Annotation of OpenXM/src/asir-doc/parts/groebner.texi, Revision 1.20
1.20 ! takayama 1: @comment $OpenXM: OpenXM/src/asir-doc/parts/groebner.texi,v 1.19 2016/08/29 04:56:58 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::
1.1 noro 22: * $B%0%l%V%J4pDl$K4X$9$kH!?t(B::
1.2 noro 23: \E
24: \BEG
25: * Distributed polynomial::
26: * Reading files::
27: * Fundamental functions::
28: * Controlling Groebner basis computations::
29: * Setting term orderings::
1.13 noro 30: * Weight::
1.2 noro 31: * Groebner basis computation with rational function coefficients::
32: * Change of ordering::
1.5 noro 33: * Weyl algebra::
1.2 noro 34: * Functions for Groebner basis computation::
35: \E
1.1 noro 36: @end menu
37:
1.2 noro 38: \BJP
1.1 noro 39: @node $BJ,;6I=8=B?9`<0(B,,, $B%0%l%V%J4pDl$N7W;;(B
40: @section $BJ,;6I=8=B?9`<0(B
1.2 noro 41: \E
42: \BEG
43: @node Distributed polynomial,,, Groebner basis computation
44: @section Distributed polynomial
45: \E
1.1 noro 46:
47: @noindent
1.2 noro 48: \BJP
1.1 noro 49: $BJ,;6I=8=B?9`<0$H$O(B, $BB?9`<0$NFbIt7A<0$N0l$D$G$"$k(B. $BDL>o$NB?9`<0(B
50: (@code{type} $B$,(B 2) $B$O(B, $B:F5"I=8=$H8F$P$l$k7A<0$GI=8=$5$l$F$$$k(B. $B$9$J$o(B
51: $B$A(B, $BFCDj$NJQ?t$r<gJQ?t$H$9$k(B 1 $BJQ?tB?9`<0$G(B, $B$=$NB>$NJQ?t$O(B, $B$=$N(B 1 $BJQ(B
52: $B?tB?9`<0$N78?t$K(B, $B<gJQ?t$r4^$^$J$$B?9`<0$H$7$F8=$l$k(B. $B$3$N78?t$,(B, $B$^$?(B,
53: $B$"$kJQ?t$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 54: \E
55: \BEG
56: A distributed polynomial is a polynomial with a special internal
57: representation different from the ordinary one.
58:
59: An ordinary polynomial (having @code{type} 2) is internally represented
60: in a format, called recursive representation.
61: In fact, it is represented as an uni-variate polynomial with respect to
62: a fixed variable, called main variable of that polynomial,
63: where the other variables appear in the coefficients which may again
64: polynomials in such variables other than the previous main variable.
65: A polynomial in the coefficients is again represented as
66: an uni-variate polynomial in a certain fixed variable,
67: the main variable. Thus, by this recursive structure of polynomial
68: representation, it is called the `recursive representation.'
69: \E
1.1 noro 70:
71: @iftex
72: @tex
1.2 noro 73: \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 ))$
74: \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 75: @end tex
76: @end iftex
77: @ifinfo
78: @example
79: (x+y+z)^2 = 1 x^2 + (2 y + (2 z)) x + ((2 z) y + (1 z^2 ))
80: @end example
81: @end ifinfo
82:
83: @noindent
1.2 noro 84: \BJP
1.1 noro 85: $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
86: $BI=8=$H8F$V(B.
1.2 noro 87: \E
88: \BEG
89: On the other hand,
90: we call a representation the distributed representation of a polynomial,
91: if a polynomial is represented, according to its original meaning,
92: as a sum of monomials,
93: where a monomial is the product of power product of variables
94: and a coefficient. We call a polynomial, represented in such an
95: internal format, a distributed polynomial. (This naming may sounds
96: something strange.)
97: \E
1.1 noro 98:
99: @iftex
100: @tex
1.2 noro 101: \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$
102: \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 103: @end tex
104: @end iftex
105: @ifinfo
106: @example
107: (x+y+z)^2 = 1 x^2 + 2 xy + 2 xz + 1 y^2 + 2 yz +1 z^2$
108: @end example
109: @end ifinfo
110:
111: @noindent
1.2 noro 112: \BJP
1.1 noro 113: $B%0%l%V%J4pDl7W;;$K$*$$$F$O(B, $BC19`<0$KCmL\$7$FA`:n$r9T$&$?$aB?9`<0$,J,;6I=8=(B
114: $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,
115: $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$$$k(B.
116: $B$3$3$G(B, $B8e$N@bL@$N$?$a$K(B, $B$$$/$D$+$N8@MU$rDj5A$7$F$*$/(B.
1.2 noro 117: \E
118: \BEG
119: For computation of Groebner basis, efficient operation is expected if
120: polynomials are represented in a distributed representation,
121: because major operations for Groebner basis are performed with respect
122: to monomials.
123: From this view point, we provide the object type distributed polynomial
124: with its object identification number 9, and objects having such a type
125: are available by @b{Asir} language.
126:
127: Here, we provide several definitions for the later description.
128: \E
1.1 noro 129:
130: @table @b
1.2 noro 131: \BJP
1.1 noro 132: @item $B9`(B (term)
133: $BJQ?t$NQQ@Q(B. $B$9$J$o$A(B, $B78?t(B 1 $B$NC19`<0$N$3$H(B. @b{Asir} $B$K$*$$$F$O(B,
1.2 noro 134: \E
135: \BEG
136: @item term
137: The power product of variables, i.e., a monomial with coefficient 1.
138: In an @b{Asir} session, it is displayed in the form like
139: \E
1.1 noro 140:
141: @example
142: <<0,1,2,3,4>>
143: @end example
144:
1.2 noro 145: \BJP
1.1 noro 146: $B$H$$$&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
147: $B$r<($9(B. $B3FJQ?t$r(B @code{a}, @code{b}, @code{c}, @code{d}, @code{e} $B$H$9$k$H(B
148: $B$3$N9`$O(B @code{b*c^2*d^3*e^4} $B$rI=$9(B.
1.2 noro 149: \E
150: \BEG
151: and also can be input in such a form.
152: This example shows a term in 5 variables. If we assume the 5 variables
153: as @code{a}, @code{b}, @code{c}, @code{d}, and @code{e},
154: the term represents @code{b*c^2*d^3*e^4} in the ordinary expression.
155: \E
1.1 noro 156:
1.2 noro 157: \BJP
1.1 noro 158: @item $B9`=g=x(B (term order)
159: $BJ,;6I=8=B?9`<0$K$*$1$k9`$O(B, $B<!$N@-<A$rK~$?$9A4=g=x$K$h$j@0Ns$5$l$k(B.
1.2 noro 160: \E
161: \BEG
162: @item term order
163: Terms are ordered according to a total order with the following properties.
164: \E
1.1 noro 165:
166: @enumerate
167: @item
1.2 noro 168: \JP $BG$0U$N9`(B @code{t} $B$KBP$7(B @code{t} > 1
169: \EG For all @code{t} @code{t} > 1.
1.1 noro 170:
171: @item
1.2 noro 172: \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}
173: \EG For all @code{t}, @code{s}, @code{u} @code{t} > @code{s} implies @code{tu} > @code{su}.
1.1 noro 174: @end enumerate
175:
1.2 noro 176: \BJP
1.1 noro 177: $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)
178: $B$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 179: \E
180: \BEG
181: Such a total order is called a term ordering. A term ordering is specified
182: by a variable ordering (a list of variables) and a type of term ordering
183: (an integer, a list or a matrix).
184: \E
1.1 noro 185:
1.2 noro 186: \BJP
1.1 noro 187: @item $BC19`<0(B (monomial)
188: $B9`$H78?t$N@Q(B.
1.2 noro 189: \E
190: \BEG
191: @item monomial
192: The product of a term and a coefficient.
193: In an @b{Asir} session, it is displayed in the form like
194: \E
1.1 noro 195:
196: @example
197: 2*<<0,1,2,3,4>>
198: @end example
199:
1.2 noro 200: \JP $B$H$$$&7A$GI=<($5$l(B, $B$^$?(B, $B$3$N7A$GF~NO2DG=$G$"$k(B.
201: \EG and also can be input in such a form.
1.1 noro 202:
1.2 noro 203: \BJP
1.19 noro 204: @item $BF,9`(B (head term)
1.1 noro 205: @itemx $BF,C19`<0(B (head monomial)
206: @itemx $BF,78?t(B (head coefficient)
207: $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
208: $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
209: $B$H8F$V(B.
1.2 noro 210: \E
211: \BEG
1.19 noro 212: @item head term
1.2 noro 213: @itemx head monomial
214: @itemx head coefficient
215:
216: Monomials in a distributed polynomial is sorted by a total order.
217: In such representation, we call the monomial that is maximum
218: with respect to the order the head monomial, and its term and coefficient
219: the head term and the head coefficient respectively.
220: \E
1.1 noro 221: @end table
222:
1.20 ! takayama 223: @noindent
! 224: ChangeLog
! 225: @itemize @bullet
! 226: \BJP
! 227: @item $BJ,;6I=8=B?9`<0$OG$0U$N%*%V%8%'%/%H$r78?t$K$b$F$k$h$&$K$J$C$?(B.
! 228: $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).
! 229: \E
! 230: \BEG
! 231: @item Distributed polynomials accept objects as coefficients.
! 232: The k-th element of a free module is expressed as <<d0,d1,...:k>> (2017-08-31).
! 233: \E
! 234: @item
! 235: 1.15 algnum.c,
! 236: 1.53 ctrl.c,
! 237: 1.66 dp-supp.c,
! 238: 1.105 dp.c,
! 239: 1.73 gr.c,
! 240: 1.4 reduct.c,
! 241: 1.16 _distm.c,
! 242: 1.17 dalg.c,
! 243: 1.52 dist.c,
! 244: 1.20 distm.c,
! 245: 1.8 gmpq.c,
! 246: 1.238 engine/nd.c,
! 247: 1.102 ca.h,
! 248: 1.411 version.h,
! 249: 1.28 cpexpr.c,
! 250: 1.42 pexpr.c,
! 251: 1.20 pexpr_body.c,
! 252: 1.40 spexpr.c,
! 253: 1.27 arith.c,
! 254: 1.77 eval.c,
! 255: 1.56 parse.h,
! 256: 1.37 parse.y,
! 257: 1.8 stdio.c,
! 258: 1.31 plotf.c
! 259: @end itemize
! 260:
1.2 noro 261: \BJP
1.1 noro 262: @node $B%U%!%$%k$NFI$_9~$_(B,,, $B%0%l%V%J4pDl$N7W;;(B
263: @section $B%U%!%$%k$NFI$_9~$_(B
1.2 noro 264: \E
265: \BEG
266: @node Reading files,,, Groebner basis computation
267: @section Reading files
268: \E
1.1 noro 269:
270: @noindent
1.2 noro 271: \BJP
1.1 noro 272: $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 273: @code{dp_gr_mod_main()}, @code{dp_gr_f_main()}
274: $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 275: $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.
276: $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
277: $B$_9~$`$3$H$K$h$j;HMQ2DG=$H$J$k(B. @samp{gr} $B$O(B, @b{Asir} $B$NI8=`(B
1.5 noro 278: $B%i%$%V%i%j%G%#%l%/%H%j$KCV$+$l$F$$$k(B.
1.2 noro 279: \E
280: \BEG
1.5 noro 281: Facilities for computing Groebner bases are
282: @code{dp_gr_main()}, @code{dp_gr_mod_main()}and @code{dp_gr_f_main()}.
283: To call these functions,
284: it is necessary to set several parameters correctly and it is convenient
285: to use a set of interface functions provided in the library file
286: @samp{gr}.
1.2 noro 287: The facilities will be ready to use after you load the package by
288: @code{load()}. The package @samp{gr} is placed in the standard library
1.5 noro 289: directory of @b{Asir}.
1.2 noro 290: \E
1.1 noro 291:
292: @example
293: [0] load("gr")$
294: @end example
295:
1.2 noro 296: \BJP
1.1 noro 297: @node $B4pK\E*$JH!?t(B,,, $B%0%l%V%J4pDl$N7W;;(B
298: @section $B4pK\E*$JH!?t(B
1.2 noro 299: \E
300: \BEG
301: @node Fundamental functions,,, Groebner basis computation
302: @section Fundamental functions
303: \E
1.1 noro 304:
305: @noindent
1.2 noro 306: \BJP
1.1 noro 307: @samp{gr} $B$G$O?tB?$/$NH!?t$,Dj5A$5$l$F$$$k$,(B, $BD>@\(B
308: $B%0%l%V%J4pDl$r7W;;$9$k$?$a$N%H%C%W%l%Y%k$O<!$N(B 3 $B$D$G$"$k(B.
309: $B0J2<$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,
310: @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 311: \E
312: \BEG
313: There are many functions and options defined in the package @samp{gr}.
314: Usually not so many of them are used. Top level functions for Groebner
315: basis computation are the following three functions.
316:
317: In the following description, @var{plist}, @var{vlist}, @var{order}
318: and @var{p} stand for a list of polynomials, a list of variables
319: (indeterminates), a type of term ordering and a prime less than
320: @code{2^27} respectively.
321: \E
1.1 noro 322:
323: @table @code
324: @item gr(@var{plist},@var{vlist},@var{order})
325:
1.2 noro 326: \BJP
1.1 noro 327: Gebauer-Moeller $B$K$h$k(B useless pair elimination criteria, sugar
328: strategy $B$*$h$S(B Traverso $B$K$h$k(B trace-lifting $B$rMQ$$$?(B Buchberger $B%"%k(B
329: $B%4%j%:%`$K$h$kM-M}?t78?t%0%l%V%J4pDl7W;;H!?t(B. $B0lHL$K$O$3$NH!?t$rMQ$$$k(B.
1.2 noro 330: \E
331: \BEG
332: Function that computes Groebner bases over the rationals. The
333: algorithm is Buchberger algorithm with useless pair elimination
334: criteria by Gebauer-Moeller, sugar strategy and trace-lifting by
335: Traverso. For ordinary computation, this function is used.
336: \E
1.1 noro 337:
338: @item hgr(@var{plist},@var{vlist},@var{order})
339:
1.2 noro 340: \BJP
1.1 noro 341: $BF~NOB?9`<0$r@F<!2=$7$?8e(B @code{gr()} $B$N%0%l%V%J4pDl8uJd@8@.It$K$h$j8u(B
342: $BJd@8@.$7(B, $BHs@F<!2=(B, interreduce $B$7$?$b$N$r(B @code{gr()} $B$N%0%l%V%J4pDl(B
343: $B%A%'%C%/It$G%A%'%C%/$9$k(B. 0 $B<!85%7%9%F%`(B ($B2r$N8D?t$,M-8B8D$NJ}Dx<07O(B)
344: $B$N>l9g(B, sugar strategy $B$,78?tKDD%$r0z$-5/$3$9>l9g$,$"$k(B. $B$3$N$h$&$J>l(B
345: $B9g(B, strategy $B$r@F<!2=$K$h$k(B strategy $B$KCV$-49$($k$3$H$K$h$j78?tKDD%$r(B
346: $BM^@)$9$k$3$H$,$G$-$k>l9g$,B?$$(B.
1.2 noro 347: \E
348: \BEG
349: After homogenizing the input polynomials a candidate of the \gr basis
350: is computed by trace-lifting. Then the candidate is dehomogenized and
351: checked whether it is indeed a Groebner basis of the input. Sugar
352: strategy often causes intermediate coefficient swells. It is
353: empirically known that the combination of homogenization and supresses
354: the swells for such cases.
355: \E
1.1 noro 356:
357: @item gr_mod(@var{plist},@var{vlist},@var{order},@var{p})
358:
1.2 noro 359: \BJP
1.1 noro 360: Gebauer-Moeller $B$K$h$k(B useless pair elimination criteria, sugar
361: strategy $B$*$h$S(B Buchberger $B%"%k%4%j%:%`$K$h$k(B GF(p) $B78?t%0%l%V%J4pDl7W(B
362: $B;;H!?t(B.
1.2 noro 363: \E
364: \BEG
365: Function that computes Groebner bases over GF(@var{p}). The same
366: algorithm as @code{gr()} is used.
367: \E
1.1 noro 368:
369: @end table
370:
1.2 noro 371: \BJP
1.1 noro 372: @node $B7W;;$*$h$SI=<($N@)8f(B,,, $B%0%l%V%J4pDl$N7W;;(B
373: @section $B7W;;$*$h$SI=<($N@)8f(B
1.2 noro 374: \E
375: \BEG
376: @node Controlling Groebner basis computations,,, Groebner basis computation
377: @section Controlling Groebner basis computations
378: \E
1.1 noro 379:
380: @noindent
1.2 noro 381: \BJP
1.1 noro 382: $B%0%l%V%J4pDl$N7W;;$K$*$$$F(B, $B$5$^$6$^$J%Q%i%a%?@_Dj$r9T$&$3$H$K$h$j7W;;(B,
383: $BI=<($r@)8f$9$k$3$H$,$G$-$k(B. $B$3$l$i$O(B, $BAH$_9~$_H!?t(B @code{dp_gr_flags()}
384: $B$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
385: $B$H(B, $B8=:_@_Dj$5$l$F$$$k%Q%i%a%?$,(B, $BL>A0$HCM$N%j%9%H$GJV$5$l$k(B.
1.2 noro 386: \E
387: \BEG
388: One can cotrol a Groebner basis computation by setting various parameters.
389: These parameters can be set and examined by a built-in function
390: @code{dp_gr_flags()}. Without argument it returns the current settings.
391: \E
1.1 noro 392:
393: @example
394: [100] dp_gr_flags();
1.5 noro 395: [Demand,0,NoSugar,0,NoCriB,0,NoGC,0,NoMC,0,NoRA,0,NoGCD,0,Top,0,
396: ShowMag,1,Print,1,Stat,0,Reverse,0,InterReduce,0,Multiple,0]
1.1 noro 397: [101]
398: @end example
399:
1.2 noro 400: \BJP
1.1 noro 401: $B0J2<$G(B, $B3F%Q%i%a%?$N0UL#$r@bL@$9$k(B. on $B$N>l9g$H$O(B, $B%Q%i%a%?$,(B 0 $B$G$J$$>l9g$r(B
402: $B$$$&(B. $B$3$l$i$N%Q%i%a%?$N=i4|CM$OA4$F(B 0 (off) $B$G$"$k(B.
1.2 noro 403: \E
404: \BEG
405: The return value is a list which contains the names of parameters and their
406: values. The meaning of the parameters are as follows. `on' means that the
407: parameter is not zero.
408: \E
1.1 noro 409:
410: @table @code
411: @item NoSugar
1.2 noro 412: \BJP
1.1 noro 413: on $B$N>l9g(B, sugar strategy $B$NBe$o$j$K(B Buchberger$B$N(B normal strategy $B$,MQ(B
414: $B$$$i$l$k(B.
1.2 noro 415: \E
416: \BEG
417: If `on', Buchberger's normal strategy is used instead of sugar strategy.
418: \E
1.1 noro 419:
420: @item NoCriB
1.2 noro 421: \JP on $B$N>l9g(B, $BITI,MWBP8!=P5,=`$N$&$A(B, $B5,=`(B B $B$rE,MQ$7$J$$(B.
422: \EG If `on', criterion B among the Gebauer-Moeller's criteria is not applied.
1.1 noro 423:
424: @item NoGC
1.2 noro 425: \JP on $B$N>l9g(B, $B7k2L$,%0%l%V%J4pDl$K$J$C$F$$$k$+$I$&$+$N%A%'%C%/$r9T$o$J$$(B.
426: \BEG
427: If `on', the check that a Groebner basis candidate is indeed a Groebner basis,
428: is not executed.
429: \E
1.1 noro 430:
431: @item NoMC
1.2 noro 432: \BJP
1.1 noro 433: on $B$N>l9g(B, $B7k2L$,F~NO%$%G%"%k$HF1Ey$N%$%G%"%k$G$"$k$+$I$&$+$N%A%'%C%/(B
434: $B$r9T$o$J$$(B.
1.2 noro 435: \E
436: \BEG
437: If `on', the check that the resulting polynomials generates the same ideal as
438: the ideal generated by the input, is not executed.
439: \E
1.1 noro 440:
441: @item NoRA
1.2 noro 442: \BJP
1.1 noro 443: on $B$N>l9g(B, $B7k2L$r(B reduced $B%0%l%V%J4pDl$K$9$k$?$a$N(B
444: interreduce $B$r9T$o$J$$(B.
1.2 noro 445: \E
446: \BEG
447: If `on', the interreduction, which makes the Groebner basis reduced, is not
448: executed.
449: \E
1.1 noro 450:
451: @item NoGCD
1.2 noro 452: \BJP
1.1 noro 453: on $B$N>l9g(B, $BM-M}<078?t$N%0%l%V%J4pDl7W;;$K$*$$$F(B, $B@8@.$5$l$?B?9`<0$N(B,
454: $B78?t$N(B content $B$r$H$i$J$$(B.
1.2 noro 455: \E
456: \BEG
457: If `on', content removals are not executed during a Groebner basis computation
458: over a rational function field.
459: \E
1.1 noro 460:
461: @item Top
1.2 noro 462: \JP on $B$N>l9g(B, normal form $B7W;;$K$*$$$FF,9`>C5n$N$_$r9T$&(B.
463: \EG If `on', Only the head term of the polynomial being reduced is reduced.
1.1 noro 464:
1.2 noro 465: @comment @item Interreduce
466: @comment \BJP
467: @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
468: @comment $B$h$k(B normal form $B$GCV$-49$($k(B.
469: @comment \E
470: @comment \BEG
471: @comment If `on', intermediate basis elements are reduced by using a newly generated
472: @comment basis element.
473: @comment \E
1.1 noro 474:
475: @item Reverse
1.2 noro 476: \BJP
1.1 noro 477: 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
478: $B@h$7$FA*$V(B.
1.2 noro 479: \E
480: \BEG
481: If `on', the selection strategy of reducer in a normal form computation
482: is such that a newer reducer is used first.
483: \E
1.1 noro 484:
485: @item Print
1.2 noro 486: \JP on $B$N>l9g(B, $B%0%l%V%J4pDl7W;;$NESCf$K$*$1$k$5$^$6$^$J>pJs$rI=<($9$k(B.
487: \BEG
488: If `on', various informations during a Groebner basis computation is
489: displayed.
490: \E
1.1 noro 491:
1.7 noro 492: @item PrintShort
493: \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.
494: \BEG
495: If `on' and Print is `off', short information during a Groebner basis computation is
496: displayed.
497: \E
498:
1.1 noro 499: @item Stat
1.2 noro 500: \BJP
1.1 noro 501: 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
502: $B$l$k%G!<%?$NFb(B, $B=87W%G!<%?$N$_$,I=<($5$l$k(B.
1.2 noro 503: \E
504: \BEG
505: If `on', a summary of informations is shown after a Groebner basis
506: computation. Note that the summary is always shown if @code{Print} is `on'.
507: \E
1.1 noro 508:
509: @item ShowMag
1.2 noro 510: \BJP
1.1 noro 511: 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
512: $B78?t$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 513: \E
514: \BEG
515: If `on' and @code{Print} is `on', the sum of bit length of
516: coefficients of a generated basis element, which we call @var{magnitude},
517: is shown after every normal computation. After comleting the
518: computation the maximal value among the sums is shown.
519: \E
1.1 noro 520:
1.7 noro 521: @item Content
522: @itemx Multiple
1.2 noro 523: \BJP
1.7 noro 524: 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
525: @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
526: $B3d$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
527: GCD $B7W;;$,9T$o$l0lHL$K$O8zN($,0-$/$J$k$,(B, @code{Content} $B$r(B 2 $BDxEY(B
1.1 noro 528: $B$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 529: backward compatibility $B$N$?$a!"(B@code{Multiple} $B$G@0?tCM$r;XDj$G$-$k(B.
1.2 noro 530: \E
531: \BEG
1.7 noro 532: If a non-zero rational number, in a normal form computation
1.2 noro 533: over the rationals, the integer content of the polynomial being
1.7 noro 534: reduced is removed when its magnitude becomes @code{Content} times
1.2 noro 535: larger than a registered value, which is set to the magnitude of the
536: input polynomial. After each content removal the registered value is
1.7 noro 537: set to the magnitude of the resulting polynomial. @code{Content} is
1.2 noro 538: equal to 1, the simiplification is done after every normal form computation.
1.7 noro 539: It is empirically known that it is often efficient to set @code{Content} to 2
1.2 noro 540: for the case where large integers appear during the computation.
1.7 noro 541: An integer value can be set by the keyword @code{Multiple} for
542: backward compatibility.
1.2 noro 543: \E
1.1 noro 544:
545: @item Demand
1.2 noro 546:
547: \BJP
1.1 noro 548: $B@5Ev$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
549: $BCf$K$*$+$l$:(B, $B$=$N%G%#%l%/%H%jCf$K%P%$%J%j%G!<%?$H$7$FCV$+$l(B, $B$=$NB?9`(B
550: $B<0$rMQ$$$k(B normal form $B7W;;$N:](B, $B<+F0E*$K%a%b%jCf$K%m!<%I$5$l$k(B. $B3FB?(B
551: $B9`<0$O(B, $BFbIt$G$N%$%s%G%C%/%9$r%U%!%$%kL>$K;}$D%U%!%$%k$K3JG<$5$l$k(B.
552: $B$3$3$G;XDj$5$l$?%G%#%l%/%H%j$K=q$+$l$?%U%!%$%k$O<+F0E*$K$O>C5n$5$l$J$$(B
553: $B$?$a(B, $B%f!<%6$,@UG$$r;}$C$F>C5n$9$kI,MW$,$"$k(B.
1.2 noro 554: \E
555: \BEG
556: If the value (a character string) is a valid directory name, then
557: generated basis elements are put in the directory and are loaded on
558: demand during normal form computations. Each elements is saved in the
559: binary form and its name coincides with the index internally used in
560: the computation. These binary files are not removed automatically
561: and one should remove them by hand.
562: \E
1.1 noro 563: @end table
564:
565: @noindent
1.2 noro 566: \JP @code{Print} $B$,(B 0 $B$G$J$$>l9g<!$N$h$&$J%G!<%?$,I=<($5$l$k(B.
567: \EG If @code{Print} is `on', the following informations are shown.
1.1 noro 568:
569: @example
570: [93] gr(cyclic(4),[c0,c1,c2,c3],0)$
571: mod= 99999989, eval = []
572: (0)(0)<<0,2,0,0>>(2,3),nb=2,nab=5,rp=2,sugar=2,mag=4
573: (0)(0)<<0,1,2,0>>(1,2),nb=3,nab=6,rp=2,sugar=3,mag=4
574: (0)(0)<<0,1,1,2>>(0,1),nb=4,nab=7,rp=3,sugar=4,mag=6
575: .
576: (0)(0)<<0,0,3,2>>(5,6),nb=5,nab=8,rp=2,sugar=5,mag=4
577: (0)(0)<<0,1,0,4>>(4,6),nb=6,nab=9,rp=3,sugar=5,mag=4
578: (0)(0)<<0,0,2,4>>(6,8),nb=7,nab=10,rp=4,sugar=6,mag=6
579: ....gb done
580: reduceall
581: .......
582: membercheck
583: (0,0)(0,0)(0,0)(0,0)
584: gbcheck total 8 pairs
585: ........
1.5 noro 586: UP=(0,0)SP=(0,0)SPM=(0,0)NF=(0,0)NFM=(0.010002,0)ZNFM=(0.010002,0)
587: 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
588: D=12 ZR=5 NZR=6 Max_mag=6
1.1 noro 589: [94]
590: @end example
591:
592: @noindent
1.2 noro 593: \BJP
1.1 noro 594: $B:G=i$KI=<($5$l$k(B @code{mod}, @code{eval} $B$O(B, trace-lifting $B$GMQ$$$i$l$kK!(B
595: $B$G$"$k(B. @code{mod} $B$OAG?t(B, @code{eval} $B$OM-M}<078?t$N>l9g$KMQ$$$i$l$k(B
596: $B?t$N%j%9%H$G$"$k(B.
1.2 noro 597: \E
598: \BEG
599: In this example @code{mod} and @code{eval} indicate moduli used in
600: trace-lifting. @code{mod} is a prime and @code{eval} is a list of integers
601: used for evaluation when the ground field is a field of rational functions.
602: \E
1.1 noro 603:
604: @noindent
1.2 noro 605: \JP $B7W;;ESCf$GB?9`<0$,@8@.$5$l$kKh$K<!$N7A$N%G!<%?$,I=<($5$l$k(B.
606: \EG The following information is shown after every normal form computation.
1.1 noro 607:
608: @example
609: (TNF)(TCONT)HT(INDEX),nb=NB,nab=NAB,rp=RP,sugar=S,mag=M
610: @end example
611:
612: @noindent
1.2 noro 613: \JP $B$=$l$i$N0UL#$O<!$NDL$j(B.
614: \EG Meaning of each component is as follows.
1.1 noro 615:
616: @table @code
1.2 noro 617: \BJP
1.1 noro 618: @item TNF
1.2 noro 619:
1.1 noro 620: normal form $B7W;;;~4V(B ($BIC(B)
621:
622: @item TCONT
1.2 noro 623:
1.1 noro 624: content $B7W;;;~4V(B ($BIC(B)
625:
626: @item HT
1.2 noro 627:
1.1 noro 628: $B@8@.$5$l$?B?9`<0$NF,9`(B
629:
630: @item INDEX
1.2 noro 631:
1.1 noro 632: S-$BB?9`<0$r9=@.$9$kB?9`<0$N%$%s%G%C%/%9$N%Z%"(B
633:
634: @item NB
1.2 noro 635:
1.1 noro 636: $B8=:_$N(B, $B>iD9@-$r=|$$$?4pDl$N?t(B
637:
638: @item NAB
1.2 noro 639:
1.1 noro 640: $B8=:_$^$G$K@8@.$5$l$?4pDl$N?t(B
641:
642: @item RP
1.2 noro 643:
1.1 noro 644: $B;D$j$N%Z%"$N?t(B
645:
646: @item S
1.2 noro 647:
1.1 noro 648: $B@8@.$5$l$?B?9`<0$N(B sugar $B$NCM(B
649:
650: @item M
1.2 noro 651:
1.1 noro 652: $B@8@.$5$l$?B?9`<0$N78?t$N%S%C%HD9$NOB(B (@code{ShowMag} $B$,(B on $B$N;~$KI=<($5$l$k(B. )
1.2 noro 653: \E
654: \BEG
655: @item TNF
656:
657: CPU time for normal form computation (second)
658:
659: @item TCONT
660:
661: CPU time for content removal(second)
662:
663: @item HT
664:
665: Head term of the generated basis element
666:
667: @item INDEX
668:
669: Pair of indices which corresponds to the reduced S-polynomial
670:
671: @item NB
672:
673: Number of basis elements after removing redundancy
674:
675: @item NAB
676:
677: Number of all the basis elements
678:
679: @item RP
680:
681: Number of remaining pairs
682:
683: @item S
684:
685: Sugar of the generated basis element
686:
687: @item M
688:
689: Magnitude of the genrated basis element (shown if @code{ShowMag} is `on'.)
