nd_weyl_gr nd_weyl_gr_trace,,, Functions for Groebner basis computation
2408: @subsection @code{nd_gr}, @code{nd_gr_trace}, @code{nd_f4}, @code{nd_f4_trace}, @code{nd_weyl_gr}, @code{nd_weyl_gr_trace}
1.15 noro 2409: @findex nd_gr
2410: @findex nd_gr_trace
2411: @findex nd_f4
1.17 noro 2412: @findex nd_f4_trace
1.15 noro 2413: @findex nd_weyl_gr
2414: @findex nd_weyl_gr_trace
2415:
2416: @table @t
1.23 ! noro 2417: @item nd_gr(@var{plist},@var{vlist},@var{p},@var{order}[|@var{option=value,...}])
! 2418: @itemx nd_gr_trace(@var{plist},@var{vlist},@var{homo},@var{p},@var{order}[|@var{option=value,...}])
! 2419: @itemx nd_f4(@var{plist},@var{vlist},@var{modular},@var{order}[|@var{option=value,...}])
! 2420: @itemx nd_f4_trace(@var{plist},@var{vlist},@var{homo},@var{p},@var{order}[|@var{option=value,...}])
! 2421: @itemx nd_weyl_gr(@var{plist},@var{vlist},@var{p},@var{order}[|@var{option=value,...}])
! 2422: @itemx nd_weyl_gr_trace(@var{plist},@var{vlist},@var{homo},@var{p},@var{order}[|@var{option=value,...}])
1.15 noro 2423: \JP :: $B%0%l%V%J4pDl$N7W;;(B ($BAH$_9~$_H!?t(B)
2424: \EG :: Groebner basis computation (built-in functions)
2425: @end table
2426:
2427: @table @var
2428: @item return
2429: \JP $B%j%9%H(B
2430: \EG list
2431: @item plist vlist
2432: \JP $B%j%9%H(B
2433: \EG list
2434: @item order
2435: \JP $B?t(B, $B%j%9%H$^$?$O9TNs(B
2436: \EG number, list or matrix
2437: @item homo
2438: \JP $B%U%i%0(B
2439: \EG flag
2440: @item modular
2441: \JP $B%U%i%0$^$?$OAG?t(B
2442: \EG flag or prime
2443: @end table
2444:
2445: \BJP
2446: @itemize @bullet
2447: @item
2448: $B$3$l$i$NH!?t$O(B, $B%0%l%V%J4pDl7W;;AH$_9~$_4X?t$N?7<BAu$G$"$k(B.
2449: @item @code{nd_gr} $B$O(B, @code{p} $B$,(B 0 $B$N$H$-M-M}?tBN>e$N(B Buchberger
2450: $B%"%k%4%j%:%`$r<B9T$9$k(B. @code{p} $B$,(B 2 $B0J>e$N<+A3?t$N$H$-(B, GF(p) $B>e$N(B
2451: Buchberger $B%"%k%4%j%:%`$r<B9T$9$k(B.
1.17 noro 2452: @item @code{nd_gr_trace} $B$*$h$S(B @code{nd_f4_trace}
2453: $B$OM-M}?tBN>e$G(B trace $B%"%k%4%j%:%`$r<B9T$9$k(B.
1.18 noro 2454: @var{p} $B$,(B 0 $B$^$?$O(B 1 $B$N$H$-(B, $B<+F0E*$KA*$P$l$?AG?t$rMQ$$$F(B, $B@.8y$9$k(B
1.15 noro 2455: $B$^$G(B trace $B%"%k%4%j%:%`$r<B9T$9$k(B.
1.18 noro 2456: @var{p} $B$,(B 2 $B0J>e$N$H$-(B, trace $B$O(BGF(p) $B>e$G7W;;$5$l$k(B. trace $B%"%k%4%j%:%`(B
2457: $B$,<:GT$7$?>l9g(B 0 $B$,JV$5$l$k(B. @var{p} $B$,Ii$N>l9g(B, $B%0%l%V%J4pDl%A%'%C%/$O(B
2458: $B9T$o$J$$(B. $B$3$N>l9g(B, @var{p} $B$,(B -1 $B$J$i$P<+F0E*$KA*$P$l$?AG?t$,(B,
1.17 noro 2459: $B$=$l0J30$O;XDj$5$l$?AG?t$rMQ$$$F%0%l%V%J4pDl8uJd$N7W;;$,9T$o$l$k(B.
2460: @code{nd_f4_trace} $B$O(B, $B3FA4<!?t$K$D$$$F(B, $B$"$kM-8BBN>e$G(B F4 $B%"%k%4%j%:%`(B
2461: $B$G9T$C$?7k2L$r$b$H$K(B, $B$=$NM-8BBN>e$G(B 0 $B$G$J$$4pDl$rM?$($k(B S-$BB?9`<0$N$_$r(B
2462: $BMQ$$$F9TNs@8@.$r9T$$(B, $B$=$NA4<!?t$K$*$1$k4pDl$r@8@.$9$kJ}K!$G$"$k(B. $BF@$i$l$k(B
2463: $BB?9`<0=89g$O$d$O$j%0%l%V%J4pDl8uJd$G$"$j(B, @code{nd_gr_trace} $B$HF1MM$N(B
2464: $B%A%'%C%/$,9T$o$l$k(B.
1.15 noro 2465: @item
1.17 noro 2466: @code{nd_f4} $B$O(B @code{modular} $B$,(B 0 $B$N$H$-M-M}?tBN>e$N(B, @code{modular} $B$,(B
2467: $B%^%7%s%5%$%:AG?t$N$H$-M-8BBN>e$N(B F4 $B%"%k%4%j%:%`$r<B9T$9$k(B.
1.15 noro 2468: @item
1.18 noro 2469: @var{plist} $B$,B?9`<0%j%9%H$N>l9g(B, @var{plist}$B$G@8@.$5$l$k%$%G%"%k$N%0%l%V%J!<4pDl$,(B
2470: $B7W;;$5$l$k(B. @var{plist} $B$,B?9`<0%j%9%H$N%j%9%H$N>l9g(B, $B3FMWAG$OB?9`<04D>e$N<+M32C72$N85$H8+$J$5$l(B,
2471: $B$3$l$i$,@8@.$9$kItJ,2C72$N%0%l%V%J!<4pDl$,7W;;$5$l$k(B. $B8e<T$N>l9g(B, $B9`=g=x$O2C72$KBP$9$k9`=g=x$r(B
2472: $B;XDj$9$kI,MW$,$"$k(B. $B$3$l$O(B @var{[s,ord]} $B$N7A$G;XDj$9$k(B. @var{s} $B$,(B 0 $B$J$i$P(B TOP (Term Over Position),
2473: 1 $B$J$i$P(B POT (Position Over Term) $B$r0UL#$7(B, @var{ord} $B$OB?9`<04D$NC19`<0$KBP$9$k9`=g=x$G$"$k(B.
2474: @item
1.15 noro 2475: @code{nd_weyl_gr}, @code{nd_weyl_gr_trace} $B$O(B Weyl $BBe?tMQ$G$"$k(B.
2476: @item
1.18 noro 2477: @code{f4} $B7O4X?t0J30$O$9$Y$FM-M}4X?t78?t$N7W;;$,2DG=$G$"$k(B.
1.15 noro 2478: @item
2479: $B0lHL$K(B @code{dp_gr_main}, @code{dp_gr_mod_main} $B$h$j9bB.$G$"$k$,(B,
2480: $BFC$KM-8BBN>e$N>l9g82Cx$G$"$k(B.
1.23 ! noro 2481: @item
! 2482: $B0J2<$N%*%W%7%g%s$,;XDj$G$-$k(B.
! 2483: @table @code
! 2484: @item homo
! 2485: 1 $B$N$H$-(B, $B@F<!2=$r7PM3$7$F7W;;$9$k(B. (@code{nd_gr}, @code{nd_f4} $B$N$_(B)
! 2486: @item dp
! 2487: 1 $B$N$H$-(B, $BJ,;6I=8=B?9`<0(B ($B2C72$N>l9g$K$O2C72B?9`<0(B) $B$r7k2L$H$7$FJV$9(B.
! 2488: @item nora
! 2489: 1 $B$N$H$-(B, $B7k2L$NAj8_4JLs$r9T$o$J$$(B.
! 2490: @end table
1.15 noro 2491: @end itemize
2492: \E
2493:
2494: \BEG
2495: @itemize @bullet
2496: @item
2497: These functions are new implementations for computing Groebner bases.
2498: @item @code{nd_gr} executes Buchberger algorithm over the rationals
2499: if @code{p} is 0, and that over GF(p) if @code{p} is a prime.
2500: @item @code{nd_gr_trace} executes the trace algorithm over the rationals.
2501: If @code{p} is 0 or 1, the trace algorithm is executed until it succeeds
2502: by using automatically chosen primes.
2503: If @code{p} a positive prime,
2504: the trace is comuted over GF(p).
2505: If the trace algorithm fails 0 is returned.
2506: If @code{p} is negative,
2507: the Groebner basis check and ideal-membership check are omitted.
2508: In this case, an automatically chosen prime if @code{p} is 1,
2509: otherwise the specified prime is used to compute a Groebner basis
2510: candidate.
1.17 noro 2511: Execution of @code{nd_f4_trace} is done as follows:
2512: For each total degree, an F4-reduction of S-polynomials over a finite field
2513: is done, and S-polynomials which give non-zero basis elements are gathered.
2514: Then F4-reduction over Q is done for the gathered S-polynomials.
2515: The obtained polynomial set is a Groebner basis candidate and the same
2516: check procedure as in the case of @code{nd_gr_trace} is done.
2517: @item
2518: @code{nd_f4} executes F4 algorithm over Q if @code{modular} is equal to 0,
2519: or over a finite field GF(@code{modular})
2520: if @code{modular} is a prime number of machine size (<2^29).
1.18 noro 2521: If @var{plist} is a list of polynomials, then a Groebner basis of the ideal generated by @var{plist}
2522: is computed. If @var{plist} is a list of lists of polynomials, then each list of polynomials are regarded
2523: as an element of a free module over a polynomial ring and a Groebner basis of the sub-module generated by @var{plist}
2524: in the free module. In the latter case a term order in the free module should be specified.
2525: This is specified by @var{[s,ord]}. If @var{s} is 0 then it means TOP (Term Over Position).
2526: If @var{s} is 1 then it means POT 1 (Position Over Term). @var{ord} is a term order in the base polynomial ring.
1.15 noro 2527: @item
2528: @code{nd_weyl_gr}, @code{nd_weyl_gr_trace} are for Weyl algebra computation.
2529: @item
1.18 noro 2530: Functions except for F4 related ones can handle rational coeffient cases.
1.15 noro 2531: @item
2532: In general these functions are more efficient than
2533: @code{dp_gr_main}, @code{dp_gr_mod_main}, especially over finite fields.
1.23 ! noro 2534: @item
! 2535: The fallowing options can be specified.
! 2536: @table @code
! 2537: @item homo
! 2538: If set to 1, the computation is done via homogenization. (only for @code{nd_gr} and @code{nd_f4})
! 2539: @item dp
! 2540: If set to 1, the functions return a list of distributed polynomials (a list of
! 2541: module polynomials when the input is a sub-module).
! 2542: @item nora
! 2543: If set to 1, the inter-reduction is not performed.
! 2544: @end table
1.15 noro 2545: @end itemize
2546: \E
2547:
2548: @example
2549: [38] load("cyclic")$
2550: [49] C=cyclic(7)$
2551: [50] V=vars(C)$
2552: [51] cputime(1)$
2553: [52] dp_gr_mod_main(C,V,0,31991,0)$
2554: 26.06sec + gc : 0.313sec(26.4sec)
2555: [53] nd_gr(C,V,31991,0)$
2556: ndv_alloc=1477188
2557: 5.737sec + gc : 0.1837sec(5.921sec)
2558: [54] dp_f4_mod_main(C,V,31991,0)$
2559: 3.51sec + gc : 0.7109sec(4.221sec)
2560: [55] nd_f4(C,V,31991,0)$
2561: 1.906sec + gc : 0.126sec(2.032sec)
2562: @end example
2563:
2564: @table @t
2565: \JP @item $B;2>H(B
2566: \EG @item References
2567: @fref{dp_ord},
2568: @fref{dp_gr_flags dp_gr_print},
1.2 noro 2569: \JP @fref{$B7W;;$*$h$SI=<($N@)8f(B}.
2570: \EG @fref{Controlling Groebner basis computations}
1.1 noro 2571: @end table
2572:
1.22 noro 2573: \JP @node nd_gr_postproc nd_weyl_gr_postproc,,, $B%0%l%V%J4pDl$K4X$9$kH!?t(B
2574: \EG @node nd_gr_postproc nd_weyl_gr_postproc,,, Functions for Groebner basis computation
2575: @subsection @code{nd_gr_postproc}, @code{nd_weyl_gr_postproc}
2576: @findex nd_gr_postproc
2577: @findex nd_weyl_gr_postproc
2578:
2579: @table @t
2580: @item nd_gr_postproc(@var{plist},@var{vlist},@var{p},@var{order},@var{check})
2581: @itemx nd_weyl_gr_postproc(@var{plist},@var{vlist},@var{p},@var{order},@var{check})
2582: \JP :: $B%0%l%V%J4pDl8uJd$N%A%'%C%/$*$h$SAj8_4JLs(B
2583: \EG :: Check of Groebner basis candidate and inter-reduction
2584: @end table
2585:
2586: @table @var
2587: @item return
2588: \JP $B%j%9%H(B $B$^$?$O(B 0
2589: \EG list or 0
2590: @item plist vlist
2591: \JP $B%j%9%H(B
2592: \EG list
2593: @item p
2594: \JP $BAG?t$^$?$O(B 0
2595: \EG prime or 0
2596: @item order
2597: \JP $B?t(B, $B%j%9%H$^$?$O9TNs(B
2598: \EG number, list or matrix
2599: @item check
2600: \JP 0 $B$^$?$O(B 1
2601: \EG 0 or 1
2602: @end table
2603:
2604: @itemize @bullet
2605: \BJP
2606: @item
2607: $B%0%l%V%J4pDl(B($B8uJd(B)$B$NAj8_4JLs$r9T$&(B.
2608: @item
2609: @code{nd_weyl_gr_postproc} $B$O(B Weyl $BBe?tMQ$G$"$k(B.
2610: @item
2611: @var{check=1} $B$N>l9g(B, @var{plist} $B$,(B, @var{vlist}, @var{p}, @var{order} $B$G;XDj$5$l$kB?9`<04D(B, $B9`=g=x$G%0%l%V%J!<4pDl$K$J$C$F$$$k$+(B
2612: $B$N%A%'%C%/$b9T$&(B.
2613: @item
2614: $B@F<!2=$7$F7W;;$7$?%0%l%V%J!<4pDl$rHs@F<!2=$7$?$b$N$rAj8_4JLs$r9T$&(B, CRT $B$G7W;;$7$?%0%l%V%J!<4pDl8uJd$N%A%'%C%/$r9T$&$J$I$N>l9g$KMQ$$$k(B.
2615: \E
2616: \BEG
2617: @item
2618: Perform the inter-reduction for a Groebner basis (candidate).
2619: @item
2620: @code{nd_weyl_gr_postproc} is for Weyl algebra.
2621: @item
2622: If @var{check=1} then the check whether @var{plist} is a Groebner basis with respect to a term order in a polynomial ring
2623: or Weyl algebra specified by @var{vlist}, @var{p} and @var{order}.
2624: @item
2625: This function is used for inter-reduction of a non-reduced Groebner basis that is obtained by dehomogenizing a Groebner basis
2626: computed via homogenization, or Groebner basis check of a Groebner basis candidate computed by CRT.
2627: \E
2628: @end itemize
2629:
2630: @example
2631: afo
2632: @end example
2633:
1.2 noro 2634: \JP @node dp_gr_flags dp_gr_print,,, $B%0%l%V%J4pDl$K4X$9$kH!?t(B
2635: \EG @node dp_gr_flags dp_gr_print,,, Functions for Groebner basis computation
1.1 noro 2636: @subsection @code{dp_gr_flags}, @code{dp_gr_print}
2637: @findex dp_gr_flags
2638: @findex dp_gr_print
2639:
2640: @table @t
2641: @item dp_gr_flags([@var{list}])
1.7 noro 2642: @itemx dp_gr_print([@var{i}])
1.2 noro 2643: \JP :: $B7W;;$*$h$SI=<(MQ%Q%i%a%?$N@_Dj(B, $B;2>H(B
2644: \BEG :: Set and show various parameters for cotrolling computations
2645: and showing informations.
2646: \E
1.1 noro 2647: @end table
2648:
2649: @table @var
2650: @item return
1.2 noro 2651: \JP $B@_DjCM(B
2652: \EG value currently set
1.1 noro 2653: @item list
1.2 noro 2654: \JP $B%j%9%H(B
2655: \EG list
1.7 noro 2656: @item i
2657: \JP $B@0?t(B
2658: \EG integer
1.1 noro 2659: @end table
2660:
2661: @itemize @bullet
1.2 noro 2662: \BJP
1.1 noro 2663: @item
1.5 noro 2664: @code{dp_gr_main()}, @code{dp_gr_mod_main()}, @code{dp_gr_f_main()} $B<B9T;~$K$*$1$k$5$^$6$^(B
1.1 noro 2665: $B$J%Q%i%a%?$r@_Dj(B, $B;2>H$9$k(B.
2666: @item
2667: $B0z?t$,$J$$>l9g(B, $B8=:_$N@_Dj$,JV$5$l$k(B.
2668: @item
2669: $B0z?t$O(B, @code{["Print",1,"NoSugar",1,...]} $B$J$k7A$N%j%9%H$G(B, $B:8$+$i=g$K(B
2670: $B@_Dj$5$l$k(B. $B%Q%i%a%?L>$OJ8;zNs$GM?$($kI,MW$,$"$k(B.
2671: @item
1.7 noro 2672: @code{dp_gr_print()} $B$O(B, $BFC$K%Q%i%a%?(B @code{Print}, @code{PrintShort} $B$NCM$rD>@\@_Dj(B, $B;2>H(B
2673: $B$G$-$k(B. $B@_Dj$5$l$kCM$O<!$NDL$j$G$"$k!#(B
2674: @table @var
2675: @item i=0
2676: @code{Print=0}, @code{PrintShort=0}
2677: @item i=1
2678: @code{Print=1}, @code{PrintShort=0}
2679: @item i=2
2680: @code{Print=0}, @code{PrintShort=1}
2681: @end table
2682: $B$3$l$O(B, @code{dp_gr_main()} $B$J$I$r%5%V%k!<%A%s$H$7$FMQ$$$k%f!<%6(B
2683: $BH!?t$K$*$$$F(B, $B$=$N%5%V%k!<%A%s$,Cf4V>pJs$NI=<((B
1.1 noro 2684: $B$r9T$&:]$K(B, $B?WB.$K%U%i%0$r8+$k$3$H$,$G$-$k$h$&$KMQ0U$5$l$F$$$k(B.
