Annotation of OpenXM/src/asir-doc/parts/groebner.texi, Revision 1.14
1.14 ! noro 1: @comment $OpenXM: OpenXM/src/asir-doc/parts/groebner.texi,v 1.13 2004/09/13 09:23:30 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$$$&7A$GI=<($5$l(B, $B$^$?(B, $B$3$N7A$GF~NO2DG=$G$"$k(B.
201: \EG and also can be input in such a form.
1.1 noro 202:
1.2 noro 203: \BJP
1.1 noro 204: @itemx $BF,C19`<0(B (head monomial)
205: @item $BF,9`(B (head term)
206: @itemx $BF,78?t(B (head coefficient)
207: $BJ,;6I=8=B?9`<0$K$*$1$k3FC19`<0$O(B, $B9`=g=x$K$h$j@0Ns$5$l$k(B. $B$3$N;~=g(B
208: $B=x:GBg$NC19`<0$rF,C19`<0(B, $B$=$l$K8=$l$k9`(B, $B78?t$r$=$l$>$lF,9`(B, $BF,78?t(B
209: $B$H8F$V(B.
1.2 noro 210: \E
211: \BEG
212: @itemx head monomial
213: @item head term
214: @itemx head coefficient
215:
216: Monomials in a distributed polynomial is sorted by a total order.
217: In such representation, we call the monomial that is maximum
218: with respect to the order the head monomial, and its term and coefficient
219: the head term and the head coefficient respectively.
220: \E
1.1 noro 221: @end table
222:
1.2 noro 223: \BJP
1.1 noro 224: @node $B%U%!%$%k$NFI$_9~$_(B,,, $B%0%l%V%J4pDl$N7W;;(B
225: @section $B%U%!%$%k$NFI$_9~$_(B
1.2 noro 226: \E
227: \BEG
228: @node Reading files,,, Groebner basis computation
229: @section Reading files
230: \E
1.1 noro 231:
232: @noindent
1.2 noro 233: \BJP
1.1 noro 234: $B%0%l%V%J4pDl$r7W;;$9$k$?$a$N4pK\E*$JH!?t$O(B @code{dp_gr_main()} $B$*$h$S(B
1.5 noro 235: @code{dp_gr_mod_main()}, @code{dp_gr_f_main()}
236: $B$J$k(B 3 $B$D$NAH$_9~$_H!?t$G$"$k$,(B, $BDL>o$O(B, $B%Q%i%a%?(B
1.1 noro 237: $B@_Dj$J$I$r9T$C$?$N$A$3$l$i$r8F$S=P$9%f!<%6H!?t$rMQ$$$k$N$,JXMx$G$"$k(B.
238: $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
239: $B$_9~$`$3$H$K$h$j;HMQ2DG=$H$J$k(B. @samp{gr} $B$O(B, @b{Asir} $B$NI8=`(B
1.5 noro 240: $B%i%$%V%i%j%G%#%l%/%H%j$KCV$+$l$F$$$k(B.
1.2 noro 241: \E
242: \BEG
1.5 noro 243: Facilities for computing Groebner bases are
244: @code{dp_gr_main()}, @code{dp_gr_mod_main()}and @code{dp_gr_f_main()}.
245: To call these functions,
246: it is necessary to set several parameters correctly and it is convenient
247: to use a set of interface functions provided in the library file
248: @samp{gr}.
1.2 noro 249: The facilities will be ready to use after you load the package by
250: @code{load()}. The package @samp{gr} is placed in the standard library
1.5 noro 251: directory of @b{Asir}.
1.2 noro 252: \E
1.1 noro 253:
254: @example
255: [0] load("gr")$
256: @end example
257:
1.2 noro 258: \BJP
1.1 noro 259: @node $B4pK\E*$JH!?t(B,,, $B%0%l%V%J4pDl$N7W;;(B
260: @section $B4pK\E*$JH!?t(B
1.2 noro 261: \E
262: \BEG
263: @node Fundamental functions,,, Groebner basis computation
264: @section Fundamental functions
265: \E
1.1 noro 266:
267: @noindent
1.2 noro 268: \BJP
1.1 noro 269: @samp{gr} $B$G$O?tB?$/$NH!?t$,Dj5A$5$l$F$$$k$,(B, $BD>@\(B
270: $B%0%l%V%J4pDl$r7W;;$9$k$?$a$N%H%C%W%l%Y%k$O<!$N(B 3 $B$D$G$"$k(B.
271: $B0J2<$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,
272: @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 273: \E
274: \BEG
275: There are many functions and options defined in the package @samp{gr}.
276: Usually not so many of them are used. Top level functions for Groebner
277: basis computation are the following three functions.
278:
279: In the following description, @var{plist}, @var{vlist}, @var{order}
280: and @var{p} stand for a list of polynomials, a list of variables
281: (indeterminates), a type of term ordering and a prime less than
282: @code{2^27} respectively.
283: \E
1.1 noro 284:
285: @table @code
286: @item gr(@var{plist},@var{vlist},@var{order})
287:
1.2 noro 288: \BJP
1.1 noro 289: Gebauer-Moeller $B$K$h$k(B useless pair elimination criteria, sugar
290: strategy $B$*$h$S(B Traverso $B$K$h$k(B trace-lifting $B$rMQ$$$?(B Buchberger $B%"%k(B
291: $B%4%j%:%`$K$h$kM-M}?t78?t%0%l%V%J4pDl7W;;H!?t(B. $B0lHL$K$O$3$NH!?t$rMQ$$$k(B.
1.2 noro 292: \E
293: \BEG
294: Function that computes Groebner bases over the rationals. The
295: algorithm is Buchberger algorithm with useless pair elimination
296: criteria by Gebauer-Moeller, sugar strategy and trace-lifting by
297: Traverso. For ordinary computation, this function is used.