690: \E
1.1 noro 691: @end table
692:
693: @noindent
1.2 noro 694: \BJP
1.1 noro 695: $B:G8e$K(B, $B=87W%G!<%?$,I=<($5$l$k(B. $B0UL#$O<!$NDL$j(B.
696: ($B;~4V$NI=<($K$*$$$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 697: \E
698: \BEG
699: The summary of the informations shown after a Groebner basis
700: computation is as follows. If a component shows timings and it
701: contains two numbers, they are a pair of time for computation and time
702: for garbage colection.
703: \E
1.1 noro 704:
705: @table @code
1.2 noro 706: \BJP
1.1 noro 707: @item UP
1.2 noro 708:
1.1 noro 709: $B%Z%"$N%j%9%H$NA`:n$K$+$+$C$?;~4V(B
710:
711: @item SP
1.2 noro 712:
1.1 noro 713: $BM-M}?t>e$N(B S-$BB?9`<07W;;;~4V(B
714:
715: @item SPM
1.2 noro 716:
1.1 noro 717: $BM-8BBN>e$N(B S-$BB?9`<07W;;;~4V(B
718:
719: @item NF
1.2 noro 720:
1.1 noro 721: $BM-M}?t>e$N(B normal form $B7W;;;~4V(B
722:
723: @item NFM
1.2 noro 724:
1.1 noro 725: $BM-8BBN>e$N(B normal form $B7W;;;~4V(B
726:
727: @item ZNFM
1.2 noro 728:
1.1 noro 729: @code{NFM} $B$NFb(B, 0 $B$X$N(B reduction $B$K$+$+$C$?;~4V(B
730:
731: @item PZ
1.2 noro 732:
1.1 noro 733: content $B7W;;;~4V(B
734:
735: @item NP
1.2 noro 736:
1.1 noro 737: $BM-M}?t78?tB?9`<0$N78?t$KBP$9$k>jM>1i;;$N7W;;;~4V(B
738:
739: @item MP
1.2 noro 740:
1.1 noro 741: S-$BB?9`<0$r@8@.$9$k%Z%"$NA*Br$K$+$+$C$?;~4V(B
742:
743: @item RA
1.2 noro 744:
1.1 noro 745: interreduce $B7W;;;~4V(B
746:
747: @item MC
1.2 noro 748:
1.1 noro 749: trace-lifting $B$K$*$1$k(B, $BF~NOB?9`<0$N%a%s%P%7%C%W7W;;;~4V(B
750:
751: @item GC
1.2 noro 752:
1.1 noro 753: $B7k2L$N%0%l%V%J4pDl8uJd$N%0%l%V%J4pDl%A%'%C%/;~4V(B
754:
755: @item T
1.2 noro 756:
1.1 noro 757: $B@8@.$5$l$?%Z%"$N?t(B
758:
759: @item B, M, F, D
1.2 noro 760:
1.1 noro 761: $B3F(B criterion $B$K$h$j=|$+$l$?%Z%"$N?t(B
762:
763: @item ZR
1.2 noro 764:
1.1 noro 765: 0 $B$K(B reduce $B$5$l$?%Z%"$N?t(B
766:
767: @item NZR
1.2 noro 768:
1.1 noro 769: 0 $B$G$J$$B?9`<0$K(B reduce $B$5$l$?%Z%"$N?t(B
770:
771: @item Max_mag
1.2 noro 772:
1.1 noro 773: $B@8@.$5$l$?B?9`<0$N(B, $B78?t$N%S%C%HD9$NOB$N:GBgCM(B
1.2 noro 774: \E
775: \BEG
776: @item UP
777:
778: Time to manipulate the list of critical pairs
779:
780: @item SP
781:
782: Time to compute S-polynomials over the rationals
783:
784: @item SPM
785:
786: Time to compute S-polynomials over a finite field
787:
788: @item NF
789:
790: Time to compute normal forms over the rationals
791:
792: @item NFM
793:
794: Time to compute normal forms over a finite field
795:
796: @item ZNFM
797:
798: Time for zero reductions in @code{NFM}
799:
800: @item PZ
801:
802: Time to remove integer contets
803:
804: @item NP
805:
806: Time to compute remainders for coefficients of polynomials with coeffieints
807: in the rationals
808:
809: @item MP
810:
811: Time to select pairs from which S-polynomials are computed
812:
813: @item RA
814:
815: Time to interreduce the Groebner basis candidate
816:
817: @item MC
1.1 noro 818:
1.2 noro 819: Time to check that each input polynomial is a member of the ideal
820: generated by the Groebner basis candidate.
821:
822: @item GC
823:
824: Time to check that the Groebner basis candidate is a Groebner basis
825:
826: @item T
827:
828: Number of critical pairs generated
829:
830: @item B, M, F, D
831:
832: Number of critical pairs removed by using each criterion
833:
834: @item ZR
835:
836: Number of S-polynomials reduced to 0
837:
838: @item NZR
839:
840: Number of S-polynomials reduced to non-zero results
841:
842: @item Max_mag
843:
844: Maximal magnitude among all the generated polynomials
845: \E
1.1 noro 846: @end table
847:
1.2 noro 848: \BJP
1.1 noro 849: @node $B9`=g=x$N@_Dj(B,,, $B%0%l%V%J4pDl$N7W;;(B
850: @section $B9`=g=x$N@_Dj(B
1.2 noro 851: \E
852: \BEG
853: @node Setting term orderings,,, Groebner basis computation
854: @section Setting term orderings
855: \E
1.1 noro 856:
857: @noindent
1.2 noro 858: \BJP
1.1 noro 859: $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
860: $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
861: $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
862: $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
863: $B9TNs$GI=8=$5$l$k(B.
1.2 noro 864: \E
865: \BEG
866: A term is internally represented as an integer vector whose components
867: are exponents with respect to variables. A variable list specifies the
868: correspondences between variables and components. A type of term ordering
869: specifies a total order for integer vectors. A type of term ordering is
870: represented by an integer, a list of integer or matrices.
871: \E
1.1 noro 872:
873: @noindent
1.2 noro 874: \JP $B4pK\E*$J9`=g=x7?$H$7$F<!$N(B 3 $B$D$,$"$k(B.
875: \EG There are following three fundamental types.
1.1 noro 876:
877: @table @code
1.2 noro 878: \JP @item 0 (DegRevLex; @b{$BA4<!?t5U<-=q<0=g=x(B})
879: \EG @item 0 (DegRevLex; @b{total degree reverse lexicographic ordering})
1.1 noro 880:
1.2 noro 881: \BJP
1.1 noro 882: $B0lHL$K(B, $B$3$N=g=x$K$h$k%0%l%V%J4pDl7W;;$,:G$b9bB.$G$"$k(B. $B$?$@$7(B,
883: $BJ}Dx<0$r2r$/$H$$$&L\E*$KMQ$$$k$3$H$O(B, $B0lHL$K$O$G$-$J$$(B. $B$3$N(B
884: $B=g=x$K$h$k%0%l%V%J4pDl$O(B, $B2r$N8D?t$N7W;;(B, $B%$%G%"%k$N%a%s%P%7%C%W$d(B,
885: $BB>$NJQ?t=g=x$X$N4pDlJQ49$N$?$a$N%=!<%9$H$7$FMQ$$$i$l$k(B.
1.2 noro 886: \E
887: \BEG
888: In general, computation by this ordering shows the fastest speed
889: in most Groebner basis computations.
890: However, for the purpose to solve polynomial equations, this type
891: of ordering is, in general, not so suitable.
892: The Groebner bases obtained by this ordering is used for computing
893: the number of solutions, solving ideal membership problem and seeds
894: for conversion to other Groebner bases under different ordering.
895: \E
1.1 noro 896:
1.2 noro 897: \JP @item 1 (DegLex; @b{$BA4<!?t<-=q<0=g=x(B})
898: \EG @item 1 (DegLex; @b{total degree lexicographic ordering})
1.1 noro 899:
1.2 noro 900: \BJP
1.1 noro 901: $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,
902: @code{DegRevLex} $B$HF1MMD>@\$=$N7k2L$rMQ$$$k$3$H$O:$Fq$G$"$k(B. $B$7$+$7(B,
903: $B<-=q<0=g=x$N%0%l%V%J4pDl$r5a$a$k:]$K(B, $B@F<!2=8e$K$3$N=g=x$G%0%l%V%J4pDl(B
904: $B$r5a$a$F$$$k(B.
1.2 noro 905: \E
906: \BEG
907: By this type term ordering, Groebner bases are obtained fairly faster
908: than Lex (lexicographic) ordering, too.
909: Alike the @code{DegRevLex} ordering, the result, in general, cannot directly
910: be used for solving polynomial equations.
911: It is used, however, in such a way
912: that a Groebner basis is computed in this ordering after homogenization
913: to obtain the final lexicographic Groebner basis.
914: \E
1.1 noro 915:
1.2 noro 916: \JP @item 2 (Lex; @b{$B<-=q<0=g=x(B})
917: \EG @item 2 (Lex; @b{lexicographic ordering})
1.1 noro 918:
1.2 noro 919: \BJP
1.1 noro 920: $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
921: $B7W;;;~4V$,$+$+$j2a$.$k$N$,FqE@$G$"$k(B. $BFC$K(B, $B2r$,M-8B8D$N>l9g(B, $B7k2L$N(B
922: $B78?t$,6K$a$FD9Bg$JB?G\D9?t$K$J$k>l9g$,B?$$(B. $B$3$N>l9g(B, @code{gr()},
923: @code{hgr()} $B$K$h$k7W;;$,6K$a$FM-8z$K$J$k>l9g$,B?$$(B.
1.2 noro 924: \E
925: \BEG
926: Groebner bases computed by this ordering give the most convenient
927: Groebner bases for solving the polynomial equations.
928: The only and serious shortcoming is the enormously long computation
929: time.
930: It is often observed that the number coefficients of the result becomes
931: very very long integers, especially if the ideal is 0-dimensional.
932: For such a case, it is empirically true for many cases
933: that i.e., computation by
934: @code{gr()} and/or @code{hgr()} may be quite effective.
935: \E
1.1 noro 936: @end table
937:
938: @noindent
1.2 noro 939: \BJP
1.1 noro 940: $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.
941: $B$3$l$O(B,
1.2 noro 942: \E
943: \BEG
944: By combining these fundamental orderingl into a list, one can make
945: various term ordering called elimination orderings.
946: \E
1.1 noro 947:
948: @code{[[O1,L1],[O2,L2],...]}
949:
950: @noindent
1.2 noro 951: \BJP
1.1 noro 952: $B$G;XDj$5$l$k(B. @code{Oi} $B$O(B 0, 1, 2 $B$N$$$:$l$+$G(B, @code{Li} $B$OJQ?t$N8D(B
953: $B?t$rI=$9(B. $B$3$N;XDj$O(B, $BJQ?t$r@hF,$+$i(B @code{L1}, @code{L2} , ...$B8D(B
954: $B$:$D$NAH$KJ,$1(B, $B$=$l$>$l$NJQ?t$K4X$7(B, $B=g$K(B @code{O1}, @code{O2},
955: ...$B$N9`=g=x7?$GBg>.$,7hDj$9$k$^$GHf3S$9$k$3$H$r0UL#$9$k(B. $B$3$N7?$N(B
956: $B=g=x$O0lHL$K>C5n=g=x$H8F$P$l$k(B.
1.2 noro 957: \E
958: \BEG
959: In this example @code{Oi} indicates 0, 1 or 2 and @code{Li} indicates
960: the number of variables subject to the correspoinding orderings.
961: This specification means the following.
962:
963: The variable list is separated into sub lists from left to right where
964: the @code{i}-th list contains @code{Li} members and it corresponds to
965: the ordering of type @code{Oi}. The result of a comparison is equal
966: to that for the leftmost different sub components. This type of ordering
967: is called an elimination ordering.
968: \E
1.1 noro 969:
970: @noindent
1.2 noro 971: \BJP
1.1 noro 972: $B$5$i$K(B, $B9TNs$K$h$j9`=g=x$r;XDj$9$k$3$H$,$G$-$k(B. $B0lHL$K(B, @code{n} $B9T(B
973: @code{m} $BNs$N<B?t9TNs(B @code{M} $B$,<!$N@-<A$r;}$D$H$9$k(B.
1.2 noro 974: \E
975: \BEG
976: Furthermore one can specify a term ordering by a matix.
977: Suppose that a real @code{n}, @code{m} matrix @code{M} has the
978: following properties.
979: \E
1.1 noro 980:
981: @enumerate
982: @item
1.2 noro 983: \JP $BD9$5(B @code{m} $B$N@0?t%Y%/%H%k(B @code{v} $B$KBP$7(B @code{Mv=0} $B$H(B @code{v=0} $B$OF1CM(B.
984: \BEG
985: For all integer verctors @code{v} of length @code{m} @code{Mv=0} is equivalent
986: to @code{v=0}.
987: \E
1.1 noro 988:
989: @item
1.2 noro 990: \BJP
1.1 noro 991: $BHsIi@.J,$r;}$DD9$5(B @code{m} $B$N(B 0 $B$G$J$$@0?t%Y%/%H%k(B @code{v} $B$KBP$7(B,
992: @code{Mv} $B$N(B 0 $B$G$J$$:G=i$N@.J,$OHsIi(B.
1.2 noro 993: \E
994: \BEG
995: For all non-negative integer vectors @code{v} the first non-zero component
996: of @code{Mv} is non-negative.
997: \E
1.1 noro 998: @end enumerate
999:
1000: @noindent
1.2 noro 1001: \BJP
1.1 noro 1002: $B$3$N;~(B, 2 $B$D$N%Y%/%H%k(B @code{t}, @code{s} $B$KBP$7(B,
1003: @code{t>s} $B$r(B, @code{M(t-s)} $B$N(B 0 $B$G$J$$:G=i$N@.J,$,HsIi(B,
1004: $B$GDj5A$9$k$3$H$K$h$j9`=g=x$,Dj5A$G$-$k(B.
1.2 noro 1005: \E
1006: \BEG
1007: Then we can define a term ordering such that, for two vectors
1008: @code{t}, @code{s}, @code{t>s} means that the first non-zero component
1009: of @code{M(t-s)} is non-negative.
1010: \E
1.1 noro 1011:
1012: @noindent
1.2 noro 1013: \BJP
1.1 noro 1014: $B9`=g=x7?$O(B, @code{gr()} $B$J$I$N0z?t$H$7$F;XDj$5$l$kB>(B, $BAH$_9~$_H!?t(B
1015: @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 1016: \E
1017: \BEG
1018: Types of term orderings are used as arguments of functions such as
1019: @code{gr()}. It is also set internally by @code{dp_ord()} and is used
1020: during executions of various functions.
1021: \E
1.1 noro 1022:
1023: @noindent
1.2 noro 1024: \BJP
1.1 noro 1025: $B$3$l$i$N=g=x$N6qBNE*$JDj5A$*$h$S%0%l%V%J4pDl$K4X$9$k99$K>\$7$$2r@b$O(B
1026: @code{[Becker,Weispfenning]} $B$J$I$r;2>H$N$3$H(B.
1.2 noro 1027: \E
1028: \BEG
1029: For concrete definitions of term ordering and more information
1030: about Groebner basis, refer to, for example, the book
1031: @code{[Becker,Weispfenning]}.
1032: \E
1.1 noro 1033:
1034: @noindent
1.2 noro 1035: \JP $B9`=g=x7?$N@_Dj$NB>$K(B, $BJQ?t$N=g=x<+BN$b7W;;;~4V$KBg$-$J1F6A$rM?$($k(B.
1036: \BEG
1037: Note that the variable ordering have strong effects on the computation
1038: time as well as the choice of types of term orderings.
1039: \E
1.1 noro 1040:
1041: @example
1042: [90] B=[x^10-t,x^8-z,x^31-x^6-x-y]$
1043: [91] gr(B,[x,y,z,t],2);
1044: [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
1045: -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
1046: +(-12*t^16+72*t^13-28*t^11-180*t^10+112*t^8+240*t^7+28*t^6-127*t^5
1047: -167*t^4-55*t^3+30*t^2+58*t-15)*z^4,
1.5 noro 1048: (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
1049: +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
1050: +(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
1051: +27*t^3-16*t^2-30*t+7)*z^4,
1052: (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
1053: -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
1054: +10*t^4-36*t^3-11*t^2-5*t+9)*z^2,
1.1 noro 1055: -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 1056: -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
1057: +20*t^19+28*t^18-120*t^16-56*t^15+14*t^14+300*t^13+70*t^12-56*t^11
1058: -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
1059: -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
1060: -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 1061: -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 1062: 2*t^2*y^3+z^2*y^2+(-2*t^5+4*t^2-6)*z^4*y
1063: +(4*t^8-t^7-8*t^5+2*t^4-4*t^3+5*t^2-t)*z,
1.1 noro 1064: z^3*y^2+2*t^3*y+(-t^7+2*t^4+t^2-t)*z^2,
1065: -t*z*y^2-2*z^3*y+t^8-2*t^5-t^3+t^2,
1.5 noro 1066: -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 1067: [93] gr(B,[t,z,y,x],2);
1068: [x^10-t,x^8-z,x^31-x^6-x-y]
1069: @end example
1070:
1071: @noindent
1.2 noro 1072: \BJP
1.1 noro 1073: $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
1074: $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
1075: $B%0%l%V%J4pDl$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
1076: $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
1077: $B$3$H$G$"$j(B, $B$3$NNc$N>l9g$O(B, @code{t}, @code{z}, @code{y} $B$,4{$K(B
1078: @code{x} $B$GI=$5$l$F$$$k$3$H$+$i$3$N$h$&$J6KC<$J7k2L$H$J$C$?$o$1$G$"$k(B.
1079: $B<B:]$K8=$l$k7W;;$K$*$$$F$O(B, $B$3$N$h$&$KA*$V$Y$-JQ?t=g=x$,L@$i$+$G$"$k(B
1080: $B$3$H$O>/$J$/(B, $B;n9T:x8m$,I,MW$J>l9g$b$"$k(B.
1.2 noro 1081: \E
1082: \BEG
1083: As you see in the above example, the Groebner base under variable
1084: ordering @code{[x,y,z,t]} has a lot of bases and each base itself is
1085: large. Under variable ordering @code{[t,z,y,x]}, however, @code{B} itself
1086: is already the Groebner basis.
1087: Roughly speaking, to obtain a Groebner base under the lexicographic
1088: ordering is to express the variables on the left (having higher order)
1089: in terms of variables on the right (having lower order).
1090: In the example, variables @code{t}, @code{z}, and @code{y} are already
1091: expressed by variable @code{x}, and the above explanation justifies
1092: such a drastic experimental results.
1093: In practice, however, optimum ordering for variables may not known
1094: beforehand, and some heuristic trial may be inevitable.
1.13 noro 1095: \E
1096:
1097: \BJP
1098: @node Weight ,,, $B%0%l%V%J4pDl$N7W;;(B
1099: @section Weight
1100: \E
1101: \BEG
1102: @node Weight,,, Groebner basis computation
1103: @section Weight
1104: \E
1105: \BJP
1106: $BA0@a$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
1107: $B$h$j0lHLE*$J$b$N$H$J$k(B.
1108: \E
1109: \BEG
1.14 noro 1110: Term orderings introduced in the previous section can be generalized
1.13 noro 1111: by setting a weight for each variable.
1112: \E
1113: @example
1114: [0] dp_td(<<1,1,1>>);
1115: 3
1116: [1] dp_set_weight([1,2,3])$
1117: [2] dp_td(<<1,1,1>>);
1118: 6
1119: @end example
1120: \BJP
1121: $BC19`<0$NA4<!?t$r7W;;$9$k:](B, $B%G%U%)%k%H$G$O(B
1122: $B3FJQ?t$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
1123: $B9M$($F$$$k$3$H$KAjEv$9$k(B. $B$3$NNc$G$O(B, $BBh0l(B, $BBhFs(B, $BBh;0JQ?t$N(B
1124: weight $B$r$=$l$>$l(B 1,2,3 $B$H;XDj$7$F$$$k(B. $B$3$N$?$a(B, @code{<<1,1,1>>}
1125: $B$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.
1126: 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.
1127: $BNc$($P(B, weight $B$r$&$^$/@_Dj$9$k$3$H$G(B, $BB?9`<0$r(B weighted homogeneous
1128: $B$K$9$k$3$H$,$G$-$k>l9g$,$"$k(B.
1129: \E
1130: \BEG
1131: By default, the total degree of a monomial is equal to
1132: the sum of all exponents. This means that the weight for each variable
1133: is set to 1.
1134: In this example, the weights for the first, the second and the third
1135: variable are set to 1, 2 and 3 respectively.
1136: Therefore the total degree of @code{<<1,1,1>>} under this weight,
1137: which is called the weight of the monomial, is @code{1*1+1*2+1*3=6}.
1.14 noro 1138: By setting weights, different term orderings can be set under a type of
1139: term ordeing. In some case a polynomial can
1140: be made weighted homogeneous by setting an appropriate weight.
1.13 noro 1141: \E
1142:
1143: \BJP
1144: $B3FJQ?t$KBP$9$k(B weight $B$r$^$H$a$?$b$N$r(B weight vector $B$H8F$V(B.
1145: $B$9$Y$F$N@.J,$,@5$G$"$j(B, $B%0%l%V%J4pDl7W;;$K$*$$$F(B, $BA4<!?t$N(B
1146: $BBe$o$j$KMQ$$$i$l$k$b$N$rFC$K(B sugar weight $B$H8F$V$3$H$K$9$k(B.
1147: sugar strategy $B$K$*$$$F(B, $BA4<!?t$NBe$o$j$K;H$o$l$k$+$i$G$"$k(B.
1148: $B0lJ}$G(B, $B3F@.J,$,I,$:$7$b@5$H$O8B$i$J$$(B weight vector $B$O(B,
1149: sugar weight $B$H$7$F@_Dj$9$k$3$H$O$G$-$J$$$,(B, $B9`=g=x$N0lHL2=$K$O(B
1150: $BM-MQ$G$"$k(B. $B$3$l$i$O(B, $B9TNs$K$h$k9`=g=x$N@_Dj$K$9$G$K8=$l$F(B
1151: $B$$$k(B. $B$9$J$o$A(B, $B9`=g=x$rDj5A$9$k9TNs$N3F9T$,(B, $B0l$D$N(B weight vector
1152: $B$H8+$J$5$l$k(B. $B$^$?(B, $B%V%m%C%/=g=x$O(B, $B3F%V%m%C%/$N(B
1153: $BJQ?t$KBP1~$9$k@.J,$N$_(B 1 $B$GB>$O(B 0 $B$N(B weight vector $B$K$h$kHf3S$r(B
1154: $B:G=i$K9T$C$F$+$i(B, $B3F%V%m%C%/Kh$N(B tie breaking $B$r9T$&$3$H$KAjEv$9$k(B.
1155: \E
1156:
1157: \BEG
1158: A list of weights for all variables is called a weight vector.
1159: A weight vector is called a sugar weight vector if
1160: its elements are all positive and it is used for computing
1161: a weighted total degree of a monomial, because such a weight
1162: is used instead of total degree in sugar strategy.
1163: On the other hand, a weight vector whose elements are not necessarily
1164: positive cannot be set as a sugar weight, but it is useful for
1165: generalizing term order. In fact, such a weight vector already
1166: appeared in a matrix order. That is, each row of a matrix defining
1167: a term order is regarded as a weight vector. A block order
1168: is also considered as a refinement of comparison by weight vectors.
1169: It compares two terms by using a weight vector whose elements
1170: corresponding to variables in a block is 1 and 0 otherwise,
1171: then it applies a tie breaker.
1.14 noro 1172: \E
1173:
1174: \BJP
1175: weight vector $B$N@_Dj$O(B @code{dp_set_weight()} $B$G9T$&$3$H$,$G$-$k(B
1176: $B$,(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
1177: $B$^$H$a$F@_Dj$G$-$k$3$H$,K>$^$7$$(B. $B$3$N$?$a(B, $B<!$N$h$&$J7A$G$b(B
1178: $B9`=g=x$,;XDj$G$-$k(B.
1179: \E
1180: \BEG
1181: A weight vector can be set by using @code{dp_set_weight()}.
1182: However it is more preferable if a weight vector can be set
1183: together with other parapmeters such as a type of term ordering
1184: and a variable order. This is realized as follows.
1185: \E
1.13 noro 1186:
1.14 noro 1187: @example
1188: [64] B=[x+y+z-6,x*y+y*z+z*x-11,x*y*z-6]$
1189: [65] dp_gr_main(B|v=[x,y,z],sugarweight=[3,2,1],order=0);
1190: [z^3-6*z^2+11*z-6,x+y+z-6,-y^2+(-z+6)*y-z^2+6*z-11]
1191: [66] dp_gr_main(B|v=[y,z,x],order=[[1,1,0],[0,1,0],[0,0,1]]);
1192: [x^3-6*x^2+11*x-6,x+y+z-6,-x^2+(-y+6)*x-y^2+6*y-11]
1193: [67] dp_gr_main(B|v=[y,z,x],order=[[x,1,y,2,z,3]]);
1194: [x+y+z-6,x^3-6*x^2+11*x-6,-x^2+(-y+6)*x-y^2+6*y-11]
1195: @end example
1196:
1197: \BJP
1198: $B$$$:$l$NNc$K$*$$$F$b(B, $B9`=g=x$O(B option $B$H$7$F;XDj$5$l$F$$$k(B.
1199: $B:G=i$NNc$G$O(B @code{v} $B$K$h$jJQ?t=g=x$r(B, @code{sugarweight} $B$K$h$j(B
1200: sugar weight vector $B$r(B, @code{order}$B$K$h$j9`=g=x7?$r;XDj$7$F$$$k(B.
1201: $BFs$DL\$NNc$K$*$1$k(B @code{order} $B$N;XDj$O(B matrix order $B$HF1MM$G$"$k(B.
1202: $B$9$J$o$A(B, $B;XDj$5$l$?(B weight vector $B$r:8$+$i=g$K;H$C$F(B weight $B$NHf3S(B
1203: $B$r9T$&(B. $B;0$DL\$NNc$bF1MM$G$"$k$,(B, $B$3$3$G$O(B weight vector $B$NMWAG$r(B
1204: $BJQ?tKh$K;XDj$7$F$$$k(B. $B;XDj$,$J$$$b$N$O(B 0 $B$H$J$k(B. $B;0$DL\$NNc$G$O(B,
1205: @code{order} $B$K$h$k;XDj$G$O9`=g=x$,7hDj$7$J$$(B. $B$3$N>l9g$K$O(B,
1206: tie breaker $B$H$7$FA4<!?t5U<-=q<0=g=x$,<+F0E*$K@_Dj$5$l$k(B.
1207: $B$3$N;XDjJ}K!$O(B, @code{dp_gr_main}, @code{dp_gr_mod_main} $B$J$I(B
1208: $B$NAH$_9~$_4X?t$G$N$_2DG=$G$"$j(B, @code{gr} $B$J$I$N%f!<%6Dj5A4X?t(B
1209: $B$G$OL$BP1~$G$"$k(B.
1210: \E
1211: \BEG
1212: In each example, a term ordering is specified as options.
1213: In the first example, a variable order, a sugar weight vector
1214: and a type of term ordering are specified by options @code{v},
1215: @code{sugarweight} and @code{order} respectively.
1216: In the second example, an option @code{order} is used
1217: to set a matrix ordering. That is, the specified weight vectors
1218: are used from left to right for comparing terms.
1219: The third example shows a variant of specifying a weight vector,
1220: where each component of a weight vector is specified variable by variable,
1221: and unspecified components are set to zero. In this example,
1222: a term order is not determined only by the specified weight vector.
1223: In such a case a tie breaker by the graded reverse lexicographic ordering
1224: is set automatically.
1225: This type of a term ordering specification can be applied only to builtin
1226: functions such as @code{dp_gr_main()}, @code{dp_gr_mod_main()}, not to
1227: user defined functions such as @code{gr()}.
1.2 noro 1228: \E
1.1 noro 1229:
1.2 noro 1230: \BJP
1.1 noro 1231: @node $BM-M}<0$r78?t$H$9$k%0%l%V%J4pDl7W;;(B,,, $B%0%l%V%J4pDl$N7W;;(B
1232: @section $BM-M}<0$r78?t$H$9$k%0%l%V%J4pDl7W;;(B
1.2 noro 1233: \E
1234: \BEG
1235: @node Groebner basis computation with rational function coefficients,,, Groebner basis computation
1236: @section Groebner basis computation with rational function coefficients
1237: \E
1.1 noro 1238:
1239: @noindent
1.2 noro 1240: \BJP
1.1 noro 1241: @code{gr()} $B$J$I$N%H%C%W%l%Y%kH!?t$O(B, $B$$$:$l$b(B, $BF~NOB?9`<0%j%9%H$K(B
1242: $B8=$l$kJQ?t(B ($BITDj85(B) $B$H(B, $BJQ?t%j%9%H$K8=$l$kJQ?t$rHf3S$7$F(B, $BJQ?t%j%9%H$K(B
1243: $B$J$$JQ?t$,F~NOB?9`<0$K8=$l$F$$$k>l9g$K$O(B, $B<+F0E*$K(B, $B$=$NJQ?t$r(B, $B78?t(B
1244: $BBN$N85$H$7$F07$&(B.
1.2 noro 1245: \E
1246: \BEG
1247: Such variables that appear within the input polynomials but
1248: not appearing in the input variable list are automatically treated
1249: as elements in the coefficient field
1250: by top level functions, such as @code{gr()}.