1.2 noro 2685: \E
2686: \BEG
2687: @item
2688: @code{dp_gr_flags()} sets and shows various parameters for Groebner basis
2689: computation.
2690: @item
2691: If no argument is specified the current settings are returned.
2692: @item
2693: Arguments must be specified as a list such as
2694: @code{["Print",1,"NoSugar",1,...]}. Names of parameters must be character
2695: strings.
2696: @item
2697: @code{dp_gr_print()} is used to set and show the value of a parameter
1.7 noro 2698: @code{Print} and @code{PrintShort}.
2699: @table @var
2700: @item i=0
2701: @code{Print=0}, @code{PrintShort=0}
2702: @item i=1
2703: @code{Print=1}, @code{PrintShort=0}
2704: @item i=2
2705: @code{Print=0}, @code{PrintShort=1}
2706: @end table
2707: This functions is prepared to get quickly the value
2708: when a user defined function calling @code{dp_gr_main()} etc.
1.2 noro 2709: uses the value as a flag for showing intermediate informations.
2710: \E
1.1 noro 2711: @end itemize
2712:
2713: @table @t
1.2 noro 2714: \JP @item $B;2>H(B
2715: \EG @item References
2716: \JP @fref{$B7W;;$*$h$SI=<($N@)8f(B}
2717: \EG @fref{Controlling Groebner basis computations}
1.1 noro 2718: @end table
2719:
1.2 noro 2720: \JP @node dp_ord,,, $B%0%l%V%J4pDl$K4X$9$kH!?t(B
2721: \EG @node dp_ord,,, Functions for Groebner basis computation
1.1 noro 2722: @subsection @code{dp_ord}
2723: @findex dp_ord
2724:
2725: @table @t
2726: @item dp_ord([@var{order}])
1.2 noro 2727: \JP :: $BJQ?t=g=x7?$N@_Dj(B, $B;2>H(B
2728: \EG :: Set and show the ordering type.
1.1 noro 2729: @end table
2730:
2731: @table @var
2732: @item return
1.2 noro 2733: \JP $BJQ?t=g=x7?(B ($B?t(B, $B%j%9%H$^$?$O9TNs(B)
2734: \EG ordering type (number, list or matrix)
1.1 noro 2735: @item order
1.2 noro 2736: \JP $B?t(B, $B%j%9%H$^$?$O9TNs(B
2737: \EG number, list or matrix
1.1 noro 2738: @end table
2739:
2740: @itemize @bullet
1.2 noro 2741: \BJP
1.1 noro 2742: @item
2743: $B0z?t$,$"$k;~(B, $BJQ?t=g=x7?$r(B @var{order} $B$K@_Dj$9$k(B. $B0z?t$,$J$$;~(B,
2744: $B8=:_@_Dj$5$l$F$$$kJQ?t=g=x7?$rJV$9(B.
2745:
2746: @item
2747: $BJ,;6I=8=B?9`<0$K4X$9$kH!?t(B, $B1i;;$O0z?t$H$7$FJQ?t=g=x7?$r$H$k$b$N$H$H$i$J$$$b$N(B
2748: $B$,$"$j(B, $B$H$i$J$$$b$N$K4X$7$F$O(B, $B$=$N;~E@$G@_Dj$5$l$F$$$kCM$rMQ$$$F7W;;$,(B
2749: $B9T$o$l$k(B.
2750:
2751: @item
2752: @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()}
2753: $B$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.
2754:
2755: @item
2756: $BJ,;6I=8=B?9`<0$N;MB'1i;;$b(B, $B@_Dj$5$l$F$$$kCM$rMQ$$$F7W;;$5$l$k(B. $B=>$C$F(B,
2757: $B$=$NB?9`<0$,@8@.$5$l$?;~E@$K$*$1$kJQ?t=g=x7?$,(B, $B;MB'1i;;;~$K@5$7$/@_Dj(B
2758: $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
2759: $B7?$K4p$E$$$F@8@.$5$l$?$b$N$G$J$1$l$P$J$i$J$$(B.
2760:
2761: @item
2762: $B%H%C%W%l%Y%kH!?t0J30$NH!?t$rD>@\8F$S=P$9>l9g$K$O(B, $B$3$NH!?t$K$h$j(B
2763: $BJQ?t=g=x7?$r@5$7$/@_Dj$7$J$1$l$P$J$i$J$$(B.
1.23 ! noro 2764:
! 2765: @item
! 2766: $B0z?t$,%j%9%H$N>l9g(B, $B<+M32C72$K$*$1$k9`=g=x7?$r@_Dj$9$k(B. $B0z?t$,(B@code{[0,Ord]} $B$N>l9g(B,
! 2767: $BB?9`<04D>e$G(B @code{Ord} $B$G;XDj$5$l$k9`=g=x$K4p$E$/(B TOP $B=g=x(B, $B0z?t$,(B @code{[1,Ord]} $B$N>l9g(B
! 2768: OPT $B=g=x$r@_Dj$9$k(B.
! 2769:
1.2 noro 2770: \E
2771: \BEG
2772: @item
2773: If an argument is specified, the function
2774: sets the current ordering type to @var{order}.
2775: If no argument is specified, the function returns the ordering
2776: type currently set.
2777:
2778: @item
2779: There are two types of functions concerning distributed polynomial,
2780: functions which take a ordering type and those which don't take it.
2781: The latter ones use the current setting.
2782:
2783: @item
2784: Functions such as @code{gr()}, which need a ordering type as an argument,
2785: call @code{dp_ord()} internally during the execution.
2786: The setting remains after the execution.
2787:
2788: Fundamental arithmetics for distributed polynomial also use the current
2789: setting. Therefore, when such arithmetics for distributed polynomials
2790: are done, the current setting must coincide with the ordering type
2791: which was used upon the creation of the polynomials. It is assumed
2792: that such polynomials were generated under the same ordering type.
2793:
2794: @item
2795: Type of term ordering must be correctly set by this function
2796: when functions other than top level functions are called directly.
1.23 ! noro 2797:
! 2798: @item
! 2799: If the argument is a list, then an ordering type in a free module is set.
! 2800: If the argument is @code{[0,Ord]} then a TOP ordering based on the ordering type specified
! 2801: by @code{Ord} is set.
! 2802: If the argument is @code{[1,Ord]} then a POT ordering is set.
1.2 noro 2803: \E
1.1 noro 2804: @end itemize
2805:
2806: @example
2807: [19] dp_ord(0)$
2808: [20] <<1,2,3>>+<<3,1,1>>;
2809: (1)*<<1,2,3>>+(1)*<<3,1,1>>
2810: [21] dp_ord(2)$
2811: [22] <<1,2,3>>+<<3,1,1>>;
2812: (1)*<<3,1,1>>+(1)*<<1,2,3>>
2813: @end example
2814:
2815: @table @t
1.2 noro 2816: \JP @item $B;2>H(B
2817: \EG @item References
2818: \JP @fref{$B9`=g=x$N@_Dj(B}
2819: \EG @fref{Setting term orderings}
1.1 noro 2820: @end table
2821:
1.18 noro 2822: \JP @node dp_set_weight dp_set_top_weight dp_weyl_set_weight,,, $B%0%l%V%J4pDl$K4X$9$kH!?t(B
2823: \EG @node dp_set_weight dp_set_top_weight dp_weyl_set_weight,,, Functions for Groebner basis computation
2824: @subsection @code{dp_set_weight}, @code{dp_set_top_weight}, @code{dp_weyl_set_weight}
2825: @findex dp_set_weight
2826: @findex dp_set_top_weight
2827: @findex dp_weyl_set_weight
2828:
2829: @table @t
2830: @item dp_set_weight([@var{weight}])
2831: \JP :: sugar weight $B$N@_Dj(B, $B;2>H(B
2832: \EG :: Set and show the sugar weight.
2833: @item dp_set_top_weight([@var{weight}])
2834: \JP :: top weight $B$N@_Dj(B, $B;2>H(B
2835: \EG :: Set and show the top weight.
2836: @item dp_weyl_set_weight([@var{weight}])
2837: \JP :: weyl weight $B$N@_Dj(B, $B;2>H(B
2838: \EG :: Set and show the weyl weight.
2839: @end table
2840:
2841: @table @var
2842: @item return
2843: \JP $B%Y%/%H%k(B
2844: \EG a vector
2845: @item weight
2846: \JP $B@0?t$N%j%9%H$^$?$O%Y%/%H%k(B
2847: \EG a list or vector of integers
2848: @end table
2849:
2850: @itemize @bullet
2851: \BJP
2852: @item
2853: @code{dp_set_weight} $B$O(B sugar weight $B$r(B @var{weight} $B$K@_Dj$9$k(B. $B0z?t$,$J$$;~(B,
2854: $B8=:_@_Dj$5$l$F$$$k(B sugar weight $B$rJV$9(B. sugar weight $B$O@5@0?t$r@.J,$H$9$k%Y%/%H%k$G(B,
2855: $B3FJQ?t$N=E$_$rI=$9(B. $B<!?t$D$-=g=x$K$*$$$F(B, $BC19`<0$N<!?t$r7W;;$9$k:]$KMQ$$$i$l$k(B.
2856: $B@F<!2=JQ?tMQ$K(B, $BKvHx$K(B 1 $B$rIU$12C$($F$*$/$H0BA4$G$"$k(B.
2857: @item
2858: @code{dp_set_top_weight} $B$O(B top weight $B$r(B @var{weight} $B$K@_Dj$9$k(B. $B0z?t$,$J$$;~(B,
2859: $B8=:_@_Dj$5$l$F$$$k(B top weight $B$rJV$9(B. top weight $B$,@_Dj$5$l$F$$$k$H$-(B,
2860: $B$^$:(B top weight $B$K$h$kC19`<0Hf3S$r@h$K9T$&(B. tie breaker $B$H$7$F8=:_@_Dj$5$l$F$$$k(B
2861: $B9`=g=x$,MQ$$$i$l$k$,(B, $B$3$NHf3S$K$O(B top weight $B$OMQ$$$i$l$J$$(B.
2862:
2863: @item
2864: @code{dp_weyl_set_weight} $B$O(B weyl weight $B$r(B @var{weight} $B$K@_Dj$9$k(B. $B0z?t$,$J$$;~(B,
2865: $B8=:_@_Dj$5$l$F$$$k(B weyl weight $B$rJV$9(B. weyl weight w $B$r@_Dj$9$k$H(B,
2866: $B9`=g=x7?(B 11 $B$G$N7W;;$K$*$$$F(B, (-w,w) $B$r(B top weight, tie breaker $B$r(B graded reverse lex
2867: $B$H$7$?9`=g=x$,@_Dj$5$l$k(B.
2868: \E
2869: \BEG
2870: @item
2871: @code{dp_set_weight} sets the sugar weight=@var{weight}. It returns the current sugar weight.
2872: A sugar weight is a vector with positive integer components and it represents the weights of variables.
2873: It is used for computing the weight of a monomial in a graded ordering.
2874: It is recommended to append a component 1 at the end of the weight vector for a homogenizing variable.
2875: @item
2876: @code{dp_set_top_weight} sets the top weight=@var{weight}. It returns the current top weight.
2877: It a top weight is set, the weights of monomials under the top weight are firstly compared.
2878: If the the weights are equal then the current term ordering is applied as a tie breaker, but
2879: the top weight is not used in the tie breaker.
2880:
2881: @item
2882: @code{dp_weyl_set_weight} sets the weyl weigh=@var{weight}. It returns the current weyl weight.
2883: If a weyl weight w is set, in the comparsion by the term order type 11, a term order with
2884: the top weight=(-w,w) and the tie breaker=graded reverse lex is applied.
2885: \E
2886: @end itemize
2887:
2888: @table @t
2889: \JP @item $B;2>H(B
2890: \EG @item References
2891: @fref{Weight}
2892: @end table
2893:
2894:
1.2 noro 2895: \JP @node dp_ptod,,, $B%0%l%V%J4pDl$K4X$9$kH!?t(B
2896: \EG @node dp_ptod,,, Functions for Groebner basis computation
1.1 noro 2897: @subsection @code{dp_ptod}
2898: @findex dp_ptod
2899:
2900: @table @t
2901: @item dp_ptod(@var{poly},@var{vlist})
1.2 noro 2902: \JP :: $BB?9`<0$rJ,;6I=8=B?9`<0$KJQ49$9$k(B.
2903: \EG :: Converts an ordinary polynomial into a distributed polynomial.
1.1 noro 2904: @end table
2905:
2906: @table @var
2907: @item return
1.2 noro 2908: \JP $BJ,;6I=8=B?9`<0(B
2909: \EG distributed polynomial
1.1 noro 2910: @item poly
1.2 noro 2911: \JP $BB?9`<0(B
2912: \EG polynomial
1.1 noro 2913: @item vlist
1.2 noro 2914: \JP $B%j%9%H(B
2915: \EG list
1.1 noro 2916: @end table
2917:
2918: @itemize @bullet
1.2 noro 2919: \BJP
1.1 noro 2920: @item
2921: $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.
2922: @item
2923: @var{vlist} $B$K4^$^$l$J$$ITDj85$O(B, $B78?tBN$KB0$9$k$H$7$FJQ49$5$l$k(B.
1.2 noro 2924: \E
2925: \BEG
2926: @item
2927: According to the variable ordering @var{vlist} and current
2928: type of term ordering, this function converts an ordinary
2929: polynomial into a distributed polynomial.
2930: @item
2931: Indeterminates not included in @var{vlist} are regarded to belong to
2932: the coefficient field.
2933: \E
1.1 noro 2934: @end itemize
2935:
2936: @example
2937: [50] dp_ord(0);
2938: 1
2939: [51] dp_ptod((x+y+z)^2,[x,y,z]);
2940: (1)*<<2,0,0>>+(2)*<<1,1,0>>+(1)*<<0,2,0>>+(2)*<<1,0,1>>+(2)*<<0,1,1>>
2941: +(1)*<<0,0,2>>
2942: [52] dp_ptod((x+y+z)^2,[x,y]);
1.5 noro 2943: (1)*<<2,0>>+(2)*<<1,1>>+(1)*<<0,2>>+(2*z)*<<1,0>>+(2*z)*<<0,1>>
2944: +(z^2)*<<0,0>>
1.1 noro 2945: @end example
2946:
2947: @table @t
1.2 noro 2948: \JP @item $B;2>H(B
2949: \EG @item References
1.1 noro 2950: @fref{dp_dtop},
2951: @fref{dp_ord}.
2952: @end table
2953:
1.23 ! noro 2954: \JP @node dpm_dptodpm,,, $B%0%l%V%J4pDl$K4X$9$kH!?t(B
! 2955: \EG @node dpm_dptodpm,,, Functions for Groebner basis computation
! 2956: @subsection @code{dpm_dptodpm}
! 2957: @findex dpm_dptodpm
! 2958:
! 2959: @table @t
! 2960: @item dpm_dptodpm(@var{dpoly},@var{pos})
! 2961: \JP :: $BJ,;6I=8=B?9`<0$r2C72B?9`<0$KJQ49$9$k(B.
! 2962: \EG :: Converts a distributed polynomial into a module polynomial.
! 2963: @end table
! 2964:
! 2965: @table @var
! 2966: @item return
! 2967: \JP $B2C72B?9`<0(B
! 2968: \EG module polynomial
! 2969: @item dpoly
! 2970: \JP $BJ,;6I=8=B?9`<0(B
! 2971: \EG distributed polynomial
! 2972: @item pos
! 2973: \JP $B@5@0?t(B
! 2974: \EG positive integer
! 2975: @end table
! 2976:
! 2977: @itemize @bullet
! 2978: \BJP
! 2979: @item
! 2980: $BJ,;6I=8=B?9`<0$r2C72B?9`<0$KJQ49$9$k(B.
! 2981: @item
! 2982: $B=PNO$O2C72B?9`<0(B @code{dpoly e_pos} $B$G$"$k(B.
! 2983: \E
! 2984: \BEG
! 2985: @item
! 2986: This function converts a distributed polynomial into a module polynomial.
! 2987: @item
! 2988: The output is @code{dpoly e_pos}.
! 2989: \E
! 2990: @end itemize
! 2991:
! 2992: @example
! 2993: [50] dp_ord([0,0])$
! 2994: [51] D=dp_ptod((x+y+z)^2,[x,y,z]);
! 2995: (1)*<<2,0,0>>+(2)*<<1,1,0>>+(1)*<<0,2,0>>+(2)*<<1,0,1>>+(2)*<<0,1,1>>
! 2996: +(1)*<<0,0,2>>
! 2997: [52] dp_dptodpm(D,2);
! 2998: (1)*<<2,0,0:2>>+(2)*<<1,1,0:2>>+(1)*<<0,2,0:2>>+(2)*<<1,0,1:2>>
! 2999: +(2)*<<0,1,1:2>>+(1)*<<0,0,2:2>>
! 3000: @end example
! 3001:
! 3002: @table @t
! 3003: \JP @item $B;2>H(B
! 3004: \EG @item References
! 3005: @fref{dp_ptod},
! 3006: @fref{dp_ord}.
! 3007: @end table
! 3008:
! 3009: \JP @node dpm_ltod,,, $B%0%l%V%J4pDl$K4X$9$kH!?t(B
! 3010: \EG @node dpm_ltod,,, Functions for Groebner basis computation
! 3011: @subsection @code{dpm_ltod}
! 3012: @findex dpm_ltod
! 3013:
! 3014: @table @t
! 3015: @item dpm_dptodpm(@var{plist},@var{vlist})
! 3016: \JP :: $BB?9`<0%j%9%H$r2C72B?9`<0$KJQ49$9$k(B.
! 3017: \EG :: Converts a list of polynomials into a module polynomial.
! 3018: @end table
! 3019:
! 3020: @table @var
! 3021: @item return
! 3022: \JP $B2C72B?9`<0(B
! 3023: \EG module polynomial
! 3024: @item plist
! 3025: \JP $BB?9`<0%j%9%H(B
! 3026: \EG list of polynomials
! 3027: @item vlist
! 3028: \JP $BJQ?t%j%9%H(B
! 3029: \EG list of variables
! 3030: @end table
! 3031:
! 3032: @itemize @bullet
! 3033: \BJP
! 3034: @item
! 3035: $BB?9`<0%j%9%H$r2C72B?9`<0$KJQ49$9$k(B.
! 3036: @item
! 3037: @code{[p1,...,pm]} $B$O(B @code{p1 e1+...+pm em} $B$KJQ49$5$l$k(B.