298: \E
1.1 noro 299:
300: @item hgr(@var{plist},@var{vlist},@var{order})
301:
1.2 noro 302: \BJP
1.1 noro 303: $BF~NOB?9`<0$r@F<!2=$7$?8e(B @code{gr()} $B$N%0%l%V%J4pDl8uJd@8@.It$K$h$j8u(B
304: $BJd@8@.$7(B, $BHs@F<!2=(B, interreduce $B$7$?$b$N$r(B @code{gr()} $B$N%0%l%V%J4pDl(B
305: $B%A%'%C%/It$G%A%'%C%/$9$k(B. 0 $B<!85%7%9%F%`(B ($B2r$N8D?t$,M-8B8D$NJ}Dx<07O(B)
306: $B$N>l9g(B, sugar strategy $B$,78?tKDD%$r0z$-5/$3$9>l9g$,$"$k(B. $B$3$N$h$&$J>l(B
307: $B9g(B, strategy $B$r@F<!2=$K$h$k(B strategy $B$KCV$-49$($k$3$H$K$h$j78?tKDD%$r(B
308: $BM^@)$9$k$3$H$,$G$-$k>l9g$,B?$$(B.
1.2 noro 309: \E
310: \BEG
311: After homogenizing the input polynomials a candidate of the \gr basis
312: is computed by trace-lifting. Then the candidate is dehomogenized and
313: checked whether it is indeed a Groebner basis of the input. Sugar
314: strategy often causes intermediate coefficient swells. It is
315: empirically known that the combination of homogenization and supresses
316: the swells for such cases.
317: \E
1.1 noro 318:
319: @item gr_mod(@var{plist},@var{vlist},@var{order},@var{p})
320:
1.2 noro 321: \BJP
1.1 noro 322: Gebauer-Moeller $B$K$h$k(B useless pair elimination criteria, sugar
323: strategy $B$*$h$S(B Buchberger $B%"%k%4%j%:%`$K$h$k(B GF(p) $B78?t%0%l%V%J4pDl7W(B
324: $B;;H!?t(B.
1.2 noro 325: \E
326: \BEG
327: Function that computes Groebner bases over GF(@var{p}). The same
328: algorithm as @code{gr()} is used.
329: \E
1.1 noro 330:
331: @end table
332:
1.2 noro 333: \BJP
1.1 noro 334: @node $B7W;;$*$h$SI=<($N@)8f(B,,, $B%0%l%V%J4pDl$N7W;;(B
335: @section $B7W;;$*$h$SI=<($N@)8f(B
1.2 noro 336: \E
337: \BEG
338: @node Controlling Groebner basis computations,,, Groebner basis computation
339: @section Controlling Groebner basis computations
340: \E
1.1 noro 341:
342: @noindent
1.2 noro 343: \BJP
1.1 noro 344: $B%0%l%V%J4pDl$N7W;;$K$*$$$F(B, $B$5$^$6$^$J%Q%i%a%?@_Dj$r9T$&$3$H$K$h$j7W;;(B,
345: $BI=<($r@)8f$9$k$3$H$,$G$-$k(B. $B$3$l$i$O(B, $BAH$_9~$_H!?t(B @code{dp_gr_flags()}
346: $B$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
347: $B$H(B, $B8=:_@_Dj$5$l$F$$$k%Q%i%a%?$,(B, $BL>A0$HCM$N%j%9%H$GJV$5$l$k(B.
1.2 noro 348: \E
349: \BEG
350: One can cotrol a Groebner basis computation by setting various parameters.
351: These parameters can be set and examined by a built-in function
352: @code{dp_gr_flags()}. Without argument it returns the current settings.
353: \E
1.1 noro 354:
355: @example
356: [100] dp_gr_flags();
1.5 noro 357: [Demand,0,NoSugar,0,NoCriB,0,NoGC,0,NoMC,0,NoRA,0,NoGCD,0,Top,0,
358: ShowMag,1,Print,1,Stat,0,Reverse,0,InterReduce,0,Multiple,0]
1.1 noro 359: [101]
360: @end example
361:
1.2 noro 362: \BJP
1.1 noro 363: $B0J2<$G(B, $B3F%Q%i%a%?$N0UL#$r@bL@$9$k(B. on $B$N>l9g$H$O(B, $B%Q%i%a%?$,(B 0 $B$G$J$$>l9g$r(B
364: $B$$$&(B. $B$3$l$i$N%Q%i%a%?$N=i4|CM$OA4$F(B 0 (off) $B$G$"$k(B.
1.2 noro 365: \E
366: \BEG
367: The return value is a list which contains the names of parameters and their
368: values. The meaning of the parameters are as follows. `on' means that the
369: parameter is not zero.
370: \E
1.1 noro 371:
372: @table @code
373: @item NoSugar
1.2 noro 374: \BJP
1.1 noro 375: on $B$N>l9g(B, sugar strategy $B$NBe$o$j$K(B Buchberger$B$N(B normal strategy $B$,MQ(B
376: $B$$$i$l$k(B.
1.2 noro 377: \E
378: \BEG
379: If `on', Buchberger's normal strategy is used instead of sugar strategy.
380: \E
1.1 noro 381:
382: @item NoCriB
1.2 noro 383: \JP on $B$N>l9g(B, $BITI,MWBP8!=P5,=`$N$&$A(B, $B5,=`(B B $B$rE,MQ$7$J$$(B.
384: \EG If `on', criterion B among the Gebauer-Moeller's criteria is not applied.
1.1 noro 385:
386: @item NoGC
1.2 noro 387: \JP on $B$N>l9g(B, $B7k2L$,%0%l%V%J4pDl$K$J$C$F$$$k$+$I$&$+$N%A%'%C%/$r9T$o$J$$(B.
388: \BEG
389: If `on', the check that a Groebner basis candidate is indeed a Groebner basis,
390: is not executed.
391: \E
1.1 noro 392:
393: @item NoMC
1.2 noro 394: \BJP
1.1 noro 395: on $B$N>l9g(B, $B7k2L$,F~NO%$%G%"%k$HF1Ey$N%$%G%"%k$G$"$k$+$I$&$+$N%A%'%C%/(B
396: $B$r9T$o$J$$(B.
1.2 noro 397: \E
398: \BEG
399: If `on', the check that the resulting polynomials generates the same ideal as
400: the ideal generated by the input, is not executed.
401: \E
1.1 noro 402:
403: @item NoRA
1.2 noro 404: \BJP
1.1 noro 405: on $B$N>l9g(B, $B7k2L$r(B reduced $B%0%l%V%J4pDl$K$9$k$?$a$N(B
406: interreduce $B$r9T$o$J$$(B.