1251: \E
1.1 noro 1252:
1253: @example
1254: [64] gr([a*x+b*y-c,d*x+e*y-f],[x,y],2);
1255: [(-e*a+d*b)*x-f*b+e*c,(-e*a+d*b)*y+f*a-d*c]
1256: @end example
1257:
1258: @noindent
1.2 noro 1259: \BJP
1.1 noro 1260: $B$3$NNc$G$O(B, @code{a}, @code{b}, @code{c}, @code{d} $B$,78?tBN$N85$H$7$F(B
1261: $B07$o$l$k(B. $B$9$J$o$A(B, $BM-M}H!?tBN(B
1262: @b{F} = @b{Q}(@code{a},@code{b},@code{c},@code{d}) $B>e$N(B 2 $BJQ?tB?9`<04D(B
1263: @b{F}[@code{x},@code{y}] $B$K$*$1$k%0%l%V%J4pDl$r5a$a$k$3$H$K$J$k(B.
1264: $BCm0U$9$Y$-$3$H$O(B,
1265: $B78?t$,BN$H$7$F07$o$l$F$$$k$3$H$G$"$k(B. $B$9$J$o$A(B, $B78?t$N4V$KB?9`<0(B
1266: $B$H$7$F$N6&DL0x;R$,$"$C$?>l9g$K$O(B, $B7k2L$+$i$=$N0x;R$O=|$+$l$F$$$k(B
1267: $B$?$a(B, $BM-M}?tBN>e$NB?9`<04D>e$NLdBj$H$7$F9M$($?>l9g$N7k2L$H$O0lHL(B
1268: $B$K$O0[$J$k(B. $B$^$?(B, $B<g$H$7$F7W;;8zN(>e$NLdBj$N$?$a(B, $BJ,;6I=8=B?9`<0(B
1269: $B$N78?t$H$7$F<B:]$K5v$5$l$k$N$OB?9`<0$^$G$G$"$k(B. $B$9$J$o$A(B, $BJ,Jl$r(B
1270: $B;}$DM-M}<0$OJ,;6I=8=B?9`<0$N78?t$H$7$F$O5v$5$l$J$$(B.
1.2 noro 1271: \E
1272: \BEG
1273: In this example, variables @code{a}, @code{b}, @code{c}, and @code{d}
1274: are treated as elements in the coefficient field.
1275: In this case, a Groebner basis is computed
1276: on a bi-variate polynomial ring
1277: @b{F}[@code{x},@code{y}]
1278: over rational function field
1279: @b{F} = @b{Q}(@code{a},@code{b},@code{c},@code{d}).
1280: Notice that coefficients are considered as a member in a field.
1281: As a consequence, polynomial factors common to the coefficients
1282: are removed so that the result, in general, is different from
1283: the result that would be obtained when the problem is considered
1284: as a computation of Groebner basis over a polynomial ring
1285: with rational function coefficients.
1286: And note that coefficients of a distributed polynomial are limited
1287: to numbers and polynomials because of efficiency.
1288: \E
1.1 noro 1289:
1.2 noro 1290: \BJP
1.1 noro 1291: @node $B4pDlJQ49(B,,, $B%0%l%V%J4pDl$N7W;;(B
1292: @section $B4pDlJQ49(B
1.2 noro 1293: \E
1294: \BEG
1295: @node Change of ordering,,, Groebner basis computation
1296: @section Change of orderng
1297: \E
1.1 noro 1298:
1299: @noindent
1.2 noro 1300: \BJP
1.1 noro 1301: $B<-=q<0=g=x$N%0%l%V%J4pDl$r5a$a$k>l9g(B, $BD>@\(B @code{gr()} $B$J$I$r5/F0$9$k(B
1302: $B$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,
1303: $B$=$l$rF~NO$H$7$F<-=q<0=g=x$N%0%l%V%J4pDl$r7W;;$9$kJ}$,8zN($,$h$$>l9g(B
1304: $B$,$"$k(B. $B$^$?(B, $BF~NO$,2?$i$+$N=g=x$G$N%0%l%V%J4pDl$K$J$C$F$$$k>l9g(B, $B4pDl(B
1305: $BJQ49$H8F$P$l$kJ}K!$K$h$j(B, Buchberger $B%"%k%4%j%:%`$K$h$i$:$K8zN(NI$/(B
1306: $B<-=q<0=g=x$N%0%l%V%J4pDl$,7W;;$G$-$k>l9g$,$"$k(B. $B$3$N$h$&$JL\E*$N$?$a$N(B
1307: $BH!?t$,(B, $B%f!<%6Dj5AH!?t$H$7$F(B @samp{gr} $B$K$$$/$D$+Dj5A$5$l$F$$$k(B.
1308: $B0J2<$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
1309: $B4{$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
1310: @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 1311: \E
1312: \BEG
1313: When we compute a lex order Groebner basis, it is often efficient to
1314: compute it via Groebner basis with respect to another order such as
1315: degree reverse lex order, rather than to compute it directory by
1316: @code{gr()} etc. If we know that an input is a Groebner basis with
1317: respect to an order, we can apply special methods called change of
1318: ordering for a Groebner basis computation with respect to another
1319: order, without using Buchberger algorithm. The following two functions
1320: are ones for change of ordering such that they convert a Groebner
1321: basis @var{gbase} with respect to the variable order @var{vlist1} and
1322: the order type @var{order} into a lex Groebner basis with respect
1323: to the variable order @var{vlist2}.
1324: \E
1.1 noro 1325:
1326: @table @code
1327: @item tolex(@var{gbase},@var{vlist1},@var{order},@var{vlist2})
1328:
1.2 noro 1329: \BJP
1.1 noro 1330: $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.
1331: $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
1332: $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 1333: \E
1334: \BEG
1335: This function can be used only when @var{gbase} is an ideal over the
1336: rationals. The input @var{gbase} must be a Groebner basis with respect
1337: to the variable order @var{vlist1} and the order type @var{order}. Moreover
1338: the ideal generated by @var{gbase} must be zero-dimensional.
1339: This computes the lex Groebner basis of @var{gbase}
1340: by using the modular change of ordering algorithm. The algorithm first
1341: computes the lex Groebner basis over a finite field. Then each element
1342: in the lex Groebner basis over the rationals is computed with undetermined
1343: coefficient method and linear equation solving by Hensel lifting.
1344: \E
1.1 noro 1345:
1346: @item tolex_tl(@var{gbase},@var{vlist1},@var{order},@var{vlist2},@var{homo})
1347:
1.2 noro 1348: \BJP
1.1 noro 1349: $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
1350: $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
1351: trace-lifting$B$K$*$1$k%0%l%V%J4pDl8uJd$NF,9`(B, $BF,78?t$N@-<A$rMxMQ$7$F(B,
1352: $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
1353: $B$k$?$a(B, $BC1$K(BBuchberger $B%"%k%4%j%:%`$r7+$jJV$9$h$j8zN($h$/7W;;$G$-$k(B.
1354: $B99$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
1355: $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,
1356: $BD>@\<-=q<0=g=x$N7W;;$r9T$&$h$j8zN($,$h$$(B. ($B$b$A$m$sNc30$"$j(B. )
1357: $B0z?t(B @var{homo} $B$,(B 0 $B$G$J$$;~(B, @code{hgr()} $B$HF1MM$K@F<!2=$r7PM3$7$F(B
1358: $B7W;;$r9T$&(B.
1.2 noro 1359: \E
1360: \BEG
1361: This function computes the lex Groebner basis of @var{gbase}. The
1362: input @var{gbase} must be a Groebner basis with respect to the
1363: variable order @var{vlist1} and the order type @var{order}.
1364: Buchberger algorithm with trace lifting is used to compute the lex
1365: Groebner basis, however the Groebner basis check and the ideal
1366: membership check can be omitted by using several properties derived
1367: from the fact that the input is a Groebner basis. So it is more
1368: efficient than simple repetition of Buchberger algorithm. If the input
1369: is zero-dimensional, this function inserts automatically a computation
1370: of Groebner basis with respect to an elimination order, which makes
1371: the whole computation more efficient for many cases. If @var{homo} is
1372: not equal to 0, homogenization is used in each step.
1373: \E
1.1 noro 1374: @end table
1375:
1376: @noindent
1.2 noro 1377: \BJP
1.1 noro 1378: $B$=$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
1379: $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
1380: @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 1381: \E
1382: \BEG
1383: For zero-dimensional systems, there are several fuctions to
1384: compute the minimal polynomial of a polynomial and or a more compact
1385: representation for zeros of the system. They are all defined in @samp{gr}.
1386: Refer to the sections for each functions.
1387: \E
1.1 noro 1388:
1.2 noro 1389: \BJP
1.6 noro 1390: @node Weyl $BBe?t(B,,, $B%0%l%V%J4pDl$N7W;;(B
1391: @section Weyl $BBe?t(B
1392: \E
1393: \BEG
1394: @node Weyl algebra,,, Groebner basis computation
1395: @section Weyl algebra
1396: \E
1397:
1398: @noindent
1399:
1400: \BJP
1401: $B$3$l$^$G$O(B, $BDL>o$N2D49$JB?9`<04D$K$*$1$k%0%l%V%J4pDl7W;;$K$D$$$F(B
1402: $B=R$Y$F$-$?$,(B, $B%0%l%V%J4pDl$NM}O@$O(B, $B$"$k>r7o$rK~$?$9Hs2D49$J(B
1403: $B4D$K$b3HD%$G$-$k(B. $B$3$N$h$&$J4D$NCf$G(B, $B1~MQ>e$b=EMW$J(B,
1404: Weyl $BBe?t(B, $B$9$J$o$AB?9`<04D>e$NHyJ,:nMQAG4D$N1i;;$*$h$S(B
1405: $B%0%l%V%J4pDl7W;;$,(B Risa/Asir $B$K<BAu$5$l$F$$$k(B.
1406:
1407: $BBN(B @code{K} $B>e$N(B @code{n} $B<!85(B Weyl $BBe?t(B
1408: @code{D=K<x1,@dots{},xn,D1,@dots{},Dn>} $B$O(B
1409: \E
1410:
1411: \BEG
1412: So far we have explained Groebner basis computation in
1413: commutative polynomial rings. However Groebner basis can be
1414: considered in more general non-commutative rings.
1415: Weyl algebra is one of such rings and
1416: Risa/Asir implements fundamental operations
1417: in Weyl algebra and Groebner basis computation in Weyl algebra.
1418:
1419: The @code{n} dimensional Weyl algebra over a field @code{K},
1420: @code{D=K<x1,@dots{},xn,D1,@dots{},Dn>} is a non-commutative
1421: algebra which has the following fundamental relations:
1422: \E
1423:
1424: @code{xi*xj-xj*xi=0}, @code{Di*Dj-Dj*Di=0}, @code{Di*xj-xj*Di=0} (@code{i!=j}),
1425: @code{Di*xi-xi*Di=1}
1426:
1427: \BJP
1428: $B$H$$$&4pK\4X78$r;}$D4D$G$"$k(B. @code{D} $B$O(B $BB?9`<04D(B @code{K[x1,@dots{},xn]} $B$r78?t(B
1429: $B$H$9$kHyJ,:nMQAG4D$G(B, @code{Di} $B$O(B @code{xi} $B$K$h$kHyJ,$rI=$9(B. $B8r494X78$K$h$j(B,
1430: @code{D} $B$N85$O(B, @code{x1^i1*@dots{}*xn^in*D1^j1*@dots{}*Dn^jn} $B$J$kC19`(B
1431: $B<0$N(B @code{K} $B@~7A7k9g$H$7$F=q$-I=$9$3$H$,$G$-$k(B.
1432: Risa/Asir $B$K$*$$$F$O(B, $B$3$NC19`<0$r(B, $B2D49$JB?9`<0$HF1MM$K(B
1433: @code{<<i1,@dots{},in,j1,@dots{},jn>>} $B$GI=$9(B. $B$9$J$o$A(B, @code{D} $B$N85$b(B
1434: $BJ,;6I=8=B?9`<0$H$7$FI=$5$l$k(B. $B2C8:;;$O(B, $B2D49$N>l9g$HF1MM$K(B, @code{+}, @code{-}
1435: $B$K$h$j(B
1436: $B<B9T$G$-$k$,(B, $B>h;;$O(B, $BHs2D49@-$r9MN8$7$F(B @code{dp_weyl_mul()} $B$H$$$&4X?t(B
1437: $B$K$h$j<B9T$9$k(B.
1438: \E
1439:
1440: \BEG
1441: @code{D} is the ring of differential operators whose coefficients
1442: are polynomials in @code{K[x1,@dots{},xn]} and
1443: @code{Di} denotes the differentiation with respect to @code{xi}.
1444: According to the commutation relation,
1445: elements of @code{D} can be represented as a @code{K}-linear combination
1446: of monomials @code{x1^i1*@dots{}*xn^in*D1^j1*@dots{}*Dn^jn}.
1447: In Risa/Asir, this type of monomial is represented
1448: by @code{<<i1,@dots{},in,j1,@dots{},jn>>} as in the case of commutative
1449: polynomial.
1450: That is, elements of @code{D} are represented by distributed polynomials.
1451: Addition and subtraction can be done by @code{+}, @code{-},
1452: but multiplication is done by calling @code{dp_weyl_mul()} because of
1453: the non-commutativity of @code{D}.
1454: \E
1455:
1456: @example
1457: [0] A=<<1,2,2,1>>;
1458: (1)*<<1,2,2,1>>
1459: [1] B=<<2,1,1,2>>;
1460: (1)*<<2,1,1,2>>
1461: [2] A*B;
1462: (1)*<<3,3,3,3>>
1463: [3] dp_weyl_mul(A,B);
1464: (1)*<<3,3,3,3>>+(1)*<<3,2,3,2>>+(4)*<<2,3,2,3>>+(4)*<<2,2,2,2>>
1465: +(2)*<<1,3,1,3>>+(2)*<<1,2,1,2>>
1466: @end example
1467:
1468: \BJP
1469: $B%0%l%V%J4pDl7W;;$K$D$$$F$b(B, Weyl $BBe?t@lMQ$N4X?t$H$7$F(B,
1470: $B<!$N4X?t$,MQ0U$7$F$"$k(B.
1471: \E
1472: \BEG
1473: The following functions are avilable for Groebner basis computation
1474: in Weyl algebra:
1475: \E
1476: @code{dp_weyl_gr_main()},
1477: @code{dp_weyl_gr_mod_main()},
1478: @code{dp_weyl_gr_f_main()},
1479: @code{dp_weyl_f4_main()},
1480: @code{dp_weyl_f4_mod_main()}.
1481: \BJP
1482: $B$^$?(B, $B1~MQ$H$7$F(B, global b $B4X?t$N7W;;$,<BAu$5$l$F$$$k(B.
1483: \E
1484: \BEG
1485: Computation of the global b function is implemented as an application.
1486: \E
1487:
1488: \BJP
1.1 noro 1489: @node $B%0%l%V%J4pDl$K4X$9$kH!?t(B,,, $B%0%l%V%J4pDl$N7W;;(B
1490: @section $B%0%l%V%J4pDl$K4X$9$kH!?t(B
1.2 noro 1491: \E
1492: \BEG
1493: @node Functions for Groebner basis computation,,, Groebner basis computation
1494: @section Functions for Groebner basis computation
1495: \E
1.1 noro 1496:
1497: @menu
1498: * gr hgr gr_mod::
1499: * lex_hensel lex_tl tolex tolex_d tolex_tl::
1500: * lex_hensel_gsl tolex_gsl tolex_gsl_d::
1501: * gr_minipoly minipoly::
1502: * tolexm minipolym::
1.6 noro 1503: * 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::
1504: * dp_f4_main dp_f4_mod_main dp_weyl_f4_main dp_weyl_f4_mod_main::
1.17 noro 1505: * nd_gr nd_gr_trace nd_f4 nd_f4_trace nd_weyl_gr nd_weyl_gr_trace::
1.1 noro 1506: * dp_gr_flags dp_gr_print::
1507: * dp_ord::
1.18 noro 1508: * dp_set_weight dp_set_top_weight dp_weyl_set_weight::
1.1 noro 1509: * dp_ptod::
1510: * dp_dtop::
1511: * dp_mod dp_rat::
1512: * dp_homo dp_dehomo::
1513: * dp_ptozp dp_prim::
1.18 noro 1514: * dp_nf dp_nf_mod dp_true_nf dp_true_nf_mod dp_weyl_nf dp_weyl_nf_mod::
1.1 noro 1515: * dp_hm dp_ht dp_hc dp_rest::
1516: * dp_td dp_sugar::
1517: * dp_lcm::
1518: * dp_redble::
1519: * dp_subd::
1520: * dp_mbase::
1521: * dp_mag::
1522: * dp_red dp_red_mod::
1523: * dp_sp dp_sp_mod::
1524: * p_nf p_nf_mod p_true_nf p_true_nf_mod ::
1525: * p_terms::
1526: * gb_comp::
1527: * katsura hkatsura cyclic hcyclic::
1528: * dp_vtoe dp_etov::
1529: * lex_hensel_gsl tolex_gsl tolex_gsl_d::
1.3 noro 1530: * primadec primedec::
1.5 noro 1531: * primedec_mod::
1.10 noro 1532: * bfunction bfct generic_bfct ann ann0::
1.1 noro 1533: @end menu
1534:
1.2 noro 1535: \JP @node gr hgr gr_mod,,, $B%0%l%V%J4pDl$K4X$9$kH!?t(B
1536: \EG @node gr hgr gr_mod,,, Functions for Groebner basis computation
1.1 noro 1537: @subsection @code{gr}, @code{hgr}, @code{gr_mod}, @code{dgr}
1538: @findex gr
1539: @findex hgr
1540: @findex gr_mod
1541: @findex dgr
1542:
1543: @table @t
1544: @item gr(@var{plist},@var{vlist},@var{order})
1545: @itemx hgr(@var{plist},@var{vlist},@var{order})
1546: @itemx gr_mod(@var{plist},@var{vlist},@var{order},@var{p})
1547: @itemx dgr(@var{plist},@var{vlist},@var{order},@var{procs})
1.2 noro 1548: \JP :: $B%0%l%V%J4pDl$N7W;;(B
1549: \EG :: Groebner basis computation
1.1 noro 1550: @end table
1551:
1552: @table @var
1553: @item return
1.2 noro 1554: \JP $B%j%9%H(B
1555: \EG list
1.4 noro 1556: @item plist vlist procs
1.2 noro 1557: \JP $B%j%9%H(B
1558: \EG list
1.1 noro 1559: @item order
1.2 noro 1560: \JP $B?t(B, $B%j%9%H$^$?$O9TNs(B
1561: \EG number, list or matrix
1.1 noro 1562: @item p
1.2 noro 1563: \JP 2^27 $BL$K~$NAG?t(B
1564: \EG prime less than 2^27
1.1 noro 1565: @end table
1566:
1567: @itemize @bullet
1.2 noro 1568: \BJP
1.1 noro 1569: @item
1570: $BI8=`%i%$%V%i%j$N(B @samp{gr} $B$GDj5A$5$l$F$$$k(B.
1571: @item
1572: $B$$$:$l$b(B, $BB?9`<0%j%9%H(B @var{plist} $B$N(B, $BJQ?t=g=x(B @var{vlist}, $B9`=g=x7?(B
1573: @var{order} $B$K4X$9$k%0%l%V%J4pDl$r5a$a$k(B. @code{gr()}, @code{hgr()}
1574: $B$O(B $BM-M}?t78?t(B, @code{gr_mod()} $B$O(B GF(@var{p}) $B78?t$H$7$F7W;;$9$k(B.
1575: @item
1576: @var{vlist} $B$OITDj85$N%j%9%H(B. @var{vlist} $B$K8=$l$J$$ITDj85$O(B,
1577: $B78?tBN$KB0$9$k$H8+$J$5$l$k(B.
1578: @item
1579: @code{gr()}, trace-lifting ($B%b%8%e%i1i;;$rMQ$$$?9bB.2=(B) $B$*$h$S(B sugar
1580: strategy $B$K$h$k7W;;(B, @code{hgr()} $B$O(B trace-lifting $B$*$h$S(B
1581: $B@F<!2=$K$h$k(B $B6:@5$5$l$?(B sugar strategy $B$K$h$k7W;;$r9T$&(B.
1582: @item
1.16 fujiwara 1583: @code{dgr()} $B$O(B, @code{gr()}, @code{hgr()} $B$r(B
1.1 noro 1584: $B;R%W%m%;%9%j%9%H(B @var{procs} $B$N(B 2 $B$D$N%W%m%;%9$K$h$jF1;~$K7W;;$5$;(B,
1585: $B@h$K7k2L$rJV$7$?J}$N7k2L$rJV$9(B. $B7k2L$OF10l$G$"$k$,(B, $B$I$A$i$NJ}K!$,(B
1586: $B9bB.$+0lHL$K$OITL@$N$?$a(B, $B<B:]$N7P2a;~4V$rC;=L$9$k$N$KM-8z$G$"$k(B.
1587: @item
1588: @code{dgr()} $B$GI=<($5$l$k;~4V$O(B, $B$3$NH!?t$,<B9T$5$l$F$$$k%W%m%;%9$G$N(B
1589: 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 1590: @item
1591: $BB?9`<0%j%9%H(B @var{plist} $B$NMWAG$,J,;6I=8=B?9`<0$N>l9g$O(B
1592: $B7k2L$bJ,;6I=8=B?9`<0$N%j%9%H$G$"$k(B.
1593: $B$3$N>l9g(B, $B0z?t$NJ,;6B?9`<0$OM?$($i$l$?=g=x$K=>$$(B @code{dp_sort} $B$G(B
1594: $B%=!<%H$5$l$F$+$i7W;;$5$l$k(B.
1595: $BB?9`<0%j%9%H$NMWAG$,J,;6I=8=B?9`<0$N>l9g$b(B
1596: $BJQ?t$N?tJ,$NITDj85$N%j%9%H$r(B @var{vlist} $B0z?t$H$7$FM?$($J$$$H$$$1$J$$(B
1597: ($B%@%_!<(B).
1.2 noro 1598: \E
1599: \BEG
1600: @item
1601: These functions are defined in @samp{gr} in the standard library
1602: directory.
1603: @item
1604: They compute a Groebner basis of a polynomial list @var{plist} with
1605: respect to the variable order @var{vlist} and the order type @var{order}.
1606: @code{gr()} and @code{hgr()} compute a Groebner basis over the rationals
1607: and @code{gr_mod} computes over GF(@var{p}).
1608: @item
1609: Variables not included in @var{vlist} are regarded as
1610: included in the ground field.
1611: @item
1612: @code{gr()} uses trace-lifting (an improvement by modular computation)
1613: and sugar strategy.
1614: @code{hgr()} uses trace-lifting and a cured sugar strategy
1615: by using homogenization.
1616: @item
1617: @code{dgr()} executes @code{gr()}, @code{dgr()} simultaneously on
1618: two process in a child process list @var{procs} and returns
1619: the result obtained first. The results returned from both the process
1620: should be equal, but it is not known in advance which method is faster.
1621: Therefore this function is useful to reduce the actual elapsed time.
1622: @item
1623: The CPU time shown after an exection of @code{dgr()} indicates
1624: that of the master process, and most of the time corresponds to the time
1625: for communication.
1.12 takayama 1626: @item
1627: When the elements of @var{plist} are distributed polynomials,
1628: the result is also a list of distributed polynomials.
1629: In this case, firstly the elements of @var{plist} is sorted by @code{dp_sort}
1630: and the Grobner basis computation is started.
1631: Variables must be given in @var{vlist} even in this case
1632: (these variables are dummy).
1.2 noro 1633: \E
1.1 noro 1634: @end itemize
1635:
1636: @example
1637: [0] load("gr")$
1638: [64] load("cyclic")$
1639: [74] G=gr(cyclic(5),[c0,c1,c2,c3,c4],2);
1640: [c4^15+122*c4^10-122*c4^5-1,...]
1641: [75] GM=gr_mod(cyclic(5),[c0,c1,c2,c3,c4],2,31991)$
1642: 24628*c4^15+29453*c4^10+2538*c4^5+7363
1643: [76] (G[0]*24628-GM[0])%31991;
1644: 0
1645: @end example
1646:
1647: @table @t
1.2 noro 1648: \JP @item $B;2>H(B
1649: \EG @item References
1.6 noro 1650: @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 1651: @fref{dp_ord}.
1652: @end table
1653:
1.2 noro 1654: \JP @node lex_hensel lex_tl tolex tolex_d tolex_tl,,, $B%0%l%V%J4pDl$K4X$9$kH!?t(B
1655: \EG @node lex_hensel lex_tl tolex tolex_d tolex_tl,,, Functions for Groebner basis computation
1.1 noro 1656: @subsection @code{lex_hensel}, @code{lex_tl}, @code{tolex}, @code{tolex_d}, @code{tolex_tl}
1657: @findex lex_hensel
1658: @findex lex_tl
1659: @findex tolex
1660: @findex tolex_d
1661: @findex tolex_tl
1662:
1663: @table @t
1664: @item lex_hensel(@var{plist},@var{vlist1},@var{order},@var{vlist2},@var{homo})
1665: @itemx lex_tl(@var{plist},@var{vlist1},@var{order},@var{vlist2},@var{homo})
1.2 noro 1666: \JP :: $B4pDlJQ49$K$h$k<-=q<0=g=x%0%l%V%J4pDl$N7W;;(B
1667: \EG:: Groebner basis computation with respect to a lex order by change of ordering
1.1 noro 1668: @item tolex(@var{plist},@var{vlist1},@var{order},@var{vlist2})
1669: @itemx tolex_d(@var{plist},@var{vlist1},@var{order},@var{vlist2},@var{procs})
1670: @itemx tolex_tl(@var{plist},@var{vlist1},@var{order},@var{vlist2},@var{homo})
1.2 noro 1671: \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
1672: \EG :: Groebner basis computation with respect to a lex order by change of ordering, starting from a Groebner basis
1.1 noro 1673: @end table
1674:
1675: @table @var
1676: @item return
1.2 noro 1677: \JP $B%j%9%H(B
1678: \EG list
1.4 noro 1679: @item plist vlist1 vlist2 procs
1.2 noro 1680: \JP $B%j%9%H(B
1681: \EG list
1.1 noro 1682: @item order
1.2 noro 1683: \JP $B?t(B, $B%j%9%H$^$?$O9TNs(B
1684: \EG number, list or matrix
1.1 noro 1685: @item homo
1.2 noro 1686: \JP $B%U%i%0(B
1687: \EG flag
1.1 noro 1688: @end table
1689:
1690: @itemize @bullet
1.2 noro 1691: \BJP
1.1 noro 1692: @item
1693: $BI8=`%i%$%V%i%j$N(B @samp{gr} $B$GDj5A$5$l$F$$$k(B.
1694: @item
1695: @code{lex_hensel()}, @code{lex_tl()} $B$O(B,
1696: $BB?9`<0%j%9%H(B @var{plist} $B$N(B, $BJQ?t=g=x(B @var{vlist1}, $B9`=g=x7?(B
1697: @var{order} $B$K4X$9$k%0%l%V%J4pDl$r5a$a(B, $B$=$l$r(B, $BJQ?t=g=x(B @var{vlist2}
1698: $B$N<-=q<0=g=x%0%l%V%J4pDl$KJQ49$9$k(B.
1699: @item
1700: @code{tolex()}, @code{tolex_tl()} $B$O(B,
1701: $BJQ?t=g=x(B @var{vlist1}, $B9`=g=x7?(B @var{order} $B$K4X$9$k%0%l%V%J4pDl$G$"$k(B
1702: $BB?9`<0%j%9%H(B @var{plist} $B$rJQ?t=g=x(B @var{vlist2} $B$N<-=q<0=g=x%0%l%V%J(B
1703: $B4pDl$KJQ49$9$k(B.
1704: @code{tolex_d()} $B$O(B, @code{tolex()} $B$K$*$1$k(B, $B3F4pDl$N7W;;$r(B, $B;R%W%m%;%9(B
1705: $B%j%9%H(B @var{procs} $B$N3F%W%m%;%9$KJ,;67W;;$5$;$k(B.
1706: @item
1707: @code{lex_hensel()}, @code{lex_tl()} $B$K$*$$$F$O(B, $B<-=q<0=g=x%0%l%V%J4pDl$N(B
1708: $B7W;;$O<!$N$h$&$K9T$o$l$k(B. (@code{[Noro,Yokoyama]} $B;2>H(B.)
1709: @enumerate
1710: @item
1711: @var{vlist1}, @var{order} $B$K4X$9$k%0%l%V%J4pDl(B @var{G0} $B$r7W;;$9$k(B.
1712: (@code{lex_hensel()} $B$N$_(B. )
1713: @item
1714: @var{G0} $B$N3F85$N(B @var{vlist2} $B$K4X$9$k<-=q<0=g=x$K$*$1$kF,78?t$r3d$i$J$$(B
1715: $B$h$&$JAG?t(B @var{p} $B$rA*$S(B, GF(@var{p}) $B>e$G$N<-=q<0=g=x%0%l%V%J4pDl(B
1716: @var{Gp} $B$r7W;;$9$k(B.
1717: @item
1718: @var{Gp} $B$K8=$l$k$9$Y$F$N9`$N(B, @var{G0} $B$K4X$9$k@55,7A(B @var{NF} $B$r7W;;$9$k(B.
1719: @item
1720: @var{Gp} $B$N3F85(B @var{f} $B$K$D$-(B, @var{f} $B$N78?t$rL$Dj78?t$G(B,
1721: @var{f} $B$N3F9`$rBP1~$9$k(B @var{NF} $B$N85$GCV$-49$((B, $B3F9`$N78?t$r(B 0 $B$HCV$$$?(B,
1722: $BL$Dj78?t$K4X$9$k@~7AJ}Dx<07O(B @var{Lf} $B$r:n$k(B.
1723: @item
1724: @var{Lf} $B$,(B, $BK!(B @var{p} $B$G0l0U2r$r;}$D$3$H$rMQ$$$F(B @var{Lf} $B$N2r$r(B
1725: $BK!(B @var{p}$B$N2r$+$i(B Hensel $B9=@.$K$h$j5a$a$k(B.