! 3038: \E
! 3039: \BEG
! 3040: @item
! 3041: This function converts a list of polynomials into a module polynomial.
! 3042: @item
! 3043: @code{[p1,...,pm]} is converted into @code{p1 e1+...+pm em}.
! 3044: \E
! 3045: @end itemize
! 3046:
! 3047: @example
! 3048: [2126] dp_ord([0,0])$
! 3049: [2127] dpm_ltod([x^2+y^2,x,y-z],[x,y,z]);
! 3050: (1)*<<2,0,0:1>>+(1)*<<0,2,0:1>>+(1)*<<1,0,0:2>>+(1)*<<0,1,0:3>>
! 3051: +(-1)*<<0,0,1:3>>
! 3052: @end example
! 3053:
! 3054: @table @t
! 3055: \JP @item $B;2>H(B
! 3056: \EG @item References
! 3057: @fref{dpm_dtol},
! 3058: @fref{dp_ord}.
! 3059: @end table
! 3060:
! 3061: \JP @node dpm_dtol,,, $B%0%l%V%J4pDl$K4X$9$kH!?t(B
! 3062: \EG @node dpm_dtol,,, Functions for Groebner basis computation
! 3063: @subsection @code{dpm_dtol}
! 3064: @findex dpm_dtol
! 3065:
! 3066: @table @t
! 3067: @item dpm_dptodpm(@var{poly},@var{vlist})
! 3068: \JP :: $B2C72B?9`<0$rB?9`<0%j%9%H$KJQ49$9$k(B.
! 3069: \EG :: Converts a module polynomial into a list of polynomials.
! 3070: @end table
! 3071:
! 3072: @table @var
! 3073: @item return
! 3074: \JP $BB?9`<0%j%9%H(B
! 3075: \EG list of polynomials
! 3076: @item poly
! 3077: \JP $B2C72B?9`<0(B
! 3078: \EG module polynomial
! 3079: @item vlist
! 3080: \JP $BJQ?t%j%9%H(B
! 3081: \EG list of variables
! 3082: @end table
! 3083:
! 3084: @itemize @bullet
! 3085: \BJP
! 3086: @item
! 3087: $B2C72B?9`<0$rB?9`<0%j%9%H$KJQ49$9$k(B.
! 3088: @item
! 3089: @code{p1 e1+...+pm em} $B$O(B @code{[p1,...,pm]} $B$KJQ49$5$l$k(B.
! 3090: @item
! 3091: $B=PNO%j%9%H$ND9$5$O(B, @code{poly} $B$K4^$^$l$kI8=`4pDl$N:GBg%$%s%G%C%/%9$H$J$k(B.
! 3092: \E
! 3093: \BEG
! 3094: @item
! 3095: This function converts a module polynomial into a list of polynomials.
! 3096: @item
! 3097: @code{p1 e1+...+pm em} is converted into @code{[p1,...,pm]}.
! 3098: @item
! 3099: The length of the output list is equal to the largest index among those of the standard bases
! 3100: containd in @code{poly}.
! 3101: \E
! 3102: @end itemize
! 3103:
! 3104: @example
! 3105: [2126] dp_ord([0,0])$
! 3106: [2127] D=(1)*<<2,0,0:1>>+(1)*<<0,2,0:1>>+(1)*<<1,0,0:2>>+(1)*<<0,1,0:3>>
! 3107: +(-1)*<<0,0,1:3>>$
! 3108: [2128] dpm_dtol(D,[x,y,z]);
! 3109: [x^2+y^2,x,y-z]
! 3110: @end example
! 3111:
! 3112: @table @t
! 3113: \JP @item $B;2>H(B
! 3114: \EG @item References
! 3115: @fref{dpm_ltod},
! 3116: @fref{dp_ord}.
! 3117: @end table
! 3118:
1.2 noro 3119: \JP @node dp_dtop,,, $B%0%l%V%J4pDl$K4X$9$kH!?t(B
3120: \EG @node dp_dtop,,, Functions for Groebner basis computation
1.1 noro 3121: @subsection @code{dp_dtop}
3122: @findex dp_dtop
3123:
3124: @table @t
3125: @item dp_dtop(@var{dpoly},@var{vlist})
1.2 noro 3126: \JP :: $BJ,;6I=8=B?9`<0$rB?9`<0$KJQ49$9$k(B.
3127: \EG :: Converts a distributed polynomial into an ordinary polynomial.
1.1 noro 3128: @end table
3129:
3130: @table @var
3131: @item return
1.2 noro 3132: \JP $BB?9`<0(B
3133: \EG polynomial
1.1 noro 3134: @item dpoly
1.2 noro 3135: \JP $BJ,;6I=8=B?9`<0(B
3136: \EG distributed polynomial
1.1 noro 3137: @item vlist
1.2 noro 3138: \JP $B%j%9%H(B
3139: \EG list
1.1 noro 3140: @end table
3141:
3142: @itemize @bullet
1.2 noro 3143: \BJP
1.1 noro 3144: @item
3145: $BJ,;6I=8=B?9`<0$r(B, $BM?$($i$l$?ITDj85%j%9%H$rMQ$$$FB?9`<0$KJQ49$9$k(B.
3146: @item
3147: $BITDj85%j%9%H$O(B, $BD9$5J,;6I=8=B?9`<0$NJQ?t$N8D?t$H0lCW$7$F$$$l$P2?$G$b$h$$(B.
1.2 noro 3148: \E
3149: \BEG
3150: @item
3151: This function converts a distributed polynomial into an ordinary polynomial
3152: according to a list of indeterminates @var{vlist}.
3153: @item
3154: @var{vlist} is such a list that its length coincides with the number of
3155: variables of @var{dpoly}.
3156: \E
1.1 noro 3157: @end itemize
3158:
3159: @example
3160: [53] T=dp_ptod((x+y+z)^2,[x,y]);
1.5 noro 3161: (1)*<<2,0>>+(2)*<<1,1>>+(1)*<<0,2>>+(2*z)*<<1,0>>+(2*z)*<<0,1>>
3162: +(z^2)*<<0,0>>
1.1 noro 3163: [54] P=dp_dtop(T,[a,b]);
3164: z^2+(2*a+2*b)*z+a^2+2*b*a+b^2
3165: @end example
3166:
1.2 noro 3167: \JP @node dp_mod dp_rat,,, $B%0%l%V%J4pDl$K4X$9$kH!?t(B
3168: \EG @node dp_mod dp_rat,,, Functions for Groebner basis computation
1.1 noro 3169: @subsection @code{dp_mod}, @code{dp_rat}
3170: @findex dp_mod
3171: @findex dp_rat
3172:
3173: @table @t
3174: @item dp_mod(@var{p},@var{mod},@var{subst})
1.2 noro 3175: \JP :: $BM-M}?t78?tJ,;6I=8=B?9`<0$NM-8BBN78?t$X$NJQ49(B
3176: \EG :: Converts a disributed polynomial into one with coefficients in a finite field.
1.1 noro 3177: @item dp_rat(@var{p})
1.2 noro 3178: \JP :: $BM-8BBN78?tJ,;6I=8=B?9`<0$NM-M}?t78?t$X$NJQ49(B
3179: \BEG
3180: :: Converts a distributed polynomial with coefficients in a finite field into
3181: one with coefficients in the rationals.
3182: \E
1.1 noro 3183: @end table
3184:
3185: @table @var
3186: @item return
1.2 noro 3187: \JP $BJ,;6I=8=B?9`<0(B
3188: \EG distributed polynomial
1.1 noro 3189: @item p
1.2 noro 3190: \JP $BJ,;6I=8=B?9`<0(B
3191: \EG distributed polynomial
1.1 noro 3192: @item mod
1.2 noro 3193: \JP $BAG?t(B
3194: \EG prime
1.1 noro 3195: @item subst
1.2 noro 3196: \JP $B%j%9%H(B
3197: \EG list
1.1 noro 3198: @end table
3199:
3200: @itemize @bullet
1.2 noro 3201: \BJP
1.1 noro 3202: @item
3203: @code{dp_nf_mod()}, @code{dp_true_nf_mod()} $B$O(B, $BF~NO$H$7$FM-8BBN78?t$N(B
3204: $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
3205: $BM-M}?t78?tJ,;6I=8=B?9`<0$rJQ49$7$FMQ$$$k$3$H$,$G$-$k(B. $B$^$?(B, $BF@$i$l$?(B
3206: $B7k2L$O(B, $BM-8BBN78?tB?9`<0$H$O1i;;$G$-$k$,(B, $BM-M}?t78?tB?9`<0$H$O1i;;$G$-$J$$(B
3207: $B$?$a(B, @code{dp_rat()} $B$K$h$jJQ49$9$kI,MW$,$"$k(B.
3208: @item
3209: $BM-8BBN78?t$N1i;;$K$*$$$F$O(B, $B$"$i$+$8$a(B @code{setmod()} $B$K$h$jM-8BBN$N85$N(B
3210: $B8D?t$r;XDj$7$F$*$/I,MW$,$"$k(B.
3211: @item
3212: @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
3213: $B$7$?8eM-8BBN78?t$KJQ49$9$k$H$$$&A`:n$r9T$&:]$N(B, $BBeF~CM$r;XDj$9$k$b$N$G(B,
3214: @code{[[@var{var},@var{value}],...]} $B$N7A$N%j%9%H$G$"$k(B.
1.2 noro 3215: \E
3216: \BEG
3217: @item
3218: @code{dp_nf_mod()} and @code{dp_true_nf_mod()} require
3219: distributed polynomials with coefficients in a finite field as arguments.
3220: @code{dp_mod()} is used to convert distributed polynomials with rational
3221: number coefficients into appropriate ones.
3222: Polynomials with coefficients in a finite field
3223: cannot be used as inputs of operations with polynomials
3224: with rational number coefficients. @code{dp_rat()} is used for such cases.
3225: @item
3226: The ground finite field must be set in advance by using @code{setmod()}.
3227: @item
3228: @var{subst} is such a list as @code{[[@var{var},@var{value}],...]}.
3229: This is valid when the ground field of the input polynomial is a
3230: rational function field. @var{var}'s are variables in the ground field and
3231: the list means that @var{value} is substituted for @var{var} before
3232: converting the coefficients into elements of a finite field.
3233: \E
1.1 noro 3234: @end itemize
3235:
3236: @example
3237: @end example
3238:
3239: @table @t
1.2 noro 3240: \JP @item $B;2>H(B
3241: \EG @item References
1.18 noro 3242: @fref{dp_nf dp_nf_mod dp_true_nf dp_true_nf_mod dp_weyl_nf dp_weyl_nf_mod},
1.1 noro 3243: @fref{subst psubst},
3244: @fref{setmod}.
3245: @end table
3246:
1.2 noro 3247: \JP @node dp_homo dp_dehomo,,, $B%0%l%V%J4pDl$K4X$9$kH!?t(B
3248: \EG @node dp_homo dp_dehomo,,, Functions for Groebner basis computation
1.1 noro 3249: @subsection @code{dp_homo}, @code{dp_dehomo}
3250: @findex dp_homo
3251: @findex dp_dehomo
3252:
3253: @table @t
3254: @item dp_homo(@var{dpoly})
1.2 noro 3255: \JP :: $BJ,;6I=8=B?9`<0$N@F<!2=(B
3256: \EG :: Homogenize a distributed polynomial
1.1 noro 3257: @item dp_dehomo(@var{dpoly})
1.2 noro 3258: \JP :: $B@F<!J,;6I=8=B?9`<0$NHs@F<!2=(B
3259: \EG :: Dehomogenize a homogenious distributed polynomial
1.1 noro 3260: @end table
3261:
3262: @table @var
3263: @item return
1.2 noro 3264: \JP $BJ,;6I=8=B?9`<0(B
3265: \EG distributed polynomial
1.1 noro 3266: @item dpoly
1.2 noro 3267: \JP $BJ,;6I=8=B?9`<0(B
3268: \EG distributed polynomial
1.1 noro 3269: @end table
3270:
3271: @itemize @bullet
1.2 noro 3272: \BJP
1.1 noro 3273: @item
3274: @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
3275: 1 $B?-$P$7(B, $B:G8e$N@.J,$NCM$r(B @var{d}-@code{deg(@var{t})}
3276: (@var{d} $B$O(B @var{dpoly} $B$NA4<!?t(B) $B$H$7$?J,;6I=8=B?9`<0$rJV$9(B.
3277: @item
3278: @code{dp_dehomo()} $B$O(B, @var{dpoly} $B$N3F9`$K$D$$$F(B, $B;X?t%Y%/%H%k$N:G8e$N@.J,(B
3279: $B$r<h$j=|$$$?J,;6B?9`<0$rJV$9(B.
3280: @item
3281: $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
3282: $B@5$7$/@_Dj$9$kI,MW$,$"$k(B.
3283: @item
3284: @code{hgr()} $B$J$I$K$*$$$F(B, $BFbItE*$KMQ$$$i$l$F$$$k(B.
1.2 noro 3285: \E
3286: \BEG
3287: @item
3288: @code{dp_homo()} makes a copy of @var{dpoly}, extends
3289: the length of the exponent vector of each term @var{t} in the copy by 1,
3290: and sets the value of the newly appended
3291: component to @var{d}-@code{deg(@var{t})}, where @var{d} is the total
3292: degree of @var{dpoly}.
3293: @item
3294: @code{dp_dehomo()} make a copy of @var{dpoly} and removes the last component
3295: of each terms in the copy.
3296: @item
3297: Appropriate term orderings must be set when the results are used as inputs
3298: of some operations.
3299: @item
3300: These are used internally in @code{hgr()} etc.
3301: \E
1.1 noro 3302: @end itemize
3303:
3304: @example
3305: [202] X=<<1,2,3>>+3*<<1,2,1>>;
3306: (1)*<<1,2,3>>+(3)*<<1,2,1>>
3307: [203] dp_homo(X);
3308: (1)*<<1,2,3,0>>+(3)*<<1,2,1,2>>
3309: [204] dp_dehomo(@@);
3310: (1)*<<1,2,3>>+(3)*<<1,2,1>>
3311: @end example
3312:
3313: @table @t
1.2 noro 3314: \JP @item $B;2>H(B
3315: \EG @item References
1.1 noro 3316: @fref{gr hgr gr_mod}.
3317: @end table
3318:
1.2 noro 3319: \JP @node dp_ptozp dp_prim,,, $B%0%l%V%J4pDl$K4X$9$kH!?t(B
3320: \EG @node dp_ptozp dp_prim,,, Functions for Groebner basis computation
1.1 noro 3321: @subsection @code{dp_ptozp}, @code{dp_prim}
3322: @findex dp_ptozp
3323: @findex dp_prim
3324:
3325: @table @t
3326: @item dp_ptozp(@var{dpoly})
1.2 noro 3327: \JP :: $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.
3328: \BEG
3329: :: Converts a distributed polynomial @var{poly} with rational coefficients
3330: into an integral distributed polynomial such that GCD of all its coefficients
3331: is 1.
3332: \E
1.19 noro 3333: @item dp_prim(@var{dpoly})
1.2 noro 3334: \JP :: $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.
3335: \BEG
3336: :: Converts a distributed polynomial @var{poly} with rational function
3337: coefficients into an integral distributed polynomial such that polynomial
3338: GCD of all its coefficients is 1.
3339: \E
1.1 noro 3340: @end table
3341:
3342: @table @var
3343: @item return
1.2 noro 3344: \JP $BJ,;6I=8=B?9`<0(B
3345: \EG distributed polynomial
1.1 noro 3346: @item dpoly
1.2 noro 3347: \JP $BJ,;6I=8=B?9`<0(B
3348: \EG distributed polynomial
1.1 noro 3349: @end table
3350:
3351: @itemize @bullet
1.2 noro 3352: \BJP
1.1 noro 3353: @item
3354: @code{dp_ptozp()} $B$O(B, @code{ptozp()} $B$KAjEv$9$kA`:n$rJ,;6I=8=B?9`<0$K(B
3355: $BBP$7$F9T$&(B. $B78?t$,B?9`<0$r4^$`>l9g(B, $B78?t$K4^$^$l$kB?9`<06&DL0x;R$O(B
3356: $B<h$j=|$+$J$$(B.
3357: @item
3358: @code{dp_prim()} $B$O(B, $B78?t$,B?9`<0$r4^$`>l9g(B, $B78?t$K4^$^$l$kB?9`<06&DL0x;R(B
3359: $B$r<h$j=|$/(B.
1.2 noro 3360: \E
3361: \BEG
3362: @item
3363: @code{dp_ptozp()} executes the same operation as @code{ptozp()} for
3364: a distributed polynomial. If the coefficients include polynomials,
3365: polynomial contents included in the coefficients are not removed.
3366: @item
3367: @code{dp_prim()} removes polynomial contents.
3368: \E
1.1 noro 3369: @end itemize
3370:
3371: @example
3372: [208] X=dp_ptod(3*(x-y)*(y-z)*(z-x),[x]);
3373: (-3*y+3*z)*<<2>>+(3*y^2-3*z^2)*<<1>>+(-3*z*y^2+3*z^2*y)*<<0>>
3374: [209] dp_ptozp(X);
3375: (-y+z)*<<2>>+(y^2-z^2)*<<1>>+(-z*y^2+z^2*y)*<<0>>
3376: [210] dp_prim(X);
3377: (1)*<<2>>+(-y-z)*<<1>>+(z*y)*<<0>>
3378: @end example
3379:
3380: @table @t
1.2 noro 3381: \JP @item $B;2>H(B
3382: \EG @item References
1.1 noro 3383: @fref{ptozp}.
3384: @end table
3385:
1.18 noro 3386: \JP @node dp_nf dp_nf_mod dp_true_nf dp_true_nf_mod dp_weyl_nf dp_weyl_nf_mod,,, $B%0%l%V%J4pDl$K4X$9$kH!?t(B
3387: \EG @node dp_nf dp_nf_mod dp_true_nf dp_true_nf_mod dp_weyl_nf dp_weyl_nf_mod,,, Functions for Groebner basis computation
1.1 noro 3388: @subsection @code{dp_nf}, @code{dp_nf_mod}, @code{dp_true_nf}, @code{dp_true_nf_mod}
3389: @findex dp_nf
3390: @findex dp_true_nf
3391: @findex dp_nf_mod
3392: @findex dp_true_nf_mod
1.18 noro 3393: @findex dp_weyl_nf
3394: @findex dp_weyl_nf_mod
1.1 noro 3395:
3396: @table @t
3397: @item dp_nf(@var{indexlist},@var{dpoly},@var{dpolyarray},@var{fullreduce})
1.18 noro 3398: @item dp_weyl_nf(@var{indexlist},@var{dpoly},@var{dpolyarray},@var{fullreduce})
1.1 noro 3399: @item dp_nf_mod(@var{indexlist},@var{dpoly},@var{dpolyarray},@var{fullreduce},@var{mod})
1.18 noro 3400: @item dp_weyl_nf_mod(@var{indexlist},@var{dpoly},@var{dpolyarray},@var{fullreduce},@var{mod})
1.2 noro 3401: \JP :: $BJ,;6I=8=B?9`<0$N@55,7A$r5a$a$k(B. ($B7k2L$ODj?tG\$5$l$F$$$k2DG=@-$"$j(B)
1.1 noro 3402:
1.2 noro 3403: \BEG
3404: :: Computes the normal form of a distributed polynomial.