1.2 noro 407: \E
408: \BEG
409: If `on', the interreduction, which makes the Groebner basis reduced, is not
410: executed.
411: \E
1.1 noro 412:
413: @item NoGCD
1.2 noro 414: \BJP
1.1 noro 415: on $B$N>l9g(B, $BM-M}<078?t$N%0%l%V%J4pDl7W;;$K$*$$$F(B, $B@8@.$5$l$?B?9`<0$N(B,
416: $B78?t$N(B content $B$r$H$i$J$$(B.
1.2 noro 417: \E
418: \BEG
419: If `on', content removals are not executed during a Groebner basis computation
420: over a rational function field.
421: \E
1.1 noro 422:
423: @item Top
1.2 noro 424: \JP on $B$N>l9g(B, normal form $B7W;;$K$*$$$FF,9`>C5n$N$_$r9T$&(B.
425: \EG If `on', Only the head term of the polynomial being reduced is reduced.
1.1 noro 426:
1.2 noro 427: @comment @item Interreduce
428: @comment \BJP
429: @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
430: @comment $B$h$k(B normal form $B$GCV$-49$($k(B.
431: @comment \E
432: @comment \BEG
433: @comment If `on', intermediate basis elements are reduced by using a newly generated
434: @comment basis element.
435: @comment \E
1.1 noro 436:
437: @item Reverse
1.2 noro 438: \BJP
1.1 noro 439: 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
440: $B@h$7$FA*$V(B.
1.2 noro 441: \E
442: \BEG
443: If `on', the selection strategy of reducer in a normal form computation
444: is such that a newer reducer is used first.
445: \E
1.1 noro 446:
447: @item Print
1.2 noro 448: \JP on $B$N>l9g(B, $B%0%l%V%J4pDl7W;;$NESCf$K$*$1$k$5$^$6$^$J>pJs$rI=<($9$k(B.
449: \BEG
450: If `on', various informations during a Groebner basis computation is
451: displayed.
452: \E
1.1 noro 453:
1.7 noro 454: @item PrintShort
455: \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.
456: \BEG
457: If `on' and Print is `off', short information during a Groebner basis computation is
458: displayed.
459: \E
460:
1.1 noro 461: @item Stat
1.2 noro 462: \BJP
1.1 noro 463: 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
464: $B$l$k%G!<%?$NFb(B, $B=87W%G!<%?$N$_$,I=<($5$l$k(B.
1.2 noro 465: \E
466: \BEG
467: If `on', a summary of informations is shown after a Groebner basis
468: computation. Note that the summary is always shown if @code{Print} is `on'.
469: \E
1.1 noro 470:
471: @item ShowMag
1.2 noro 472: \BJP
1.1 noro 473: 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
474: $B78?t$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 475: \E
476: \BEG
477: If `on' and @code{Print} is `on', the sum of bit length of
478: coefficients of a generated basis element, which we call @var{magnitude},
479: is shown after every normal computation. After comleting the
480: computation the maximal value among the sums is shown.
481: \E
1.1 noro 482:
1.7 noro 483: @item Content
484: @itemx Multiple
1.2 noro 485: \BJP
1.7 noro 486: 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
487: @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
488: $B3d$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
489: GCD $B7W;;$,9T$o$l0lHL$K$O8zN($,0-$/$J$k$,(B, @code{Content} $B$r(B 2 $BDxEY(B
1.1 noro 490: $B$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 491: backward compatibility $B$N$?$a!"(B@code{Multiple} $B$G@0?tCM$r;XDj$G$-$k(B.
1.2 noro 492: \E
493: \BEG
1.7 noro 494: If a non-zero rational number, in a normal form computation
1.2 noro 495: over the rationals, the integer content of the polynomial being
1.7 noro 496: reduced is removed when its magnitude becomes @code{Content} times
1.2 noro 497: larger than a registered value, which is set to the magnitude of the
498: input polynomial. After each content removal the registered value is
1.7 noro 499: set to the magnitude of the resulting polynomial. @code{Content} is
1.2 noro 500: equal to 1, the simiplification is done after every normal form computation.
1.7 noro 501: It is empirically known that it is often efficient to set @code{Content} to 2
1.2 noro 502: for the case where large integers appear during the computation.
1.7 noro 503: An integer value can be set by the keyword @code{Multiple} for
504: backward compatibility.
1.2 noro 505: \E
1.1 noro 506:
507: @item Demand
1.2 noro 508:
509: \BJP
1.1 noro 510: $B@5Ev$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
511: $BCf$K$*$+$l$:(B, $B$=$N%G%#%l%/%H%jCf$K%P%$%J%j%G!<%?$H$7$FCV$+$l(B, $B$=$NB?9`(B
512: $B<0$rMQ$$$k(B normal form $B7W;;$N:](B, $B<+F0E*$K%a%b%jCf$K%m!<%I$5$l$k(B. $B3FB?(B
513: $B9`<0$O(B, $BFbIt$G$N%$%s%G%C%/%9$r%U%!%$%kL>$K;}$D%U%!%$%k$K3JG<$5$l$k(B.
514: $B$3$3$G;XDj$5$l$?%G%#%l%/%H%j$K=q$+$l$?%U%!%$%k$O<+F0E*$K$O>C5n$5$l$J$$(B
515: $B$?$a(B, $B%f!<%6$,@UG$$r;}$C$F>C5n$9$kI,MW$,$"$k(B.
1.2 noro 516: \E
517: \BEG
518: If the value (a character string) is a valid directory name, then
519: generated basis elements are put in the directory and are loaded on
520: demand during normal form computations. Each elements is saved in the
521: binary form and its name coincides with the index internally used in
522: the computation. These binary files are not removed automatically
523: and one should remove them by hand.
524: \E
1.1 noro 525: @end table
526:
527: @noindent
1.2 noro 528: \JP @code{Print} $B$,(B 0 $B$G$J$$>l9g<!$N$h$&$J%G!<%?$,I=<($5$l$k(B.
529: \EG If @code{Print} is `on', the following informations are shown.