1726: @item
1727: $B$9$Y$F$N(B @var{Gp} $B$N85$K$D$-@~7AJ}Dx<0$,2r$1$?$i$=$N2rA4BN$,5a$a$k(B
1728: $B<-=q<0=g=x$G$N%0%l%V%J4pDl(B. $B$b$7$I$l$+$N@~7AJ}Dx<0$N5a2r$K<:GT$7$?$i(B,
1729: @var{p} $B$r$H$jD>$7$F$d$jD>$9(B.
1730: @end enumerate
1731:
1732: @item
1733: @code{lex_tl()}, @code{tolex_tl()} $B$K$*$$$F$O(B, $B<-=q<0=g=x%0%l%V%J4pDl$N(B
1734: $B7W;;$O<!$N$h$&$K9T$o$l$k(B.
1735:
1736: @enumerate
1737: @item
1738: @var{vlist1}, @var{order} $B$K4X$9$k%0%l%V%J4pDl(B @var{G0} $B$r7W;;$9$k(B.
1739: (@code{lex_hensel()} $B$N$_(B. )
1740: @item
1741: @var{G0} $B$,(B 0 $B<!85%7%9%F%`$G$J$$$H$-(B, @var{G0} $B$rF~NO$H$7$F(B,
1742: @var{G0} $B$N3F85$N(B @var{vlist2} $B$K4X$9$k<-=q<0=g=x$K$*$1$kF,78?t$r3d$i$J$$(B
1743: $B$h$&$JAG?t(B @var{p} $B$rA*$S(B, @var{p} $B$rMQ$$$?(B trace-lifting $B$K$h$j<-=q<0(B
1744: $B=g=x$N%0%l%V%J4pDl8uJd$r5a$a(B, $B$b$75a$^$C$?$J$i%A%'%C%/$J$7$K$=$l$,5a$a$k(B
1745: $B%0%l%V%J4pDl$H$J$k(B. $B$b$7<:GT$7$?$i(B, @var{p} $B$r$H$jD>$7$F$d$jD>$9(B.
1746: @item
1747: @var{G0} $B$,(B 0 $B<!85%7%9%F%`$N$H$-(B, @var{G0} $B$rF~NO$H$7$F(B,
1748: $B$^$:(B, @var{vlist2} $B$N:G8e$NJQ?t0J30$r>C5n$9$k>C5n=g=x$K$h$j(B
1749: $B%0%l%V%J4pDl(B @var{G1} $B$r7W;;$7(B, $B$=$l$+$i<-=q<0=g=x$N%0%l%V%J4pDl$r(B
1750: $B7W;;$9$k(B. $B$=$N:](B, $B3F%9%F%C%W$G$O(B, $BF~NO$N3F85$N(B, $B5a$a$k=g=x$K$*$1$k(B
1751: $BF,78?t$r3d$i$J$$AG?t$rMQ$$$?(B trace-lifting $B$G%0%l%V%J4pDl8uJd$r5a$a(B,
1752: $B$b$75a$^$C$?$i%A%'%C%/$J$7$K$=$l$,$=$N=g=x$G$N%0%l%V%J4pDl$H$J$k(B.
1753: @end enumerate
1754:
1755: @item
1756: $BM-M}<078?t$N7W;;$O(B, @code{lex_tl()}, @code{tolex_tl()} $B$N$_<u$1IU$1$k(B.
1757: @item
1758: @code{homo} $B$,(B 0 $B$G$J$$>l9g(B, $BFbIt$G5/F0$5$l$k(B Buchberger $B%"%k%4%j%:%`$K(B
1759: $B$*$$$F(B, $B@F<!2=$,9T$o$l$k(B.
1760: @item
1761: @code{tolex_d()} $B$GI=<($5$l$k;~4V$O(B, $B$3$NH!?t$,<B9T$5$l$F$$$k%W%m%;%9$K(B
1762: $B$*$$$F9T$o$l$?7W;;$KBP1~$7$F$$$F(B, $B;R%W%m%;%9$K$*$1$k;~4V$O4^$^$l$J$$(B.
1.2 noro 1763: \E
1764: \BEG
1765: @item
1766: These functions are defined in @samp{gr} in the standard library
1767: directory.
1768: @item
1769: @code{lex_hensel()} and @code{lex_tl()} first compute a Groebner basis
1770: with respect to the variable order @var{vlist1} and the order type @var{order}.
1771: Then the Groebner basis is converted into a lex order Groebner basis
1772: with respect to the varable order @var{vlist2}.
1773: @item
1774: @code{tolex()} and @code{tolex_tl()} convert a Groebner basis @var{plist}
1775: with respect to the variable order @var{vlist1} and the order type @var{order}
1776: into a lex order Groebner basis
1777: with respect to the varable order @var{vlist2}.
1778: @code{tolex_d()} does computations of basis elements in @code{tolex()}
1779: in parallel on the processes in a child process list @var{procs}.
1780: @item
1781: In @code{lex_hensel()} and @code{tolex_hensel()} a lex order Groebner basis
1782: is computed as follows.(Refer to @code{[Noro,Yokoyama]}.)
1783: @enumerate
1784: @item
1785: Compute a Groebner basis @var{G0} with respect to @var{vlist1} and @var{order}.
1786: (Only in @code{lex_hensel()}. )
1787: @item
1788: Choose a prime which does not divide head coefficients of elements in @var{G0}
1789: with respect to @var{vlist1} and @var{order}. Then compute a lex order
1790: Groebner basis @var{Gp} over GF(@var{p}) with respect to @var{vlist2}.
1791: @item
1792: Compute @var{NF}, the set of all the normal forms with respect to
1793: @var{G0} of terms appearing in @var{Gp}.
1794: @item
1795: For each element @var{f} in @var{Gp}, replace coefficients and terms in @var{f}
1796: with undetermined coefficients and the corresponding polynomials in @var{NF}
1797: respectively, and generate a system of liear equation @var{Lf} by equating
1798: the coefficients of terms in the replaced polynomial with 0.
1799: @item
1800: Solve @var{Lf} by Hensel lifting, starting from the unique mod @var{p}
1801: solution.
1802: @item
1803: If all the linear equations generated from the elements in @var{Gp}
1804: could be solved, then the set of solutions corresponds to a lex order
1805: Groebner basis. Otherwise redo the whole process with another @var{p}.
1806: @end enumerate
1807:
1808: @item
1809: In @code{lex_tl()} and @code{tolex_tl()} a lex order Groebner basis
1810: is computed as follows.(Refer to @code{[Noro,Yokoyama]}.)
1811:
1812: @enumerate
1813: @item
1814: Compute a Groebner basis @var{G0} with respect to @var{vlist1} and @var{order}.
1815: (Only in @code{lex_tl()}. )
1816: @item
1817: If @var{G0} is not zero-dimensional, choose a prime which does not divide
1818: head coefficients of elements in @var{G0} with respect to @var{vlist1} and
1819: @var{order}. Then compute a candidate of a lex order Groebner basis
1820: via trace lifting with @var{p}. If it succeeds the candidate is indeed
1821: a lex order Groebner basis without any check. Otherwise redo the whole
1822: process with another @var{p}.
1823: @item
1824:
1825: If @var{G0} is zero-dimensional, starting from @var{G0},
1826: compute a Groebner basis @var{G1} with respect to an elimination order
1827: to eliminate variables other than the last varibale in @var{vlist2}.
1828: Then compute a lex order Groebner basis stating from @var{G1}. These
1829: computations are done by trace lifting and the selection of a mudulus
1830: @var{p} is the same as in non zero-dimensional cases.
1831: @end enumerate
1832:
1833: @item
1834: Computations with rational function coefficients can be done only by
1835: @code{lex_tl()} and @code{tolex_tl()}.
1836: @item
1837: If @code{homo} is not equal to 0, homogenization is used in Buchberger
1838: algorithm.
1839: @item
1840: The CPU time shown after an execution of @code{tolex_d()} indicates
1841: that of the master process, and it does not include the time in child
1842: processes.
1843: \E
1.1 noro 1844: @end itemize
1845:
1846: @example
1847: [78] K=katsura(5)$
1848: 30msec + gc : 20msec
1849: [79] V=[u5,u4,u3,u2,u1,u0]$
1850: 0msec
1851: [80] G0=hgr(K,V,2)$
1852: 91.558sec + gc : 15.583sec
1853: [81] G1=lex_hensel(K,V,0,V,0)$
1854: 49.049sec + gc : 9.961sec
1855: [82] G2=lex_tl(K,V,0,V,1)$
1856: 31.186sec + gc : 3.500sec
1857: [83] gb_comp(G0,G1);
1858: 1
1859: 10msec
1860: [84] gb_comp(G0,G2);
1861: 1
1862: @end example
1863:
1864: @table @t
1.2 noro 1865: \JP @item $B;2>H(B
1866: \EG @item References
1.6 noro 1867: @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 1868: \JP @fref{dp_ord}, @fref{$BJ,;67W;;(B}
1869: \EG @fref{dp_ord}, @fref{Distributed computation}
1.1 noro 1870: @end table
1871:
1.2 noro 1872: \JP @node lex_hensel_gsl tolex_gsl tolex_gsl_d,,, $B%0%l%V%J4pDl$K4X$9$kH!?t(B
1873: \EG @node lex_hensel_gsl tolex_gsl tolex_gsl_d,,, Functions for Groebner basis computation
1.1 noro 1874: @subsection @code{lex_hensel_gsl}, @code{tolex_gsl}, @code{tolex_gsl_d}
1875: @findex lex_hensel_gsl
1876: @findex tolex_gsl
1877: @findex tolex_gsl_d
1878:
1879: @table @t
1880: @item lex_hensel_gsl(@var{plist},@var{vlist1},@var{order},@var{vlist2},@var{homo})
1.2 noro 1881: \JP :: GSL $B7A<0$N%$%G%"%k4pDl$N7W;;(B
1882: \EG ::Computation of an GSL form ideal basis
1.8 noro 1883: @item tolex_gsl(@var{plist},@var{vlist1},@var{order},@var{vlist2})
1884: @itemx tolex_gsl_d(@var{plist},@var{vlist1},@var{order},@var{vlist2},@var{procs})
1.2 noro 1885: \JP :: $B%0%l%V%J4pDl$rF~NO$H$9$k(B, GSL $B7A<0$N%$%G%"%k4pDl$N7W;;(B
1886: \EG :: Computation of an GSL form ideal basis stating from a Groebner basis
1.1 noro 1887: @end table
1888:
1889: @table @var
1890: @item return
1.2 noro 1891: \JP $B%j%9%H(B
1892: \EG list
1.4 noro 1893: @item plist vlist1 vlist2 procs
1.2 noro 1894: \JP $B%j%9%H(B
1895: \EG list
1.1 noro 1896: @item order
1.2 noro 1897: \JP $B?t(B, $B%j%9%H$^$?$O9TNs(B
1898: \EG number, list or matrix
1.1 noro 1899: @item homo
1.2 noro 1900: \JP $B%U%i%0(B
1901: \EG flag
1.1 noro 1902: @end table
1903:
1904: @itemize @bullet
1.2 noro 1905: \BJP
1.1 noro 1906: @item
1907: @code{lex_hensel_gsl()} $B$O(B @code{lex_hensel()} $B$N(B, @code{tolex_gsl()} $B$O(B
1908: @code{tolex()} $B$NJQ<o$G(B, $B7k2L$N$_$,0[$J$k(B.
1909: @code{tolex_gsl_d()} $B$O(B, $B4pDl7W;;$r(B, @code{procs} $B$G;XDj$5$l$k;R%W%m%;%9$K(B
1910: $BJ,;67W;;$5$;$k(B.
1911: @item
1912: $BF~NO$,(B 0 $B<!85%7%9%F%`$G(B, $B$=$N<-=q<0=g=x%0%l%V%J4pDl$,(B
1913: @code{[f0,x1-f1,...,xn-fn]} (@code{f0},...,@code{fn} $B$O(B
1914: @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,
1915: @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)
1916: $B$rJV$9(B.
1.2 noro 1917: $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 1918: @code{x0} $B$N(B1 $BJQ?tB?9`<0$G(B,
1919: $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')]}
1920: $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
1921: $B$h$kDL>o$N%0%l%V%J4pDl$rJV$9(B.
1922: @item
1923: GSL $B7A<0$K$h$jI=$5$l$k4pDl$O%0%l%V%J4pDl$G$O$J$$$,(B, $B0lHL$K78?t$,(B SL $B7A<0(B
1924: $B$N%0%l%V%J4pDl$h$jHs>o$K>.$5$$$?$a7W;;$bB.$/(B, $B2r$b5a$a$d$9$$(B.
1925: @code{tolex_gsl_d()} $B$GI=<($5$l$k;~4V$O(B, $B$3$NH!?t$,<B9T$5$l$F$$$k%W%m%;%9$K(B
1926: $B$*$$$F9T$o$l$?7W;;$KBP1~$7$F$$$F(B, $B;R%W%m%;%9$K$*$1$k;~4V$O4^$^$l$J$$(B.
1.2 noro 1927: \E
1928: \BEG
1929: @item
1930: @code{lex_hensel_gsl()} and @code{lex_hensel()} are variants of
1931: @code{tolex_gsl()} and @code{tolex()} respectively. The results are
1932: Groebner basis or a kind of ideal basis, called GSL form.
1933: @code{tolex_gsl_d()} does basis computations in parallel on child
1934: processes specified in @code{procs}.
1935:
1936: @item
1937: If the input is zero-dimensional and a lex order Groebner basis has
1938: the form @code{[f0,x1-f1,...,xn-fn]} (@code{f0},...,@code{fn} are
1939: univariate polynomials of @code{x0}; SL form), then this these
1940: functions return a list such as
1941: @code{[[x1,g1,d1],...,[xn,gn,dn],[x0,f0,f0']]} (GSL form). In this list
1942: @code{gi} is a univariate polynomial of @code{x0} such that
1943: @code{di*f0'*fi-gi} divides @code{f0} and the roots of the input ideal is
1944: @code{[x1=g1/(d1*f0'),...,xn=gn/(dn*f0')]} for @code{x0}
1945: such that @code{f0(x0)=0}.
1946: If the lex order Groebner basis does not have the above form,
1947: these functions return
1948: a lex order Groebner basis computed by @code{tolex()}.
1949: @item
1950: Though an ideal basis represented as GSL form is not a Groebner basis
1951: we can expect that the coefficients are much smaller than those in a Groebner
1952: basis and that the computation is efficient.
1953: The CPU time shown after an execution of @code{tolex_gsl_d()} indicates
1954: that of the master process, and it does not include the time in child
1955: processes.
1956: \E
1.1 noro 1957: @end itemize
1958:
1959: @example
1960: [103] K=katsura(5)$
1961: [104] V=[u5,u4,u3,u2,u1,u0]$
1962: [105] G0=gr(K,V,0)$
1963: [106] GSL=tolex_gsl(G0,V,0,V)$
1964: [107] GSL[0];
1965: [u1,8635837421130477667200000000*u0^31-...]
1966: [108] GSL[1];
1967: [u2,10352277157007342793600000000*u0^31-...]
1968: [109] GSL[5];
1.5 noro 1969: [u0,11771021876193064124640000000*u0^32-...,
1970: 376672700038178051988480000000*u0^31-...]
1.1 noro 1971: @end example
1972:
1973: @table @t
1.2 noro 1974: \JP @item $B;2>H(B
1975: \EG @item References
1.1 noro 1976: @fref{lex_hensel lex_tl tolex tolex_d tolex_tl},
1.2 noro 1977: \JP @fref{$BJ,;67W;;(B}
1978: \EG @fref{Distributed computation}
1.1 noro 1979: @end table
1980:
1.2 noro 1981: \JP @node gr_minipoly minipoly,,, $B%0%l%V%J4pDl$K4X$9$kH!?t(B
1982: \EG @node gr_minipoly minipoly,,, Functions for Groebner basis computation
1.1 noro 1983: @subsection @code{gr_minipoly}, @code{minipoly}
1984: @findex gr_minipoly
1985: @findex minipoly
1986:
1987: @table @t
1988: @item gr_minipoly(@var{plist},@var{vlist},@var{order},@var{poly},@var{v},@var{homo})
1.2 noro 1989: \JP :: $BB?9`<0$N(B, $B%$%G%"%k$rK!$H$7$?:G>.B?9`<0$N7W;;(B
1990: \EG :: Computation of the minimal polynomial of a polynomial modulo an ideal
1.1 noro 1991: @item minipoly(@var{plist},@var{vlist},@var{order},@var{poly},@var{v})
1.2 noro 1992: \JP :: $B%0%l%V%J4pDl$rF~NO$H$9$k(B, $BB?9`<0$N:G>.B?9`<0$N7W;;(B
1993: \EG :: Computation of the minimal polynomial of a polynomial modulo an ideal
1.1 noro 1994: @end table
1995:
1996: @table @var
1997: @item return
1.2 noro 1998: \JP $BB?9`<0(B
1999: \EG polynomial
1.4 noro 2000: @item plist vlist
1.2 noro 2001: \JP $B%j%9%H(B
2002: \EG list
1.1 noro 2003: @item order
1.2 noro 2004: \JP $B?t(B, $B%j%9%H$^$?$O9TNs(B
2005: \EG number, list or matrix
1.1 noro 2006: @item poly
1.2 noro 2007: \JP $BB?9`<0(B
2008: \EG polynomial
1.1 noro 2009: @item v
1.2 noro 2010: \JP $BITDj85(B
2011: \EG indeterminate
1.1 noro 2012: @item homo
1.2 noro 2013: \JP $B%U%i%0(B
2014: \EG flag
1.1 noro 2015: @end table
2016:
2017: @itemize @bullet
1.2 noro 2018: \BJP
1.1 noro 2019: @item
2020: @code{gr_minipoly()} $B$O%0%l%V%J4pDl$N7W;;$+$i9T$$(B, @code{minipoly()} $B$O(B
2021: $BF~NO$r%0%l%V%J4pDl$H$_$J$9(B.
2022: @item
2023: $B%$%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,
2024: K[@var{v}] $B$N85(B f(@var{v}) $B$K(B f(@var{p}) mod I $B$rBP1~$5$;$k(B
2025: $B4D=`F17?$N3K$O(B 0 $B$G$J$$B?9`<0$K$h$j@8@.$5$l$k(B. $B$3$N@8@.85$r(B @var{p}
2026: $B$N(B, $BK!(B @var{I} $B$G$N:G>.B?9`<0$H8F$V(B.
2027: @item
2028: @code{gr_minipoly()}, @code{minipoly()} $B$O(B, $BB?9`<0(B @var{p} $B$N:G>.B?9`<0(B
2029: $B$r5a$a(B, @var{v} $B$rJQ?t$H$9$kB?9`<0$H$7$FJV$9(B.
2030: @item
2031: $B:G>.B?9`<0$O(B, $B%0%l%V%J4pDl$N(B 1 $B$D$N85$H$7$F7W;;$9$k$3$H$b$G$-$k$,(B,
2032: $B:G>.B?9`<0$N$_$r5a$a$?$$>l9g(B, @code{minipoly()}, @code{gr_minipoly()} $B$O(B
2033: $B%0%l%V%J4pDl$rMQ$$$kJ}K!$KHf$Y$F8zN($,$h$$(B.
2034: @item
2035: @code{gr_minipoly()} $B$K;XDj$9$k9`=g=x$H$7$F$O(B, $BDL>oA4<!?t5U<-=q<0=g=x$r(B
2036: $BMQ$$$k(B.
1.2 noro 2037: \E
2038: \BEG
2039: @item
2040: @code{gr_minipoly()} begins by computing a Groebner basis.
2041: @code{minipoly()} regards an input as a Groebner basis with respect to
2042: the variable order @var{vlist} and the order type @var{order}.
2043: @item
2044: Let K be a field. If an ideal @var{I} in K[X] is zero-dimensional, then, for
2045: a polynomial @var{p} in K[X], the kernel of a homomorphism from
2046: K[@var{v}] to K[X]/@var{I} which maps f(@var{v}) to f(@var{p}) mod @var{I}
2047: is generated by a polynomial. The generator is called the minimal polynomial
2048: of @var{p} modulo @var{I}.
2049: @item
2050: @code{gr_minipoly()} and @code{minipoly()} computes the minimal polynomial
2051: of a polynomial @var{p} and returns it as a polynomial of @var{v}.
2052: @item
2053: The minimal polynomial can be computed as an element of a Groebner basis.
2054: But if we are only interested in the minimal polynomial,
2055: @code{minipoly()} and @code{gr_minipoly()} can compute it more efficiently
2056: than methods using Groebner basis computation.
2057: @item
2058: It is recommended to use a degree reverse lex order as a term order
2059: for @code{gr_minipoly()}.
2060: \E
1.1 noro 2061: @end itemize
2062:
2063: @example
2064: [117] G=tolex(G0,V,0,V)$
2065: 43.818sec + gc : 11.202sec
2066: [118] GSL=tolex_gsl(G0,V,0,V)$
2067: 17.123sec + gc : 2.590sec
2068: [119] MP=minipoly(G0,V,0,u0,z)$
2069: 4.370sec + gc : 780msec
2070: @end example
2071:
2072: @table @t
1.2 noro 2073: \JP @item $B;2>H(B
2074: \EG @item References
1.1 noro 2075: @fref{lex_hensel lex_tl tolex tolex_d tolex_tl}.
2076: @end table
2077:
1.2 noro 2078: \JP @node tolexm minipolym,,, $B%0%l%V%J4pDl$K4X$9$kH!?t(B
2079: \EG @node tolexm minipolym,,, Functions for Groebner basis computation
1.1 noro 2080: @subsection @code{tolexm}, @code{minipolym}
2081: @findex tolexm
2082: @findex minipolym
2083:
2084: @table @t
2085: @item tolexm(@var{plist},@var{vlist1},@var{order},@var{vlist2},@var{mod})
1.2 noro 2086: \JP :: $BK!(B @var{mod} $B$G$N4pDlJQ49$K$h$k%0%l%V%J4pDl7W;;(B
2087: \EG :: Groebner basis computation modulo @var{mod} by change of ordering.
1.1 noro 2088: @item minipolym(@var{plist},@var{vlist1},@var{order},@var{poly},@var{v},@var{mod})
1.2 noro 2089: \JP :: $BK!(B @var{mod} $B$G$N%0%l%V%J4pDl$K$h$kB?9`<0$N:G>.B?9`<0$N7W;;(B
2090: \EG :: Minimal polynomial computation modulo @var{mod} the same method as
1.1 noro 2091: @end table
2092:
2093: @table @var
2094: @item return
1.2 noro 2095: \JP @code{tolexm()} : $B%j%9%H(B, @code{minipolym()} : $BB?9`<0(B
2096: \EG @code{tolexm()} : list, @code{minipolym()} : polynomial
1.4 noro 2097: @item plist vlist1 vlist2
1.2 noro 2098: \JP $B%j%9%H(B
2099: \EG list
1.1 noro 2100: @item order
1.2 noro 2101: \JP $B?t(B, $B%j%9%H$^$?$O9TNs(B
2102: \EG number, list or matrix
1.1 noro 2103: @item mod
1.2 noro 2104: \JP $BAG?t(B
2105: \EG prime
1.1 noro 2106: @end table
2107:
2108: @itemize @bullet
1.2 noro 2109: \BJP
1.1 noro 2110: @item
2111: $BF~NO(B @var{plist} $B$O$$$:$l$b(B $BJQ?t=g=x(B @var{vlist1}, $B9`=g=x7?(B @var{order},
2112: $BK!(B @var{mod} $B$K$*$1$k%0%l%V%J4pDl$G$J$1$l$P$J$i$J$$(B.
2113: @item
2114: @code{minipolym()} $B$O(B @code{minipoly} $B$KBP1~$9$k7W;;$rK!(B @var{mod}$B$G9T$&(B.
2115: @item
2116: @code{tolexm()} $B$O(B FGLM $BK!$K$h$k4pDlJQ49$K$h$j(B @var{vlist2},
2117: $B<-=q<0=g=x$K$h$k%0%l%V%J4pDl$r7W;;$9$k(B.
1.2 noro 2118: \E
2119: \BEG
2120: @item
2121: An input @var{plist} must be a Groebner basis modulo @var{mod}
2122: with respect to the variable order @var{vlist1} and the order type @var{order}.
2123: @item
2124: @code{minipolym()} executes the same computation as in @code{minipoly}.
2125: @item
2126: @code{tolexm()} computes a lex order Groebner basis modulo @var{mod}
2127: with respect to the variable order @var{vlist2}, by using FGLM algorithm.
2128: \E
1.1 noro 2129: @end itemize
2130:
2131: @example
2132: [197] tolexm(G0,V,0,V,31991);
2133: [8271*u0^31+10435*u0^30+816*u0^29+26809*u0^28+...,...]
2134: [198] minipolym(G0,V,0,u0,z,31991);
2135: z^32+11405*z^31+20868*z^30+21602*z^29+...
2136: @end example
2137:
2138: @table @t
1.2 noro 2139: \JP @item $B;2>H(B
2140: \EG @item References
1.1 noro 2141: @fref{lex_hensel lex_tl tolex tolex_d tolex_tl},
2142: @fref{gr_minipoly minipoly}.
2143: @end table
2144:
1.6 noro 2145: \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
2146: \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
2147: @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 2148: @findex dp_gr_main
2149: @findex dp_gr_mod_main
1.5 noro 2150: @findex dp_gr_f_main
1.6 noro 2151: @findex dp_weyl_gr_main
2152: @findex dp_weyl_gr_mod_main
2153: @findex dp_weyl_gr_f_main
1.1 noro 2154:
2155: @table @t
2156: @item dp_gr_main(@var{plist},@var{vlist},@var{homo},@var{modular},@var{order})
2157: @itemx dp_gr_mod_main(@var{plist},@var{vlist},@var{homo},@var{modular},@var{order})
1.5 noro 2158: @itemx dp_gr_f_main(@var{plist},@var{vlist},@var{homo},@var{order})
1.6 noro 2159: @itemx dp_weyl_gr_main(@var{plist},@var{vlist},@var{homo},@var{modular},@var{order})
2160: @itemx dp_weyl_gr_mod_main(@var{plist},@var{vlist},@var{homo},@var{modular},@var{order})
2161: @itemx dp_weyl_gr_f_main(@var{plist},@var{vlist},@var{homo},@var{order})
1.2 noro 2162: \JP :: $B%0%l%V%J4pDl$N7W;;(B ($BAH$_9~$_H!?t(B)
2163: \EG :: Groebner basis computation (built-in functions)
1.1 noro 2164: @end table
2165:
2166: @table @var
2167: @item return
1.2 noro 2168: \JP $B%j%9%H(B
2169: \EG list
1.4 noro 2170: @item plist vlist
1.2 noro 2171: \JP $B%j%9%H(B
2172: \EG list
1.1 noro 2173: @item order
1.2 noro 2174: \JP $B?t(B, $B%j%9%H$^$?$O9TNs(B
2175: \EG number, list or matrix
1.1 noro 2176: @item homo
1.2 noro 2177: \JP $B%U%i%0(B
2178: \EG flag
1.1 noro 2179: @item modular
1.2 noro 2180: \JP $B%U%i%0$^$?$OAG?t(B
2181: \EG flag or prime
1.1 noro 2182: @end table
2183:
2184: @itemize @bullet
1.2 noro 2185: \BJP
1.1 noro 2186: @item
2187: $B$3$l$i$NH!?t$O(B, $B%0%l%V%J4pDl7W;;$N4pK\E*AH$_9~$_H!?t$G$"$j(B, @code{gr()},
2188: @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 2189: $B$r9T$C$F$$$k(B. $B4X?tL>$K(B weyl $B$,F~$C$F$$$k$b$N$O(B, Weyl $BBe?t>e$N7W;;(B
2190: $B$N$?$a$N4X?t$G$"$k(B.
1.1 noro 2191: @item
1.6 noro 2192: @code{dp_gr_f_main()}, @code{dp_weyl_f_main()} $B$O(B, $B<o!9$NM-8BBN>e$N%0%l%V%J4pDl$r7W;;$9$k(B
1.5 noro 2193: $B>l9g$KMQ$$$k(B. $BF~NO$O(B, $B$"$i$+$8$a(B, @code{simp_ff()} $B$J$I$G(B,
2194: $B9M$($kM-8BBN>e$K<M1F$5$l$F$$$kI,MW$,$"$k(B.
2195: @item
1.1 noro 2196: $B%U%i%0(B @var{homo} $B$,(B 0 $B$G$J$$;~(B, $BF~NO$r@F<!2=$7$F$+$i(B Buchberger $B%"%k%4%j%:%`(B
2197: $B$r<B9T$9$k(B.
2198: @item
2199: @code{dp_gr_mod_main()} $B$KBP$7$F$O(B, @var{modular} $B$O(B, GF(@var{modular}) $B>e(B
2200: $B$G$N7W;;$r0UL#$9$k(B.
2201: @code{dp_gr_main()} $B$KBP$7$F$O(B, @var{modular} $B$O<!$N$h$&$J0UL#$r;}$D(B.
2202: @enumerate
2203: @item
2204: @var{modular} $B$,(B 1 $B$N;~(B, trace-lifting $B$K$h$k7W;;$r9T$&(B. $BAG?t$O(B
2205: @code{lprime(0)} $B$+$i=g$K@.8y$9$k$^$G(B @code{lprime()} $B$r8F$S=P$7$F@8@.$9$k(B.
2206: @item
2207: @var{modular} $B$,(B 2 $B0J>e$N<+A3?t$N;~(B, $B$=$NCM$rAG?t$H$_$J$7$F(B trace-lifting
2208: $B$r9T$&(B. $B$=$NAG?t$G<:GT$7$?>l9g(B, 0 $B$rJV$9(B.
2209: @item
2210: @var{modular} $B$,Ii$N>l9g(B,
2211: @var{-modular} $B$KBP$7$F>e=R$N5,B'$,E,MQ$5$l$k$,(B, trace-lifting $B$N:G=*(B
2212: $BCJ3,$N%0%l%V%J4pDl%A%'%C%/$H%$%G%"%k%a%s%P%7%C%W%A%'%C%/$,>JN,$5$l$k(B.
2213: @end enumerate
2214:
2215: @item
2216: @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
2217: @code{dp_gr_main(P,V,1,1,O)}, @code{gr_mod(P,V,O,M)} $B$O(B
2218: @code{dp_gr_mod_main(P,V,0,M,O)} $B$r$=$l$>$l<B9T$9$k(B.