3405: (The result may be multiplied by a constant in the ground field.)
3406: \E
1.1 noro 3407: @item dp_true_nf(@var{indexlist},@var{dpoly},@var{dpolyarray},@var{fullreduce})
3408: @item dp_true_nf_mod(@var{indexlist},@var{dpoly},@var{dpolyarray},@var{fullreduce},@var{mod})
1.2 noro 3409: \JP :: $BJ,;6I=8=B?9`<0$N@55,7A$r5a$a$k(B. ($B??$N7k2L$r(B @code{[$BJ,;R(B, $BJ,Jl(B]} $B$N7A$GJV$9(B)
3410: \BEG
3411: :: Computes the normal form of a distributed polynomial. (The true result
3412: is returned in such a list as @code{[numerator, denominator]})
3413: \E
1.1 noro 3414: @end table
3415:
3416: @table @var
3417: @item return
1.2 noro 3418: \JP @code{dp_nf()} : $BJ,;6I=8=B?9`<0(B, @code{dp_true_nf()} : $B%j%9%H(B
3419: \EG @code{dp_nf()} : distributed polynomial, @code{dp_true_nf()} : list
1.1 noro 3420: @item indexlist
1.2 noro 3421: \JP $B%j%9%H(B
3422: \EG list
1.1 noro 3423: @item dpoly
1.2 noro 3424: \JP $BJ,;6I=8=B?9`<0(B
3425: \EG distributed polynomial
1.1 noro 3426: @item dpolyarray
1.2 noro 3427: \JP $BG[Ns(B
3428: \EG array of distributed polynomial
1.1 noro 3429: @item fullreduce
1.2 noro 3430: \JP $B%U%i%0(B
3431: \EG flag
1.1 noro 3432: @item mod
1.2 noro 3433: \JP $BAG?t(B
3434: \EG prime
1.1 noro 3435: @end table
3436:
3437: @itemize @bullet
1.2 noro 3438: \BJP
1.1 noro 3439: @item
3440: $BJ,;6I=8=B?9`<0(B @var{dpoly} $B$N@55,7A$r5a$a$k(B.
3441: @item
1.18 noro 3442: $BL>A0$K(B weyl $B$r4^$`4X?t$O%o%$%kBe?t$K$*$1$k@55,7A7W;;$r9T$&(B. $B0J2<$N@bL@$O(B weyl $B$r4^$`$b$N$KBP$7$F$bF1MM$K@.N)$9$k(B.
3443: @item
1.1 noro 3444: @code{dp_nf_mod()}, @code{dp_true_nf_mod()} $B$NF~NO$O(B, @code{dp_mod()} $B$J$I(B
3445: $B$K$h$j(B, $BM-8BBN>e$NJ,;6I=8=B?9`<0$K$J$C$F$$$J$1$l$P$J$i$J$$(B.
3446: @item
3447: $B7k2L$KM-M}?t(B, $BM-M}<0$,4^$^$l$k$N$rHr$1$k$?$a(B, @code{dp_nf()} $B$O(B
3448: $B??$NCM$NDj?tG\$NCM$rJV$9(B. $BM-M}<078?t$N>l9g$N(B @code{dp_nf_mod()} $B$bF1MM(B
3449: $B$G$"$k$,(B, $B78?tBN$,M-8BBN$N>l9g(B @code{dp_nf_mod()} $B$O??$NCM$rJV$9(B.
3450: @item
3451: @code{dp_true_nf()}, @code{dp_true_nf_mod()} $B$O(B,
3452: @code{[@var{nm},@var{dn}]} $B$J$k7A$N%j%9%H$rJV$9(B.
3453: $B$?$@$7(B, @var{nm} $B$O78?t$KJ,?t(B, $BM-M}<0$r4^$^$J$$J,;6I=8=B?9`<0(B, @var{dn} $B$O(B
3454: $B?t$^$?$OB?9`<0$G(B @var{nm}/@var{dn} $B$,??$NCM$H$J$k(B.
3455: @item
3456: @var{dpolyarray} $B$OJ,;6I=8=B?9`<0$rMWAG$H$9$k%Y%/%H%k(B,
3457: @var{indexlist} $B$O@55,2=7W;;$KMQ$$$k(B @var{dpolyarray} $B$NMWAG$N%$%s%G%C%/%9(B
3458: $B$N%j%9%H(B.
3459: @item
3460: @var{fullreduce} $B$,(B 0 $B$G$J$$$H$-A4$F$N9`$KBP$7$F4JLs$r9T$&(B. @var{fullreduce}
3461: $B$,(B 0 $B$N$H$-F,9`$N$_$KBP$7$F4JLs$r9T$&(B.
3462: @item
3463: @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.
3464: @item
3465: $B0lHL$K$O(B @var{indexlist} $B$NM?$(J}$K$h$jH!?t$NCM$O0[$J$k2DG=@-$,$"$k$,(B,
3466: $B%0%l%V%J4pDl$KBP$7$F$O0l0UE*$KDj$^$k(B.
3467: @item
3468: $BJ,;6I=8=$G$J$$8GDj$5$l$?B?9`<0=89g$K$h$k@55,7A$rB??t5a$a$kI,MW$,$"$k>l9g(B
3469: $B$KJXMx$G$"$k(B. $BC10l$N1i;;$K4X$7$F$O(B, @code{p_nf}, @code{p_true_nf} $B$r(B
3470: $BMQ$$$k$H$h$$(B.
1.2 noro 3471: \E
3472: \BEG
3473: @item
3474: Computes the normal form of a distributed polynomial.
3475: @item
1.18 noro 3476: Functions whose name contain @code{weyl} compute normal forms in Weyl algebra. The description below also applies to
3477: the functions for Weyl algebra.
3478: @item
1.2 noro 3479: @code{dp_nf_mod()} and @code{dp_true_nf_mod()} require
3480: distributed polynomials with coefficients in a finite field as arguments.
3481: @item
3482: The result of @code{dp_nf()} may be multiplied by a constant in the
3483: ground field in order to make the result integral. The same is true
3484: for @code{dp_nf_mod()}, but it returns the true normal form if
3485: the ground field is a finite field.
3486: @item
3487: @code{dp_true_nf()} and @code{dp_true_nf_mod()} return
3488: such a list as @code{[@var{nm},@var{dn}]}.
3489: Here @var{nm} is a distributed polynomial whose coefficients are integral
3490: in the ground field, @var{dn} is an integral element in the ground
3491: field and @var{nm}/@var{dn} is the true normal form.
3492: @item
3493: @var{dpolyarray} is a vector whose components are distributed polynomials
3494: and @var{indexlist} is a list of indices which is used for the normal form
3495: computation.
3496: @item
3497: When argument @var{fullreduce} has non-zero value,
3498: all terms are reduced. When it has value 0,
3499: only the head term is reduced.
3500: @item
3501: As for the polynomials specified by @var{indexlist}, one specified by
3502: an index placed at the preceding position has priority to be selected.
3503: @item
3504: In general, the result of the function may be different depending on
3505: @var{indexlist}. However, the result is unique for Groebner bases.
3506: @item
3507: These functions are useful when a fixed non-distributed polynomial set
3508: is used as a set of reducers to compute normal forms of many polynomials.
3509: For single computation @code{p_nf} and @code{p_true_nf} are sufficient.
3510: \E
1.1 noro 3511: @end itemize
3512:
3513: @example
3514: [0] load("gr")$
3515: [64] load("katsura")$
3516: [69] K=katsura(4)$
3517: [70] dp_ord(2)$
3518: [71] V=[u0,u1,u2,u3,u4]$
3519: [72] DP1=newvect(length(K),map(dp_ptod,K,V))$
3520: [73] G=gr(K,V,2)$
3521: [74] DP2=newvect(length(G),map(dp_ptod,G,V))$
3522: [75] T=dp_ptod((u0-u1+u2-u3+u4)^2,V)$
3523: [76] dp_dtop(dp_nf([0,1,2,3,4],T,DP1,1),V);
1.5 noro 3524: u4^2+(6*u3+2*u2+6*u1-2)*u4+9*u3^2+(6*u2+18*u1-6)*u3+u2^2
3525: +(6*u1-2)*u2+9*u1^2-6*u1+1
1.1 noro 3526: [77] dp_dtop(dp_nf([4,3,2,1,0],T,DP1,1),V);
3527: -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
3528: [78] dp_dtop(dp_nf([0,1,2,3,4],T,DP2,1),V);
1.5 noro 3529: -11380879768451657780886122972730785203470970010204714556333530492210
3530: 456775930005716505560062087150928400876150217079820311439477560587583
3531: 488*u4^15+...
1.1 noro 3532: [79] dp_dtop(dp_nf([4,3,2,1,0],T,DP2,1),V);
1.5 noro 3533: -11380879768451657780886122972730785203470970010204714556333530492210
3534: 456775930005716505560062087150928400876150217079820311439477560587583
3535: 488*u4^15+...
1.1 noro 3536: [80] @@78==@@79;
3537: 1
3538: @end example
3539:
3540: @table @t
1.2 noro 3541: \JP @item $B;2>H(B
3542: \EG @item References
1.1 noro 3543: @fref{dp_dtop},
3544: @fref{dp_ord},
3545: @fref{dp_mod dp_rat},
3546: @fref{p_nf p_nf_mod p_true_nf p_true_nf_mod}.
3547: @end table
3548:
1.23 ! noro 3549: \JP @node dpm_nf dpm_nf_and_quotient,,, $B%0%l%V%J4pDl$K4X$9$kH!?t(B
! 3550: \EG @node dpm_nf dpm_nf_and_quotient,,, Functions for Groebner basis computation
! 3551: @subsection @code{dpm_nf}, @code{dpm_nf_and_quotient}
! 3552: @findex dpm_nf
! 3553: @findex dpm_nf_and_quotient
! 3554:
! 3555: @table @t
! 3556: @item dpm_nf([@var{indexlist},]@var{dpoly},@var{dpolyarray},@var{fullreduce})
! 3557: \JP :: $B2C72B?9`<0$N@55,7A$r5a$a$k(B. ($B7k2L$ODj?tG\$5$l$F$$$k2DG=@-$"$j(B)
! 3558:
! 3559: \BEG
! 3560: :: Computes the normal form of a module polynomial.
! 3561: (The result may be multiplied by a constant in the ground field.)
! 3562: \E
! 3563: @item dpm_nf_and_quotient([@var{indexlist},]@var{dpoly},@var{dpolyarray})
! 3564: \JP :: $B2C72B?9`<0$N@55,7A$H>&$r5a$a$k(B.
! 3565: \BEG
! 3566: :: Computes the normal form of a module polynomial and the quotient.
! 3567: \E
! 3568: @end table
! 3569:
! 3570: @table @var
! 3571: @item return
! 3572: \JP @code{dpm_nf()} : $B2C72B?9`<0(B, @code{dpm_nf_and_quotient()} : $B%j%9%H(B
! 3573: \EG @code{dpm_nf()} : module polynomial, @code{dpm_nf_and_quotient()} : list
! 3574: @item indexlist
! 3575: \JP $B%j%9%H(B
! 3576: \EG list
! 3577: @item dpoly
! 3578: \JP $B2C72B?9`<0(B
! 3579: \EG module polynomial
! 3580: @item dpolyarray
! 3581: \JP $BG[Ns(B
! 3582: \EG array of module polynomial
! 3583: @end table
! 3584:
! 3585: @itemize @bullet
! 3586: \BJP
! 3587: @item
! 3588: $B2C72B?9`<0(B @var{dpoly} $B$N@55,7A$r5a$a$k(B.
! 3589: @item
! 3590: $B7k2L$KM-M}?t(B, $BM-M}<0$,4^$^$l$k$N$rHr$1$k$?$a(B, @code{dpm_nf()} $B$O(B
! 3591: $B??$NCM$NDj?tG\$NCM$rJV$9(B.
! 3592: @item
! 3593: @var{dpolyarray} $B$O2C72B?9`<0$rMWAG$H$9$k%Y%/%H%k(B,
! 3594: @var{indexlist} $B$O@55,2=7W;;$KMQ$$$k(B @var{dpolyarray} $B$NMWAG$N%$%s%G%C%/%9(B
! 3595: @item
! 3596: @var{indexlist} $B$,M?$($i$l$F$$$k>l9g(B, @var{dpolyarray} $B$NCf$G(B, @var{indexlist} $B$G;XDj$5$l$?$b$N$N$_$,(B, $BA0$NJ}$+$iM%@hE*$K;H$o$l$k(B.
! 3597: @var{indexlist} $B$,M?$($i$l$F$$$J$$>l9g$K$O(B, @var{dpolyarray} $B$NCf$NA4$F$NB?9`<0$,A0$NJ}$+$iM%@hE*$K;H$o$l$k(B.
! 3598: @item
! 3599: @code{dpm_nf_and_quotient()} $B$O(B,
! 3600: @code{[@var{nm},@var{dn},@var{quo}]} $B$J$k7A$N%j%9%H$rJV$9(B.
! 3601: $B$?$@$7(B, @var{nm} $B$O78?t$KJ,?t$r4^$^$J$$2C72B?9`<0(B, @var{dn} $B$O(B
! 3602: $B?t$^$?$OB?9`<0$G(B @var{nm}/@var{dn} $B$,??$NCM$H$J$k(B.
! 3603: @var{quo} $B$O=|;;$N>&$rI=$9G[Ns$G(B, @var{dn}@var{dpoly}=@var{nm}+@var{quo[0]dpolyarray[0]+...} $B$,@.$jN)$D(B.
! 3604: $B$N%j%9%H(B.
! 3605: @item
! 3606: @var{fullreduce} $B$,(B 0 $B$G$J$$$H$-A4$F$N9`$KBP$7$F4JLs$r9T$&(B. @var{fullreduce}
! 3607: $B$,(B 0 $B$N$H$-F,9`$N$_$KBP$7$F4JLs$r9T$&(B.
! 3608: \E
! 3609: \BEG
! 3610: @item
! 3611: Computes the normal form of a module polynomial.
! 3612: @item
! 3613: The result of @code{dpm_nf()} may be multiplied by a constant in the
! 3614: ground field in order to make the result integral.
! 3615: @item
! 3616: @var{dpolyarray} is a vector whose components are module polynomials
! 3617: and @var{indexlist} is a list of indices which is used for the normal form
! 3618: computation.
! 3619: @item
! 3620: If @var{indexlist} is given, only the polynomials in @var{dpolyarray} specified in @var{indexlist}
! 3621: is used in the division. An index placed at the preceding position has priority to be selected.
! 3622: If @var{indexlist} is not given, all the polynomials in @var{dpolyarray} are used.
! 3623: @item
! 3624: @code{dpm_nf_and_quotient()} returns
! 3625: such a list as @code{[@var{nm},@var{dn},@var{quo}]}.
! 3626: Here @var{nm} is a module polynomial whose coefficients are integral
! 3627: in the ground field, @var{dn} is an integral element in the ground
! 3628: field and @var{nm}/@var{dn} is the true normal form.
! 3629: @var{quo} is an array containing the quotients of the division satisfying
! 3630: @var{dn}@var{dpoly}=@var{nm}+@var{quo[0]dpolyarray[0]+...}.
! 3631: @item
! 3632: When argument @var{fullreduce} has non-zero value,
! 3633: all terms are reduced. When it has value 0,
! 3634: only the head term is reduced.
! 3635: \E
! 3636: @end itemize
! 3637:
! 3638: @example
! 3639: [2126] dp_ord([1,0])$
! 3640: [2127] S=ltov([(1)*<<0,0,2,0:1>>+(1)*<<0,0,1,1:1>>+(1)*<<0,0,0,2:1>>
! 3641: +(-1)*<<3,0,0,0:2>>+(-1)*<<0,0,2,1:2>>+(-1)*<<0,0,1,2:2>>
! 3642: +(1)*<<3,0,1,0:3>>+(1)*<<3,0,0,1:3>>+(1)*<<0,0,2,2:3>>,
! 3643: (-1)*<<0,1,0,0:1>>+(-1)*<<0,0,1,0:1>>+(-1)*<<0,0,0,1:1>>
! 3644: +(-1)*<<3,0,0,0:3>>+(1)*<<0,1,1,1:3>>,(1)*<<0,1,0,0:2>>
! 3645: +(1)*<<0,0,1,0:2>>+(1)*<<0,0,0,1:2>>+(-1)*<<0,1,1,0:3>>
! 3646: +(-1)*<<0,1,0,1:3>>+(-1)*<<0,0,1,1:3>>])$
! 3647: [2128] U=dpm_sp(S[0],S[1]);
! 3648: (1)*<<0,0,3,0:1>>+(-1)*<<0,1,1,1:1>>+(1)*<<0,0,2,1:1>>
! 3649: +(-1)*<<0,1,0,2:1>>+(1)*<<3,1,0,0:2>>+(1)*<<0,1,2,1:2>>
! 3650: +(1)*<<0,1,1,2:2>>+(-1)*<<3,1,1,0:3>>+(1)*<<3,0,2,0:3>>
! 3651: +(-1)*<<3,1,0,1:3>>+(-1)*<<0,1,3,1:3>>+(-1)*<<0,1,2,2:3>>
! 3652: [2129] dpm_nf(U,S,1);
! 3653: 0
! 3654: [2130] L=dpm_nf_and_quotient(U,S)$
! 3655: [2131] Q=L[2]$
! 3656: [2132] D=L[1]$
! 3657: [2133] D*U-(Q[1]*S[1]+Q[2]*S[2]);
! 3658: 0
! 3659: @end example
! 3660:
! 3661: @table @t
! 3662: \JP @item $B;2>H(B
! 3663: \EG @item References
! 3664: @fref{dpm_sp},
! 3665: @fref{dp_ord}.