1.1 noro 530:
531: @example
532: [93] gr(cyclic(4),[c0,c1,c2,c3],0)$
533: mod= 99999989, eval = []
534: (0)(0)<<0,2,0,0>>(2,3),nb=2,nab=5,rp=2,sugar=2,mag=4
535: (0)(0)<<0,1,2,0>>(1,2),nb=3,nab=6,rp=2,sugar=3,mag=4
536: (0)(0)<<0,1,1,2>>(0,1),nb=4,nab=7,rp=3,sugar=4,mag=6
537: .
538: (0)(0)<<0,0,3,2>>(5,6),nb=5,nab=8,rp=2,sugar=5,mag=4
539: (0)(0)<<0,1,0,4>>(4,6),nb=6,nab=9,rp=3,sugar=5,mag=4
540: (0)(0)<<0,0,2,4>>(6,8),nb=7,nab=10,rp=4,sugar=6,mag=6
541: ....gb done
542: reduceall
543: .......
544: membercheck
545: (0,0)(0,0)(0,0)(0,0)
546: gbcheck total 8 pairs
547: ........
1.5 noro 548: UP=(0,0)SP=(0,0)SPM=(0,0)NF=(0,0)NFM=(0.010002,0)ZNFM=(0.010002,0)
549: 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
550: D=12 ZR=5 NZR=6 Max_mag=6
1.1 noro 551: [94]
552: @end example
553:
554: @noindent
1.2 noro 555: \BJP
1.1 noro 556: $B:G=i$KI=<($5$l$k(B @code{mod}, @code{eval} $B$O(B, trace-lifting $B$GMQ$$$i$l$kK!(B
557: $B$G$"$k(B. @code{mod} $B$OAG?t(B, @code{eval} $B$OM-M}<078?t$N>l9g$KMQ$$$i$l$k(B
558: $B?t$N%j%9%H$G$"$k(B.
1.2 noro 559: \E
560: \BEG
561: In this example @code{mod} and @code{eval} indicate moduli used in
562: trace-lifting. @code{mod} is a prime and @code{eval} is a list of integers
563: used for evaluation when the ground field is a field of rational functions.
564: \E
1.1 noro 565:
566: @noindent
1.2 noro 567: \JP $B7W;;ESCf$GB?9`<0$,@8@.$5$l$kKh$K<!$N7A$N%G!<%?$,I=<($5$l$k(B.
568: \EG The following information is shown after every normal form computation.
1.1 noro 569:
570: @example
571: (TNF)(TCONT)HT(INDEX),nb=NB,nab=NAB,rp=RP,sugar=S,mag=M
572: @end example
573:
574: @noindent
1.2 noro 575: \JP $B$=$l$i$N0UL#$O<!$NDL$j(B.
576: \EG Meaning of each component is as follows.
1.1 noro 577:
578: @table @code
1.2 noro 579: \BJP
1.1 noro 580: @item TNF
1.2 noro 581:
1.1 noro 582: normal form $B7W;;;~4V(B ($BIC(B)
583:
584: @item TCONT
1.2 noro 585:
1.1 noro 586: content $B7W;;;~4V(B ($BIC(B)
587:
588: @item HT
1.2 noro 589:
1.1 noro 590: $B@8@.$5$l$?B?9`<0$NF,9`(B
591:
592: @item INDEX
1.2 noro 593:
1.1 noro 594: S-$BB?9`<0$r9=@.$9$kB?9`<0$N%$%s%G%C%/%9$N%Z%"(B
595:
596: @item NB
1.2 noro 597:
1.1 noro 598: $B8=:_$N(B, $B>iD9@-$r=|$$$?4pDl$N?t(B
599:
600: @item NAB
1.2 noro 601:
1.1 noro 602: $B8=:_$^$G$K@8@.$5$l$?4pDl$N?t(B
603:
604: @item RP
1.2 noro 605:
1.1 noro 606: $B;D$j$N%Z%"$N?t(B
607:
608: @item S
1.2 noro 609:
1.1 noro 610: $B@8@.$5$l$?B?9`<0$N(B sugar $B$NCM(B
611:
612: @item M
1.2 noro 613:
1.1 noro 614: $B@8@.$5$l$?B?9`<0$N78?t$N%S%C%HD9$NOB(B (@code{ShowMag} $B$,(B on $B$N;~$KI=<($5$l$k(B. )
1.2 noro 615: \E
616: \BEG
617: @item TNF
618:
619: CPU time for normal form computation (second)
620:
621: @item TCONT
622:
623: CPU time for content removal(second)
624:
625: @item HT
626:
627: Head term of the generated basis element
628:
629: @item INDEX
630:
631: Pair of indices which corresponds to the reduced S-polynomial
632:
633: @item NB
634:
635: Number of basis elements after removing redundancy
636:
637: @item NAB
638:
639: Number of all the basis elements
640:
641: @item RP
642:
643: Number of remaining pairs
644:
645: @item S
646:
647: Sugar of the generated basis element
648:
649: @item M
650:
651: Magnitude of the genrated basis element (shown if @code{ShowMag} is `on'.)
652: \E
1.1 noro 653: @end table
654:
655: @noindent
1.2 noro 656: \BJP
1.1 noro 657: $B:G8e$K(B, $B=87W%G!<%?$,I=<($5$l$k(B. $B0UL#$O<!$NDL$j(B.
658: ($B;~4V$NI=<($K$*$$$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 659: \E
660: \BEG
661: The summary of the informations shown after a Groebner basis
662: computation is as follows. If a component shows timings and it
663: contains two numbers, they are a pair of time for computation and time
664: for garbage colection.