2219: @item
2220: @var{homo}, @var{modular} $B$NB>$K(B, @code{dp_gr_flags()} $B$G@_Dj$5$l$k(B
2221: $B$5$^$6$^$J%U%i%0$K$h$j7W;;$,@)8f$5$l$k(B.
1.2 noro 2222: \E
2223: \BEG
2224: @item
2225: These functions are fundamental built-in functions for Groebner basis
2226: computation and @code{gr()},@code{hgr()} and @code{gr_mod()}
1.6 noro 2227: are all interfaces to these functions. Functions whose names
2228: contain weyl are those for computation in Weyl algebra.
1.2 noro 2229: @item
1.6 noro 2230: @code{dp_gr_f_main()} and @code{dp_weyl_gr_f_main()}
2231: are functions for Groebner basis computation
1.5 noro 2232: over various finite fields. Coefficients of input polynomials
2233: must be converted to elements of a finite field
2234: currently specified by @code{setmod_ff()}.
2235: @item
1.2 noro 2236: If @var{homo} is not equal to 0, homogenization is applied before entering
2237: Buchberger algorithm
2238: @item
2239: For @code{dp_gr_mod_main()}, @var{modular} means a computation over
2240: GF(@var{modular}).
2241: For @code{dp_gr_main()}, @var{modular} has the following mean.
2242: @enumerate
2243: @item
2244: If @var{modular} is 1 , trace lifting is used. Primes for trace lifting
2245: are generated by @code{lprime()}, starting from @code{lprime(0)}, until
2246: the computation succeeds.
2247: @item
2248: If @var{modular} is an integer greater than 1, the integer is regarded as a
2249: prime and trace lifting is executed by using the prime. If the computation
2250: fails then 0 is returned.
2251: @item
2252: If @var{modular} is negative, the above rule is applied for @var{-modular}
2253: but the Groebner basis check and ideal-membership check are omitted in
2254: the last stage of trace lifting.
2255: @end enumerate
2256:
2257: @item
2258: @code{gr(P,V,O)}, @code{hgr(P,V,O)} and @code{gr_mod(P,V,O,M)} execute
2259: @code{dp_gr_main(P,V,0,1,O)}, @code{dp_gr_main(P,V,1,1,O)}
2260: and @code{dp_gr_mod_main(P,V,0,M,O)} respectively.
2261: @item
2262: Actual computation is controlled by various parameters set by
2263: @code{dp_gr_flags()}, other then by @var{homo} and @var{modular}.
2264: \E
1.1 noro 2265: @end itemize
2266:
2267: @table @t
1.2 noro 2268: \JP @item $B;2>H(B
2269: \EG @item References
1.1 noro 2270: @fref{dp_ord},
2271: @fref{dp_gr_flags dp_gr_print},
2272: @fref{gr hgr gr_mod},
1.5 noro 2273: @fref{setmod_ff},
1.2 noro 2274: \JP @fref{$B7W;;$*$h$SI=<($N@)8f(B}.
2275: \EG @fref{Controlling Groebner basis computations}
1.1 noro 2276: @end table
2277:
1.6 noro 2278: \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
2279: \EG @node dp_f4_main dp_f4_mod_main dp_weyl_f4_main dp_weyl_f4_mod_main,,, Functions for Groebner basis computation
2280: @subsection @code{dp_f4_main}, @code{dp_f4_mod_main}, @code{dp_weyl_f4_main}, @code{dp_weyl_f4_mod_main}
1.1 noro 2281: @findex dp_f4_main
2282: @findex dp_f4_mod_main
1.6 noro 2283: @findex dp_weyl_f4_main
2284: @findex dp_weyl_f4_mod_main
1.1 noro 2285:
2286: @table @t
2287: @item dp_f4_main(@var{plist},@var{vlist},@var{order})
2288: @itemx dp_f4_mod_main(@var{plist},@var{vlist},@var{order})
1.6 noro 2289: @itemx dp_weyl_f4_main(@var{plist},@var{vlist},@var{order})
2290: @itemx dp_weyl_f4_mod_main(@var{plist},@var{vlist},@var{order})
1.2 noro 2291: \JP :: F4 $B%"%k%4%j%:%`$K$h$k%0%l%V%J4pDl$N7W;;(B ($BAH$_9~$_H!?t(B)
2292: \EG :: Groebner basis computation by F4 algorithm (built-in functions)
1.1 noro 2293: @end table
2294:
2295: @table @var
2296: @item return
1.2 noro 2297: \JP $B%j%9%H(B
2298: \EG list
1.4 noro 2299: @item plist vlist
1.2 noro 2300: \JP $B%j%9%H(B
2301: \EG list
1.1 noro 2302: @item order
1.2 noro 2303: \JP $B?t(B, $B%j%9%H$^$?$O9TNs(B
2304: \EG number, list or matrix
1.1 noro 2305: @end table
2306:
2307: @itemize @bullet
1.2 noro 2308: \BJP
1.1 noro 2309: @item
2310: F4 $B%"%k%4%j%:%`$K$h$j%0%l%V%J4pDl$N7W;;$r9T$&(B.
2311: @item
2312: F4 $B%"%k%4%j%:%`$O(B, J.C. Faugere $B$K$h$jDs>'$5$l$??7@$Be%0%l%V%J4pDl(B
2313: $B;;K!$G$"$j(B, $BK\<BAu$O(B, $BCf9q>jM>DjM}$K$h$k@~7AJ}Dx<05a2r$rMQ$$$?(B
2314: $B;n83E*$J<BAu$G$"$k(B.
2315: @item
1.6 noro 2316: $B@F<!2=$N0z?t$,$J$$$3$H$r=|$1$P(B, $B0z?t$*$h$SF0:n$O$=$l$>$l(B
2317: @code{dp_gr_main()}, @code{dp_gr_mod_main()},
2318: @code{dp_weyl_gr_main()}, @code{dp_weyl_gr_mod_main()}
1.1 noro 2319: $B$HF1MM$G$"$k(B.
1.2 noro 2320: \E
2321: \BEG
2322: @item
2323: These functions compute Groebner bases by F4 algorithm.
2324: @item
2325: F4 is a new generation algorithm for Groebner basis computation
2326: invented by J.C. Faugere. The current implementation of @code{dp_f4_main()}
2327: uses Chinese Remainder theorem and not highly optimized.
2328: @item
2329: Arguments and actions are the same as those of
1.6 noro 2330: @code{dp_gr_main()}, @code{dp_gr_mod_main()},
2331: @code{dp_weyl_gr_main()}, @code{dp_weyl_gr_mod_main()},
2332: except for lack of the argument for controlling homogenization.
1.2 noro 2333: \E
1.1 noro 2334: @end itemize
2335:
2336: @table @t
1.2 noro 2337: \JP @item $B;2>H(B
2338: \EG @item References
1.1 noro 2339: @fref{dp_ord},
2340: @fref{dp_gr_flags dp_gr_print},
2341: @fref{gr hgr gr_mod},
1.15 noro 2342: \JP @fref{$B7W;;$*$h$SI=<($N@)8f(B}.
2343: \EG @fref{Controlling Groebner basis computations}
2344: @end table
2345:
1.17 noro 2346: \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
2347: \EG @node nd_gr nd_gr_trace nd_f4 nd_f4_trace nd_weyl_gr nd_weyl_gr_trace,,, Functions for Groebner basis computation
2348: @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 2349: @findex nd_gr
2350: @findex nd_gr_trace
2351: @findex nd_f4
1.17 noro 2352: @findex nd_f4_trace
1.15 noro 2353: @findex nd_weyl_gr
2354: @findex nd_weyl_gr_trace
2355:
2356: @table @t
2357: @item nd_gr(@var{plist},@var{vlist},@var{p},@var{order})
2358: @itemx nd_gr_trace(@var{plist},@var{vlist},@var{homo},@var{p},@var{order})
2359: @itemx nd_f4(@var{plist},@var{vlist},@var{modular},@var{order})
1.17 noro 2360: @itemx nd_f4_trace(@var{plist},@var{vlist},@var{homo},@var{p},@var{order})
1.19 noro 2361: @itemx nd_weyl_gr(@var{plist},@var{vlist},@var{p},@var{order})
1.15 noro 2362: @itemx nd_weyl_gr_trace(@var{plist},@var{vlist},@var{homo},@var{p},@var{order})
2363: \JP :: $B%0%l%V%J4pDl$N7W;;(B ($BAH$_9~$_H!?t(B)
2364: \EG :: Groebner basis computation (built-in functions)
2365: @end table
2366:
2367: @table @var
2368: @item return
2369: \JP $B%j%9%H(B
2370: \EG list
2371: @item plist vlist
2372: \JP $B%j%9%H(B
2373: \EG list
2374: @item order
2375: \JP $B?t(B, $B%j%9%H$^$?$O9TNs(B
2376: \EG number, list or matrix
2377: @item homo
2378: \JP $B%U%i%0(B
2379: \EG flag
2380: @item modular
2381: \JP $B%U%i%0$^$?$OAG?t(B
2382: \EG flag or prime
2383: @end table
2384:
2385: \BJP
2386: @itemize @bullet
2387: @item
2388: $B$3$l$i$NH!?t$O(B, $B%0%l%V%J4pDl7W;;AH$_9~$_4X?t$N?7<BAu$G$"$k(B.
2389: @item @code{nd_gr} $B$O(B, @code{p} $B$,(B 0 $B$N$H$-M-M}?tBN>e$N(B Buchberger
2390: $B%"%k%4%j%:%`$r<B9T$9$k(B. @code{p} $B$,(B 2 $B0J>e$N<+A3?t$N$H$-(B, GF(p) $B>e$N(B
2391: Buchberger $B%"%k%4%j%:%`$r<B9T$9$k(B.
1.17 noro 2392: @item @code{nd_gr_trace} $B$*$h$S(B @code{nd_f4_trace}
2393: $B$OM-M}?tBN>e$G(B trace $B%"%k%4%j%:%`$r<B9T$9$k(B.
1.18 noro 2394: @var{p} $B$,(B 0 $B$^$?$O(B 1 $B$N$H$-(B, $B<+F0E*$KA*$P$l$?AG?t$rMQ$$$F(B, $B@.8y$9$k(B
1.15 noro 2395: $B$^$G(B trace $B%"%k%4%j%:%`$r<B9T$9$k(B.
1.18 noro 2396: @var{p} $B$,(B 2 $B0J>e$N$H$-(B, trace $B$O(BGF(p) $B>e$G7W;;$5$l$k(B. trace $B%"%k%4%j%:%`(B
2397: $B$,<:GT$7$?>l9g(B 0 $B$,JV$5$l$k(B. @var{p} $B$,Ii$N>l9g(B, $B%0%l%V%J4pDl%A%'%C%/$O(B
2398: $B9T$o$J$$(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 2399: $B$=$l0J30$O;XDj$5$l$?AG?t$rMQ$$$F%0%l%V%J4pDl8uJd$N7W;;$,9T$o$l$k(B.
2400: @code{nd_f4_trace} $B$O(B, $B3FA4<!?t$K$D$$$F(B, $B$"$kM-8BBN>e$G(B F4 $B%"%k%4%j%:%`(B
2401: $B$G9T$C$?7k2L$r$b$H$K(B, $B$=$NM-8BBN>e$G(B 0 $B$G$J$$4pDl$rM?$($k(B S-$BB?9`<0$N$_$r(B
2402: $BMQ$$$F9TNs@8@.$r9T$$(B, $B$=$NA4<!?t$K$*$1$k4pDl$r@8@.$9$kJ}K!$G$"$k(B. $BF@$i$l$k(B
2403: $BB?9`<0=89g$O$d$O$j%0%l%V%J4pDl8uJd$G$"$j(B, @code{nd_gr_trace} $B$HF1MM$N(B
2404: $B%A%'%C%/$,9T$o$l$k(B.
1.15 noro 2405: @item
1.17 noro 2406: @code{nd_f4} $B$O(B @code{modular} $B$,(B 0 $B$N$H$-M-M}?tBN>e$N(B, @code{modular} $B$,(B
2407: $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 2408: @item
1.18 noro 2409: @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
2410: $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,
2411: $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
2412: $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),
2413: 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.
2414: @item
1.15 noro 2415: @code{nd_weyl_gr}, @code{nd_weyl_gr_trace} $B$O(B Weyl $BBe?tMQ$G$"$k(B.
2416: @item
1.18 noro 2417: @code{f4} $B7O4X?t0J30$O$9$Y$FM-M}4X?t78?t$N7W;;$,2DG=$G$"$k(B.
1.15 noro 2418: @item
2419: $B0lHL$K(B @code{dp_gr_main}, @code{dp_gr_mod_main} $B$h$j9bB.$G$"$k$,(B,
2420: $BFC$KM-8BBN>e$N>l9g82Cx$G$"$k(B.
2421: @end itemize
2422: \E
2423:
2424: \BEG
2425: @itemize @bullet
2426: @item
2427: These functions are new implementations for computing Groebner bases.
2428: @item @code{nd_gr} executes Buchberger algorithm over the rationals
2429: if @code{p} is 0, and that over GF(p) if @code{p} is a prime.
2430: @item @code{nd_gr_trace} executes the trace algorithm over the rationals.
2431: If @code{p} is 0 or 1, the trace algorithm is executed until it succeeds
2432: by using automatically chosen primes.
2433: If @code{p} a positive prime,
2434: the trace is comuted over GF(p).
2435: If the trace algorithm fails 0 is returned.
2436: If @code{p} is negative,
2437: the Groebner basis check and ideal-membership check are omitted.
2438: In this case, an automatically chosen prime if @code{p} is 1,
2439: otherwise the specified prime is used to compute a Groebner basis
2440: candidate.
1.17 noro 2441: Execution of @code{nd_f4_trace} is done as follows:
2442: For each total degree, an F4-reduction of S-polynomials over a finite field
2443: is done, and S-polynomials which give non-zero basis elements are gathered.
2444: Then F4-reduction over Q is done for the gathered S-polynomials.
2445: The obtained polynomial set is a Groebner basis candidate and the same
2446: check procedure as in the case of @code{nd_gr_trace} is done.
2447: @item
2448: @code{nd_f4} executes F4 algorithm over Q if @code{modular} is equal to 0,
2449: or over a finite field GF(@code{modular})
2450: if @code{modular} is a prime number of machine size (<2^29).
1.18 noro 2451: If @var{plist} is a list of polynomials, then a Groebner basis of the ideal generated by @var{plist}
2452: is computed. If @var{plist} is a list of lists of polynomials, then each list of polynomials are regarded
2453: as an element of a free module over a polynomial ring and a Groebner basis of the sub-module generated by @var{plist}
2454: in the free module. In the latter case a term order in the free module should be specified.
2455: This is specified by @var{[s,ord]}. If @var{s} is 0 then it means TOP (Term Over Position).
2456: 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 2457: @item
2458: @code{nd_weyl_gr}, @code{nd_weyl_gr_trace} are for Weyl algebra computation.
2459: @item
1.18 noro 2460: Functions except for F4 related ones can handle rational coeffient cases.
1.15 noro 2461: @item
2462: In general these functions are more efficient than
2463: @code{dp_gr_main}, @code{dp_gr_mod_main}, especially over finite fields.
2464: @end itemize
2465: \E
2466:
2467: @example
2468: [38] load("cyclic")$
2469: [49] C=cyclic(7)$
2470: [50] V=vars(C)$
2471: [51] cputime(1)$
2472: [52] dp_gr_mod_main(C,V,0,31991,0)$
2473: 26.06sec + gc : 0.313sec(26.4sec)
2474: [53] nd_gr(C,V,31991,0)$
2475: ndv_alloc=1477188
2476: 5.737sec + gc : 0.1837sec(5.921sec)
2477: [54] dp_f4_mod_main(C,V,31991,0)$
2478: 3.51sec + gc : 0.7109sec(4.221sec)
2479: [55] nd_f4(C,V,31991,0)$
2480: 1.906sec + gc : 0.126sec(2.032sec)
2481: @end example
2482:
2483: @table @t
2484: \JP @item $B;2>H(B
2485: \EG @item References
2486: @fref{dp_ord},
2487: @fref{dp_gr_flags dp_gr_print},
1.2 noro 2488: \JP @fref{$B7W;;$*$h$SI=<($N@)8f(B}.
2489: \EG @fref{Controlling Groebner basis computations}
1.1 noro 2490: @end table
2491:
1.2 noro 2492: \JP @node dp_gr_flags dp_gr_print,,, $B%0%l%V%J4pDl$K4X$9$kH!?t(B
2493: \EG @node dp_gr_flags dp_gr_print,,, Functions for Groebner basis computation
1.1 noro 2494: @subsection @code{dp_gr_flags}, @code{dp_gr_print}
2495: @findex dp_gr_flags
2496: @findex dp_gr_print
2497:
2498: @table @t
2499: @item dp_gr_flags([@var{list}])
1.7 noro 2500: @itemx dp_gr_print([@var{i}])
1.2 noro 2501: \JP :: $B7W;;$*$h$SI=<(MQ%Q%i%a%?$N@_Dj(B, $B;2>H(B
2502: \BEG :: Set and show various parameters for cotrolling computations
2503: and showing informations.
2504: \E
1.1 noro 2505: @end table
2506:
2507: @table @var
2508: @item return
1.2 noro 2509: \JP $B@_DjCM(B
2510: \EG value currently set
1.1 noro 2511: @item list
1.2 noro 2512: \JP $B%j%9%H(B
2513: \EG list
1.7 noro 2514: @item i
2515: \JP $B@0?t(B
2516: \EG integer
1.1 noro 2517: @end table
2518:
2519: @itemize @bullet
1.2 noro 2520: \BJP
1.1 noro 2521: @item
1.5 noro 2522: @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 2523: $B$J%Q%i%a%?$r@_Dj(B, $B;2>H$9$k(B.
2524: @item
2525: $B0z?t$,$J$$>l9g(B, $B8=:_$N@_Dj$,JV$5$l$k(B.
2526: @item
2527: $B0z?t$O(B, @code{["Print",1,"NoSugar",1,...]} $B$J$k7A$N%j%9%H$G(B, $B:8$+$i=g$K(B
2528: $B@_Dj$5$l$k(B. $B%Q%i%a%?L>$OJ8;zNs$GM?$($kI,MW$,$"$k(B.
2529: @item
1.7 noro 2530: @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
2531: $B$G$-$k(B. $B@_Dj$5$l$kCM$O<!$NDL$j$G$"$k!#(B
2532: @table @var
2533: @item i=0
2534: @code{Print=0}, @code{PrintShort=0}
2535: @item i=1
2536: @code{Print=1}, @code{PrintShort=0}
2537: @item i=2
2538: @code{Print=0}, @code{PrintShort=1}
2539: @end table
2540: $B$3$l$O(B, @code{dp_gr_main()} $B$J$I$r%5%V%k!<%A%s$H$7$FMQ$$$k%f!<%6(B
2541: $BH!?t$K$*$$$F(B, $B$=$N%5%V%k!<%A%s$,Cf4V>pJs$NI=<((B
1.1 noro 2542: $B$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 2543: \E
2544: \BEG
2545: @item
2546: @code{dp_gr_flags()} sets and shows various parameters for Groebner basis
2547: computation.
2548: @item
2549: If no argument is specified the current settings are returned.
2550: @item
2551: Arguments must be specified as a list such as
2552: @code{["Print",1,"NoSugar",1,...]}. Names of parameters must be character
2553: strings.
2554: @item
2555: @code{dp_gr_print()} is used to set and show the value of a parameter
1.7 noro 2556: @code{Print} and @code{PrintShort}.
2557: @table @var
2558: @item i=0
2559: @code{Print=0}, @code{PrintShort=0}
2560: @item i=1
2561: @code{Print=1}, @code{PrintShort=0}
2562: @item i=2
2563: @code{Print=0}, @code{PrintShort=1}
2564: @end table
2565: This functions is prepared to get quickly the value
2566: when a user defined function calling @code{dp_gr_main()} etc.
1.2 noro 2567: uses the value as a flag for showing intermediate informations.
2568: \E
1.1 noro 2569: @end itemize
2570:
2571: @table @t
1.2 noro 2572: \JP @item $B;2>H(B
2573: \EG @item References
2574: \JP @fref{$B7W;;$*$h$SI=<($N@)8f(B}
2575: \EG @fref{Controlling Groebner basis computations}
1.1 noro 2576: @end table
2577:
1.2 noro 2578: \JP @node dp_ord,,, $B%0%l%V%J4pDl$K4X$9$kH!?t(B
2579: \EG @node dp_ord,,, Functions for Groebner basis computation
1.1 noro 2580: @subsection @code{dp_ord}
2581: @findex dp_ord
2582:
2583: @table @t
2584: @item dp_ord([@var{order}])
1.2 noro 2585: \JP :: $BJQ?t=g=x7?$N@_Dj(B, $B;2>H(B
2586: \EG :: Set and show the ordering type.
1.1 noro 2587: @end table
2588:
2589: @table @var
2590: @item return
1.2 noro 2591: \JP $BJQ?t=g=x7?(B ($B?t(B, $B%j%9%H$^$?$O9TNs(B)
2592: \EG ordering type (number, list or matrix)
1.1 noro 2593: @item order
1.2 noro 2594: \JP $B?t(B, $B%j%9%H$^$?$O9TNs(B
2595: \EG number, list or matrix
1.1 noro 2596: @end table
2597:
2598: @itemize @bullet
1.2 noro 2599: \BJP
1.1 noro 2600: @item
2601: $B0z?t$,$"$k;~(B, $BJQ?t=g=x7?$r(B @var{order} $B$K@_Dj$9$k(B. $B0z?t$,$J$$;~(B,
2602: $B8=:_@_Dj$5$l$F$$$kJQ?t=g=x7?$rJV$9(B.
2603:
2604: @item
2605: $BJ,;6I=8=B?9`<0$K4X$9$kH!?t(B, $B1i;;$O0z?t$H$7$FJQ?t=g=x7?$r$H$k$b$N$H$H$i$J$$$b$N(B
2606: $B$,$"$j(B, $B$H$i$J$$$b$N$K4X$7$F$O(B, $B$=$N;~E@$G@_Dj$5$l$F$$$kCM$rMQ$$$F7W;;$,(B
2607: $B9T$o$l$k(B.
2608:
2609: @item
2610: @code{gr()} $B$J$I(B, $B0z?t$H$7$FJQ?t=g=x7?$r$H$k$b$N$O(B, $BFbIt$G(B @code{dp_ord()}
2611: $B$r8F$S=P$7(B, $BJQ?t=g=x7?$r@_Dj$9$k(B. $B$3$N@_Dj$O(B, $B7W;;=*N;8e$b@8$-;D$k(B.
2612:
2613: @item
2614: $BJ,;6I=8=B?9`<0$N;MB'1i;;$b(B, $B@_Dj$5$l$F$$$kCM$rMQ$$$F7W;;$5$l$k(B. $B=>$C$F(B,
2615: $B$=$NB?9`<0$,@8@.$5$l$?;~E@$K$*$1$kJQ?t=g=x7?$,(B, $B;MB'1i;;;~$K@5$7$/@_Dj(B
2616: $B$5$l$F$$$J$1$l$P$J$i$J$$(B. $B$^$?(B, $B1i;;BP>]$H$J$kB?9`<0$O(B, $BF10l$NJQ?t=g=x(B
2617: $B7?$K4p$E$$$F@8@.$5$l$?$b$N$G$J$1$l$P$J$i$J$$(B.
2618:
2619: @item
2620: $B%H%C%W%l%Y%kH!?t0J30$NH!?t$rD>@\8F$S=P$9>l9g$K$O(B, $B$3$NH!?t$K$h$j(B
2621: $BJQ?t=g=x7?$r@5$7$/@_Dj$7$J$1$l$P$J$i$J$$(B.
1.2 noro 2622: \E
2623: \BEG
2624: @item
2625: If an argument is specified, the function
2626: sets the current ordering type to @var{order}.
2627: If no argument is specified, the function returns the ordering
2628: type currently set.
2629:
2630: @item
2631: There are two types of functions concerning distributed polynomial,
2632: functions which take a ordering type and those which don't take it.
2633: The latter ones use the current setting.
2634:
2635: @item
2636: Functions such as @code{gr()}, which need a ordering type as an argument,
2637: call @code{dp_ord()} internally during the execution.
2638: The setting remains after the execution.
2639:
2640: Fundamental arithmetics for distributed polynomial also use the current
2641: setting. Therefore, when such arithmetics for distributed polynomials
2642: are done, the current setting must coincide with the ordering type
2643: which was used upon the creation of the polynomials. It is assumed
2644: that such polynomials were generated under the same ordering type.
2645:
2646: @item
2647: Type of term ordering must be correctly set by this function
2648: when functions other than top level functions are called directly.
2649: \E
1.1 noro 2650: @end itemize
2651:
2652: @example
2653: [19] dp_ord(0)$
2654: [20] <<1,2,3>>+<<3,1,1>>;
2655: (1)*<<1,2,3>>+(1)*<<3,1,1>>
2656: [21] dp_ord(2)$
2657: [22] <<1,2,3>>+<<3,1,1>>;
2658: (1)*<<3,1,1>>+(1)*<<1,2,3>>
2659: @end example
2660:
2661: @table @t
1.2 noro 2662: \JP @item $B;2>H(B
2663: \EG @item References
2664: \JP @fref{$B9`=g=x$N@_Dj(B}
2665: \EG @fref{Setting term orderings}
1.1 noro 2666: @end table
2667:
1.18 noro 2668: \JP @node dp_set_weight dp_set_top_weight dp_weyl_set_weight,,, $B%0%l%V%J4pDl$K4X$9$kH!?t(B
2669: \EG @node dp_set_weight dp_set_top_weight dp_weyl_set_weight,,, Functions for Groebner basis computation
2670: @subsection @code{dp_set_weight}, @code{dp_set_top_weight}, @code{dp_weyl_set_weight}
2671: @findex dp_set_weight
2672: @findex dp_set_top_weight
2673: @findex dp_weyl_set_weight
2674:
2675: @table @t
2676: @item dp_set_weight([@var{weight}])
2677: \JP :: sugar weight $B$N@_Dj(B, $B;2>H(B
2678: \EG :: Set and show the sugar weight.
2679: @item dp_set_top_weight([@var{weight}])
2680: \JP :: top weight $B$N@_Dj(B, $B;2>H(B
2681: \EG :: Set and show the top weight.
2682: @item dp_weyl_set_weight([@var{weight}])
2683: \JP :: weyl weight $B$N@_Dj(B, $B;2>H(B
2684: \EG :: Set and show the weyl weight.
2685: @end table
2686:
2687: @table @var
2688: @item return
2689: \JP $B%Y%/%H%k(B
2690: \EG a vector
2691: @item weight
2692: \JP $B@0?t$N%j%9%H$^$?$O%Y%/%H%k(B
2693: \EG a list or vector of integers
2694: @end table
2695:
2696: @itemize @bullet
2697: \BJP
2698: @item
2699: @code{dp_set_weight} $B$O(B sugar weight $B$r(B @var{weight} $B$K@_Dj$9$k(B. $B0z?t$,$J$$;~(B,
2700: $B8=:_@_Dj$5$l$F$$$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,
2701: $B3FJQ?t$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.
2702: $B@F<!2=JQ?tMQ$K(B, $BKvHx$K(B 1 $B$rIU$12C$($F$*$/$H0BA4$G$"$k(B.
2703: @item
2704: @code{dp_set_top_weight} $B$O(B top weight $B$r(B @var{weight} $B$K@_Dj$9$k(B. $B0z?t$,$J$$;~(B,
2705: $B8=:_@_Dj$5$l$F$$$k(B top weight $B$rJV$9(B. top weight $B$,@_Dj$5$l$F$$$k$H$-(B,
2706: $B$^$:(B top weight $B$K$h$kC19`<0Hf3S$r@h$K9T$&(B. tie breaker $B$H$7$F8=:_@_Dj$5$l$F$$$k(B
2707: $B9`=g=x$,MQ$$$i$l$k$,(B, $B$3$NHf3S$K$O(B top weight $B$OMQ$$$i$l$J$$(B.
2708:
2709: @item
2710: @code{dp_weyl_set_weight} $B$O(B weyl weight $B$r(B @var{weight} $B$K@_Dj$9$k(B. $B0z?t$,$J$$;~(B,
2711: $B8=:_@_Dj$5$l$F$$$k(B weyl weight $B$rJV$9(B. weyl weight w $B$r@_Dj$9$k$H(B,
2712: $B9`=g=x7?(B 11 $B$G$N7W;;$K$*$$$F(B, (-w,w) $B$r(B top weight, tie breaker $B$r(B graded reverse lex
2713: $B$H$7$?9`=g=x$,@_Dj$5$l$k(B.
2714: \E
2715: \BEG
2716: @item
2717: @code{dp_set_weight} sets the sugar weight=@var{weight}. It returns the current sugar weight.
2718: A sugar weight is a vector with positive integer components and it represents the weights of variables.
2719: It is used for computing the weight of a monomial in a graded ordering.
2720: It is recommended to append a component 1 at the end of the weight vector for a homogenizing variable.
2721: @item
2722: @code{dp_set_top_weight} sets the top weight=@var{weight}. It returns the current top weight.
2723: It a top weight is set, the weights of monomials under the top weight are firstly compared.
2724: If the the weights are equal then the current term ordering is applied as a tie breaker, but
2725: the top weight is not used in the tie breaker.
2726:
2727: @item
2728: @code{dp_weyl_set_weight} sets the weyl weigh=@var{weight}. It returns the current weyl weight.
2729: If a weyl weight w is set, in the comparsion by the term order type 11, a term order with
2730: the top weight=(-w,w) and the tie breaker=graded reverse lex is applied.
2731: \E
2732: @end itemize
2733:
2734: @table @t
2735: \JP @item $B;2>H(B
2736: \EG @item References
2737: @fref{Weight}
2738: @end table
2739:
2740:
1.2 noro 2741: \JP @node dp_ptod,,, $B%0%l%V%J4pDl$K4X$9$kH!?t(B
2742: \EG @node dp_ptod,,, Functions for Groebner basis computation
1.1 noro 2743: @subsection @code{dp_ptod}
2744: @findex dp_ptod
2745:
2746: @table @t
2747: @item dp_ptod(@var{poly},@var{vlist})
1.2 noro 2748: \JP :: $BB?9`<0$rJ,;6I=8=B?9`<0$KJQ49$9$k(B.