! 3666: @end table
! 3667:
! 3668:
1.2 noro 3669: \JP @node dp_hm dp_ht dp_hc dp_rest,,, $B%0%l%V%J4pDl$K4X$9$kH!?t(B
3670: \EG @node dp_hm dp_ht dp_hc dp_rest,,, Functions for Groebner basis computation
1.1 noro 3671: @subsection @code{dp_hm}, @code{dp_ht}, @code{dp_hc}, @code{dp_rest}
3672: @findex dp_hm
3673: @findex dp_ht
3674: @findex dp_hc
3675: @findex dp_rest
3676:
3677: @table @t
3678: @item dp_hm(@var{dpoly})
1.2 noro 3679: \JP :: $BF,C19`<0$r<h$j=P$9(B.
3680: \EG :: Gets the head monomial.
1.1 noro 3681: @item dp_ht(@var{dpoly})
1.2 noro 3682: \JP :: $BF,9`$r<h$j=P$9(B.
3683: \EG :: Gets the head term.
1.1 noro 3684: @item dp_hc(@var{dpoly})
1.2 noro 3685: \JP :: $BF,78?t$r<h$j=P$9(B.
3686: \EG :: Gets the head coefficient.
1.1 noro 3687: @item dp_rest(@var{dpoly})
1.2 noro 3688: \JP :: $BF,C19`<0$r<h$j=|$$$?;D$j$rJV$9(B.
3689: \EG :: Gets the remainder of the polynomial where the head monomial is removed.
1.1 noro 3690: @end table
3691:
3692: @table @var
1.2 noro 3693: \BJP
1.1 noro 3694: @item return
3695: @code{dp_hm()}, @code{dp_ht()}, @code{dp_rest()} : $BJ,;6I=8=B?9`<0(B,
3696: @code{dp_hc()} : $B?t$^$?$OB?9`<0(B
3697: @item dpoly
3698: $BJ,;6I=8=B?9`<0(B
1.2 noro 3699: \E
3700: \BEG
3701: @item return
3702: @code{dp_hm()}, @code{dp_ht()}, @code{dp_rest()} : distributed polynomial
3703: @code{dp_hc()} : number or polynomial
3704: @item dpoly
3705: distributed polynomial
3706: \E
1.1 noro 3707: @end table
3708:
3709: @itemize @bullet
1.2 noro 3710: \BJP
1.1 noro 3711: @item
3712: $B$3$l$i$O(B, $BJ,;6I=8=B?9`<0$N3FItJ,$r<h$j=P$9$?$a$NH!?t$G$"$k(B.
3713: @item
3714: $BJ,;6I=8=B?9`<0(B @var{p} $B$KBP$7<!$,@.$jN)$D(B.
1.2 noro 3715: \E
3716: \BEG
3717: @item
3718: These are used to get various parts of a distributed polynomial.
3719: @item
3720: The next equations hold for a distributed polynomial @var{p}.
3721: \E
1.1 noro 3722: @table @code
3723: @item @var{p} = dp_hm(@var{p}) + dp_rest(@var{p})
3724: @item dp_hm(@var{p}) = dp_hc(@var{p}) dp_ht(@var{p})
3725: @end table
3726: @end itemize
3727:
3728: @example
3729: [87] dp_ord(0)$
3730: [88] X=ptozp((a46^2+7/10*a46+7/48)*u3^4-50/27*a46^2-35/27*a46-49/216)$
3731: [89] T=dp_ptod(X,[u3,u4,a46])$
3732: [90] dp_hm(T);
3733: (2160)*<<4,0,2>>
3734: [91] dp_ht(T);
3735: (1)*<<4,0,2>>
3736: [92] dp_hc(T);
3737: 2160
3738: [93] dp_rest(T);
3739: (1512)*<<4,0,1>>+(315)*<<4,0,0>>+(-4000)*<<0,0,2>>+(-2800)*<<0,0,1>>
3740: +(-490)*<<0,0,0>>
3741: @end example
3742:
1.23 ! noro 3743: \JP @node dpm_hm dpm_ht dpm_hc dpm_hp dpm_rest,,, $B%0%l%V%J4pDl$K4X$9$kH!?t(B
! 3744: \EG @node dpm_hm dpm_ht dpm_hc dpm_hp dpm_rest,,, Functions for Groebner basis computation
! 3745: @subsection @code{dpm_hm}, @code{dpm_ht}, @code{dpm_hc}, @code{dpm_hp}, @code{dpm_rest}
! 3746: @findex dpm_hm
! 3747: @findex dpm_ht
! 3748: @findex dpm_hc
! 3749: @findex dpm_hp
! 3750: @findex dpm_rest
! 3751:
! 3752: @table @t
! 3753: @item dpm_hm(@var{dpoly})
! 3754: \JP :: $B2C72B?9`<0$NF,C19`<0$r<h$j=P$9(B.
! 3755: \EG :: Gets the head monomial of a module polynomial.
! 3756: @item dpm_ht(@var{dpoly})
! 3757: \JP :: $B2C72B?9`<0$NF,9`$r<h$j=P$9(B.
! 3758: \EG :: Gets the head term of a module polynomial.
! 3759: @item dpm_hc(@var{dpoly})
! 3760: \JP :: $B2C72B?9`<0$NF,78?t$r<h$j=P$9(B.
! 3761: \EG :: Gets the head coefficient of a module polynomial.
! 3762: @item dpm_hp(@var{dpoly})
! 3763: \JP :: $B2C72B?9`<0$NF,0LCV$r<h$j=P$9(B.
! 3764: \EG :: Gets the head position of a module polynomial.
! 3765: @item dpm_rest(@var{dpoly})
! 3766: \JP :: $B2C72B?9`<0$NF,C19`<0$r<h$j=|$$$?;D$j$rJV$9(B.
! 3767: \EG :: Gets the remainder of a module polynomial where the head monomial is removed.
! 3768: @end table
! 3769:
! 3770: @table @var
! 3771: \BJP
! 3772: @item return
! 3773: @code{dp_hm()}, @code{dp_ht()}, @code{dp_rest()} : $B2C72B?9`<0(B,
! 3774: @code{dp_hc()} : $B?t$^$?$OB?9`<0(B
! 3775: @item dpoly
! 3776: $B2C72B?9`<0(B
! 3777: \E
! 3778: \BEG
! 3779: @item return
! 3780: @code{dpm_hm()}, @code{dpm_ht()}, @code{dpm_rest()} : module polynomial
! 3781: @code{dpm_hc()} : monomial
! 3782: @item dpoly
! 3783: distributed polynomial
! 3784: \E
! 3785: @end table
! 3786:
! 3787: @itemize @bullet
! 3788: \BJP
! 3789: @item
! 3790: $B$3$l$i$O(B, $B2C72B?9`<0$N3FItJ,$r<h$j=P$9$?$a$NH!?t$G$"$k(B.
! 3791: @item
! 3792: @code{dpm_hc()} $B$O(B, @code{dpm_hm()} $B$N(B, $BI8=`4pDl$K4X$9$k78?t$G$"$kC19`<0$rJV$9(B.
! 3793: $B%9%+%i!<78?t$r<h$j=P$9$K$O(B, $B$5$i$K(B @code{dp_hc()} $B$r<B9T$9$k(B.
! 3794: @item
! 3795: @code{dpm_hp()} $B$O(B, $BF,2C72C19`<0$K4^$^$l$kI8=`4pDl$N%$%s%G%C%/%9$rJV$9(B.
! 3796: \E
! 3797: \BEG
! 3798: @item
! 3799: These are used to get various parts of a module polynomial.
! 3800: @item
! 3801: @code{dpm_hc()} returns the monomial that is the coefficient of @code{dpm_hm()} with respect to the
! 3802: standard base.
! 3803: For getting its scalar coefficient apply @code{dp_hc()}.
! 3804: @item
! 3805: @code{dpm_hp()} returns the index of the standard base conteind in the head module monomial.
! 3806: \E
! 3807: @end itemize
! 3808:
! 3809: @example
! 3810: [2126] dp_ord([1,0]);
! 3811: [1,0]
! 3812: [2127] F=2*<<1,2,0:2>>-3*<<1,0,2:3>>+<<2,1,0:2>>;
! 3813: (1)*<<2,1,0:2>>+(2)*<<1,2,0:2>>+(-3)*<<1,0,2:3>>
! 3814: [2128] M=dpm_hm(F);
! 3815: (1)*<<2,1,0:2>>
! 3816: [2129] C=dpm_hc(F);
! 3817: (1)*<<2,1,0>>
! 3818: [2130] R=dpm_rest(F);
! 3819: (2)*<<1,2,0:2>>+(-3)*<<1,0,2:3>>
! 3820: [2131] dpm_hp(F);
! 3821: 2
! 3822: @end example
! 3823:
! 3824:
1.2 noro 3825: \JP @node dp_td dp_sugar,,, $B%0%l%V%J4pDl$K4X$9$kH!?t(B
3826: \EG @node dp_td dp_sugar,,, Functions for Groebner basis computation
1.1 noro 3827: @subsection @code{dp_td}, @code{dp_sugar}
3828: @findex dp_td
3829: @findex dp_sugar
3830:
3831: @table @t
3832: @item dp_td(@var{dpoly})
1.2 noro 3833: \JP :: $BF,9`$NA4<!?t$rJV$9(B.
3834: \EG :: Gets the total degree of the head term.
1.1 noro 3835: @item dp_sugar(@var{dpoly})
1.2 noro 3836: \JP :: $BB?9`<0$N(B @code{sugar} $B$rJV$9(B.
3837: \EG :: Gets the @code{sugar} of a polynomial.
1.1 noro 3838: @end table
3839:
3840: @table @var
3841: @item return
1.2 noro 3842: \JP $B<+A3?t(B
3843: \EG non-negative integer
1.1 noro 3844: @item dpoly
1.2 noro 3845: \JP $BJ,;6I=8=B?9`<0(B
3846: \EG distributed polynomial
1.1 noro 3847: @item onoff
1.2 noro 3848: \JP $B%U%i%0(B
3849: \EG flag
1.1 noro 3850: @end table
3851:
3852: @itemize @bullet
1.2 noro 3853: \BJP
1.1 noro 3854: @item
3855: @code{dp_td()} $B$O(B, $BF,9`$NA4<!?t(B, $B$9$J$o$A3FJQ?t$N;X?t$NOB$rJV$9(B.
3856: @item
3857: $BJ,;6I=8=B?9`<0$,@8@.$5$l$k$H(B, @code{sugar} $B$H8F$P$l$k$"$k@0?t$,IUM?(B
3858: $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.
3859: @item
3860: @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
3861: $B7hDj$9$k$?$a$N=EMW$J;X?K$H$J$k(B.
1.2 noro 3862: \E
3863: \BEG
3864: @item
3865: Function @code{dp_td()} returns the total degree of the head term,
3866: i.e., the sum of all exponent of variables in that term.
3867: @item
3868: Upon creation of a distributed polynomial, an integer called @code{sugar}
3869: is associated. This value is
3870: the total degree of the virtually homogenized one of the original
3871: polynomial.
3872: @item
3873: The quantity @code{sugar} is an important guide to determine the
3874: selection strategy of critical pairs in Groebner basis computation.
3875: \E
1.1 noro 3876: @end itemize
3877:
3878: @example
3879: [74] dp_ord(0)$
3880: [75] X=<<1,2>>+<<0,1>>$
3881: [76] Y=<<1,2>>+<<1,0>>$
3882: [77] Z=X-Y;
3883: (-1)*<<1,0>>+(1)*<<0,1>>
3884: [78] dp_sugar(T);
3885: 3
3886: @end example
3887:
1.2 noro 3888: \JP @node dp_lcm,,, $B%0%l%V%J4pDl$K4X$9$kH!?t(B
3889: \EG @node dp_lcm,,, Functions for Groebner basis computation
1.1 noro 3890: @subsection @code{dp_lcm}
3891: @findex dp_lcm
3892:
3893: @table @t
3894: @item dp_lcm(@var{dpoly1},@var{dpoly2})
1.2 noro 3895: \JP :: $B:G>.8xG\9`$rJV$9(B.
3896: \EG :: Returns the least common multiple of the head terms of the given two polynomials.
1.1 noro 3897: @end table
3898:
3899: @table @var
3900: @item return
1.2 noro 3901: \JP $BJ,;6I=8=B?9`<0(B
3902: \EG distributed polynomial
1.4 noro 3903: @item dpoly1 dpoly2
1.2 noro 3904: \JP $BJ,;6I=8=B?9`<0(B
3905: \EG distributed polynomial
1.1 noro 3906: @end table
3907:
3908: @itemize @bullet
1.2 noro 3909: \BJP
1.1 noro 3910: @item
3911: $B$=$l$>$l$N0z?t$NF,9`$N:G>.8xG\9`$rJV$9(B. $B78?t$O(B 1 $B$G$"$k(B.
1.2 noro 3912: \E
3913: \BEG
3914: @item
3915: Returns the least common multiple of the head terms of the given
3916: two polynomials, where coefficient is always set to 1.
3917: \E
1.1 noro 3918: @end itemize
3919:
3920: @example
3921: [100] dp_lcm(<<1,2,3,4,5>>,<<5,4,3,2,1>>);
3922: (1)*<<5,4,3,4,5>>
3923: @end example
3924:
3925: @table @t
1.2 noro 3926: \JP @item $B;2>H(B
3927: \EG @item References
1.1 noro 3928: @fref{p_nf p_nf_mod p_true_nf p_true_nf_mod}.
3929: @end table
3930:
1.2 noro 3931: \JP @node dp_redble,,, $B%0%l%V%J4pDl$K4X$9$kH!?t(B
3932: \EG @node dp_redble,,, Functions for Groebner basis computation
1.1 noro 3933: @subsection @code{dp_redble}
3934: @findex dp_redble
3935:
3936: @table @t
3937: @item dp_redble(@var{dpoly1},@var{dpoly2})
1.2 noro 3938: \JP :: $BF,9`$I$&$7$,@0=|2DG=$+$I$&$+D4$Y$k(B.
3939: \EG :: Checks whether one head term is divisible by the other head term.
1.1 noro 3940: @end table
3941:
3942: @table @var
3943: @item return
1.2 noro 3944: \JP $B@0?t(B
3945: \EG integer
1.4 noro 3946: @item dpoly1 dpoly2
1.2 noro 3947: \JP $BJ,;6I=8=B?9`<0(B
3948: \EG distributed polynomial
1.1 noro 3949: @end table
3950:
3951: @itemize @bullet
1.2 noro 3952: \BJP
1.1 noro 3953: @item
3954: @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
3955: 0 $B$rJV$9(B.
3956: @item
3957: $BB?9`<0$N4JLs$r9T$&:](B, $B$I$N9`$r4JLs$G$-$k$+$rC5$9$N$KMQ$$$k(B.
1.2 noro 3958: \E
3959: \BEG
3960: @item
3961: Returns 1 if the head term of @var{dpoly2} divides the head term of
3962: @var{dpoly1}; otherwise 0.
3963: @item
3964: Used for finding candidate terms at reduction of polynomials.
3965: \E
1.1 noro 3966: @end itemize
3967:
3968: @example
3969: [148] C;
3970: (1)*<<1,1,1,0,0>>+(1)*<<0,1,1,1,0>>+(1)*<<1,1,0,0,1>>+(1)*<<1,0,0,1,1>>
3971: [149] T;
3972: (3)*<<2,1,0,0,0>>+(3)*<<1,2,0,0,0>>+(1)*<<0,3,0,0,0>>+(6)*<<1,1,1,0,0>>
3973: [150] for ( ; T; T = dp_rest(T)) print(dp_redble(T,C));
3974: 0
3975: 0
3976: 0
3977: 1
3978: @end example
3979:
3980: @table @t
1.2 noro 3981: \JP @item $B;2>H(B
3982: \EG @item References
1.1 noro 3983: @fref{dp_red dp_red_mod}.
3984: @end table
3985:
1.23 ! noro 3986: \JP @node dpm_redble,,, $B%0%l%V%J4pDl$K4X$9$kH!?t(B
! 3987: \EG @node dpm_redble,,, Functions for Groebner basis computation
! 3988: @subsection @code{dpm_redble}
! 3989: @findex dpm_redble
! 3990:
! 3991: @table @t
! 3992: @item dpm_redble(@var{dpoly1},@var{dpoly2})
! 3993: \JP :: $BF,9`$I$&$7$,@0=|2DG=$+$I$&$+D4$Y$k(B.
! 3994: \EG :: Checks whether one head term is divisible by the other head term.
! 3995: @end table
! 3996:
! 3997: @table @var
! 3998: @item return
! 3999: \JP $B@0?t(B
! 4000: \EG integer
! 4001: @item dpoly1 dpoly2
! 4002: \JP $B2C72B?9`<0(B
! 4003: \EG module polynomial
! 4004: @end table
! 4005:
! 4006: @itemize @bullet
! 4007: \BJP
! 4008: @item
! 4009: @var{dpoly1} $B$NF,9`$,(B @var{dpoly2} $B$NF,9`$G3d$j@Z$l$l$P(B 1, $B3d$j@Z$l$J$1$l$P(B
! 4010: 0 $B$rJV$9(B.
! 4011: @item
! 4012: $BB?9`<0$N4JLs$r9T$&:](B, $B$I$N9`$r4JLs$G$-$k$+$rC5$9$N$KMQ$$$k(B.
! 4013: \E
! 4014: \BEG
! 4015: @item
! 4016: Returns 1 if the head term of @var{dpoly2} divides the head term of
! 4017: @var{dpoly1}; otherwise 0.
! 4018: @item
! 4019: Used for finding candidate terms at reduction of polynomials.
! 4020: \E
! 4021: @end itemize
! 4022:
1.2 noro 4023: \JP @node dp_subd,,, $B%0%l%V%J4pDl$K4X$9$kH!?t(B
4024: \EG @node dp_subd,,, Functions for Groebner basis computation
1.1 noro 4025: @subsection @code{dp_subd}
4026: @findex dp_subd
4027:
4028: @table @t
4029: @item dp_subd(@var{dpoly1},@var{dpoly2})
1.2 noro 4030: \JP :: $BF,9`$N>&C19`<0$rJV$9(B.
4031: \EG :: Returns the quotient monomial of the head terms.
1.1 noro 4032: @end table
4033:
4034: @table @var
4035: @item return
1.2 noro 4036: \JP $BJ,;6I=8=B?9`<0(B
4037: \EG distributed polynomial
1.4 noro 4038: @item dpoly1 dpoly2
1.2 noro 4039: \JP $BJ,;6I=8=B?9`<0(B
4040: \EG distributed polynomial
1.1 noro 4041: @end table
4042:
4043: @itemize @bullet
1.2 noro 4044: \BJP
1.1 noro 4045: @item
4046: @code{dp_ht(@var{dpoly1})/dp_ht(@var{dpoly2})} $B$r5a$a$k(B. $B7k2L$N78?t$O(B 1
4047: $B$G$"$k(B.
4048: @item
4049: $B3d$j@Z$l$k$3$H$,$"$i$+$8$a$o$+$C$F$$$kI,MW$,$"$k(B.