665: \E
1.1 noro 666:
667: @table @code
1.2 noro 668: \BJP
1.1 noro 669: @item UP
1.2 noro 670:
1.1 noro 671: $B%Z%"$N%j%9%H$NA`:n$K$+$+$C$?;~4V(B
672:
673: @item SP
1.2 noro 674:
1.1 noro 675: $BM-M}?t>e$N(B S-$BB?9`<07W;;;~4V(B
676:
677: @item SPM
1.2 noro 678:
1.1 noro 679: $BM-8BBN>e$N(B S-$BB?9`<07W;;;~4V(B
680:
681: @item NF
1.2 noro 682:
1.1 noro 683: $BM-M}?t>e$N(B normal form $B7W;;;~4V(B
684:
685: @item NFM
1.2 noro 686:
1.1 noro 687: $BM-8BBN>e$N(B normal form $B7W;;;~4V(B
688:
689: @item ZNFM
1.2 noro 690:
1.1 noro 691: @code{NFM} $B$NFb(B, 0 $B$X$N(B reduction $B$K$+$+$C$?;~4V(B
692:
693: @item PZ
1.2 noro 694:
1.1 noro 695: content $B7W;;;~4V(B
696:
697: @item NP
1.2 noro 698:
1.1 noro 699: $BM-M}?t78?tB?9`<0$N78?t$KBP$9$k>jM>1i;;$N7W;;;~4V(B
700:
701: @item MP
1.2 noro 702:
1.1 noro 703: S-$BB?9`<0$r@8@.$9$k%Z%"$NA*Br$K$+$+$C$?;~4V(B
704:
705: @item RA
1.2 noro 706:
1.1 noro 707: interreduce $B7W;;;~4V(B
708:
709: @item MC
1.2 noro 710:
1.1 noro 711: trace-lifting $B$K$*$1$k(B, $BF~NOB?9`<0$N%a%s%P%7%C%W7W;;;~4V(B
712:
713: @item GC
1.2 noro 714:
1.1 noro 715: $B7k2L$N%0%l%V%J4pDl8uJd$N%0%l%V%J4pDl%A%'%C%/;~4V(B
716:
717: @item T
1.2 noro 718:
1.1 noro 719: $B@8@.$5$l$?%Z%"$N?t(B
720:
721: @item B, M, F, D
1.2 noro 722:
1.1 noro 723: $B3F(B criterion $B$K$h$j=|$+$l$?%Z%"$N?t(B
724:
725: @item ZR
1.2 noro 726:
1.1 noro 727: 0 $B$K(B reduce $B$5$l$?%Z%"$N?t(B
728:
729: @item NZR
1.2 noro 730:
1.1 noro 731: 0 $B$G$J$$B?9`<0$K(B reduce $B$5$l$?%Z%"$N?t(B
732:
733: @item Max_mag
1.2 noro 734:
1.1 noro 735: $B@8@.$5$l$?B?9`<0$N(B, $B78?t$N%S%C%HD9$NOB$N:GBgCM(B
1.2 noro 736: \E
737: \BEG
738: @item UP
739:
740: Time to manipulate the list of critical pairs
741:
742: @item SP
743:
744: Time to compute S-polynomials over the rationals
745:
746: @item SPM
747:
748: Time to compute S-polynomials over a finite field
749:
750: @item NF
751:
752: Time to compute normal forms over the rationals
753:
754: @item NFM
755:
756: Time to compute normal forms over a finite field
757:
758: @item ZNFM
759:
760: Time for zero reductions in @code{NFM}
761:
762: @item PZ
763:
764: Time to remove integer contets
765:
766: @item NP
767:
768: Time to compute remainders for coefficients of polynomials with coeffieints
769: in the rationals
770:
771: @item MP
772:
773: Time to select pairs from which S-polynomials are computed
774:
775: @item RA
776:
777: Time to interreduce the Groebner basis candidate
778:
779: @item MC
1.1 noro 780:
1.2 noro 781: Time to check that each input polynomial is a member of the ideal
782: generated by the Groebner basis candidate.
783:
784: @item GC
785:
786: Time to check that the Groebner basis candidate is a Groebner basis
787:
788: @item T
789:
790: Number of critical pairs generated
791:
792: @item B, M, F, D
793:
794: Number of critical pairs removed by using each criterion
795:
796: @item ZR
797:
798: Number of S-polynomials reduced to 0
799:
800: @item NZR
801:
802: Number of S-polynomials reduced to non-zero results
803:
804: @item Max_mag
805:
806: Maximal magnitude among all the generated polynomials
807: \E
1.1 noro 808: @end table
809:
1.2 noro 810: \BJP
1.1 noro 811: @node $B9`=g=x$N@_Dj(B,,, $B%0%l%V%J4pDl$N7W;;(B
812: @section $B9`=g=x$N@_Dj(B
1.2 noro 813: \E
814: \BEG
815: @node Setting term orderings,,, Groebner basis computation
816: @section Setting term orderings
817: \E
1.1 noro 818:
819: @noindent
1.2 noro 820: \BJP
1.1 noro 821: $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
822: $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
823: $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
824: $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
825: $B9TNs$GI=8=$5$l$k(B.
1.2 noro 826: \E
827: \BEG
828: A term is internally represented as an integer vector whose components
829: are exponents with respect to variables. A variable list specifies the
830: correspondences between variables and components. A type of term ordering
831: specifies a total order for integer vectors. A type of term ordering is
832: represented by an integer, a list of integer or matrices.
833: \E
1.1 noro 834:
835: @noindent
1.2 noro 836: \JP $B4pK\E*$J9`=g=x7?$H$7$F<!$N(B 3 $B$D$,$"$k(B.
837: \EG There are following three fundamental types.
1.1 noro 838:
839: @table @code
1.2 noro 840: \JP @item 0 (DegRevLex; @b{$BA4<!?t5U<-=q<0=g=x(B})
841: \EG @item 0 (DegRevLex; @b{total degree reverse lexicographic ordering})
1.1 noro 842:
1.2 noro 843: \BJP
1.1 noro 844: $B0lHL$K(B, $B$3$N=g=x$K$h$k%0%l%V%J4pDl7W;;$,:G$b9bB.$G$"$k(B. $B$?$@$7(B,
845: $BJ}Dx<0$r2r$/$H$$$&L\E*$KMQ$$$k$3$H$O(B, $B0lHL$K$O$G$-$J$$(B. $B$3$N(B
846: $B=g=x$K$h$k%0%l%V%J4pDl$O(B, $B2r$N8D?t$N7W;;(B, $B%$%G%"%k$N%a%s%P%7%C%W$d(B,
847: $BB>$NJQ?t=g=x$X$N4pDlJQ49$N$?$a$N%=!<%9$H$7$FMQ$$$i$l$k(B.
1.2 noro 848: \E
849: \BEG
850: In general, computation by this ordering shows the fastest speed
851: in most Groebner basis computations.
852: However, for the purpose to solve polynomial equations, this type
853: of ordering is, in general, not so suitable.
854: The Groebner bases obtained by this ordering is used for computing
855: the number of solutions, solving ideal membership problem and seeds
856: for conversion to other Groebner bases under different ordering.