2749: \EG :: Converts an ordinary polynomial into a distributed polynomial.
1.1 noro 2750: @end table
2751:
2752: @table @var
2753: @item return
1.2 noro 2754: \JP $BJ,;6I=8=B?9`<0(B
2755: \EG distributed polynomial
1.1 noro 2756: @item poly
1.2 noro 2757: \JP $BB?9`<0(B
2758: \EG polynomial
1.1 noro 2759: @item vlist
1.2 noro 2760: \JP $B%j%9%H(B
2761: \EG list
1.1 noro 2762: @end table
2763:
2764: @itemize @bullet
1.2 noro 2765: \BJP
1.1 noro 2766: @item
2767: $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.
2768: @item
2769: @var{vlist} $B$K4^$^$l$J$$ITDj85$O(B, $B78?tBN$KB0$9$k$H$7$FJQ49$5$l$k(B.
1.2 noro 2770: \E
2771: \BEG
2772: @item
2773: According to the variable ordering @var{vlist} and current
2774: type of term ordering, this function converts an ordinary
2775: polynomial into a distributed polynomial.
2776: @item
2777: Indeterminates not included in @var{vlist} are regarded to belong to
2778: the coefficient field.
2779: \E
1.1 noro 2780: @end itemize
2781:
2782: @example
2783: [50] dp_ord(0);
2784: 1
2785: [51] dp_ptod((x+y+z)^2,[x,y,z]);
2786: (1)*<<2,0,0>>+(2)*<<1,1,0>>+(1)*<<0,2,0>>+(2)*<<1,0,1>>+(2)*<<0,1,1>>
2787: +(1)*<<0,0,2>>
2788: [52] dp_ptod((x+y+z)^2,[x,y]);
1.5 noro 2789: (1)*<<2,0>>+(2)*<<1,1>>+(1)*<<0,2>>+(2*z)*<<1,0>>+(2*z)*<<0,1>>
2790: +(z^2)*<<0,0>>
1.1 noro 2791: @end example
2792:
2793: @table @t
1.2 noro 2794: \JP @item $B;2>H(B
2795: \EG @item References
1.1 noro 2796: @fref{dp_dtop},
2797: @fref{dp_ord}.
2798: @end table
2799:
1.2 noro 2800: \JP @node dp_dtop,,, $B%0%l%V%J4pDl$K4X$9$kH!?t(B
2801: \EG @node dp_dtop,,, Functions for Groebner basis computation
1.1 noro 2802: @subsection @code{dp_dtop}
2803: @findex dp_dtop
2804:
2805: @table @t
2806: @item dp_dtop(@var{dpoly},@var{vlist})
1.2 noro 2807: \JP :: $BJ,;6I=8=B?9`<0$rB?9`<0$KJQ49$9$k(B.
2808: \EG :: Converts a distributed polynomial into an ordinary polynomial.
1.1 noro 2809: @end table
2810:
2811: @table @var
2812: @item return
1.2 noro 2813: \JP $BB?9`<0(B
2814: \EG polynomial
1.1 noro 2815: @item dpoly
1.2 noro 2816: \JP $BJ,;6I=8=B?9`<0(B
2817: \EG distributed polynomial
1.1 noro 2818: @item vlist
1.2 noro 2819: \JP $B%j%9%H(B
2820: \EG list
1.1 noro 2821: @end table
2822:
2823: @itemize @bullet
1.2 noro 2824: \BJP
1.1 noro 2825: @item
2826: $BJ,;6I=8=B?9`<0$r(B, $BM?$($i$l$?ITDj85%j%9%H$rMQ$$$FB?9`<0$KJQ49$9$k(B.
2827: @item
2828: $BITDj85%j%9%H$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 2829: \E
2830: \BEG
2831: @item
2832: This function converts a distributed polynomial into an ordinary polynomial
2833: according to a list of indeterminates @var{vlist}.
2834: @item
2835: @var{vlist} is such a list that its length coincides with the number of
2836: variables of @var{dpoly}.
2837: \E
1.1 noro 2838: @end itemize
2839:
2840: @example
2841: [53] T=dp_ptod((x+y+z)^2,[x,y]);
1.5 noro 2842: (1)*<<2,0>>+(2)*<<1,1>>+(1)*<<0,2>>+(2*z)*<<1,0>>+(2*z)*<<0,1>>
2843: +(z^2)*<<0,0>>
1.1 noro 2844: [54] P=dp_dtop(T,[a,b]);
2845: z^2+(2*a+2*b)*z+a^2+2*b*a+b^2
2846: @end example
2847:
1.2 noro 2848: \JP @node dp_mod dp_rat,,, $B%0%l%V%J4pDl$K4X$9$kH!?t(B
2849: \EG @node dp_mod dp_rat,,, Functions for Groebner basis computation
1.1 noro 2850: @subsection @code{dp_mod}, @code{dp_rat}
2851: @findex dp_mod
2852: @findex dp_rat
2853:
2854: @table @t
2855: @item dp_mod(@var{p},@var{mod},@var{subst})
1.2 noro 2856: \JP :: $BM-M}?t78?tJ,;6I=8=B?9`<0$NM-8BBN78?t$X$NJQ49(B
2857: \EG :: Converts a disributed polynomial into one with coefficients in a finite field.
1.1 noro 2858: @item dp_rat(@var{p})
1.2 noro 2859: \JP :: $BM-8BBN78?tJ,;6I=8=B?9`<0$NM-M}?t78?t$X$NJQ49(B
2860: \BEG
2861: :: Converts a distributed polynomial with coefficients in a finite field into
2862: one with coefficients in the rationals.
2863: \E
1.1 noro 2864: @end table
2865:
2866: @table @var
2867: @item return
1.2 noro 2868: \JP $BJ,;6I=8=B?9`<0(B
2869: \EG distributed polynomial
1.1 noro 2870: @item p
1.2 noro 2871: \JP $BJ,;6I=8=B?9`<0(B
2872: \EG distributed polynomial
1.1 noro 2873: @item mod
1.2 noro 2874: \JP $BAG?t(B
2875: \EG prime
1.1 noro 2876: @item subst
1.2 noro 2877: \JP $B%j%9%H(B
2878: \EG list
1.1 noro 2879: @end table
2880:
2881: @itemize @bullet
1.2 noro 2882: \BJP
1.1 noro 2883: @item
2884: @code{dp_nf_mod()}, @code{dp_true_nf_mod()} $B$O(B, $BF~NO$H$7$FM-8BBN78?t$N(B
2885: $BJ,;6I=8=B?9`<0$rI,MW$H$9$k(B. $B$3$N$h$&$J>l9g(B, @code{dp_mod()} $B$K$h$j(B
2886: $BM-M}?t78?tJ,;6I=8=B?9`<0$rJQ49$7$FMQ$$$k$3$H$,$G$-$k(B. $B$^$?(B, $BF@$i$l$?(B
2887: $B7k2L$O(B, $BM-8BBN78?tB?9`<0$H$O1i;;$G$-$k$,(B, $BM-M}?t78?tB?9`<0$H$O1i;;$G$-$J$$(B
2888: $B$?$a(B, @code{dp_rat()} $B$K$h$jJQ49$9$kI,MW$,$"$k(B.
2889: @item
2890: $BM-8BBN78?t$N1i;;$K$*$$$F$O(B, $B$"$i$+$8$a(B @code{setmod()} $B$K$h$jM-8BBN$N85$N(B
2891: $B8D?t$r;XDj$7$F$*$/I,MW$,$"$k(B.
2892: @item
2893: @var{subst} $B$O(B, $B78?t$,M-M}<0$N>l9g(B, $B$=$NM-M}<0$NJQ?t$K$"$i$+$8$a?t$rBeF~(B
2894: $B$7$?8eM-8BBN78?t$KJQ49$9$k$H$$$&A`:n$r9T$&:]$N(B, $BBeF~CM$r;XDj$9$k$b$N$G(B,
2895: @code{[[@var{var},@var{value}],...]} $B$N7A$N%j%9%H$G$"$k(B.
1.2 noro 2896: \E
2897: \BEG
2898: @item
2899: @code{dp_nf_mod()} and @code{dp_true_nf_mod()} require
2900: distributed polynomials with coefficients in a finite field as arguments.
2901: @code{dp_mod()} is used to convert distributed polynomials with rational
2902: number coefficients into appropriate ones.
2903: Polynomials with coefficients in a finite field
2904: cannot be used as inputs of operations with polynomials
2905: with rational number coefficients. @code{dp_rat()} is used for such cases.
2906: @item
2907: The ground finite field must be set in advance by using @code{setmod()}.
2908: @item
2909: @var{subst} is such a list as @code{[[@var{var},@var{value}],...]}.
2910: This is valid when the ground field of the input polynomial is a
2911: rational function field. @var{var}'s are variables in the ground field and
2912: the list means that @var{value} is substituted for @var{var} before
2913: converting the coefficients into elements of a finite field.
2914: \E
1.1 noro 2915: @end itemize
2916:
2917: @example
2918: @end example
2919:
2920: @table @t
1.2 noro 2921: \JP @item $B;2>H(B
2922: \EG @item References
1.18 noro 2923: @fref{dp_nf dp_nf_mod dp_true_nf dp_true_nf_mod dp_weyl_nf dp_weyl_nf_mod},
1.1 noro 2924: @fref{subst psubst},
2925: @fref{setmod}.
2926: @end table
2927:
1.2 noro 2928: \JP @node dp_homo dp_dehomo,,, $B%0%l%V%J4pDl$K4X$9$kH!?t(B
2929: \EG @node dp_homo dp_dehomo,,, Functions for Groebner basis computation
1.1 noro 2930: @subsection @code{dp_homo}, @code{dp_dehomo}
2931: @findex dp_homo
2932: @findex dp_dehomo
2933:
2934: @table @t
2935: @item dp_homo(@var{dpoly})
1.2 noro 2936: \JP :: $BJ,;6I=8=B?9`<0$N@F<!2=(B
2937: \EG :: Homogenize a distributed polynomial
1.1 noro 2938: @item dp_dehomo(@var{dpoly})
1.2 noro 2939: \JP :: $B@F<!J,;6I=8=B?9`<0$NHs@F<!2=(B
2940: \EG :: Dehomogenize a homogenious distributed polynomial
1.1 noro 2941: @end table
2942:
2943: @table @var
2944: @item return
1.2 noro 2945: \JP $BJ,;6I=8=B?9`<0(B
2946: \EG distributed polynomial
1.1 noro 2947: @item dpoly
1.2 noro 2948: \JP $BJ,;6I=8=B?9`<0(B
2949: \EG distributed polynomial
1.1 noro 2950: @end table
2951:
2952: @itemize @bullet
1.2 noro 2953: \BJP
1.1 noro 2954: @item
2955: @code{dp_homo()} $B$O(B, @var{dpoly} $B$N(B $B3F9`(B @var{t} $B$K$D$$$F(B, $B;X?t%Y%/%H%k$ND9$5$r(B
2956: 1 $B?-$P$7(B, $B:G8e$N@.J,$NCM$r(B @var{d}-@code{deg(@var{t})}
2957: (@var{d} $B$O(B @var{dpoly} $B$NA4<!?t(B) $B$H$7$?J,;6I=8=B?9`<0$rJV$9(B.
2958: @item
2959: @code{dp_dehomo()} $B$O(B, @var{dpoly} $B$N3F9`$K$D$$$F(B, $B;X?t%Y%/%H%k$N:G8e$N@.J,(B
2960: $B$r<h$j=|$$$?J,;6B?9`<0$rJV$9(B.
2961: @item
2962: $B$$$:$l$b(B, $B@8@.$5$l$?B?9`<0$rMQ$$$?1i;;$r9T$&>l9g(B, $B$=$l$i$KE,9g$9$k9`=g=x$r(B
2963: $B@5$7$/@_Dj$9$kI,MW$,$"$k(B.
2964: @item
2965: @code{hgr()} $B$J$I$K$*$$$F(B, $BFbItE*$KMQ$$$i$l$F$$$k(B.
1.2 noro 2966: \E
2967: \BEG
2968: @item
2969: @code{dp_homo()} makes a copy of @var{dpoly}, extends
2970: the length of the exponent vector of each term @var{t} in the copy by 1,
2971: and sets the value of the newly appended
2972: component to @var{d}-@code{deg(@var{t})}, where @var{d} is the total
2973: degree of @var{dpoly}.
2974: @item
2975: @code{dp_dehomo()} make a copy of @var{dpoly} and removes the last component
2976: of each terms in the copy.
2977: @item
2978: Appropriate term orderings must be set when the results are used as inputs
2979: of some operations.
2980: @item
2981: These are used internally in @code{hgr()} etc.
2982: \E
1.1 noro 2983: @end itemize
2984:
2985: @example
2986: [202] X=<<1,2,3>>+3*<<1,2,1>>;
2987: (1)*<<1,2,3>>+(3)*<<1,2,1>>
2988: [203] dp_homo(X);
2989: (1)*<<1,2,3,0>>+(3)*<<1,2,1,2>>
2990: [204] dp_dehomo(@@);
2991: (1)*<<1,2,3>>+(3)*<<1,2,1>>
2992: @end example
2993:
2994: @table @t
1.2 noro 2995: \JP @item $B;2>H(B
2996: \EG @item References
1.1 noro 2997: @fref{gr hgr gr_mod}.
2998: @end table
2999:
1.2 noro 3000: \JP @node dp_ptozp dp_prim,,, $B%0%l%V%J4pDl$K4X$9$kH!?t(B
3001: \EG @node dp_ptozp dp_prim,,, Functions for Groebner basis computation
1.1 noro 3002: @subsection @code{dp_ptozp}, @code{dp_prim}
3003: @findex dp_ptozp
3004: @findex dp_prim
3005:
3006: @table @t
3007: @item dp_ptozp(@var{dpoly})
1.2 noro 3008: \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.
3009: \BEG
3010: :: Converts a distributed polynomial @var{poly} with rational coefficients
3011: into an integral distributed polynomial such that GCD of all its coefficients
3012: is 1.
3013: \E
1.19 noro 3014: @item dp_prim(@var{dpoly})
1.2 noro 3015: \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.
3016: \BEG
3017: :: Converts a distributed polynomial @var{poly} with rational function
3018: coefficients into an integral distributed polynomial such that polynomial
3019: GCD of all its coefficients is 1.
3020: \E
1.1 noro 3021: @end table
3022:
3023: @table @var
3024: @item return
1.2 noro 3025: \JP $BJ,;6I=8=B?9`<0(B
3026: \EG distributed polynomial
1.1 noro 3027: @item dpoly
1.2 noro 3028: \JP $BJ,;6I=8=B?9`<0(B
3029: \EG distributed polynomial
1.1 noro 3030: @end table
3031:
3032: @itemize @bullet
1.2 noro 3033: \BJP
1.1 noro 3034: @item
3035: @code{dp_ptozp()} $B$O(B, @code{ptozp()} $B$KAjEv$9$kA`:n$rJ,;6I=8=B?9`<0$K(B
3036: $BBP$7$F9T$&(B. $B78?t$,B?9`<0$r4^$`>l9g(B, $B78?t$K4^$^$l$kB?9`<06&DL0x;R$O(B
3037: $B<h$j=|$+$J$$(B.
3038: @item
3039: @code{dp_prim()} $B$O(B, $B78?t$,B?9`<0$r4^$`>l9g(B, $B78?t$K4^$^$l$kB?9`<06&DL0x;R(B
3040: $B$r<h$j=|$/(B.
1.2 noro 3041: \E
3042: \BEG
3043: @item
3044: @code{dp_ptozp()} executes the same operation as @code{ptozp()} for
3045: a distributed polynomial. If the coefficients include polynomials,
3046: polynomial contents included in the coefficients are not removed.
3047: @item
3048: @code{dp_prim()} removes polynomial contents.
3049: \E
1.1 noro 3050: @end itemize
3051:
3052: @example
3053: [208] X=dp_ptod(3*(x-y)*(y-z)*(z-x),[x]);
3054: (-3*y+3*z)*<<2>>+(3*y^2-3*z^2)*<<1>>+(-3*z*y^2+3*z^2*y)*<<0>>
3055: [209] dp_ptozp(X);
3056: (-y+z)*<<2>>+(y^2-z^2)*<<1>>+(-z*y^2+z^2*y)*<<0>>
3057: [210] dp_prim(X);
3058: (1)*<<2>>+(-y-z)*<<1>>+(z*y)*<<0>>
3059: @end example
3060:
3061: @table @t
1.2 noro 3062: \JP @item $B;2>H(B
3063: \EG @item References
1.1 noro 3064: @fref{ptozp}.
3065: @end table
3066:
1.18 noro 3067: \JP @node dp_nf dp_nf_mod dp_true_nf dp_true_nf_mod dp_weyl_nf dp_weyl_nf_mod,,, $B%0%l%V%J4pDl$K4X$9$kH!?t(B
3068: \EG @node dp_nf dp_nf_mod dp_true_nf dp_true_nf_mod dp_weyl_nf dp_weyl_nf_mod,,, Functions for Groebner basis computation
1.1 noro 3069: @subsection @code{dp_nf}, @code{dp_nf_mod}, @code{dp_true_nf}, @code{dp_true_nf_mod}
3070: @findex dp_nf
3071: @findex dp_true_nf
3072: @findex dp_nf_mod
3073: @findex dp_true_nf_mod
1.18 noro 3074: @findex dp_weyl_nf
3075: @findex dp_weyl_nf_mod
1.1 noro 3076:
3077: @table @t
3078: @item dp_nf(@var{indexlist},@var{dpoly},@var{dpolyarray},@var{fullreduce})
1.18 noro 3079: @item dp_weyl_nf(@var{indexlist},@var{dpoly},@var{dpolyarray},@var{fullreduce})
1.1 noro 3080: @item dp_nf_mod(@var{indexlist},@var{dpoly},@var{dpolyarray},@var{fullreduce},@var{mod})
1.18 noro 3081: @item dp_weyl_nf_mod(@var{indexlist},@var{dpoly},@var{dpolyarray},@var{fullreduce},@var{mod})
1.2 noro 3082: \JP :: $BJ,;6I=8=B?9`<0$N@55,7A$r5a$a$k(B. ($B7k2L$ODj?tG\$5$l$F$$$k2DG=@-$"$j(B)
1.1 noro 3083:
1.2 noro 3084: \BEG
3085: :: Computes the normal form of a distributed polynomial.
3086: (The result may be multiplied by a constant in the ground field.)
3087: \E
1.1 noro 3088: @item dp_true_nf(@var{indexlist},@var{dpoly},@var{dpolyarray},@var{fullreduce})
3089: @item dp_true_nf_mod(@var{indexlist},@var{dpoly},@var{dpolyarray},@var{fullreduce},@var{mod})
1.2 noro 3090: \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)
3091: \BEG
3092: :: Computes the normal form of a distributed polynomial. (The true result
3093: is returned in such a list as @code{[numerator, denominator]})
3094: \E
1.1 noro 3095: @end table
3096:
3097: @table @var
3098: @item return
1.2 noro 3099: \JP @code{dp_nf()} : $BJ,;6I=8=B?9`<0(B, @code{dp_true_nf()} : $B%j%9%H(B
3100: \EG @code{dp_nf()} : distributed polynomial, @code{dp_true_nf()} : list
1.1 noro 3101: @item indexlist
1.2 noro 3102: \JP $B%j%9%H(B
3103: \EG list
1.1 noro 3104: @item dpoly
1.2 noro 3105: \JP $BJ,;6I=8=B?9`<0(B
3106: \EG distributed polynomial
1.1 noro 3107: @item dpolyarray
1.2 noro 3108: \JP $BG[Ns(B
3109: \EG array of distributed polynomial
1.1 noro 3110: @item fullreduce
1.2 noro 3111: \JP $B%U%i%0(B
3112: \EG flag
1.1 noro 3113: @item mod
1.2 noro 3114: \JP $BAG?t(B
3115: \EG prime
1.1 noro 3116: @end table
3117:
3118: @itemize @bullet
1.2 noro 3119: \BJP
1.1 noro 3120: @item
3121: $BJ,;6I=8=B?9`<0(B @var{dpoly} $B$N@55,7A$r5a$a$k(B.
3122: @item
1.18 noro 3123: $BL>A0$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.
3124: @item
1.1 noro 3125: @code{dp_nf_mod()}, @code{dp_true_nf_mod()} $B$NF~NO$O(B, @code{dp_mod()} $B$J$I(B
3126: $B$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.
3127: @item
3128: $B7k2L$KM-M}?t(B, $BM-M}<0$,4^$^$l$k$N$rHr$1$k$?$a(B, @code{dp_nf()} $B$O(B
3129: $B??$NCM$NDj?tG\$NCM$rJV$9(B. $BM-M}<078?t$N>l9g$N(B @code{dp_nf_mod()} $B$bF1MM(B
3130: $B$G$"$k$,(B, $B78?tBN$,M-8BBN$N>l9g(B @code{dp_nf_mod()} $B$O??$NCM$rJV$9(B.
3131: @item
3132: @code{dp_true_nf()}, @code{dp_true_nf_mod()} $B$O(B,
3133: @code{[@var{nm},@var{dn}]} $B$J$k7A$N%j%9%H$rJV$9(B.
3134: $B$?$@$7(B, @var{nm} $B$O78?t$KJ,?t(B, $BM-M}<0$r4^$^$J$$J,;6I=8=B?9`<0(B, @var{dn} $B$O(B
3135: $B?t$^$?$OB?9`<0$G(B @var{nm}/@var{dn} $B$,??$NCM$H$J$k(B.
3136: @item
3137: @var{dpolyarray} $B$OJ,;6I=8=B?9`<0$rMWAG$H$9$k%Y%/%H%k(B,
3138: @var{indexlist} $B$O@55,2=7W;;$KMQ$$$k(B @var{dpolyarray} $B$NMWAG$N%$%s%G%C%/%9(B
3139: $B$N%j%9%H(B.
3140: @item
3141: @var{fullreduce} $B$,(B 0 $B$G$J$$$H$-A4$F$N9`$KBP$7$F4JLs$r9T$&(B. @var{fullreduce}
3142: $B$,(B 0 $B$N$H$-F,9`$N$_$KBP$7$F4JLs$r9T$&(B.
3143: @item
3144: @var{indexlist} $B$G;XDj$5$l$?B?9`<0$O(B, $BA0$NJ}$N$b$N$,M%@hE*$K;H$o$l$k(B.
3145: @item
3146: $B0lHL$K$O(B @var{indexlist} $B$NM?$(J}$K$h$jH!?t$NCM$O0[$J$k2DG=@-$,$"$k$,(B,
3147: $B%0%l%V%J4pDl$KBP$7$F$O0l0UE*$KDj$^$k(B.
3148: @item
3149: $BJ,;6I=8=$G$J$$8GDj$5$l$?B?9`<0=89g$K$h$k@55,7A$rB??t5a$a$kI,MW$,$"$k>l9g(B
3150: $B$KJXMx$G$"$k(B. $BC10l$N1i;;$K4X$7$F$O(B, @code{p_nf}, @code{p_true_nf} $B$r(B
3151: $BMQ$$$k$H$h$$(B.
1.2 noro 3152: \E
3153: \BEG
3154: @item
3155: Computes the normal form of a distributed polynomial.
3156: @item
1.18 noro 3157: Functions whose name contain @code{weyl} compute normal forms in Weyl algebra. The description below also applies to
3158: the functions for Weyl algebra.
3159: @item
1.2 noro 3160: @code{dp_nf_mod()} and @code{dp_true_nf_mod()} require
3161: distributed polynomials with coefficients in a finite field as arguments.
3162: @item
3163: The result of @code{dp_nf()} may be multiplied by a constant in the
3164: ground field in order to make the result integral. The same is true
3165: for @code{dp_nf_mod()}, but it returns the true normal form if
3166: the ground field is a finite field.
3167: @item
3168: @code{dp_true_nf()} and @code{dp_true_nf_mod()} return
3169: such a list as @code{[@var{nm},@var{dn}]}.
3170: Here @var{nm} is a distributed polynomial whose coefficients are integral
3171: in the ground field, @var{dn} is an integral element in the ground
3172: field and @var{nm}/@var{dn} is the true normal form.
3173: @item
3174: @var{dpolyarray} is a vector whose components are distributed polynomials
3175: and @var{indexlist} is a list of indices which is used for the normal form
3176: computation.
3177: @item
3178: When argument @var{fullreduce} has non-zero value,
3179: all terms are reduced. When it has value 0,
3180: only the head term is reduced.
3181: @item
3182: As for the polynomials specified by @var{indexlist}, one specified by
3183: an index placed at the preceding position has priority to be selected.
3184: @item
3185: In general, the result of the function may be different depending on
3186: @var{indexlist}. However, the result is unique for Groebner bases.
3187: @item
3188: These functions are useful when a fixed non-distributed polynomial set
3189: is used as a set of reducers to compute normal forms of many polynomials.
3190: For single computation @code{p_nf} and @code{p_true_nf} are sufficient.
3191: \E
1.1 noro 3192: @end itemize
3193:
3194: @example
3195: [0] load("gr")$
3196: [64] load("katsura")$
3197: [69] K=katsura(4)$
3198: [70] dp_ord(2)$
3199: [71] V=[u0,u1,u2,u3,u4]$
3200: [72] DP1=newvect(length(K),map(dp_ptod,K,V))$
3201: [73] G=gr(K,V,2)$
3202: [74] DP2=newvect(length(G),map(dp_ptod,G,V))$
3203: [75] T=dp_ptod((u0-u1+u2-u3+u4)^2,V)$
3204: [76] dp_dtop(dp_nf([0,1,2,3,4],T,DP1,1),V);
1.5 noro 3205: u4^2+(6*u3+2*u2+6*u1-2)*u4+9*u3^2+(6*u2+18*u1-6)*u3+u2^2
3206: +(6*u1-2)*u2+9*u1^2-6*u1+1
1.1 noro 3207: [77] dp_dtop(dp_nf([4,3,2,1,0],T,DP1,1),V);
3208: -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
3209: [78] dp_dtop(dp_nf([0,1,2,3,4],T,DP2,1),V);
1.5 noro 3210: -11380879768451657780886122972730785203470970010204714556333530492210
3211: 456775930005716505560062087150928400876150217079820311439477560587583
3212: 488*u4^15+...
1.1 noro 3213: [79] dp_dtop(dp_nf([4,3,2,1,0],T,DP2,1),V);
1.5 noro 3214: -11380879768451657780886122972730785203470970010204714556333530492210
3215: 456775930005716505560062087150928400876150217079820311439477560587583
3216: 488*u4^15+...
1.1 noro 3217: [80] @@78==@@79;
3218: 1
3219: @end example
3220:
3221: @table @t
1.2 noro 3222: \JP @item $B;2>H(B
3223: \EG @item References
1.1 noro 3224: @fref{dp_dtop},
3225: @fref{dp_ord},
3226: @fref{dp_mod dp_rat},
3227: @fref{p_nf p_nf_mod p_true_nf p_true_nf_mod}.
3228: @end table
3229:
1.2 noro 3230: \JP @node dp_hm dp_ht dp_hc dp_rest,,, $B%0%l%V%J4pDl$K4X$9$kH!?t(B
3231: \EG @node dp_hm dp_ht dp_hc dp_rest,,, Functions for Groebner basis computation
1.1 noro 3232: @subsection @code{dp_hm}, @code{dp_ht}, @code{dp_hc}, @code{dp_rest}
3233: @findex dp_hm
3234: @findex dp_ht
3235: @findex dp_hc
3236: @findex dp_rest
3237:
3238: @table @t
3239: @item dp_hm(@var{dpoly})
1.2 noro 3240: \JP :: $BF,C19`<0$r<h$j=P$9(B.
3241: \EG :: Gets the head monomial.
1.1 noro 3242: @item dp_ht(@var{dpoly})
1.2 noro 3243: \JP :: $BF,9`$r<h$j=P$9(B.
3244: \EG :: Gets the head term.
1.1 noro 3245: @item dp_hc(@var{dpoly})
1.2 noro 3246: \JP :: $BF,78?t$r<h$j=P$9(B.
3247: \EG :: Gets the head coefficient.
1.1 noro 3248: @item dp_rest(@var{dpoly})
1.2 noro 3249: \JP :: $BF,C19`<0$r<h$j=|$$$?;D$j$rJV$9(B.
3250: \EG :: Gets the remainder of the polynomial where the head monomial is removed.
1.1 noro 3251: @end table
3252:
3253: @table @var
1.2 noro 3254: \BJP
1.1 noro 3255: @item return
3256: @code{dp_hm()}, @code{dp_ht()}, @code{dp_rest()} : $BJ,;6I=8=B?9`<0(B,
3257: @code{dp_hc()} : $B?t$^$?$OB?9`<0(B
3258: @item dpoly
3259: $BJ,;6I=8=B?9`<0(B
1.2 noro 3260: \E
3261: \BEG
3262: @item return
3263: @code{dp_hm()}, @code{dp_ht()}, @code{dp_rest()} : distributed polynomial
3264: @code{dp_hc()} : number or polynomial
3265: @item dpoly
3266: distributed polynomial
3267: \E
1.1 noro 3268: @end table
3269:
3270: @itemize @bullet
1.2 noro 3271: \BJP
1.1 noro 3272: @item
3273: $B$3$l$i$O(B, $BJ,;6I=8=B?9`<0$N3FItJ,$r<h$j=P$9$?$a$NH!?t$G$"$k(B.
3274: @item
3275: $BJ,;6I=8=B?9`<0(B @var{p} $B$KBP$7<!$,@.$jN)$D(B.
1.2 noro 3276: \E
3277: \BEG
3278: @item
3279: These are used to get various parts of a distributed polynomial.
3280: @item
3281: The next equations hold for a distributed polynomial @var{p}.