1.2 noro 4050: \E
4051: \BEG
4052: @item
4053: Gets @code{dp_ht(@var{dpoly1})/dp_ht(@var{dpoly2})}.
4054: The coefficient of the result is always set to 1.
4055: @item
4056: Divisibility assumed.
4057: \E
1.1 noro 4058: @end itemize
4059:
4060: @example
4061: [162] dp_subd(<<1,2,3,4,5>>,<<1,1,2,3,4>>);
4062: (1)*<<0,1,1,1,1>>
4063: @end example
4064:
4065: @table @t
1.2 noro 4066: \JP @item $B;2>H(B
4067: \EG @item References
1.1 noro 4068: @fref{dp_red dp_red_mod}.
4069: @end table
4070:
1.2 noro 4071: \JP @node dp_vtoe dp_etov,,, $B%0%l%V%J4pDl$K4X$9$kH!?t(B
4072: \EG @node dp_vtoe dp_etov,,, Functions for Groebner basis computation
1.1 noro 4073: @subsection @code{dp_vtoe}, @code{dp_etov}
4074: @findex dp_vtoe
4075: @findex dp_etov
4076:
4077: @table @t
4078: @item dp_vtoe(@var{vect})
1.2 noro 4079: \JP :: $B;X?t%Y%/%H%k$r9`$KJQ49(B
4080: \EG :: Converts an exponent vector into a term.
1.1 noro 4081: @item dp_etov(@var{dpoly})
1.2 noro 4082: \JP :: $BF,9`$r;X?t%Y%/%H%k$KJQ49(B
4083: \EG :: Convert the head term of a distributed polynomial into an exponent vector.
1.1 noro 4084: @end table
4085:
4086: @table @var
4087: @item return
1.2 noro 4088: \JP @code{dp_vtoe} : $BJ,;6I=8=B?9`<0(B, @code{dp_etov} : $B%Y%/%H%k(B
4089: \EG @code{dp_vtoe} : distributed polynomial, @code{dp_etov} : vector
1.1 noro 4090: @item vect
1.2 noro 4091: \JP $B%Y%/%H%k(B
4092: \EG vector
1.1 noro 4093: @item dpoly
1.2 noro 4094: \JP $BJ,;6I=8=B?9`<0(B
4095: \EG distributed polynomial
1.1 noro 4096: @end table
4097:
4098: @itemize @bullet
1.2 noro 4099: \BJP
1.1 noro 4100: @item
4101: @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.
4102: @item
4103: @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
4104: $B%Y%/%H%k$KJQ49$9$k(B.
1.2 noro 4105: \E
4106: \BEG
4107: @item
4108: @code{dp_vtoe()} generates a term whose exponent vector is @var{vect}.
4109: @item
4110: @code{dp_etov()} generates a vector which is the exponent vector of the
4111: head term of @code{dpoly}.
4112: \E
1.1 noro 4113: @end itemize
4114:
4115: @example
4116: [211] X=<<1,2,3>>;
4117: (1)*<<1,2,3>>
4118: [212] V=dp_etov(X);
4119: [ 1 2 3 ]
4120: [213] V[2]++$
4121: [214] Y=dp_vtoe(V);
4122: (1)*<<1,2,4>>
4123: @end example
4124:
1.2 noro 4125: \JP @node dp_mbase,,, $B%0%l%V%J4pDl$K4X$9$kH!?t(B
4126: \EG @node dp_mbase,,, Functions for Groebner basis computation
1.1 noro 4127: @subsection @code{dp_mbase}
4128: @findex dp_mbase
4129:
4130: @table @t
4131: @item dp_mbase(@var{dplist})
1.2 noro 4132: \JP :: monomial $B4pDl$N7W;;(B
4133: \EG :: Computes the monomial basis
1.1 noro 4134: @end table
4135:
4136: @table @var
4137: @item return
1.2 noro 4138: \JP $BJ,;6I=8=B?9`<0$N%j%9%H(B
4139: \EG list of distributed polynomial
1.1 noro 4140: @item dplist
1.2 noro 4141: \JP $BJ,;6I=8=B?9`<0$N%j%9%H(B
4142: \EG list of distributed polynomial
1.1 noro 4143: @end table
4144:
4145: @itemize @bullet
1.2 noro 4146: \BJP
1.1 noro 4147: @item
4148: $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
4149: $B$G$"$k(B @var{dplist} $B$K$D$$$F(B,
4150: @var{dplist} $B$,(B K[X] $BCf$G@8@.$9$k%$%G%"%k(B I $B$,(B 0 $B<!85$N;~(B,
4151: K $B>eM-8B<!85@~7A6u4V$G$"$k(B K[X]/I $B$N(B monomial $B$K$h$k4pDl$r5a$a$k(B.
4152: @item
4153: $BF@$i$l$?4pDl$N8D?t$,(B, K[X]/I $B$N(B K-$B@~7A6u4V$H$7$F$N<!85$KEy$7$$(B.
1.2 noro 4154: \E
4155: \BEG
4156: @item
4157: Assuming that @var{dplist} is a list of distributed polynomials which
4158: is a Groebner basis with respect to the current ordering type and
4159: that the ideal @var{I} generated by @var{dplist} in K[X] is zero-dimensional,
4160: this function computes the monomial basis of a finite dimenstional K-vector
4161: space K[X]/I.
4162: @item
4163: The number of elements in the monomial basis is equal to the
4164: K-dimenstion of K[X]/I.
4165: \E
1.1 noro 4166: @end itemize
4167:
4168: @example
4169: [215] K=katsura(5)$
4170: [216] V=[u5,u4,u3,u2,u1,u0]$
4171: [217] G0=gr(K,V,0)$
4172: [218] H=map(dp_ptod,G0,V)$
4173: [219] map(dp_ptod,dp_mbase(H),V)$
4174: [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,
4175: 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,
4176: u1*u2,u1^2,u4*u0,u3*u0,u2*u0,u1*u0,u0^2,u4,u3,u2,u1,u0,1]
4177: @end example
4178:
4179: @table @t
1.2 noro 4180: \JP @item $B;2>H(B
4181: \EG @item References
1.1 noro 4182: @fref{gr hgr gr_mod}.
4183: @end table
4184:
1.2 noro 4185: \JP @node dp_mag,,, $B%0%l%V%J4pDl$K4X$9$kH!?t(B
4186: \EG @node dp_mag,,, Functions for Groebner basis computation
1.1 noro 4187: @subsection @code{dp_mag}
4188: @findex dp_mag
4189:
4190: @table @t
4191: @item dp_mag(@var{p})
1.2 noro 4192: \JP :: $B78?t$N%S%C%HD9$NOB$rJV$9(B
4193: \EG :: Computes the sum of bit lengths of coefficients of a distributed polynomial.
1.1 noro 4194: @end table
4195:
4196: @table @var
4197: @item return
1.2 noro 4198: \JP $B?t(B
4199: \EG integer
1.1 noro 4200: @item p
1.2 noro 4201: \JP $BJ,;6I=8=B?9`<0(B
4202: \EG distributed polynomial
1.1 noro 4203: @end table
4204:
4205: @itemize @bullet
1.2 noro 4206: \BJP
1.1 noro 4207: @item
4208: $BJ,;6I=8=B?9`<0$N78?t$K8=$l$kM-M}?t$K$D$-(B, $B$=$NJ,JlJ,;R(B ($B@0?t$N>l9g$OJ,;R(B)
4209: $B$N%S%C%HD9$NAmOB$rJV$9(B.
4210: @item
4211: $BBP>]$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$*$$$F$O(B
4212: $B78?tKDD%$,LdBj$H$J$j(B, $BESCf@8@.$5$l$kB?9`<0$,78?tKDD%$r5/$3$7$F$$$k$+$I$&$+(B
4213: $B$NH=Dj$KLrN)$D(B.
4214: @item
4215: @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
4216: $BESCf@8@.$5$l$kB?9`<0$K$?$$$9$k(B @code{dp_mag()} $B$NCM$r8+$k$3$H$,$G$-$k(B.
1.2 noro 4217: \E
4218: \BEG
4219: @item
4220: This function computes the sum of bit lengths of coefficients of a
4221: distributed polynomial @var{p}. If a coefficient is non integral,
4222: the sum of bit lengths of the numerator and the denominator is taken.
4223: @item
4224: This is a measure of the size of a polynomial. Especially for
4225: zero-dimensional system coefficient swells are often serious and
4226: the returned value is useful to detect such swells.
4227: @item
4228: If @code{ShowMag} and @code{Print} for @code{dp_gr_flags()} are on,
4229: values of @code{dp_mag()} for intermediate basis elements are shown.
4230: \E
1.1 noro 4231: @end itemize
4232:
4233: @example
4234: [221] X=dp_ptod((x+2*y)^10,[x,y])$
4235: [222] dp_mag(X);
4236: 115
4237: @end example
4238:
4239: @table @t
1.2 noro 4240: \JP @item $B;2>H(B
4241: \EG @item References
1.1 noro 4242: @fref{dp_gr_flags dp_gr_print}.
4243: @end table
4244:
1.2 noro 4245: \JP @node dp_red dp_red_mod,,, $B%0%l%V%J4pDl$K4X$9$kH!?t(B
4246: \EG @node dp_red dp_red_mod,,, Functions for Groebner basis computation
1.1 noro 4247: @subsection @code{dp_red}, @code{dp_red_mod}
4248: @findex dp_red
4249: @findex dp_red_mod
4250:
4251: @table @t
4252: @item dp_red(@var{dpoly1},@var{dpoly2},@var{dpoly3})
4253: @item dp_red_mod(@var{dpoly1},@var{dpoly2},@var{dpoly3},@var{mod})
1.2 noro 4254: \JP :: $B0l2s$N4JLsA`:n(B
4255: \EG :: Single reduction operation
1.1 noro 4256: @end table
4257:
4258: @table @var
4259: @item return
1.2 noro 4260: \JP $B%j%9%H(B
4261: \EG list
1.4 noro 4262: @item dpoly1 dpoly2 dpoly3
1.2 noro 4263: \JP $BJ,;6I=8=B?9`<0(B
4264: \EG distributed polynomial
1.1 noro 4265: @item vlist
1.2 noro 4266: \JP $B%j%9%H(B
4267: \EG list
1.1 noro 4268: @item mod
1.2 noro 4269: \JP $BAG?t(B
4270: \EG prime
1.1 noro 4271: @end table
4272:
4273: @itemize @bullet
1.2 noro 4274: \BJP
1.1 noro 4275: @item
4276: @var{dpoly1} + @var{dpoly2} $B$J$kJ,;6I=8=B?9`<0$r(B @var{dpoly3} $B$G(B
4277: 1 $B2s4JLs$9$k(B.
4278: @item
4279: @code{dp_red_mod()} $B$NF~NO$O(B, $BA4$FM-8BBN78?t$KJQ49$5$l$F$$$kI,MW$,$"$k(B.
4280: @item
4281: $B4JLs$5$l$k9`$O(B @var{dpoly2} $B$NF,9`$G$"$k(B. $B=>$C$F(B, @var{dpoly2} $B$N(B
4282: $BF,9`$,(B @var{dpoly3} $B$NF,9`$G3d$j@Z$l$k$3$H$,$"$i$+$8$a$o$+$C$F$$$J$1$l$P(B
4283: $B$J$i$J$$(B.
4284: @item
4285: $B0z?t$,@0?t78?t$N;~(B, $B4JLs$O(B, $BJ,?t$,8=$l$J$$$h$&(B, $B@0?t(B @var{a}, @var{b},
1.4 noro 4286: $B9`(B @var{t} $B$K$h$j(B @var{a}(@var{dpoly1} + @var{dpoly2})-@var{bt} @var{dpoly3} $B$H$7$F7W;;$5$l$k(B.
1.1 noro 4287: @item
4288: $B7k2L$O(B, @code{[@var{a dpoly1},@var{a dpoly2 - bt dpoly3}]} $B$J$k%j%9%H$G$"$k(B.
1.2 noro 4289: \E
4290: \BEG
4291: @item
4292: Reduces a distributed polynomial, @var{dpoly1} + @var{dpoly2},
4293: by @var{dpoly3} for single time.
4294: @item
4295: An input for @code{dp_red_mod()} must be converted into a distributed
4296: polynomial with coefficients in a finite field.
4297: @item
4298: This implies that
4299: the divisibility of the head term of @var{dpoly2} by the head term of
4300: @var{dpoly3} is assumed.
4301: @item
4302: When integral coefficients, computation is so carefully performed that
4303: no rational operations appear in the reduction procedure.
4304: It is computed for integers @var{a} and @var{b}, and a term @var{t} as:
1.4 noro 4305: @var{a}(@var{dpoly1} + @var{dpoly2})-@var{bt} @var{dpoly3}.
1.2 noro 4306: @item
4307: The result is a list @code{[@var{a dpoly1},@var{a dpoly2 - bt dpoly3}]}.
4308: \E
1.1 noro 4309: @end itemize
4310:
4311: @example
4312: [157] D=(3)*<<2,1,0,0,0>>+(3)*<<1,2,0,0,0>>+(1)*<<0,3,0,0,0>>;
4313: (3)*<<2,1,0,0,0>>+(3)*<<1,2,0,0,0>>+(1)*<<0,3,0,0,0>>
4314: [158] R=(6)*<<1,1,1,0,0>>;
4315: (6)*<<1,1,1,0,0>>
4316: [159] C=12*<<1,1,1,0,0>>+(1)*<<0,1,1,1,0>>+(1)*<<1,1,0,0,1>>;
4317: (12)*<<1,1,1,0,0>>+(1)*<<0,1,1,1,0>>+(1)*<<1,1,0,0,1>>
4318: [160] dp_red(D,R,C);
1.5 noro 4319: [(6)*<<2,1,0,0,0>>+(6)*<<1,2,0,0,0>>+(2)*<<0,3,0,0,0>>,
4320: (-1)*<<0,1,1,1,0>>+(-1)*<<1,1,0,0,1>>]
1.1 noro 4321: @end example
4322:
4323: @table @t
1.2 noro 4324: \JP @item $B;2>H(B
4325: \EG @item References
1.1 noro 4326: @fref{dp_mod dp_rat}.
4327: @end table
4328:
1.2 noro 4329: \JP @node dp_sp dp_sp_mod,,, $B%0%l%V%J4pDl$K4X$9$kH!?t(B
4330: \EG @node dp_sp dp_sp_mod,,, Functions for Groebner basis computation
1.1 noro 4331: @subsection @code{dp_sp}, @code{dp_sp_mod}
4332: @findex dp_sp
4333: @findex dp_sp_mod
4334:
4335: @table @t
4336: @item dp_sp(@var{dpoly1},@var{dpoly2})
4337: @item dp_sp_mod(@var{dpoly1},@var{dpoly2},@var{mod})
1.2 noro 4338: \JP :: S-$BB?9`<0$N7W;;(B
4339: \EG :: Computation of an S-polynomial
1.1 noro 4340: @end table
4341:
4342: @table @var
4343: @item return
1.2 noro 4344: \JP $BJ,;6I=8=B?9`<0(B
4345: \EG distributed polynomial
1.4 noro 4346: @item dpoly1 dpoly2
1.2 noro 4347: \JP $BJ,;6I=8=B?9`<0(B
4348: \EG distributed polynomial
1.1 noro 4349: @item mod
1.2 noro 4350: \JP $BAG?t(B
4351: \EG prime
1.1 noro 4352: @end table
4353:
4354: @itemize @bullet
1.2 noro 4355: \BJP
1.1 noro 4356: @item
4357: @var{dpoly1}, @var{dpoly2} $B$N(B S-$BB?9`<0$r7W;;$9$k(B.
4358: @item
4359: @code{dp_sp_mod()} $B$NF~NO$O(B, $BA4$FM-8BBN78?t$KJQ49$5$l$F$$$kI,MW$,$"$k(B.
4360: @item
4361: $B7k2L$KM-M}?t(B, $BM-M}<0$,F~$k$N$rHr$1$k$?$a(B, $B7k2L$,Dj?tG\(B, $B$"$k$$$OB?9`<0(B
4362: $BG\$5$l$F$$$k2DG=@-$,$"$k(B.
1.2 noro 4363: \E
4364: \BEG
4365: @item
4366: This function computes the S-polynomial of @var{dpoly1} and @var{dpoly2}.
4367: @item
4368: Inputs of @code{dp_sp_mod()} must be polynomials with coefficients in a
4369: finite field.
4370: @item
4371: The result may be multiplied by a constant in the ground field in order to
4372: make the result integral.
4373: \E
1.1 noro 4374: @end itemize
4375:
4376: @example
4377: [227] X=dp_ptod(x^2*y+x*y,[x,y]);
4378: (1)*<<2,1>>+(1)*<<1,1>>
4379: [228] Y=dp_ptod(x*y^2+x*y,[x,y]);
4380: (1)*<<1,2>>+(1)*<<1,1>>
4381: [229] dp_sp(X,Y);
4382: (-1)*<<2,1>>+(1)*<<1,2>>
4383: @end example
4384:
4385: @table @t
1.2 noro 4386: \JP @item $B;2>H(B
4387: \EG @item References
1.1 noro 4388: @fref{dp_mod dp_rat}.
4389: @end table
1.23 ! noro 4390:
! 4391: \JP @node dpm_sp,,, $B%0%l%V%J4pDl$K4X$9$kH!?t(B
! 4392: \EG @node dmp_sp,,, Functions for Groebner basis computation
! 4393: @subsection @code{dpm_sp}
! 4394: @findex dpm_sp
! 4395:
! 4396: @table @t
! 4397: @item dpm_sp(@var{dpoly1},@var{dpoly2}[|coef=1])
! 4398: \JP :: S-$BB?9`<0$N7W;;(B
! 4399: \EG :: Computation of an S-polynomial
! 4400: @end table
! 4401:
! 4402: @table @var
! 4403: @item return
! 4404: \JP $B2C72B?9`<0$^$?$O%j%9%H(B
! 4405: \EG module polynomial or list
! 4406: @item dpoly1 dpoly2
! 4407: \JP $B2C72B?9`<0(B
! 4408: \EG module polynomial
! 4409: \JP $BJ,;6I=8=B?9`<0(B
! 4410: @end table
! 4411:
! 4412: @itemize @bullet
! 4413: \BJP
! 4414: @item
! 4415: @var{dpoly1}, @var{dpoly2} $B$N(B S-$BB?9`<0$r7W;;$9$k(B.
! 4416: @item
! 4417: $B%*%W%7%g%s(B @var{coef=1} $B$,;XDj$5$l$F$$$k>l9g(B, @code{[S,t1,t2]} $B$J$k%j%9%H$rJV$9(B.