857: \E
1.1 noro 858:
1.2 noro 859: \JP @item 1 (DegLex; @b{$BA4<!?t<-=q<0=g=x(B})
860: \EG @item 1 (DegLex; @b{total degree lexicographic ordering})
1.1 noro 861:
1.2 noro 862: \BJP
1.1 noro 863: $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,
864: @code{DegRevLex} $B$HF1MMD>@\$=$N7k2L$rMQ$$$k$3$H$O:$Fq$G$"$k(B. $B$7$+$7(B,
865: $B<-=q<0=g=x$N%0%l%V%J4pDl$r5a$a$k:]$K(B, $B@F<!2=8e$K$3$N=g=x$G%0%l%V%J4pDl(B
866: $B$r5a$a$F$$$k(B.
1.2 noro 867: \E
868: \BEG
869: By this type term ordering, Groebner bases are obtained fairly faster
870: than Lex (lexicographic) ordering, too.
871: Alike the @code{DegRevLex} ordering, the result, in general, cannot directly
872: be used for solving polynomial equations.
873: It is used, however, in such a way
874: that a Groebner basis is computed in this ordering after homogenization
875: to obtain the final lexicographic Groebner basis.
876: \E
1.1 noro 877:
1.2 noro 878: \JP @item 2 (Lex; @b{$B<-=q<0=g=x(B})
879: \EG @item 2 (Lex; @b{lexicographic ordering})
1.1 noro 880:
1.2 noro 881: \BJP
1.1 noro 882: $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
883: $B7W;;;~4V$,$+$+$j2a$.$k$N$,FqE@$G$"$k(B. $BFC$K(B, $B2r$,M-8B8D$N>l9g(B, $B7k2L$N(B
884: $B78?t$,6K$a$FD9Bg$JB?G\D9?t$K$J$k>l9g$,B?$$(B. $B$3$N>l9g(B, @code{gr()},
885: @code{hgr()} $B$K$h$k7W;;$,6K$a$FM-8z$K$J$k>l9g$,B?$$(B.
1.2 noro 886: \E
887: \BEG
888: Groebner bases computed by this ordering give the most convenient
889: Groebner bases for solving the polynomial equations.
890: The only and serious shortcoming is the enormously long computation
891: time.
892: It is often observed that the number coefficients of the result becomes
893: very very long integers, especially if the ideal is 0-dimensional.
894: For such a case, it is empirically true for many cases
895: that i.e., computation by
896: @code{gr()} and/or @code{hgr()} may be quite effective.
897: \E
1.1 noro 898: @end table
899:
900: @noindent
1.2 noro 901: \BJP
1.1 noro 902: $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.
903: $B$3$l$O(B,
1.2 noro 904: \E
905: \BEG
906: By combining these fundamental orderingl into a list, one can make
907: various term ordering called elimination orderings.
908: \E
1.1 noro 909:
910: @code{[[O1,L1],[O2,L2],...]}
911:
912: @noindent
1.2 noro 913: \BJP
1.1 noro 914: $B$G;XDj$5$l$k(B. @code{Oi} $B$O(B 0, 1, 2 $B$N$$$:$l$+$G(B, @code{Li} $B$OJQ?t$N8D(B
915: $B?t$rI=$9(B. $B$3$N;XDj$O(B, $BJQ?t$r@hF,$+$i(B @code{L1}, @code{L2} , ...$B8D(B
916: $B$:$D$NAH$KJ,$1(B, $B$=$l$>$l$NJQ?t$K4X$7(B, $B=g$K(B @code{O1}, @code{O2},
917: ...$B$N9`=g=x7?$GBg>.$,7hDj$9$k$^$GHf3S$9$k$3$H$r0UL#$9$k(B. $B$3$N7?$N(B
918: $B=g=x$O0lHL$K>C5n=g=x$H8F$P$l$k(B.
1.2 noro 919: \E
920: \BEG
921: In this example @code{Oi} indicates 0, 1 or 2 and @code{Li} indicates
922: the number of variables subject to the correspoinding orderings.
923: This specification means the following.
924:
925: The variable list is separated into sub lists from left to right where
926: the @code{i}-th list contains @code{Li} members and it corresponds to
927: the ordering of type @code{Oi}. The result of a comparison is equal
928: to that for the leftmost different sub components. This type of ordering
929: is called an elimination ordering.
930: \E
1.1 noro 931:
932: @noindent
1.2 noro 933: \BJP
1.1 noro 934: $B$5$i$K(B, $B9TNs$K$h$j9`=g=x$r;XDj$9$k$3$H$,$G$-$k(B. $B0lHL$K(B, @code{n} $B9T(B
935: @code{m} $BNs$N<B?t9TNs(B @code{M} $B$,<!$N@-<A$r;}$D$H$9$k(B.
1.2 noro 936: \E
937: \BEG
938: Furthermore one can specify a term ordering by a matix.
939: Suppose that a real @code{n}, @code{m} matrix @code{M} has the
940: following properties.
941: \E
1.1 noro 942:
943: @enumerate
944: @item
1.2 noro 945: \JP $BD9$5(B @code{m} $B$N@0?t%Y%/%H%k(B @code{v} $B$KBP$7(B @code{Mv=0} $B$H(B @code{v=0} $B$OF1CM(B.
946: \BEG
947: For all integer verctors @code{v} of length @code{m} @code{Mv=0} is equivalent
948: to @code{v=0}.
949: \E
1.1 noro 950:
951: @item
1.2 noro 952: \BJP
1.1 noro 953: $BHsIi@.J,$r;}$DD9$5(B @code{m} $B$N(B 0 $B$G$J$$@0?t%Y%/%H%k(B @code{v} $B$KBP$7(B,
954: @code{Mv} $B$N(B 0 $B$G$J$$:G=i$N@.J,$OHsIi(B.
1.2 noro 955: \E
956: \BEG
957: For all non-negative integer vectors @code{v} the first non-zero component
958: of @code{Mv} is non-negative.