3282: \E
1.1 noro 3283: @table @code
3284: @item @var{p} = dp_hm(@var{p}) + dp_rest(@var{p})
3285: @item dp_hm(@var{p}) = dp_hc(@var{p}) dp_ht(@var{p})
3286: @end table
3287: @end itemize
3288:
3289: @example
3290: [87] dp_ord(0)$
3291: [88] X=ptozp((a46^2+7/10*a46+7/48)*u3^4-50/27*a46^2-35/27*a46-49/216)$
3292: [89] T=dp_ptod(X,[u3,u4,a46])$
3293: [90] dp_hm(T);
3294: (2160)*<<4,0,2>>
3295: [91] dp_ht(T);
3296: (1)*<<4,0,2>>
3297: [92] dp_hc(T);
3298: 2160
3299: [93] dp_rest(T);
3300: (1512)*<<4,0,1>>+(315)*<<4,0,0>>+(-4000)*<<0,0,2>>+(-2800)*<<0,0,1>>
3301: +(-490)*<<0,0,0>>
3302: @end example
3303:
1.2 noro 3304: \JP @node dp_td dp_sugar,,, $B%0%l%V%J4pDl$K4X$9$kH!?t(B
3305: \EG @node dp_td dp_sugar,,, Functions for Groebner basis computation
1.1 noro 3306: @subsection @code{dp_td}, @code{dp_sugar}
3307: @findex dp_td
3308: @findex dp_sugar
3309:
3310: @table @t
3311: @item dp_td(@var{dpoly})
1.2 noro 3312: \JP :: $BF,9`$NA4<!?t$rJV$9(B.
3313: \EG :: Gets the total degree of the head term.
1.1 noro 3314: @item dp_sugar(@var{dpoly})
1.2 noro 3315: \JP :: $BB?9`<0$N(B @code{sugar} $B$rJV$9(B.
3316: \EG :: Gets the @code{sugar} of a polynomial.
1.1 noro 3317: @end table
3318:
3319: @table @var
3320: @item return
1.2 noro 3321: \JP $B<+A3?t(B
3322: \EG non-negative integer
1.1 noro 3323: @item dpoly
1.2 noro 3324: \JP $BJ,;6I=8=B?9`<0(B
3325: \EG distributed polynomial
1.1 noro 3326: @item onoff
1.2 noro 3327: \JP $B%U%i%0(B
3328: \EG flag
1.1 noro 3329: @end table
3330:
3331: @itemize @bullet
1.2 noro 3332: \BJP
1.1 noro 3333: @item
3334: @code{dp_td()} $B$O(B, $BF,9`$NA4<!?t(B, $B$9$J$o$A3FJQ?t$N;X?t$NOB$rJV$9(B.
3335: @item
3336: $BJ,;6I=8=B?9`<0$,@8@.$5$l$k$H(B, @code{sugar} $B$H8F$P$l$k$"$k@0?t$,IUM?(B
3337: $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.
3338: @item
3339: @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
3340: $B7hDj$9$k$?$a$N=EMW$J;X?K$H$J$k(B.
1.2 noro 3341: \E
3342: \BEG
3343: @item
3344: Function @code{dp_td()} returns the total degree of the head term,
3345: i.e., the sum of all exponent of variables in that term.
3346: @item
3347: Upon creation of a distributed polynomial, an integer called @code{sugar}
3348: is associated. This value is
3349: the total degree of the virtually homogenized one of the original
3350: polynomial.
3351: @item
3352: The quantity @code{sugar} is an important guide to determine the
3353: selection strategy of critical pairs in Groebner basis computation.
3354: \E
1.1 noro 3355: @end itemize
3356:
3357: @example
3358: [74] dp_ord(0)$
3359: [75] X=<<1,2>>+<<0,1>>$
3360: [76] Y=<<1,2>>+<<1,0>>$
3361: [77] Z=X-Y;
3362: (-1)*<<1,0>>+(1)*<<0,1>>
3363: [78] dp_sugar(T);
3364: 3
3365: @end example
3366:
1.2 noro 3367: \JP @node dp_lcm,,, $B%0%l%V%J4pDl$K4X$9$kH!?t(B
3368: \EG @node dp_lcm,,, Functions for Groebner basis computation
1.1 noro 3369: @subsection @code{dp_lcm}
3370: @findex dp_lcm
3371:
3372: @table @t
3373: @item dp_lcm(@var{dpoly1},@var{dpoly2})
1.2 noro 3374: \JP :: $B:G>.8xG\9`$rJV$9(B.
3375: \EG :: Returns the least common multiple of the head terms of the given two polynomials.
1.1 noro 3376: @end table
3377:
3378: @table @var
3379: @item return
1.2 noro 3380: \JP $BJ,;6I=8=B?9`<0(B
3381: \EG distributed polynomial
1.4 noro 3382: @item dpoly1 dpoly2
1.2 noro 3383: \JP $BJ,;6I=8=B?9`<0(B
3384: \EG distributed polynomial
1.1 noro 3385: @end table
3386:
3387: @itemize @bullet
1.2 noro 3388: \BJP
1.1 noro 3389: @item
3390: $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 3391: \E
3392: \BEG
3393: @item
3394: Returns the least common multiple of the head terms of the given
3395: two polynomials, where coefficient is always set to 1.
3396: \E
1.1 noro 3397: @end itemize
3398:
3399: @example
3400: [100] dp_lcm(<<1,2,3,4,5>>,<<5,4,3,2,1>>);
3401: (1)*<<5,4,3,4,5>>
3402: @end example
3403:
3404: @table @t
1.2 noro 3405: \JP @item $B;2>H(B
3406: \EG @item References
1.1 noro 3407: @fref{p_nf p_nf_mod p_true_nf p_true_nf_mod}.
3408: @end table
3409:
1.2 noro 3410: \JP @node dp_redble,,, $B%0%l%V%J4pDl$K4X$9$kH!?t(B
3411: \EG @node dp_redble,,, Functions for Groebner basis computation
1.1 noro 3412: @subsection @code{dp_redble}
3413: @findex dp_redble
3414:
3415: @table @t
3416: @item dp_redble(@var{dpoly1},@var{dpoly2})
1.2 noro 3417: \JP :: $BF,9`$I$&$7$,@0=|2DG=$+$I$&$+D4$Y$k(B.
3418: \EG :: Checks whether one head term is divisible by the other head term.
1.1 noro 3419: @end table
3420:
3421: @table @var
3422: @item return
1.2 noro 3423: \JP $B@0?t(B
3424: \EG integer
1.4 noro 3425: @item dpoly1 dpoly2
1.2 noro 3426: \JP $BJ,;6I=8=B?9`<0(B
3427: \EG distributed polynomial
1.1 noro 3428: @end table
3429:
3430: @itemize @bullet
1.2 noro 3431: \BJP
1.1 noro 3432: @item
3433: @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
3434: 0 $B$rJV$9(B.
3435: @item
3436: $BB?9`<0$N4JLs$r9T$&:](B, $B$I$N9`$r4JLs$G$-$k$+$rC5$9$N$KMQ$$$k(B.
1.2 noro 3437: \E
3438: \BEG
3439: @item
3440: Returns 1 if the head term of @var{dpoly2} divides the head term of
3441: @var{dpoly1}; otherwise 0.
3442: @item
3443: Used for finding candidate terms at reduction of polynomials.
3444: \E
1.1 noro 3445: @end itemize
3446:
3447: @example
3448: [148] C;
3449: (1)*<<1,1,1,0,0>>+(1)*<<0,1,1,1,0>>+(1)*<<1,1,0,0,1>>+(1)*<<1,0,0,1,1>>
3450: [149] T;
3451: (3)*<<2,1,0,0,0>>+(3)*<<1,2,0,0,0>>+(1)*<<0,3,0,0,0>>+(6)*<<1,1,1,0,0>>
3452: [150] for ( ; T; T = dp_rest(T)) print(dp_redble(T,C));
3453: 0
3454: 0
3455: 0
3456: 1
3457: @end example
3458:
3459: @table @t
1.2 noro 3460: \JP @item $B;2>H(B
3461: \EG @item References
1.1 noro 3462: @fref{dp_red dp_red_mod}.
3463: @end table
3464:
1.2 noro 3465: \JP @node dp_subd,,, $B%0%l%V%J4pDl$K4X$9$kH!?t(B
3466: \EG @node dp_subd,,, Functions for Groebner basis computation
1.1 noro 3467: @subsection @code{dp_subd}
3468: @findex dp_subd
3469:
3470: @table @t
3471: @item dp_subd(@var{dpoly1},@var{dpoly2})
1.2 noro 3472: \JP :: $BF,9`$N>&C19`<0$rJV$9(B.
3473: \EG :: Returns the quotient monomial of the head terms.
1.1 noro 3474: @end table
3475:
3476: @table @var
3477: @item return
1.2 noro 3478: \JP $BJ,;6I=8=B?9`<0(B
3479: \EG distributed polynomial
1.4 noro 3480: @item dpoly1 dpoly2
1.2 noro 3481: \JP $BJ,;6I=8=B?9`<0(B
3482: \EG distributed polynomial
1.1 noro 3483: @end table
3484:
3485: @itemize @bullet
1.2 noro 3486: \BJP
1.1 noro 3487: @item
3488: @code{dp_ht(@var{dpoly1})/dp_ht(@var{dpoly2})} $B$r5a$a$k(B. $B7k2L$N78?t$O(B 1
3489: $B$G$"$k(B.
3490: @item
3491: $B3d$j@Z$l$k$3$H$,$"$i$+$8$a$o$+$C$F$$$kI,MW$,$"$k(B.
1.2 noro 3492: \E
3493: \BEG
3494: @item
3495: Gets @code{dp_ht(@var{dpoly1})/dp_ht(@var{dpoly2})}.
3496: The coefficient of the result is always set to 1.
3497: @item
3498: Divisibility assumed.
3499: \E
1.1 noro 3500: @end itemize
3501:
3502: @example
3503: [162] dp_subd(<<1,2,3,4,5>>,<<1,1,2,3,4>>);
3504: (1)*<<0,1,1,1,1>>
3505: @end example
3506:
3507: @table @t
1.2 noro 3508: \JP @item $B;2>H(B
3509: \EG @item References
1.1 noro 3510: @fref{dp_red dp_red_mod}.
3511: @end table
3512:
1.2 noro 3513: \JP @node dp_vtoe dp_etov,,, $B%0%l%V%J4pDl$K4X$9$kH!?t(B
3514: \EG @node dp_vtoe dp_etov,,, Functions for Groebner basis computation
1.1 noro 3515: @subsection @code{dp_vtoe}, @code{dp_etov}
3516: @findex dp_vtoe
3517: @findex dp_etov
3518:
3519: @table @t
3520: @item dp_vtoe(@var{vect})
1.2 noro 3521: \JP :: $B;X?t%Y%/%H%k$r9`$KJQ49(B
3522: \EG :: Converts an exponent vector into a term.
1.1 noro 3523: @item dp_etov(@var{dpoly})
1.2 noro 3524: \JP :: $BF,9`$r;X?t%Y%/%H%k$KJQ49(B
3525: \EG :: Convert the head term of a distributed polynomial into an exponent vector.
1.1 noro 3526: @end table
3527:
3528: @table @var
3529: @item return
1.2 noro 3530: \JP @code{dp_vtoe} : $BJ,;6I=8=B?9`<0(B, @code{dp_etov} : $B%Y%/%H%k(B
3531: \EG @code{dp_vtoe} : distributed polynomial, @code{dp_etov} : vector
1.1 noro 3532: @item vect
1.2 noro 3533: \JP $B%Y%/%H%k(B
3534: \EG vector
1.1 noro 3535: @item dpoly
1.2 noro 3536: \JP $BJ,;6I=8=B?9`<0(B
3537: \EG distributed polynomial
1.1 noro 3538: @end table
3539:
3540: @itemize @bullet
1.2 noro 3541: \BJP
1.1 noro 3542: @item
3543: @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.
3544: @item
3545: @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
3546: $B%Y%/%H%k$KJQ49$9$k(B.
1.2 noro 3547: \E
3548: \BEG
3549: @item
3550: @code{dp_vtoe()} generates a term whose exponent vector is @var{vect}.
3551: @item
3552: @code{dp_etov()} generates a vector which is the exponent vector of the
3553: head term of @code{dpoly}.
3554: \E
1.1 noro 3555: @end itemize
3556:
3557: @example
3558: [211] X=<<1,2,3>>;
3559: (1)*<<1,2,3>>
3560: [212] V=dp_etov(X);
3561: [ 1 2 3 ]
3562: [213] V[2]++$
3563: [214] Y=dp_vtoe(V);
3564: (1)*<<1,2,4>>
3565: @end example
3566:
1.2 noro 3567: \JP @node dp_mbase,,, $B%0%l%V%J4pDl$K4X$9$kH!?t(B
3568: \EG @node dp_mbase,,, Functions for Groebner basis computation
1.1 noro 3569: @subsection @code{dp_mbase}
3570: @findex dp_mbase
3571:
3572: @table @t
3573: @item dp_mbase(@var{dplist})
1.2 noro 3574: \JP :: monomial $B4pDl$N7W;;(B
3575: \EG :: Computes the monomial basis
1.1 noro 3576: @end table
3577:
3578: @table @var
3579: @item return
1.2 noro 3580: \JP $BJ,;6I=8=B?9`<0$N%j%9%H(B
3581: \EG list of distributed polynomial
1.1 noro 3582: @item dplist
1.2 noro 3583: \JP $BJ,;6I=8=B?9`<0$N%j%9%H(B
3584: \EG list of distributed polynomial
1.1 noro 3585: @end table
3586:
3587: @itemize @bullet
1.2 noro 3588: \BJP
1.1 noro 3589: @item
3590: $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
3591: $B$G$"$k(B @var{dplist} $B$K$D$$$F(B,
3592: @var{dplist} $B$,(B K[X] $BCf$G@8@.$9$k%$%G%"%k(B I $B$,(B 0 $B<!85$N;~(B,
3593: K $B>eM-8B<!85@~7A6u4V$G$"$k(B K[X]/I $B$N(B monomial $B$K$h$k4pDl$r5a$a$k(B.
3594: @item
3595: $BF@$i$l$?4pDl$N8D?t$,(B, K[X]/I $B$N(B K-$B@~7A6u4V$H$7$F$N<!85$KEy$7$$(B.
1.2 noro 3596: \E
3597: \BEG
3598: @item
3599: Assuming that @var{dplist} is a list of distributed polynomials which
3600: is a Groebner basis with respect to the current ordering type and
3601: that the ideal @var{I} generated by @var{dplist} in K[X] is zero-dimensional,
3602: this function computes the monomial basis of a finite dimenstional K-vector
3603: space K[X]/I.
3604: @item
3605: The number of elements in the monomial basis is equal to the
3606: K-dimenstion of K[X]/I.
3607: \E
1.1 noro 3608: @end itemize
3609:
3610: @example
3611: [215] K=katsura(5)$
3612: [216] V=[u5,u4,u3,u2,u1,u0]$
3613: [217] G0=gr(K,V,0)$
3614: [218] H=map(dp_ptod,G0,V)$
3615: [219] map(dp_ptod,dp_mbase(H),V)$
3616: [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,
3617: 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,
3618: u1*u2,u1^2,u4*u0,u3*u0,u2*u0,u1*u0,u0^2,u4,u3,u2,u1,u0,1]
3619: @end example
3620:
3621: @table @t
1.2 noro 3622: \JP @item $B;2>H(B
3623: \EG @item References
1.1 noro 3624: @fref{gr hgr gr_mod}.
3625: @end table
3626:
1.2 noro 3627: \JP @node dp_mag,,, $B%0%l%V%J4pDl$K4X$9$kH!?t(B
3628: \EG @node dp_mag,,, Functions for Groebner basis computation
1.1 noro 3629: @subsection @code{dp_mag}
3630: @findex dp_mag
3631:
3632: @table @t
3633: @item dp_mag(@var{p})
1.2 noro 3634: \JP :: $B78?t$N%S%C%HD9$NOB$rJV$9(B
3635: \EG :: Computes the sum of bit lengths of coefficients of a distributed polynomial.
1.1 noro 3636: @end table
3637:
3638: @table @var
3639: @item return
1.2 noro 3640: \JP $B?t(B
3641: \EG integer
1.1 noro 3642: @item p
1.2 noro 3643: \JP $BJ,;6I=8=B?9`<0(B
3644: \EG distributed polynomial
1.1 noro 3645: @end table
3646:
3647: @itemize @bullet
1.2 noro 3648: \BJP
1.1 noro 3649: @item
3650: $BJ,;6I=8=B?9`<0$N78?t$K8=$l$kM-M}?t$K$D$-(B, $B$=$NJ,JlJ,;R(B ($B@0?t$N>l9g$OJ,;R(B)
3651: $B$N%S%C%HD9$NAmOB$rJV$9(B.
3652: @item
3653: $BBP>]$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$*$$$F$O(B
3654: $B78?tKDD%$,LdBj$H$J$j(B, $BESCf@8@.$5$l$kB?9`<0$,78?tKDD%$r5/$3$7$F$$$k$+$I$&$+(B
3655: $B$NH=Dj$KLrN)$D(B.
3656: @item
3657: @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
3658: $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 3659: \E
3660: \BEG
3661: @item
3662: This function computes the sum of bit lengths of coefficients of a
3663: distributed polynomial @var{p}. If a coefficient is non integral,
3664: the sum of bit lengths of the numerator and the denominator is taken.
3665: @item
3666: This is a measure of the size of a polynomial. Especially for
3667: zero-dimensional system coefficient swells are often serious and
3668: the returned value is useful to detect such swells.
3669: @item
3670: If @code{ShowMag} and @code{Print} for @code{dp_gr_flags()} are on,
3671: values of @code{dp_mag()} for intermediate basis elements are shown.
3672: \E
1.1 noro 3673: @end itemize
3674:
3675: @example
3676: [221] X=dp_ptod((x+2*y)^10,[x,y])$
3677: [222] dp_mag(X);
3678: 115
3679: @end example
3680:
3681: @table @t
1.2 noro 3682: \JP @item $B;2>H(B
3683: \EG @item References
1.1 noro 3684: @fref{dp_gr_flags dp_gr_print}.
3685: @end table
3686:
1.2 noro 3687: \JP @node dp_red dp_red_mod,,, $B%0%l%V%J4pDl$K4X$9$kH!?t(B
3688: \EG @node dp_red dp_red_mod,,, Functions for Groebner basis computation
1.1 noro 3689: @subsection @code{dp_red}, @code{dp_red_mod}
3690: @findex dp_red
3691: @findex dp_red_mod
3692:
3693: @table @t
3694: @item dp_red(@var{dpoly1},@var{dpoly2},@var{dpoly3})
3695: @item dp_red_mod(@var{dpoly1},@var{dpoly2},@var{dpoly3},@var{mod})
1.2 noro 3696: \JP :: $B0l2s$N4JLsA`:n(B
3697: \EG :: Single reduction operation
1.1 noro 3698: @end table
3699:
3700: @table @var
3701: @item return
1.2 noro 3702: \JP $B%j%9%H(B
3703: \EG list
1.4 noro 3704: @item dpoly1 dpoly2 dpoly3
1.2 noro 3705: \JP $BJ,;6I=8=B?9`<0(B
3706: \EG distributed polynomial
1.1 noro 3707: @item vlist
1.2 noro 3708: \JP $B%j%9%H(B
3709: \EG list
1.1 noro 3710: @item mod
1.2 noro 3711: \JP $BAG?t(B
3712: \EG prime
1.1 noro 3713: @end table
3714:
3715: @itemize @bullet
1.2 noro 3716: \BJP
1.1 noro 3717: @item
3718: @var{dpoly1} + @var{dpoly2} $B$J$kJ,;6I=8=B?9`<0$r(B @var{dpoly3} $B$G(B
3719: 1 $B2s4JLs$9$k(B.
3720: @item
3721: @code{dp_red_mod()} $B$NF~NO$O(B, $BA4$FM-8BBN78?t$KJQ49$5$l$F$$$kI,MW$,$"$k(B.
3722: @item
3723: $B4JLs$5$l$k9`$O(B @var{dpoly2} $B$NF,9`$G$"$k(B. $B=>$C$F(B, @var{dpoly2} $B$N(B
3724: $BF,9`$,(B @var{dpoly3} $B$NF,9`$G3d$j@Z$l$k$3$H$,$"$i$+$8$a$o$+$C$F$$$J$1$l$P(B
3725: $B$J$i$J$$(B.
3726: @item
3727: $B0z?t$,@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 3728: $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 3729: @item
3730: $B7k2L$O(B, @code{[@var{a dpoly1},@var{a dpoly2 - bt dpoly3}]} $B$J$k%j%9%H$G$"$k(B.
1.2 noro 3731: \E
3732: \BEG
3733: @item
3734: Reduces a distributed polynomial, @var{dpoly1} + @var{dpoly2},
3735: by @var{dpoly3} for single time.
3736: @item
3737: An input for @code{dp_red_mod()} must be converted into a distributed
3738: polynomial with coefficients in a finite field.
3739: @item
3740: This implies that
3741: the divisibility of the head term of @var{dpoly2} by the head term of
3742: @var{dpoly3} is assumed.
3743: @item
3744: When integral coefficients, computation is so carefully performed that
3745: no rational operations appear in the reduction procedure.
3746: It is computed for integers @var{a} and @var{b}, and a term @var{t} as:
1.4 noro 3747: @var{a}(@var{dpoly1} + @var{dpoly2})-@var{bt} @var{dpoly3}.
1.2 noro 3748: @item
3749: The result is a list @code{[@var{a dpoly1},@var{a dpoly2 - bt dpoly3}]}.
3750: \E
1.1 noro 3751: @end itemize
3752:
3753: @example
3754: [157] D=(3)*<<2,1,0,0,0>>+(3)*<<1,2,0,0,0>>+(1)*<<0,3,0,0,0>>;
3755: (3)*<<2,1,0,0,0>>+(3)*<<1,2,0,0,0>>+(1)*<<0,3,0,0,0>>
3756: [158] R=(6)*<<1,1,1,0,0>>;
3757: (6)*<<1,1,1,0,0>>
3758: [159] C=12*<<1,1,1,0,0>>+(1)*<<0,1,1,1,0>>+(1)*<<1,1,0,0,1>>;
3759: (12)*<<1,1,1,0,0>>+(1)*<<0,1,1,1,0>>+(1)*<<1,1,0,0,1>>
3760: [160] dp_red(D,R,C);
1.5 noro 3761: [(6)*<<2,1,0,0,0>>+(6)*<<1,2,0,0,0>>+(2)*<<0,3,0,0,0>>,
3762: (-1)*<<0,1,1,1,0>>+(-1)*<<1,1,0,0,1>>]
1.1 noro 3763: @end example
3764:
3765: @table @t
1.2 noro 3766: \JP @item $B;2>H(B
3767: \EG @item References
1.1 noro 3768: @fref{dp_mod dp_rat}.
3769: @end table
3770:
1.2 noro 3771: \JP @node dp_sp dp_sp_mod,,, $B%0%l%V%J4pDl$K4X$9$kH!?t(B
3772: \EG @node dp_sp dp_sp_mod,,, Functions for Groebner basis computation
1.1 noro 3773: @subsection @code{dp_sp}, @code{dp_sp_mod}
3774: @findex dp_sp
3775: @findex dp_sp_mod
3776:
3777: @table @t
3778: @item dp_sp(@var{dpoly1},@var{dpoly2})
3779: @item dp_sp_mod(@var{dpoly1},@var{dpoly2},@var{mod})
1.2 noro 3780: \JP :: S-$BB?9`<0$N7W;;(B
3781: \EG :: Computation of an S-polynomial
1.1 noro 3782: @end table
3783:
3784: @table @var
3785: @item return
1.2 noro 3786: \JP $BJ,;6I=8=B?9`<0(B
3787: \EG distributed polynomial
1.4 noro 3788: @item dpoly1 dpoly2
1.2 noro 3789: \JP $BJ,;6I=8=B?9`<0(B
3790: \EG distributed polynomial
1.1 noro 3791: @item mod
1.2 noro 3792: \JP $BAG?t(B
3793: \EG prime
1.1 noro 3794: @end table
3795:
3796: @itemize @bullet
1.2 noro 3797: \BJP
1.1 noro 3798: @item
3799: @var{dpoly1}, @var{dpoly2} $B$N(B S-$BB?9`<0$r7W;;$9$k(B.
3800: @item
3801: @code{dp_sp_mod()} $B$NF~NO$O(B, $BA4$FM-8BBN78?t$KJQ49$5$l$F$$$kI,MW$,$"$k(B.
3802: @item
3803: $B7k2L$KM-M}?t(B, $BM-M}<0$,F~$k$N$rHr$1$k$?$a(B, $B7k2L$,Dj?tG\(B, $B$"$k$$$OB?9`<0(B
3804: $BG\$5$l$F$$$k2DG=@-$,$"$k(B.
1.2 noro 3805: \E
3806: \BEG
3807: @item
3808: This function computes the S-polynomial of @var{dpoly1} and @var{dpoly2}.
3809: @item
3810: Inputs of @code{dp_sp_mod()} must be polynomials with coefficients in a
3811: finite field.
3812: @item
3813: The result may be multiplied by a constant in the ground field in order to
3814: make the result integral.
3815: \E
1.1 noro 3816: @end itemize
3817:
3818: @example
3819: [227] X=dp_ptod(x^2*y+x*y,[x,y]);
3820: (1)*<<2,1>>+(1)*<<1,1>>
3821: [228] Y=dp_ptod(x*y^2+x*y,[x,y]);
3822: (1)*<<1,2>>+(1)*<<1,1>>
3823: [229] dp_sp(X,Y);
3824: (-1)*<<2,1>>+(1)*<<1,2>>
3825: @end example
3826:
3827: @table @t
1.2 noro 3828: \JP @item $B;2>H(B
3829: \EG @item References
1.1 noro 3830: @fref{dp_mod dp_rat}.
3831: @end table
1.2 noro 3832: \JP @node p_nf p_nf_mod p_true_nf p_true_nf_mod,,, $B%0%l%V%J4pDl$K4X$9$kH!?t(B
3833: \EG @node p_nf p_nf_mod p_true_nf p_true_nf_mod,,, Functions for Groebner basis computation
1.1 noro 3834: @subsection @code{p_nf}, @code{p_nf_mod}, @code{p_true_nf}, @code{p_true_nf_mod}
3835: @findex p_nf
3836: @findex p_nf_mod
3837: @findex p_true_nf
3838: @findex p_true_nf_mod
3839:
3840: @table @t
3841: @item p_nf(@var{poly},@var{plist},@var{vlist},@var{order})
3842: @itemx p_nf_mod(@var{poly},@var{plist},@var{vlist},@var{order},@var{mod})
1.2 noro 3843: \JP :: $BI=8=B?9`<0$N@55,7A$r5a$a$k(B. ($B7k2L$ODj?tG\$5$l$F$$$k2DG=@-$"$j(B)
3844: \BEG
3845: :: Computes the normal form of the given polynomial.
3846: (The result may be multiplied by a constant.)
3847: \E
1.1 noro 3848: @item p_true_nf(@var{poly},@var{plist},@var{vlist},@var{order})
3849: @itemx p_true_nf_mod(@var{poly},@var{plist},@var{vlist},@var{order},@var{mod})
1.2 noro 3850: \JP :: $BI=8=B?9`<0$N@55,7A$r5a$a$k(B. ($B??$N7k2L$r(B @code{[$BJ,;R(B, $BJ,Jl(B]} $B$N7A$GJV$9(B)
3851: \BEG
3852: :: Computes the normal form of the given polynomial. (The result is returned
3853: as a form of @code{[numerator, denominator]})
3854: \E
1.1 noro 3855: @end table
3856:
3857: @table @var
3858: @item return
1.2 noro 3859: \JP @code{p_nf} : $BB?9`<0(B, @code{p_true_nf} : $B%j%9%H(B
3860: \EG @code{p_nf} : polynomial, @code{p_true_nf} : list
1.1 noro 3861: @item poly
1.2 noro 3862: \JP $BB?9`<0(B
3863: \EG polynomial
1.4 noro 3864: @item plist vlist
1.2 noro 3865: \JP $B%j%9%H(B
3866: \EG list
1.1 noro 3867: @item order
1.2 noro 3868: \JP $B?t(B, $B%j%9%H$^$?$O9TNs(B
3869: \EG number, list or matrix
1.1 noro 3870: @item mod
1.2 noro 3871: \JP $BAG?t(B
3872: \EG prime
1.1 noro 3873: @end table
3874:
3875: @itemize @bullet
1.2 noro 3876: \BJP
1.1 noro 3877: @item
3878: @samp{gr} $B$GDj5A$5$l$F$$$k(B.
3879: @item
3880: $BB?9`<0$N(B, $BB?9`<0%j%9%H$K$h$k@55,7A$r5a$a$k(B.
3881: @item
3882: @code{dp_nf()}, @code{dp_true_nf()}, @code{dp_nf_mod()}, @code{dp_true_nf_mod}
3883: $B$KBP$9$k%$%s%?%U%'!<%9$G$"$k(B.
3884: @item
3885: @var{poly} $B$*$h$S(B @var{plist} $B$O(B, $BJQ?t=g=x(B @var{vlist} $B$*$h$S(B
3886: $BJQ?t=g=x7?(B @var{otype} $B$K=>$C$FJ,;6I=8=B?9`<0$KJQ49$5$l(B,
3887: @code{dp_nf()}, @code{dp_true_nf()}, @code{dp_nf_mod()},
3888: @code{dp_true_nf_mod()} $B$KEO$5$l$k(B.
3889: @item
3890: @code{dp_nf()}, @code{dp_true_nf()}, @code{dp_nf_mod()},
3891: @code{dp_true_nf_mod()} $B$O(B @var{fullreduce} $B$,(B 1 $B$G8F$S=P$5$l$k(B.
3892: @item
3893: $B7k2L$OB?9`<0$KJQ49$5$l$F=PNO$5$l$k(B.
3894: @item
3895: @code{p_true_nf()}, @code{p_true_nf_mod()} $B$N=PNO$K4X$7$F$O(B,
3896: @code{dp_true_nf()}, @code{dp_true_nf_mod()} $B$N9`$r;2>H(B.
1.2 noro 3897: \E
3898: \BEG
3899: @item
3900: Defined in the package @samp{gr}.
3901: @item
3902: Obtains the normal form of a polynomial by a polynomial list.
3903: @item
3904: These are interfaces to @code{dp_nf()}, @code{dp_true_nf()}, @code{dp_nf_mod()},
3905: @code{dp_true_nf_mod}
3906: @item
3907: The polynomial @var{poly} and the polynomials in @var{plist} is
3908: converted, according to the variable ordering @var{vlist} and
3909: type of term ordering @var{otype}, into their distributed polynomial
3910: counterparts and passed to @code{dp_nf()}.