! 4418: $B$3$3$G(B, @code{t1}, @code{t2} $B$O(BS-$BB?9`<0$r:n$k:]$N78?tC19`<0$G(B @code{S=t1 dpoly1-t2 dpoly2}
! 4419: $B$rK~$?$9(B.
! 4420: \E
! 4421: \BEG
! 4422: @item
! 4423: This function computes the S-polynomial of @var{dpoly1} and @var{dpoly2}.
! 4424: @item
! 4425: If an option @var{coef=1} is specified, it returns a list @code{[S,t1,t2]},
! 4426: where @code{S} is the S-polynmial and @code{t1}, @code{t2} are monomials satisfying @code{S=t1 dpoly1-t2 dpoly2}.
! 4427: \E
! 4428: @end itemize
! 4429:
1.2 noro 4430: \JP @node p_nf p_nf_mod p_true_nf p_true_nf_mod,,, $B%0%l%V%J4pDl$K4X$9$kH!?t(B
4431: \EG @node p_nf p_nf_mod p_true_nf p_true_nf_mod,,, Functions for Groebner basis computation
1.1 noro 4432: @subsection @code{p_nf}, @code{p_nf_mod}, @code{p_true_nf}, @code{p_true_nf_mod}
4433: @findex p_nf
4434: @findex p_nf_mod
4435: @findex p_true_nf
4436: @findex p_true_nf_mod
4437:
4438: @table @t
4439: @item p_nf(@var{poly},@var{plist},@var{vlist},@var{order})
4440: @itemx p_nf_mod(@var{poly},@var{plist},@var{vlist},@var{order},@var{mod})
1.2 noro 4441: \JP :: $BI=8=B?9`<0$N@55,7A$r5a$a$k(B. ($B7k2L$ODj?tG\$5$l$F$$$k2DG=@-$"$j(B)
4442: \BEG
4443: :: Computes the normal form of the given polynomial.
4444: (The result may be multiplied by a constant.)
4445: \E
1.1 noro 4446: @item p_true_nf(@var{poly},@var{plist},@var{vlist},@var{order})
4447: @itemx p_true_nf_mod(@var{poly},@var{plist},@var{vlist},@var{order},@var{mod})
1.2 noro 4448: \JP :: $BI=8=B?9`<0$N@55,7A$r5a$a$k(B. ($B??$N7k2L$r(B @code{[$BJ,;R(B, $BJ,Jl(B]} $B$N7A$GJV$9(B)
4449: \BEG
4450: :: Computes the normal form of the given polynomial. (The result is returned
4451: as a form of @code{[numerator, denominator]})
4452: \E
1.1 noro 4453: @end table
4454:
4455: @table @var
4456: @item return
1.2 noro 4457: \JP @code{p_nf} : $BB?9`<0(B, @code{p_true_nf} : $B%j%9%H(B
4458: \EG @code{p_nf} : polynomial, @code{p_true_nf} : list
1.1 noro 4459: @item poly
1.2 noro 4460: \JP $BB?9`<0(B
4461: \EG polynomial
1.4 noro 4462: @item plist vlist
1.2 noro 4463: \JP $B%j%9%H(B
4464: \EG list
1.1 noro 4465: @item order
1.2 noro 4466: \JP $B?t(B, $B%j%9%H$^$?$O9TNs(B
4467: \EG number, list or matrix
1.1 noro 4468: @item mod
1.2 noro 4469: \JP $BAG?t(B
4470: \EG prime
1.1 noro 4471: @end table
4472:
4473: @itemize @bullet
1.2 noro 4474: \BJP
1.1 noro 4475: @item
4476: @samp{gr} $B$GDj5A$5$l$F$$$k(B.
4477: @item
4478: $BB?9`<0$N(B, $BB?9`<0%j%9%H$K$h$k@55,7A$r5a$a$k(B.
4479: @item
4480: @code{dp_nf()}, @code{dp_true_nf()}, @code{dp_nf_mod()}, @code{dp_true_nf_mod}
4481: $B$KBP$9$k%$%s%?%U%'!<%9$G$"$k(B.
4482: @item
4483: @var{poly} $B$*$h$S(B @var{plist} $B$O(B, $BJQ?t=g=x(B @var{vlist} $B$*$h$S(B
4484: $BJQ?t=g=x7?(B @var{otype} $B$K=>$C$FJ,;6I=8=B?9`<0$KJQ49$5$l(B,
4485: @code{dp_nf()}, @code{dp_true_nf()}, @code{dp_nf_mod()},
4486: @code{dp_true_nf_mod()} $B$KEO$5$l$k(B.
4487: @item
4488: @code{dp_nf()}, @code{dp_true_nf()}, @code{dp_nf_mod()},
4489: @code{dp_true_nf_mod()} $B$O(B @var{fullreduce} $B$,(B 1 $B$G8F$S=P$5$l$k(B.
4490: @item
4491: $B7k2L$OB?9`<0$KJQ49$5$l$F=PNO$5$l$k(B.
4492: @item
4493: @code{p_true_nf()}, @code{p_true_nf_mod()} $B$N=PNO$K4X$7$F$O(B,
4494: @code{dp_true_nf()}, @code{dp_true_nf_mod()} $B$N9`$r;2>H(B.
1.2 noro 4495: \E
4496: \BEG
4497: @item
4498: Defined in the package @samp{gr}.
4499: @item
4500: Obtains the normal form of a polynomial by a polynomial list.
4501: @item
4502: These are interfaces to @code{dp_nf()}, @code{dp_true_nf()}, @code{dp_nf_mod()},
4503: @code{dp_true_nf_mod}
4504: @item
4505: The polynomial @var{poly} and the polynomials in @var{plist} is
4506: converted, according to the variable ordering @var{vlist} and
4507: type of term ordering @var{otype}, into their distributed polynomial
4508: counterparts and passed to @code{dp_nf()}.
4509: @item
4510: @code{dp_nf()}, @code{dp_true_nf()}, @code{dp_nf_mod()} and
4511: @code{dp_true_nf_mod()}
4512: is called with value 1 for @var{fullreduce}.
4513: @item
4514: The result is converted back into an ordinary polynomial.
4515: @item
4516: As for @code{p_true_nf()}, @code{p_true_nf_mod()}
4517: refer to @code{dp_true_nf()} and @code{dp_true_nf_mod()}.
4518: \E
1.1 noro 4519: @end itemize
4520:
4521: @example
4522: [79] K = katsura(5)$
4523: [80] V = [u5,u4,u3,u2,u1,u0]$
4524: [81] G = hgr(K,V,2)$
4525: [82] p_nf(K[1],G,V,2);
4526: 0
4527: [83] L = p_true_nf(K[1]+1,G,V,2);
4528: [-1503...,-1503...]
4529: [84] L[0]/L[1];
4530: 1
4531: @end example
4532:
4533: @table @t
1.2 noro 4534: \JP @item $B;2>H(B
4535: \EG @item References
1.1 noro 4536: @fref{dp_ptod},
4537: @fref{dp_dtop},
4538: @fref{dp_ord},
1.19 noro 4539: @fref{dp_nf dp_nf_mod dp_true_nf dp_true_nf_mod dp_weyl_nf dp_weyl_nf_mod}.
1.1 noro 4540: @end table
4541:
1.2 noro 4542: \JP @node p_terms,,, $B%0%l%V%J4pDl$K4X$9$kH!?t(B
4543: \EG @node p_terms,,, Functions for Groebner basis computation
1.1 noro 4544: @subsection @code{p_terms}
4545: @findex p_terms
4546:
4547: @table @t
4548: @item p_terms(@var{poly},@var{vlist},@var{order})
1.2 noro 4549: \JP :: $BB?9`<0$K$"$i$o$l$kC19`$r%j%9%H$K$9$k(B.
4550: \EG :: Monomials appearing in the given polynomial is collected into a list.
1.1 noro 4551: @end table
4552:
4553: @table @var
4554: @item return
1.2 noro 4555: \JP $B%j%9%H(B
4556: \EG list
1.1 noro 4557: @item poly
1.2 noro 4558: \JP $BB?9`<0(B
4559: \EG polynomial
1.1 noro 4560: @item vlist
1.2 noro 4561: \JP $B%j%9%H(B
4562: \EG list
1.1 noro 4563: @item order
1.2 noro 4564: \JP $B?t(B, $B%j%9%H$^$?$O9TNs(B
4565: \EG number, list or matrix
1.1 noro 4566: @end table
4567:
4568: @itemize @bullet
1.2 noro 4569: \BJP
1.1 noro 4570: @item
4571: @samp{gr} $B$GDj5A$5$l$F$$$k(B.
4572: @item
4573: $BB?9`<0$rC19`$KE83+$7$?;~$K8=$l$k9`$r%j%9%H$K$7$FJV$9(B.
4574: @var{vlist} $B$*$h$S(B @var{order} $B$K$h$jDj$^$k9`=g=x$K$h$j(B, $B=g=x$N9b$$$b$N(B
4575: $B$,%j%9%H$N@hF,$KMh$k$h$&$K%=!<%H$5$l$k(B.
4576: @item
4577: $B%0%l%V%J4pDl$O$7$P$7$P78?t$,5pBg$K$J$k$?$a(B, $B<B:]$K$I$N9`$,8=$l$F(B
4578: $B$$$k$N$+$r8+$k$?$a$J$I$KMQ$$$k(B.
1.2 noro 4579: \E
4580: \BEG
4581: @item
4582: Defined in the package @samp{gr}.
4583: @item
4584: This returns a list which contains all non-zero monomials in the given
4585: polynomial. The monomials are ordered according to the current
4586: type of term ordering and @var{vlist}.
4587: @item
4588: Since polynomials in a Groebner base often have very large coefficients,
4589: examining a polynomial as it is may sometimes be difficult to perform.
4590: For such a case, this function enables to examine which term is really
4591: exists.
4592: \E
1.1 noro 4593: @end itemize
4594:
4595: @example
4596: [233] G=gr(katsura(5),[u5,u4,u3,u2,u1,u0],2)$
4597: [234] p_terms(G[0],[u5,u4,u3,u2,u1,u0],2);
1.5 noro 4598: [u5,u0^31,u0^30,u0^29,u0^28,u0^27,u0^26,u0^25,u0^24,u0^23,u0^22,
4599: u0^21,u0^20,u0^19,u0^18,u0^17,u0^16,u0^15,u0^14,u0^13,u0^12,u0^11,
4600: u0^10,u0^9,u0^8,u0^7,u0^6,u0^5,u0^4,u0^3,u0^2,u0,1]
1.1 noro 4601: @end example
4602:
1.2 noro 4603: \JP @node gb_comp,,, $B%0%l%V%J4pDl$K4X$9$kH!?t(B
4604: \EG @node gb_comp,,, Functions for Groebner basis computation
1.1 noro 4605: @subsection @code{gb_comp}
4606: @findex gb_comp
4607:
4608: @table @t
4609: @item gb_comp(@var{plist1}, @var{plist2})
1.2 noro 4610: \JP :: $BB?9`<0%j%9%H$,(B, $BId9f$r=|$$$F=89g$H$7$FEy$7$$$+$I$&$+D4$Y$k(B.
4611: \EG :: Checks whether two polynomial lists are equal or not as a set
1.1 noro 4612: @end table
4613:
4614: @table @var
1.2 noro 4615: \JP @item return 0 $B$^$?$O(B 1
4616: \EG @item return 0 or 1
1.4 noro 4617: @item plist1 plist2
1.1 noro 4618: @end table
4619:
4620: @itemize @bullet
1.2 noro 4621: \BJP
1.1 noro 4622: @item
4623: @var{plist1}, @var{plist2} $B$K$D$$$F(B, $BId9f$r=|$$$F=89g$H$7$FEy$7$$$+$I$&$+(B
4624: $BD4$Y$k(B.
4625: @item
4626: $B0[$J$kJ}K!$G5a$a$?%0%l%V%J4pDl$O(B, $B4pDl$N=g=x(B, $BId9f$,0[$J$k>l9g$,$"$j(B,
4627: $B$=$l$i$,Ey$7$$$+$I$&$+$rD4$Y$k$?$a$KMQ$$$k(B.
1.2 noro 4628: \E
4629: \BEG
4630: @item
4631: This function checks whether @var{plist1} and @var{plist2} are equal or
4632: not as a set .
4633: @item
4634: For the same input and the same term ordering different
4635: functions for Groebner basis computations may produce different outputs
4636: as lists. This function compares such lists whether they are equal
4637: as a generating set of an ideal.
4638: \E
1.1 noro 4639: @end itemize
4640:
4641: @example
4642: [243] C=cyclic(6)$
4643: [244] V=[c0,c1,c2,c3,c4,c5]$
4644: [245] G0=gr(C,V,0)$
4645: [246] G=tolex(G0,V,0,V)$
4646: [247] GG=lex_tl(C,V,0,V,0)$
4647: [248] gb_comp(G,GG);
4648: 1
4649: @end example
4650:
1.2 noro 4651: \JP @node katsura hkatsura cyclic hcyclic,,, $B%0%l%V%J4pDl$K4X$9$kH!?t(B
4652: \EG @node katsura hkatsura cyclic hcyclic,,, Functions for Groebner basis computation
1.1 noro 4653: @subsection @code{katsura}, @code{hkatsura}, @code{cyclic}, @code{hcyclic}
4654: @findex katsura
4655: @findex hkatsura
4656: @findex cyclic
4657: @findex hcyclic
4658:
4659: @table @t
4660: @item katsura(@var{n})
4661: @item hkatsura(@var{n})
4662: @item cyclic(@var{n})
4663: @item hcyclic(@var{n})
1.2 noro 4664: \JP :: $BB?9`<0%j%9%H$N@8@.(B
4665: \EG :: Generates a polynomial list of standard benchmark.
1.1 noro 4666: @end table
4667:
4668: @table @var
4669: @item return
1.2 noro 4670: \JP $B%j%9%H(B
4671: \EG list
1.1 noro 4672: @item n
1.2 noro 4673: \JP $B@0?t(B
4674: \EG integer
1.1 noro 4675: @end table
4676:
4677: @itemize @bullet
1.2 noro 4678: \BJP
1.1 noro 4679: @item
4680: @code{katsura()} $B$O(B @samp{katsura}, @code{cyclic()} $B$O(B @samp{cyclic}
4681: $B$GDj5A$5$l$F$$$k(B.
4682: @item
4683: $B%0%l%V%J4pDl7W;;$G$7$P$7$P%F%9%H(B, $B%Y%s%A%^!<%/$KMQ$$$i$l$k(B @code{katsura},
4684: @code{cyclic} $B$*$h$S$=$N@F<!2=$r@8@.$9$k(B.
4685: @item
4686: @code{cyclic} $B$O(B @code{Arnborg}, @code{Lazard}, @code{Davenport} $B$J$I$N(B
4687: $BL>$G8F$P$l$k$3$H$b$"$k(B.
1.2 noro 4688: \E
4689: \BEG
4690: @item
4691: Function @code{katsura()} is defined in @samp{katsura}, and
4692: function @code{cyclic()} in @samp{cyclic}.
4693: @item
4694: These functions generate a series of polynomial sets, respectively,
4695: which are often used for testing and bench marking:
4696: @code{katsura}, @code{cyclic} and their homogenized versions.
4697: @item
4698: Polynomial set @code{cyclic} is sometimes called by other name:
4699: @code{Arnborg}, @code{Lazard}, and @code{Davenport}.
4700: \E
1.1 noro 4701: @end itemize
4702:
4703: @example
4704: [74] load("katsura")$
4705: [79] load("cyclic")$
4706: [89] katsura(5);
4707: [u0+2*u4+2*u3+2*u2+2*u1+2*u5-1,2*u4*u0-u4+2*u1*u3+u2^2+2*u5*u1,
1.5 noro 4708: 2*u3*u0+2*u1*u4-u3+(2*u1+2*u5)*u2,2*u2*u0+2*u2*u4+(2*u1+2*u5)*u3
4709: -u2+u1^2,2*u1*u0+(2*u3+2*u5)*u4+2*u2*u3+2*u1*u2-u1,
1.1 noro 4710: u0^2-u0+2*u4^2+2*u3^2+2*u2^2+2*u1^2+2*u5^2]
4711: [90] hkatsura(5);
4712: [-t+u0+2*u4+2*u3+2*u2+2*u1+2*u5,
4713: -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,
4714: -u2*t+2*u2*u0+2*u2*u4+(2*u1+2*u5)*u3+u1^2,
4715: -u1*t+2*u1*u0+(2*u3+2*u5)*u4+2*u2*u3+2*u1*u2,
4716: -u0*t+u0^2+2*u4^2+2*u3^2+2*u2^2+2*u1^2+2*u5^2]
4717: [91] cyclic(6);
4718: [c5*c4*c3*c2*c1*c0-1,
4719: ((((c4+c5)*c3+c5*c4)*c2+c5*c4*c3)*c1+c5*c4*c3*c2)*c0+c5*c4*c3*c2*c1,
4720: (((c3+c5)*c2+c5*c4)*c1+c5*c4*c3)*c0+c4*c3*c2*c1+c5*c4*c3*c2,
4721: ((c2+c5)*c1+c5*c4)*c0+c3*c2*c1+c4*c3*c2+c5*c4*c3,
4722: (c1+c5)*c0+c2*c1+c3*c2+c4*c3+c5*c4,c0+c1+c2+c3+c4+c5]
4723: [92] hcyclic(6);
4724: [-c^6+c5*c4*c3*c2*c1*c0,
4725: ((((c4+c5)*c3+c5*c4)*c2+c5*c4*c3)*c1+c5*c4*c3*c2)*c0+c5*c4*c3*c2*c1,
4726: (((c3+c5)*c2+c5*c4)*c1+c5*c4*c3)*c0+c4*c3*c2*c1+c5*c4*c3*c2,
4727: ((c2+c5)*c1+c5*c4)*c0+c3*c2*c1+c4*c3*c2+c5*c4*c3,
4728: (c1+c5)*c0+c2*c1+c3*c2+c4*c3+c5*c4,c0+c1+c2+c3+c4+c5]
4729: @end example
4730:
4731: @table @t
1.2 noro 4732: \JP @item $B;2>H(B
4733: \EG @item References
1.1 noro 4734: @fref{dp_dtop}.
4735: @end table
4736:
1.3 noro 4737: \JP @node primadec primedec,,, $B%0%l%V%J4pDl$K4X$9$kH!?t(B
4738: \EG @node primadec primedec,,, Functions for Groebner basis computation
4739: @subsection @code{primadec}, @code{primedec}
4740: @findex primadec
4741: @findex primedec
4742:
4743: @table @t
4744: @item primadec(@var{plist},@var{vlist})
4745: @item primedec(@var{plist},@var{vlist})
4746: \JP :: $B%$%G%"%k$NJ,2r(B
4747: \EG :: Computes decompositions of ideals.