959: \E
1.1 noro 960: @end enumerate
961:
962: @noindent
1.2 noro 963: \BJP
1.1 noro 964: $B$3$N;~(B, 2 $B$D$N%Y%/%H%k(B @code{t}, @code{s} $B$KBP$7(B,
965: @code{t>s} $B$r(B, @code{M(t-s)} $B$N(B 0 $B$G$J$$:G=i$N@.J,$,HsIi(B,
966: $B$GDj5A$9$k$3$H$K$h$j9`=g=x$,Dj5A$G$-$k(B.
1.2 noro 967: \E
968: \BEG
969: Then we can define a term ordering such that, for two vectors
970: @code{t}, @code{s}, @code{t>s} means that the first non-zero component
971: of @code{M(t-s)} is non-negative.
972: \E
1.1 noro 973:
974: @noindent
1.2 noro 975: \BJP
1.1 noro 976: $B9`=g=x7?$O(B, @code{gr()} $B$J$I$N0z?t$H$7$F;XDj$5$l$kB>(B, $BAH$_9~$_H!?t(B
977: @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 978: \E
979: \BEG
980: Types of term orderings are used as arguments of functions such as
981: @code{gr()}. It is also set internally by @code{dp_ord()} and is used
982: during executions of various functions.
983: \E
1.1 noro 984:
985: @noindent
1.2 noro 986: \BJP
1.1 noro 987: $B$3$l$i$N=g=x$N6qBNE*$JDj5A$*$h$S%0%l%V%J4pDl$K4X$9$k99$K>\$7$$2r@b$O(B
988: @code{[Becker,Weispfenning]} $B$J$I$r;2>H$N$3$H(B.
1.2 noro 989: \E
990: \BEG
991: For concrete definitions of term ordering and more information
992: about Groebner basis, refer to, for example, the book
993: @code{[Becker,Weispfenning]}.
994: \E
1.1 noro 995:
996: @noindent
1.2 noro 997: \JP $B9`=g=x7?$N@_Dj$NB>$K(B, $BJQ?t$N=g=x<+BN$b7W;;;~4V$KBg$-$J1F6A$rM?$($k(B.
998: \BEG
999: Note that the variable ordering have strong effects on the computation
1000: time as well as the choice of types of term orderings.
1001: \E
1.1 noro 1002:
1003: @example
1004: [90] B=[x^10-t,x^8-z,x^31-x^6-x-y]$
1005: [91] gr(B,[x,y,z,t],2);
1006: [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
1007: -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
1008: +(-12*t^16+72*t^13-28*t^11-180*t^10+112*t^8+240*t^7+28*t^6-127*t^5
1009: -167*t^4-55*t^3+30*t^2+58*t-15)*z^4,
1.5 noro 1010: (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
1011: +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
1012: +(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
1013: +27*t^3-16*t^2-30*t+7)*z^4,
1014: (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
1015: -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
1016: +10*t^4-36*t^3-11*t^2-5*t+9)*z^2,
1.1 noro 1017: -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 1018: -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
1019: +20*t^19+28*t^18-120*t^16-56*t^15+14*t^14+300*t^13+70*t^12-56*t^11
1020: -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
1021: -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
1022: -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 1023: -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 1024: 2*t^2*y^3+z^2*y^2+(-2*t^5+4*t^2-6)*z^4*y
1025: +(4*t^8-t^7-8*t^5+2*t^4-4*t^3+5*t^2-t)*z,
1.1 noro 1026: z^3*y^2+2*t^3*y+(-t^7+2*t^4+t^2-t)*z^2,
1027: -t*z*y^2-2*z^3*y+t^8-2*t^5-t^3+t^2,
1.5 noro 1028: -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 1029: [93] gr(B,[t,z,y,x],2);
1030: [x^10-t,x^8-z,x^31-x^6-x-y]
1031: @end example
1032:
1033: @noindent
1.2 noro 1034: \BJP
1.1 noro 1035: $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
1036: $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
1037: $B%0%l%V%J4pDl$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
1038: $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
1039: $B$3$H$G$"$j(B, $B$3$NNc$N>l9g$O(B, @code{t}, @code{z}, @code{y} $B$,4{$K(B
1040: @code{x} $B$GI=$5$l$F$$$k$3$H$+$i$3$N$h$&$J6KC<$J7k2L$H$J$C$?$o$1$G$"$k(B.
1041: $B<B:]$K8=$l$k7W;;$K$*$$$F$O(B, $B$3$N$h$&$KA*$V$Y$-JQ?t=g=x$,L@$i$+$G$"$k(B
1042: $B$3$H$O>/$J$/(B, $B;n9T:x8m$,I,MW$J>l9g$b$"$k(B.
1.2 noro 1043: \E
1044: \BEG
1045: As you see in the above example, the Groebner base under variable
1046: ordering @code{[x,y,z,t]} has a lot of bases and each base itself is
1047: large. Under variable ordering @code{[t,z,y,x]}, however, @code{B} itself
1048: is already the Groebner basis.
1049: Roughly speaking, to obtain a Groebner base under the lexicographic
1050: ordering is to express the variables on the left (having higher order)
1051: in terms of variables on the right (having lower order).
1052: In the example, variables @code{t}, @code{z}, and @code{y} are already
1053: expressed by variable @code{x}, and the above explanation justifies
1054: such a drastic experimental results.
1055: In practice, however, optimum ordering for variables may not known
1056: beforehand, and some heuristic trial may be inevitable.
1.13 noro 1057: \E
1058:
1059: \BJP
1060: @node Weight ,,, $B%0%l%V%J4pDl$N7W;;(B
1061: @section Weight
1062: \E
1063: \BEG
1064: @node Weight,,, Groebner basis computation
1065: @section Weight
1066: \E
1067: \BJP
1068: $BA0@a$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
1069: $B$h$j0lHLE*$J$b$N$H$J$k(B.
1070: \E
1071: \BEG
1.14 ! noro 1072: Term orderings introduced in the previous section can be generalized
1.13 noro 1073: by setting a weight for each variable.
1074: \E
1075: @example
1076: [0] dp_td(<<1,1,1>>);
1077: 3
1078: [1] dp_set_weight([1,2,3])$
1079: [2] dp_td(<<1,1,1>>);
1080: 6
1081: @end example
1082: \BJP
1083: $BC19`<0$NA4<!?t$r7W;;$9$k:](B, $B%G%U%)%k%H$G$O(B
1084: $B3FJQ?t$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
1085: $B9M$($F$$$k$3$H$KAjEv$9$k(B. $B$3$NNc$G$O(B, $BBh0l(B, $BBhFs(B, $BBh;0JQ?t$N(B
1086: weight $B$r$=$l$>$l(B 1,2,3 $B$H;XDj$7$F$$$k(B. $B$3$N$?$a(B, @code{<<1,1,1>>}
1087: $B$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.