3911: @item
3912: @code{dp_nf()}, @code{dp_true_nf()}, @code{dp_nf_mod()} and
3913: @code{dp_true_nf_mod()}
3914: is called with value 1 for @var{fullreduce}.
3915: @item
3916: The result is converted back into an ordinary polynomial.
3917: @item
3918: As for @code{p_true_nf()}, @code{p_true_nf_mod()}
3919: refer to @code{dp_true_nf()} and @code{dp_true_nf_mod()}.
3920: \E
1.1 noro 3921: @end itemize
3922:
3923: @example
3924: [79] K = katsura(5)$
3925: [80] V = [u5,u4,u3,u2,u1,u0]$
3926: [81] G = hgr(K,V,2)$
3927: [82] p_nf(K[1],G,V,2);
3928: 0
3929: [83] L = p_true_nf(K[1]+1,G,V,2);
3930: [-1503...,-1503...]
3931: [84] L[0]/L[1];
3932: 1
3933: @end example
3934:
3935: @table @t
1.2 noro 3936: \JP @item $B;2>H(B
3937: \EG @item References
1.1 noro 3938: @fref{dp_ptod},
3939: @fref{dp_dtop},
3940: @fref{dp_ord},
1.19 noro 3941: @fref{dp_nf dp_nf_mod dp_true_nf dp_true_nf_mod dp_weyl_nf dp_weyl_nf_mod}.
1.1 noro 3942: @end table
3943:
1.2 noro 3944: \JP @node p_terms,,, $B%0%l%V%J4pDl$K4X$9$kH!?t(B
3945: \EG @node p_terms,,, Functions for Groebner basis computation
1.1 noro 3946: @subsection @code{p_terms}
3947: @findex p_terms
3948:
3949: @table @t
3950: @item p_terms(@var{poly},@var{vlist},@var{order})
1.2 noro 3951: \JP :: $BB?9`<0$K$"$i$o$l$kC19`$r%j%9%H$K$9$k(B.
3952: \EG :: Monomials appearing in the given polynomial is collected into a list.
1.1 noro 3953: @end table
3954:
3955: @table @var
3956: @item return
1.2 noro 3957: \JP $B%j%9%H(B
3958: \EG list
1.1 noro 3959: @item poly
1.2 noro 3960: \JP $BB?9`<0(B
3961: \EG polynomial
1.1 noro 3962: @item vlist
1.2 noro 3963: \JP $B%j%9%H(B
3964: \EG list
1.1 noro 3965: @item order
1.2 noro 3966: \JP $B?t(B, $B%j%9%H$^$?$O9TNs(B
3967: \EG number, list or matrix
1.1 noro 3968: @end table
3969:
3970: @itemize @bullet
1.2 noro 3971: \BJP
1.1 noro 3972: @item
3973: @samp{gr} $B$GDj5A$5$l$F$$$k(B.
3974: @item
3975: $BB?9`<0$rC19`$KE83+$7$?;~$K8=$l$k9`$r%j%9%H$K$7$FJV$9(B.
3976: @var{vlist} $B$*$h$S(B @var{order} $B$K$h$jDj$^$k9`=g=x$K$h$j(B, $B=g=x$N9b$$$b$N(B
3977: $B$,%j%9%H$N@hF,$KMh$k$h$&$K%=!<%H$5$l$k(B.
3978: @item
3979: $B%0%l%V%J4pDl$O$7$P$7$P78?t$,5pBg$K$J$k$?$a(B, $B<B:]$K$I$N9`$,8=$l$F(B
3980: $B$$$k$N$+$r8+$k$?$a$J$I$KMQ$$$k(B.
1.2 noro 3981: \E
3982: \BEG
3983: @item
3984: Defined in the package @samp{gr}.
3985: @item
3986: This returns a list which contains all non-zero monomials in the given
3987: polynomial. The monomials are ordered according to the current
3988: type of term ordering and @var{vlist}.
3989: @item
3990: Since polynomials in a Groebner base often have very large coefficients,
3991: examining a polynomial as it is may sometimes be difficult to perform.
3992: For such a case, this function enables to examine which term is really
3993: exists.
3994: \E
1.1 noro 3995: @end itemize
3996:
3997: @example
3998: [233] G=gr(katsura(5),[u5,u4,u3,u2,u1,u0],2)$
3999: [234] p_terms(G[0],[u5,u4,u3,u2,u1,u0],2);
1.5 noro 4000: [u5,u0^31,u0^30,u0^29,u0^28,u0^27,u0^26,u0^25,u0^24,u0^23,u0^22,
4001: u0^21,u0^20,u0^19,u0^18,u0^17,u0^16,u0^15,u0^14,u0^13,u0^12,u0^11,
4002: u0^10,u0^9,u0^8,u0^7,u0^6,u0^5,u0^4,u0^3,u0^2,u0,1]
1.1 noro 4003: @end example
4004:
1.2 noro 4005: \JP @node gb_comp,,, $B%0%l%V%J4pDl$K4X$9$kH!?t(B
4006: \EG @node gb_comp,,, Functions for Groebner basis computation
1.1 noro 4007: @subsection @code{gb_comp}
4008: @findex gb_comp
4009:
4010: @table @t
4011: @item gb_comp(@var{plist1}, @var{plist2})
1.2 noro 4012: \JP :: $BB?9`<0%j%9%H$,(B, $BId9f$r=|$$$F=89g$H$7$FEy$7$$$+$I$&$+D4$Y$k(B.
4013: \EG :: Checks whether two polynomial lists are equal or not as a set
1.1 noro 4014: @end table
4015:
4016: @table @var
1.2 noro 4017: \JP @item return 0 $B$^$?$O(B 1
4018: \EG @item return 0 or 1
1.4 noro 4019: @item plist1 plist2
1.1 noro 4020: @end table
4021:
4022: @itemize @bullet
1.2 noro 4023: \BJP
1.1 noro 4024: @item
4025: @var{plist1}, @var{plist2} $B$K$D$$$F(B, $BId9f$r=|$$$F=89g$H$7$FEy$7$$$+$I$&$+(B
4026: $BD4$Y$k(B.
4027: @item
4028: $B0[$J$kJ}K!$G5a$a$?%0%l%V%J4pDl$O(B, $B4pDl$N=g=x(B, $BId9f$,0[$J$k>l9g$,$"$j(B,
4029: $B$=$l$i$,Ey$7$$$+$I$&$+$rD4$Y$k$?$a$KMQ$$$k(B.
1.2 noro 4030: \E
4031: \BEG
4032: @item
4033: This function checks whether @var{plist1} and @var{plist2} are equal or
4034: not as a set .
4035: @item
4036: For the same input and the same term ordering different
4037: functions for Groebner basis computations may produce different outputs
4038: as lists. This function compares such lists whether they are equal
4039: as a generating set of an ideal.
4040: \E
1.1 noro 4041: @end itemize
4042:
4043: @example
4044: [243] C=cyclic(6)$
4045: [244] V=[c0,c1,c2,c3,c4,c5]$
4046: [245] G0=gr(C,V,0)$
4047: [246] G=tolex(G0,V,0,V)$
4048: [247] GG=lex_tl(C,V,0,V,0)$
4049: [248] gb_comp(G,GG);
4050: 1
4051: @end example
4052:
1.2 noro 4053: \JP @node katsura hkatsura cyclic hcyclic,,, $B%0%l%V%J4pDl$K4X$9$kH!?t(B
4054: \EG @node katsura hkatsura cyclic hcyclic,,, Functions for Groebner basis computation
1.1 noro 4055: @subsection @code{katsura}, @code{hkatsura}, @code{cyclic}, @code{hcyclic}
4056: @findex katsura
4057: @findex hkatsura
4058: @findex cyclic
4059: @findex hcyclic
4060:
4061: @table @t
4062: @item katsura(@var{n})
4063: @item hkatsura(@var{n})
4064: @item cyclic(@var{n})
4065: @item hcyclic(@var{n})
1.2 noro 4066: \JP :: $BB?9`<0%j%9%H$N@8@.(B
4067: \EG :: Generates a polynomial list of standard benchmark.
1.1 noro 4068: @end table
4069:
4070: @table @var
4071: @item return
1.2 noro 4072: \JP $B%j%9%H(B
4073: \EG list
1.1 noro 4074: @item n
1.2 noro 4075: \JP $B@0?t(B
4076: \EG integer
1.1 noro 4077: @end table
4078:
4079: @itemize @bullet
1.2 noro 4080: \BJP
1.1 noro 4081: @item
4082: @code{katsura()} $B$O(B @samp{katsura}, @code{cyclic()} $B$O(B @samp{cyclic}
4083: $B$GDj5A$5$l$F$$$k(B.
4084: @item
4085: $B%0%l%V%J4pDl7W;;$G$7$P$7$P%F%9%H(B, $B%Y%s%A%^!<%/$KMQ$$$i$l$k(B @code{katsura},
4086: @code{cyclic} $B$*$h$S$=$N@F<!2=$r@8@.$9$k(B.
4087: @item
4088: @code{cyclic} $B$O(B @code{Arnborg}, @code{Lazard}, @code{Davenport} $B$J$I$N(B
4089: $BL>$G8F$P$l$k$3$H$b$"$k(B.
1.2 noro 4090: \E
4091: \BEG
4092: @item
4093: Function @code{katsura()} is defined in @samp{katsura}, and
4094: function @code{cyclic()} in @samp{cyclic}.
4095: @item
4096: These functions generate a series of polynomial sets, respectively,
4097: which are often used for testing and bench marking:
4098: @code{katsura}, @code{cyclic} and their homogenized versions.
4099: @item
4100: Polynomial set @code{cyclic} is sometimes called by other name:
4101: @code{Arnborg}, @code{Lazard}, and @code{Davenport}.
4102: \E
1.1 noro 4103: @end itemize
4104:
4105: @example
4106: [74] load("katsura")$
4107: [79] load("cyclic")$
4108: [89] katsura(5);
4109: [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 4110: 2*u3*u0+2*u1*u4-u3+(2*u1+2*u5)*u2,2*u2*u0+2*u2*u4+(2*u1+2*u5)*u3
4111: -u2+u1^2,2*u1*u0+(2*u3+2*u5)*u4+2*u2*u3+2*u1*u2-u1,
1.1 noro 4112: u0^2-u0+2*u4^2+2*u3^2+2*u2^2+2*u1^2+2*u5^2]
4113: [90] hkatsura(5);
4114: [-t+u0+2*u4+2*u3+2*u2+2*u1+2*u5,
4115: -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,
4116: -u2*t+2*u2*u0+2*u2*u4+(2*u1+2*u5)*u3+u1^2,
4117: -u1*t+2*u1*u0+(2*u3+2*u5)*u4+2*u2*u3+2*u1*u2,
4118: -u0*t+u0^2+2*u4^2+2*u3^2+2*u2^2+2*u1^2+2*u5^2]
4119: [91] cyclic(6);
4120: [c5*c4*c3*c2*c1*c0-1,
4121: ((((c4+c5)*c3+c5*c4)*c2+c5*c4*c3)*c1+c5*c4*c3*c2)*c0+c5*c4*c3*c2*c1,
4122: (((c3+c5)*c2+c5*c4)*c1+c5*c4*c3)*c0+c4*c3*c2*c1+c5*c4*c3*c2,
4123: ((c2+c5)*c1+c5*c4)*c0+c3*c2*c1+c4*c3*c2+c5*c4*c3,
4124: (c1+c5)*c0+c2*c1+c3*c2+c4*c3+c5*c4,c0+c1+c2+c3+c4+c5]
4125: [92] hcyclic(6);
4126: [-c^6+c5*c4*c3*c2*c1*c0,
4127: ((((c4+c5)*c3+c5*c4)*c2+c5*c4*c3)*c1+c5*c4*c3*c2)*c0+c5*c4*c3*c2*c1,
4128: (((c3+c5)*c2+c5*c4)*c1+c5*c4*c3)*c0+c4*c3*c2*c1+c5*c4*c3*c2,
4129: ((c2+c5)*c1+c5*c4)*c0+c3*c2*c1+c4*c3*c2+c5*c4*c3,
4130: (c1+c5)*c0+c2*c1+c3*c2+c4*c3+c5*c4,c0+c1+c2+c3+c4+c5]
4131: @end example
4132:
4133: @table @t
1.2 noro 4134: \JP @item $B;2>H(B
4135: \EG @item References
1.1 noro 4136: @fref{dp_dtop}.
4137: @end table
4138:
1.3 noro 4139: \JP @node primadec primedec,,, $B%0%l%V%J4pDl$K4X$9$kH!?t(B
4140: \EG @node primadec primedec,,, Functions for Groebner basis computation
4141: @subsection @code{primadec}, @code{primedec}
4142: @findex primadec
4143: @findex primedec
4144:
4145: @table @t
4146: @item primadec(@var{plist},@var{vlist})
4147: @item primedec(@var{plist},@var{vlist})
4148: \JP :: $B%$%G%"%k$NJ,2r(B
4149: \EG :: Computes decompositions of ideals.
4150: @end table
4151:
4152: @table @var
4153: @item return
4154: @itemx plist
4155: \JP $BB?9`<0%j%9%H(B
4156: \EG list of polynomials
4157: @item vlist
4158: \JP $BJQ?t%j%9%H(B
4159: \EG list of variables
4160: @end table
4161:
4162: @itemize @bullet
4163: \BJP
4164: @item
4165: @code{primadec()}, @code{primedec} $B$O(B @samp{primdec} $B$GDj5A$5$l$F$$$k(B.
4166: @item
4167: @code{primadec()}, @code{primedec()} $B$O$=$l$>$lM-M}?tBN>e$G$N%$%G%"%k$N(B
4168: $B=`AGJ,2r(B, $B:,4p$NAG%$%G%"%kJ,2r$r9T$&(B.
4169: @item
4170: $B0z?t$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.
4171: @item
4172: @code{primadec} $B$O(B @code{[$B=`AG@.J,(B, $BIUB0AG%$%G%"%k(B]} $B$N%j%9%H$rJV$9(B.
4173: @item
4174: @code{primadec} $B$O(B $BAG0x;R$N%j%9%H$rJV$9(B.
4175: @item
4176: $B7k2L$K$*$$$F(B, $BB?9`<0%j%9%H$H$7$FI=<($5$l$F$$$k3F%$%G%"%k$OA4$F(B
4177: $B%0%l%V%J4pDl$G$"$k(B. $BBP1~$9$k9`=g=x$O(B, $B$=$l$>$l(B
4178: $BJQ?t(B @code{PRIMAORD}, @code{PRIMEORD} $B$K3JG<$5$l$F$$$k(B.
4179: @item
4180: @code{primadec} $B$O(B @code{[Shimoyama,Yokoyama]} $B$N=`AGJ,2r%"%k%4%j%:%`(B
4181: $B$r<BAu$7$F$$$k(B.
4182: @item
4183: $B$b$7AG0x;R$N$_$r5a$a$?$$$J$i(B, @code{primedec} $B$r;H$&J}$,$h$$(B.
4184: $B$3$l$O(B, $BF~NO%$%G%"%k$,:,4p%$%G%"%k$G$J$$>l9g$K(B, @code{primadec}
4185: $B$N7W;;$KM>J,$J%3%9%H$,I,MW$H$J$k>l9g$,$"$k$+$i$G$"$k(B.
4186: \E
4187: \BEG
4188: @item
4189: Function @code{primadec()} and @code{primedec} are defined in @samp{primdec}.
4190: @item
4191: @code{primadec()}, @code{primedec()} are the function for primary
4192: ideal decomposition and prime decomposition of the radical over the
4193: rationals respectively.
4194: @item
4195: The arguments are a list of polynomials and a list of variables.
4196: These functions accept ideals with rational function coefficients only.
4197: @item
4198: @code{primadec} returns the list of pair lists consisting a primary component
4199: and its associated prime.
4200: @item
4201: @code{primedec} returns the list of prime components.
4202: @item
4203: Each component is a Groebner basis and the corresponding term order
4204: is indicated by the global variables @code{PRIMAORD}, @code{PRIMEORD}
4205: respectively.
4206: @item
4207: @code{primadec} implements the primary decompostion algorithm
4208: in @code{[Shimoyama,Yokoyama]}.
4209: @item
4210: If one only wants to know the prime components of an ideal, then
4211: use @code{primedec} because @code{primadec} may need additional costs
4212: if an input ideal is not radical.
4213: \E
4214: @end itemize
4215:
4216: @example
4217: [84] load("primdec")$
4218: [102] primedec([p*q*x-q^2*y^2+q^2*y,-p^2*x^2+p^2*x+p*q*y,
4219: (q^3*y^4-2*q^3*y^3+q^3*y^2)*x-q^3*y^4+q^3*y^3,
4220: -q^3*y^4+2*q^3*y^3+(-q^3+p*q^2)*y^2],[p,q,x,y]);
4221: [[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]]
4222: [103] primadec([x,z*y,w*y^2,w^2*y-z^3,y^3],[x,y,z,w]);
4223: [[[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]]]
4224: @end example
4225:
4226: @table @t
4227: \JP @item $B;2>H(B
4228: \EG @item References
4229: @fref{fctr sqfr},
4230: \JP @fref{$B9`=g=x$N@_Dj(B}.
4231: \EG @fref{Setting term orderings}.
4232: @end table
1.5 noro 4233:
4234: \JP @node primedec_mod,,, $B%0%l%V%J4pDl$K4X$9$kH!?t(B
4235: \EG @node primedec_mod,,, Functions for Groebner basis computation
4236: @subsection @code{primedec_mod}
4237: @findex primedec_mod
4238:
4239: @table @t
4240: @item primedec_mod(@var{plist},@var{vlist},@var{ord},@var{mod},@var{strategy})
4241: \JP :: $B%$%G%"%k$NJ,2r(B
4242: \EG :: Computes decompositions of ideals over small finite fields.
4243: @end table
4244:
4245: @table @var
4246: @item return
4247: @itemx plist
4248: \JP $BB?9`<0%j%9%H(B
4249: \EG list of polynomials
4250: @item vlist
4251: \JP $BJQ?t%j%9%H(B
4252: \EG list of variables
4253: @item ord
4254: \JP $B?t(B, $B%j%9%H$^$?$O9TNs(B
4255: \EG number, list or matrix
4256: @item mod
4257: \JP $B@5@0?t(B
4258: \EG positive integer
4259: @item strategy
4260: \JP $B@0?t(B
4261: \EG integer
4262: @end table
4263:
4264: @itemize @bullet
4265: \BJP
4266: @item
4267: @code{primedec_mod()} $B$O(B @samp{primdec_mod}
4268: $B$GDj5A$5$l$F$$$k(B. @code{[Yokoyama]} $B$NAG%$%G%"%kJ,2r%"%k%4%j%:%`(B
4269: $B$r<BAu$7$F$$$k(B.
4270: @item
4271: @code{primedec_mod()} $B$OM-8BBN>e$G$N%$%G%"%k$N(B
4272: $B:,4p$NAG%$%G%"%kJ,2r$r9T$$(B, $BAG%$%G%"%k$N%j%9%H$rJV$9(B.
4273: @item
4274: @code{primedec_mod()} $B$O(B, GF(@var{mod}) $B>e$G$NJ,2r$rM?$($k(B.
4275: $B7k2L$N3F@.J,$N@8@.85$O(B, $B@0?t78?tB?9`<0$G$"$k(B.
4276: @item
4277: $B7k2L$K$*$$$F(B, $BB?9`<0%j%9%H$H$7$FI=<($5$l$F$$$k3F%$%G%"%k$OA4$F(B
4278: [@var{vlist},@var{ord}] $B$G;XDj$5$l$k9`=g=x$K4X$9$k%0%l%V%J4pDl$G$"$k(B.
4279: @item
4280: @var{strategy} $B$,(B 0 $B$G$J$$$H$-(B, incremental $B$K(B component $B$N6&DL(B
4281: $BItJ,$r7W;;$9$k$3$H$K$h$k(B early termination $B$r9T$&(B. $B0lHL$K(B,
4282: $B%$%G%"%k$N<!85$,9b$$>l9g$KM-8z$@$,(B, 0 $B<!85$N>l9g$J$I(B, $B<!85$,>.$5$$(B
4283: $B>l9g$K$O(B overhead $B$,Bg$-$$>l9g$,$"$k(B.
1.7 noro 4284: @item
4285: $B7W;;ESCf$GFbIt>pJs$r8+$?$$>l9g$K$O!"(B
4286: $BA0$b$C$F(B @code{dp_gr_print(2)} $B$r<B9T$7$F$*$1$P$h$$(B.
1.5 noro 4287: \E
4288: \BEG
4289: @item
4290: Function @code{primedec_mod()}
4291: is defined in @samp{primdec_mod} and implements the prime decomposition
4292: algorithm in @code{[Yokoyama]}.
4293: @item
4294: @code{primedec_mod()}
4295: is the function for prime ideal decomposition
4296: of the radical of a polynomial ideal over small finite field,
4297: and they return a list of prime ideals, which are associated primes
4298: of the input ideal.
4299: @item
4300: @code{primedec_mod()} gives the decomposition over GF(@var{mod}).
4301: The generators of each resulting component consists of integral polynomials.
4302: @item
4303: Each resulting component is a Groebner basis with respect to
4304: a term order specified by [@var{vlist},@var{ord}].
4305: @item
4306: If @var{strategy} is non zero, then the early termination strategy
4307: is tried by computing the intersection of obtained components
4308: incrementally. In general, this strategy is useful when the krull
4309: dimension of the ideal is high, but it may add some overhead
4310: if the dimension is small.
1.7 noro 4311: @item
4312: If you want to see internal information during the computation,
4313: execute @code{dp_gr_print(2)} in advance.
1.5 noro 4314: \E
4315: @end itemize
4316:
4317: @example
4318: [0] load("primdec_mod")$
4319: [246] PP444=[x^8+x^2+t,y^8+y^2+t,z^8+z^2+t]$
4320: [247] primedec_mod(PP444,[x,y,z,t],0,2,1);
4321: [[y+z,x+z,z^8+z^2+t],[x+y,y^2+y+z^2+z+1,z^8+z^2+t],
4322: [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],
4323: [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],
4324: [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],
4325: [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]]
4326: [248]
4327: @end example
4328:
4329: @table @t
4330: \JP @item $B;2>H(B
4331: \EG @item References
4332: @fref{modfctr},
1.6 noro 4333: @fref{dp_gr_main dp_gr_mod_main dp_gr_f_main dp_weyl_gr_main dp_weyl_gr_mod_main dp_weyl_gr_f_main},
1.5 noro 4334: \JP @fref{$B9`=g=x$N@_Dj(B}.
1.7 noro 4335: \EG @fref{Setting term orderings},
4336: @fref{dp_gr_flags dp_gr_print}.
1.5 noro 4337: @end table
4338:
1.10 noro 4339: \JP @node bfunction bfct generic_bfct ann ann0,,, $B%0%l%V%J4pDl$K4X$9$kH!?t(B
4340: \EG @node bfunction bfct generic_bfct ann ann0,,, Functions for Groebner basis computation
4341: @subsection @code{bfunction}, @code{bfct}, @code{generic_bfct}, @code{ann}, @code{ann0}
1.6 noro 4342: @findex bfunction
1.9 noro 4343: @findex bfct
1.6 noro 4344: @findex generic_bfct
1.10 noro 4345: @findex ann
4346: @findex ann0
1.5 noro 4347:
1.6 noro 4348: @table @t
4349: @item bfunction(@var{f})
1.10 noro 4350: @itemx bfct(@var{f})
4351: @itemx generic_bfct(@var{plist},@var{vlist},@var{dvlist},@var{weight})
4352: \JP :: @var{b} $B4X?t$N7W;;(B
4353: \EG :: Computes the global @var{b} function of a polynomial or an ideal
4354: @item ann(@var{f})
4355: @itemx ann0(@var{f})
4356: \JP :: $BB?9`<0$N%Y%-$N(B annihilator $B$N7W;;(B
4357: \EG :: Computes the annihilator of a power of polynomial
1.6 noro 4358: @end table
1.10 noro 4359:
1.6 noro 4360: @table @var
4361: @item return
1.10 noro 4362: \JP $BB?9`<0$^$?$O%j%9%H(B
4363: \EG polynomial or list
4364: @item f
1.6 noro 4365: \JP $BB?9`<0(B
4366: \EG polynomial
4367: @item plist
4368: \JP $BB?9`<0%j%9%H(B
4369: \EG list of polynomials
4370: @item vlist dvlist
4371: \JP $BJQ?t%j%9%H(B
4372: \EG list of variables
4373: @end table
1.5 noro 4374:
1.6 noro 4375: @itemize @bullet
4376: \BJP
4377: @item @samp{bfct} $B$GDj5A$5$l$F$$$k(B.
1.10 noro 4378: @item @code{bfunction(@var{f})}, @code{bfct(@var{f})} $B$OB?9`<0(B @var{f} $B$N(B global @var{b} $B4X?t(B @code{b(s)} $B$r(B
1.6 noro 4379: $B7W;;$9$k(B. @code{b(s)} $B$O(B, Weyl $BBe?t(B @code{D} $B>e$N0lJQ?tB?9`<04D(B @code{D[s]}
4380: $B$N85(B @code{P(x,s)} $B$,B8:_$7$F(B, @code{P(x,s)f^(s+1)=b(s)f^s} $B$rK~$?$9$h$&$J(B
4381: $BB?9`<0(B @code{b(s)} $B$NCf$G(B, $B<!?t$,:G$bDc$$$b$N$G$"$k(B.
4382: @item @code{generic_bfct(@var{f},@var{vlist},@var{dvlist},@var{weight})}
4383: $B$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 4384: $B%&%'%$%H(B @var{weight} $B$K4X$9$k(B global @var{b} $B4X?t$r7W;;$9$k(B.
1.6 noro 4385: @var{vlist} $B$O(B @code{x}-$BJQ?t(B, @var{vlist} $B$OBP1~$9$k(B @code{D}-$BJQ?t(B
4386: $B$r=g$KJB$Y$k(B.
1.9 noro 4387: @item @code{bfunction} $B$H(B @code{bfct} $B$G$OMQ$$$F$$$k%"%k%4%j%:%`$,(B
1.11 noro 4388: $B0[$J$k(B. $B$I$A$i$,9bB.$+$OF~NO$K$h$k(B.
1.10 noro 4389: @item @code{ann(@var{f})} $B$O(B, @code{@var{f}^s} $B$N(B annihilator ideal
4390: $B$N@8@.7O$rJV$9(B. @code{ann(@var{f})} $B$O(B, @code{[@var{a},@var{list}]}
4391: $B$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,
4392: @var{list} $B$O(B @code{ann(@var{f})} $B$N7k2L$N(B @code{s}$ $B$K(B, @var{a} $B$r(B
4393: $BBeF~$7$?$b$N$G$"$k(B.
1.7 noro 4394: @item $B>\:Y$K$D$$$F$O(B, [Saito,Sturmfels,Takayama] $B$r8+$h(B.
1.6 noro 4395: \E
4396: \BEG
4397: @item These functions are defined in @samp{bfct}.
1.10 noro 4398: @item @code{bfunction(@var{f})} and @code{bfct(@var{f})} compute the global @var{b}-function @code{b(s)} of
1.6 noro 4399: a polynomial @var{f}.
4400: @code{b(s)} is a polynomial of the minimal degree
4401: such that there exists @code{P(x,s)} in D[s], which is a polynomial
4402: ring over Weyl algebra @code{D}, and @code{P(x,s)f^(s+1)=b(s)f^s} holds.
4403: @item @code{generic_bfct(@var{f},@var{vlist},@var{dvlist},@var{weight})}
1.10 noro 4404: computes the global @var{b}-function of a left ideal @code{I} in @code{D}
1.6 noro 4405: generated by @var{plist}, with respect to @var{weight}.
4406: @var{vlist} is the list of @code{x}-variables,
4407: @var{vlist} is the list of corresponding @code{D}-variables.
1.9 noro 4408: @item @code{bfunction(@var{f})} and @code{bfct(@var{f})} implement
4409: different algorithms and the efficiency depends on inputs.
1.10 noro 4410: @item @code{ann(@var{f})} returns the generator set of the annihilator
4411: ideal of @code{@var{f}^s}.
4412: @code{ann(@var{f})} returns a list @code{[@var{a},@var{list}]},
4413: where @var{a} is the minimal integral root of the global @var{b}-function
4414: of @var{f}, and @var{list} is a list of polynomials obtained by
4415: substituting @code{s} in @code{ann(@var{f})} with @var{a}.
1.7 noro 4416: @item See [Saito,Sturmfels,Takayama] for the details.
1.6 noro 4417: \E
4418: @end itemize
4419:
4420: @example
4421: [0] load("bfct")$
4422: [216] bfunction(x^3+y^3+z^3+x^2*y^2*z^2+x*y*z);
4423: -9*s^5-63*s^4-173*s^3-233*s^2-154*s-40
4424: [217] fctr(@@);
4425: [[-1,1],[s+2,1],[3*s+4,1],[3*s+5,1],[s+1,2]]
4426: [218] F = [4*x^3*dt+y*z*dt+dx,x*z*dt+4*y^3*dt+dy,
4427: x*y*dt+5*z^4*dt+dz,-x^4-z*y*x-y^4-z^5+t]$
4428: [219] generic_bfct(F,[t,z,y,x],[dt,dz,dy,dx],[1,0,0,0]);
4429: 20000*s^10-70000*s^9+101750*s^8-79375*s^7+35768*s^6-9277*s^5
4430: +1278*s^4-72*s^3
1.10 noro 4431: [220] P=x^3-y^2$
4432: [221] ann(P);
4433: [2*dy*x+3*dx*y^2,-3*dx*x-2*dy*y+6*s]
4434: [222] ann0(P);
4435: [-1,[2*dy*x+3*dx*y^2,-3*dx*x-2*dy*y-6]]
1.6 noro 4436: @end example
4437:
4438: @table @t
4439: \JP @item $B;2>H(B
4440: \EG @item References
4441: \JP @fref{Weyl $BBe?t(B}.
4442: \EG @fref{Weyl algebra}.
4443: @end table
1.5 noro 4444:
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