4748: @end table
4749:
4750: @table @var
4751: @item return
4752: @itemx plist
4753: \JP $BB?9`<0%j%9%H(B
4754: \EG list of polynomials
4755: @item vlist
4756: \JP $BJQ?t%j%9%H(B
4757: \EG list of variables
4758: @end table
4759:
4760: @itemize @bullet
4761: \BJP
4762: @item
4763: @code{primadec()}, @code{primedec} $B$O(B @samp{primdec} $B$GDj5A$5$l$F$$$k(B.
4764: @item
4765: @code{primadec()}, @code{primedec()} $B$O$=$l$>$lM-M}?tBN>e$G$N%$%G%"%k$N(B
4766: $B=`AGJ,2r(B, $B:,4p$NAG%$%G%"%kJ,2r$r9T$&(B.
4767: @item
4768: $B0z?t$OB?9`<0%j%9%H$*$h$SJQ?t%j%9%H$G$"$k(B. $BB?9`<0$OM-M}?t78?t$N$_$,5v$5$l$k(B.
4769: @item
4770: @code{primadec} $B$O(B @code{[$B=`AG@.J,(B, $BIUB0AG%$%G%"%k(B]} $B$N%j%9%H$rJV$9(B.
4771: @item
4772: @code{primadec} $B$O(B $BAG0x;R$N%j%9%H$rJV$9(B.
4773: @item
4774: $B7k2L$K$*$$$F(B, $BB?9`<0%j%9%H$H$7$FI=<($5$l$F$$$k3F%$%G%"%k$OA4$F(B
4775: $B%0%l%V%J4pDl$G$"$k(B. $BBP1~$9$k9`=g=x$O(B, $B$=$l$>$l(B
4776: $BJQ?t(B @code{PRIMAORD}, @code{PRIMEORD} $B$K3JG<$5$l$F$$$k(B.
4777: @item
4778: @code{primadec} $B$O(B @code{[Shimoyama,Yokoyama]} $B$N=`AGJ,2r%"%k%4%j%:%`(B
4779: $B$r<BAu$7$F$$$k(B.
4780: @item
4781: $B$b$7AG0x;R$N$_$r5a$a$?$$$J$i(B, @code{primedec} $B$r;H$&J}$,$h$$(B.
4782: $B$3$l$O(B, $BF~NO%$%G%"%k$,:,4p%$%G%"%k$G$J$$>l9g$K(B, @code{primadec}
4783: $B$N7W;;$KM>J,$J%3%9%H$,I,MW$H$J$k>l9g$,$"$k$+$i$G$"$k(B.
4784: \E
4785: \BEG
4786: @item
4787: Function @code{primadec()} and @code{primedec} are defined in @samp{primdec}.
4788: @item
4789: @code{primadec()}, @code{primedec()} are the function for primary
4790: ideal decomposition and prime decomposition of the radical over the
4791: rationals respectively.
4792: @item
4793: The arguments are a list of polynomials and a list of variables.
4794: These functions accept ideals with rational function coefficients only.
4795: @item
4796: @code{primadec} returns the list of pair lists consisting a primary component
4797: and its associated prime.
4798: @item
4799: @code{primedec} returns the list of prime components.
4800: @item
4801: Each component is a Groebner basis and the corresponding term order
4802: is indicated by the global variables @code{PRIMAORD}, @code{PRIMEORD}
4803: respectively.
4804: @item
4805: @code{primadec} implements the primary decompostion algorithm
4806: in @code{[Shimoyama,Yokoyama]}.
4807: @item
4808: If one only wants to know the prime components of an ideal, then
4809: use @code{primedec} because @code{primadec} may need additional costs
4810: if an input ideal is not radical.
4811: \E
4812: @end itemize
4813:
4814: @example
4815: [84] load("primdec")$
4816: [102] primedec([p*q*x-q^2*y^2+q^2*y,-p^2*x^2+p^2*x+p*q*y,
4817: (q^3*y^4-2*q^3*y^3+q^3*y^2)*x-q^3*y^4+q^3*y^3,
4818: -q^3*y^4+2*q^3*y^3+(-q^3+p*q^2)*y^2],[p,q,x,y]);
4819: [[y,x],[y,p],[x,q],[q,p],[x-1,q],[y-1,p],[(y-1)*x-y,q*y^2-2*q*y-p+q]]
4820: [103] primadec([x,z*y,w*y^2,w^2*y-z^3,y^3],[x,y,z,w]);
4821: [[[x,z*y,y^2,w^2*y-z^3],[z,y,x]],[[w,x,z*y,z^3,y^3],[w,z,y,x]]]
4822: @end example
4823:
4824: @table @t
4825: \JP @item $B;2>H(B
4826: \EG @item References
4827: @fref{fctr sqfr},
4828: \JP @fref{$B9`=g=x$N@_Dj(B}.
4829: \EG @fref{Setting term orderings}.
4830: @end table
1.5 noro 4831:
4832: \JP @node primedec_mod,,, $B%0%l%V%J4pDl$K4X$9$kH!?t(B
4833: \EG @node primedec_mod,,, Functions for Groebner basis computation
4834: @subsection @code{primedec_mod}
4835: @findex primedec_mod
4836:
4837: @table @t
4838: @item primedec_mod(@var{plist},@var{vlist},@var{ord},@var{mod},@var{strategy})
4839: \JP :: $B%$%G%"%k$NJ,2r(B
4840: \EG :: Computes decompositions of ideals over small finite fields.
4841: @end table
4842:
4843: @table @var
4844: @item return
4845: @itemx plist
4846: \JP $BB?9`<0%j%9%H(B
4847: \EG list of polynomials
4848: @item vlist
4849: \JP $BJQ?t%j%9%H(B
4850: \EG list of variables
4851: @item ord
4852: \JP $B?t(B, $B%j%9%H$^$?$O9TNs(B
4853: \EG number, list or matrix
4854: @item mod
4855: \JP $B@5@0?t(B
4856: \EG positive integer
4857: @item strategy
4858: \JP $B@0?t(B
4859: \EG integer
4860: @end table
4861:
4862: @itemize @bullet
4863: \BJP
4864: @item
4865: @code{primedec_mod()} $B$O(B @samp{primdec_mod}
4866: $B$GDj5A$5$l$F$$$k(B. @code{[Yokoyama]} $B$NAG%$%G%"%kJ,2r%"%k%4%j%:%`(B
4867: $B$r<BAu$7$F$$$k(B.
4868: @item
4869: @code{primedec_mod()} $B$OM-8BBN>e$G$N%$%G%"%k$N(B
4870: $B:,4p$NAG%$%G%"%kJ,2r$r9T$$(B, $BAG%$%G%"%k$N%j%9%H$rJV$9(B.
4871: @item
4872: @code{primedec_mod()} $B$O(B, GF(@var{mod}) $B>e$G$NJ,2r$rM?$($k(B.
4873: $B7k2L$N3F@.J,$N@8@.85$O(B, $B@0?t78?tB?9`<0$G$"$k(B.
4874: @item
4875: $B7k2L$K$*$$$F(B, $BB?9`<0%j%9%H$H$7$FI=<($5$l$F$$$k3F%$%G%"%k$OA4$F(B
4876: [@var{vlist},@var{ord}] $B$G;XDj$5$l$k9`=g=x$K4X$9$k%0%l%V%J4pDl$G$"$k(B.
4877: @item
4878: @var{strategy} $B$,(B 0 $B$G$J$$$H$-(B, incremental $B$K(B component $B$N6&DL(B
4879: $BItJ,$r7W;;$9$k$3$H$K$h$k(B early termination $B$r9T$&(B. $B0lHL$K(B,
4880: $B%$%G%"%k$N<!85$,9b$$>l9g$KM-8z$@$,(B, 0 $B<!85$N>l9g$J$I(B, $B<!85$,>.$5$$(B
4881: $B>l9g$K$O(B overhead $B$,Bg$-$$>l9g$,$"$k(B.
1.7 noro 4882: @item
4883: $B7W;;ESCf$GFbIt>pJs$r8+$?$$>l9g$K$O!"(B
4884: $BA0$b$C$F(B @code{dp_gr_print(2)} $B$r<B9T$7$F$*$1$P$h$$(B.
1.5 noro 4885: \E
4886: \BEG
4887: @item
4888: Function @code{primedec_mod()}
4889: is defined in @samp{primdec_mod} and implements the prime decomposition
4890: algorithm in @code{[Yokoyama]}.
4891: @item
4892: @code{primedec_mod()}
4893: is the function for prime ideal decomposition
4894: of the radical of a polynomial ideal over small finite field,
4895: and they return a list of prime ideals, which are associated primes
4896: of the input ideal.
4897: @item
4898: @code{primedec_mod()} gives the decomposition over GF(@var{mod}).
4899: The generators of each resulting component consists of integral polynomials.
4900: @item
4901: Each resulting component is a Groebner basis with respect to
4902: a term order specified by [@var{vlist},@var{ord}].
4903: @item
4904: If @var{strategy} is non zero, then the early termination strategy
4905: is tried by computing the intersection of obtained components
4906: incrementally. In general, this strategy is useful when the krull
4907: dimension of the ideal is high, but it may add some overhead
4908: if the dimension is small.
1.7 noro 4909: @item
4910: If you want to see internal information during the computation,
4911: execute @code{dp_gr_print(2)} in advance.
1.5 noro 4912: \E
4913: @end itemize
4914:
4915: @example
4916: [0] load("primdec_mod")$
4917: [246] PP444=[x^8+x^2+t,y^8+y^2+t,z^8+z^2+t]$
4918: [247] primedec_mod(PP444,[x,y,z,t],0,2,1);
4919: [[y+z,x+z,z^8+z^2+t],[x+y,y^2+y+z^2+z+1,z^8+z^2+t],
4920: [y+z+1,x+z+1,z^8+z^2+t],[x+z,y^2+y+z^2+z+1,z^8+z^2+t],
4921: [y+z,x^2+x+z^2+z+1,z^8+z^2+t],[y+z+1,x^2+x+z^2+z+1,z^8+z^2+t],
4922: [x+z+1,y^2+y+z^2+z+1,z^8+z^2+t],[y+z+1,x+z,z^8+z^2+t],
4923: [x+y+1,y^2+y+z^2+z+1,z^8+z^2+t],[y+z,x+z+1,z^8+z^2+t]]
4924: [248]
4925: @end example
4926:
4927: @table @t
4928: \JP @item $B;2>H(B
4929: \EG @item References
4930: @fref{modfctr},
1.6 noro 4931: @fref{dp_gr_main dp_gr_mod_main dp_gr_f_main dp_weyl_gr_main dp_weyl_gr_mod_main dp_weyl_gr_f_main},
1.5 noro 4932: \JP @fref{$B9`=g=x$N@_Dj(B}.
1.7 noro 4933: \EG @fref{Setting term orderings},
4934: @fref{dp_gr_flags dp_gr_print}.
1.5 noro 4935: @end table
4936:
1.10 noro 4937: \JP @node bfunction bfct generic_bfct ann ann0,,, $B%0%l%V%J4pDl$K4X$9$kH!?t(B
4938: \EG @node bfunction bfct generic_bfct ann ann0,,, Functions for Groebner basis computation
4939: @subsection @code{bfunction}, @code{bfct}, @code{generic_bfct}, @code{ann}, @code{ann0}
1.6 noro 4940: @findex bfunction
1.9 noro 4941: @findex bfct
1.6 noro 4942: @findex generic_bfct
1.10 noro 4943: @findex ann
4944: @findex ann0
1.5 noro 4945:
1.6 noro 4946: @table @t
4947: @item bfunction(@var{f})
1.10 noro 4948: @itemx bfct(@var{f})
4949: @itemx generic_bfct(@var{plist},@var{vlist},@var{dvlist},@var{weight})
4950: \JP :: @var{b} $B4X?t$N7W;;(B
4951: \EG :: Computes the global @var{b} function of a polynomial or an ideal
4952: @item ann(@var{f})
4953: @itemx ann0(@var{f})
4954: \JP :: $BB?9`<0$N%Y%-$N(B annihilator $B$N7W;;(B
4955: \EG :: Computes the annihilator of a power of polynomial
1.6 noro 4956: @end table
1.10 noro 4957:
1.6 noro 4958: @table @var
4959: @item return
1.10 noro 4960: \JP $BB?9`<0$^$?$O%j%9%H(B
4961: \EG polynomial or list
4962: @item f
1.6 noro 4963: \JP $BB?9`<0(B
4964: \EG polynomial
4965: @item plist
4966: \JP $BB?9`<0%j%9%H(B
4967: \EG list of polynomials
4968: @item vlist dvlist
4969: \JP $BJQ?t%j%9%H(B
4970: \EG list of variables
4971: @end table
1.5 noro 4972:
1.6 noro 4973: @itemize @bullet
4974: \BJP
4975: @item @samp{bfct} $B$GDj5A$5$l$F$$$k(B.
1.10 noro 4976: @item @code{bfunction(@var{f})}, @code{bfct(@var{f})} $B$OB?9`<0(B @var{f} $B$N(B global @var{b} $B4X?t(B @code{b(s)} $B$r(B
1.6 noro 4977: $B7W;;$9$k(B. @code{b(s)} $B$O(B, Weyl $BBe?t(B @code{D} $B>e$N0lJQ?tB?9`<04D(B @code{D[s]}
4978: $B$N85(B @code{P(x,s)} $B$,B8:_$7$F(B, @code{P(x,s)f^(s+1)=b(s)f^s} $B$rK~$?$9$h$&$J(B
4979: $BB?9`<0(B @code{b(s)} $B$NCf$G(B, $B<!?t$,:G$bDc$$$b$N$G$"$k(B.
4980: @item @code{generic_bfct(@var{f},@var{vlist},@var{dvlist},@var{weight})}
4981: $B$O(B, @var{plist} $B$G@8@.$5$l$k(B @code{D} $B$N:8%$%G%"%k(B @code{I} $B$N(B,
1.10 noro 4982: $B%&%'%$%H(B @var{weight} $B$K4X$9$k(B global @var{b} $B4X?t$r7W;;$9$k(B.
1.6 noro 4983: @var{vlist} $B$O(B @code{x}-$BJQ?t(B, @var{vlist} $B$OBP1~$9$k(B @code{D}-$BJQ?t(B
4984: $B$r=g$KJB$Y$k(B.
1.9 noro 4985: @item @code{bfunction} $B$H(B @code{bfct} $B$G$OMQ$$$F$$$k%"%k%4%j%:%`$,(B
1.11 noro 4986: $B0[$J$k(B. $B$I$A$i$,9bB.$+$OF~NO$K$h$k(B.
1.10 noro 4987: @item @code{ann(@var{f})} $B$O(B, @code{@var{f}^s} $B$N(B annihilator ideal
4988: $B$N@8@.7O$rJV$9(B. @code{ann(@var{f})} $B$O(B, @code{[@var{a},@var{list}]}
4989: $B$J$k%j%9%H$rJV$9(B. $B$3$3$G(B, @var{a} $B$O(B @var{f} $B$N(B @var{b} $B4X?t$N:G>.@0?t:,(B,
4990: @var{list} $B$O(B @code{ann(@var{f})} $B$N7k2L$N(B @code{s}$ $B$K(B, @var{a} $B$r(B
4991: $BBeF~$7$?$b$N$G$"$k(B.
1.7 noro 4992: @item $B>\:Y$K$D$$$F$O(B, [Saito,Sturmfels,Takayama] $B$r8+$h(B.
1.6 noro 4993: \E
4994: \BEG
4995: @item These functions are defined in @samp{bfct}.
1.10 noro 4996: @item @code{bfunction(@var{f})} and @code{bfct(@var{f})} compute the global @var{b}-function @code{b(s)} of
1.6 noro 4997: a polynomial @var{f}.
4998: @code{b(s)} is a polynomial of the minimal degree
4999: such that there exists @code{P(x,s)} in D[s], which is a polynomial
5000: ring over Weyl algebra @code{D}, and @code{P(x,s)f^(s+1)=b(s)f^s} holds.
5001: @item @code{generic_bfct(@var{f},@var{vlist},@var{dvlist},@var{weight})}
1.10 noro 5002: computes the global @var{b}-function of a left ideal @code{I} in @code{D}
1.6 noro 5003: generated by @var{plist}, with respect to @var{weight}.
5004: @var{vlist} is the list of @code{x}-variables,
5005: @var{vlist} is the list of corresponding @code{D}-variables.
1.9 noro 5006: @item @code{bfunction(@var{f})} and @code{bfct(@var{f})} implement
5007: different algorithms and the efficiency depends on inputs.
1.10 noro 5008: @item @code{ann(@var{f})} returns the generator set of the annihilator
5009: ideal of @code{@var{f}^s}.
5010: @code{ann(@var{f})} returns a list @code{[@var{a},@var{list}]},
5011: where @var{a} is the minimal integral root of the global @var{b}-function
5012: of @var{f}, and @var{list} is a list of polynomials obtained by
5013: substituting @code{s} in @code{ann(@var{f})} with @var{a}.
1.7 noro 5014: @item See [Saito,Sturmfels,Takayama] for the details.
1.6 noro 5015: \E
5016: @end itemize
5017:
5018: @example
5019: [0] load("bfct")$
5020: [216] bfunction(x^3+y^3+z^3+x^2*y^2*z^2+x*y*z);
5021: -9*s^5-63*s^4-173*s^3-233*s^2-154*s-40
5022: [217] fctr(@@);
5023: [[-1,1],[s+2,1],[3*s+4,1],[3*s+5,1],[s+1,2]]
5024: [218] F = [4*x^3*dt+y*z*dt+dx,x*z*dt+4*y^3*dt+dy,
5025: x*y*dt+5*z^4*dt+dz,-x^4-z*y*x-y^4-z^5+t]$
5026: [219] generic_bfct(F,[t,z,y,x],[dt,dz,dy,dx],[1,0,0,0]);
5027: 20000*s^10-70000*s^9+101750*s^8-79375*s^7+35768*s^6-9277*s^5
5028: +1278*s^4-72*s^3
1.10 noro 5029: [220] P=x^3-y^2$
5030: [221] ann(P);
5031: [2*dy*x+3*dx*y^2,-3*dx*x-2*dy*y+6*s]
5032: [222] ann0(P);
5033: [-1,[2*dy*x+3*dx*y^2,-3*dx*x-2*dy*y-6]]
1.6 noro 5034: @end example
5035:
5036: @table @t
5037: \JP @item $B;2>H(B
5038: \EG @item References
5039: \JP @fref{Weyl $BBe?t(B}.
5040: \EG @fref{Weyl algebra}.
5041: @end table
1.5 noro 5042:
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