1088: 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.
1089: $BNc$($P(B, weight $B$r$&$^$/@_Dj$9$k$3$H$G(B, $BB?9`<0$r(B weighted homogeneous
1090: $B$K$9$k$3$H$,$G$-$k>l9g$,$"$k(B.
1091: \E
1092: \BEG
1093: By default, the total degree of a monomial is equal to
1094: the sum of all exponents. This means that the weight for each variable
1095: is set to 1.
1096: In this example, the weights for the first, the second and the third
1097: variable are set to 1, 2 and 3 respectively.
1098: Therefore the total degree of @code{<<1,1,1>>} under this weight,
1099: which is called the weight of the monomial, is @code{1*1+1*2+1*3=6}.
1.14 ! noro 1100: By setting weights, different term orderings can be set under a type of
! 1101: term ordeing. In some case a polynomial can
! 1102: be made weighted homogeneous by setting an appropriate weight.
1.13 noro 1103: \E
1104:
1105: \BJP
1106: $B3FJQ?t$KBP$9$k(B weight $B$r$^$H$a$?$b$N$r(B weight vector $B$H8F$V(B.
1107: $B$9$Y$F$N@.J,$,@5$G$"$j(B, $B%0%l%V%J4pDl7W;;$K$*$$$F(B, $BA4<!?t$N(B
1108: $BBe$o$j$KMQ$$$i$l$k$b$N$rFC$K(B sugar weight $B$H8F$V$3$H$K$9$k(B.
1109: sugar strategy $B$K$*$$$F(B, $BA4<!?t$NBe$o$j$K;H$o$l$k$+$i$G$"$k(B.
1110: $B0lJ}$G(B, $B3F@.J,$,I,$:$7$b@5$H$O8B$i$J$$(B weight vector $B$O(B,
1111: sugar weight $B$H$7$F@_Dj$9$k$3$H$O$G$-$J$$$,(B, $B9`=g=x$N0lHL2=$K$O(B
1112: $BM-MQ$G$"$k(B. $B$3$l$i$O(B, $B9TNs$K$h$k9`=g=x$N@_Dj$K$9$G$K8=$l$F(B
1113: $B$$$k(B. $B$9$J$o$A(B, $B9`=g=x$rDj5A$9$k9TNs$N3F9T$,(B, $B0l$D$N(B weight vector
1114: $B$H8+$J$5$l$k(B. $B$^$?(B, $B%V%m%C%/=g=x$O(B, $B3F%V%m%C%/$N(B
1115: $BJQ?t$KBP1~$9$k@.J,$N$_(B 1 $B$GB>$O(B 0 $B$N(B weight vector $B$K$h$kHf3S$r(B
1116: $B:G=i$K9T$C$F$+$i(B, $B3F%V%m%C%/Kh$N(B tie breaking $B$r9T$&$3$H$KAjEv$9$k(B.
1117: \E
1118:
1119: \BEG
1120: A list of weights for all variables is called a weight vector.
1121: A weight vector is called a sugar weight vector if
1122: its elements are all positive and it is used for computing
1123: a weighted total degree of a monomial, because such a weight
1124: is used instead of total degree in sugar strategy.
1125: On the other hand, a weight vector whose elements are not necessarily
1126: positive cannot be set as a sugar weight, but it is useful for
1127: generalizing term order. In fact, such a weight vector already
1128: appeared in a matrix order. That is, each row of a matrix defining
1129: a term order is regarded as a weight vector. A block order
1130: is also considered as a refinement of comparison by weight vectors.
1131: It compares two terms by using a weight vector whose elements
1132: corresponding to variables in a block is 1 and 0 otherwise,
1133: then it applies a tie breaker.
1.14 ! noro 1134: \E
! 1135:
! 1136: \BJP
! 1137: weight vector $B$N@_Dj$O(B @code{dp_set_weight()} $B$G9T$&$3$H$,$G$-$k(B
! 1138: $B$,(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
! 1139: $B$^$H$a$F@_Dj$G$-$k$3$H$,K>$^$7$$(B. $B$3$N$?$a(B, $B<!$N$h$&$J7A$G$b(B
! 1140: $B9`=g=x$,;XDj$G$-$k(B.
! 1141: \E
! 1142: \BEG
! 1143: A weight vector can be set by using @code{dp_set_weight()}.
! 1144: However it is more preferable if a weight vector can be set
! 1145: together with other parapmeters such as a type of term ordering
! 1146: and a variable order. This is realized as follows.
! 1147: \E
1.13 noro 1148:
1.14 ! noro 1149: @example
! 1150: [64] B=[x+y+z-6,x*y+y*z+z*x-11,x*y*z-6]$
! 1151: [65] dp_gr_main(B|v=[x,y,z],sugarweight=[3,2,1],order=0);
! 1152: [z^3-6*z^2+11*z-6,x+y+z-6,-y^2+(-z+6)*y-z^2+6*z-11]
! 1153: [66] dp_gr_main(B|v=[y,z,x],order=[[1,1,0],[0,1,0],[0,0,1]]);
! 1154: [x^3-6*x^2+11*x-6,x+y+z-6,-x^2+(-y+6)*x-y^2+6*y-11]
! 1155: [67] dp_gr_main(B|v=[y,z,x],order=[[x,1,y,2,z,3]]);
! 1156: [x+y+z-6,x^3-6*x^2+11*x-6,-x^2+(-y+6)*x-y^2+6*y-11]
! 1157: @end example
! 1158:
! 1159: \BJP
! 1160: $B$$$:$l$NNc$K$*$$$F$b(B, $B9`=g=x$O(B option $B$H$7$F;XDj$5$l$F$$$k(B.
! 1161: $B:G=i$NNc$G$O(B @code{v} $B$K$h$jJQ?t=g=x$r(B, @code{sugarweight} $B$K$h$j(B
! 1162: sugar weight vector $B$r(B, @code{order}$B$K$h$j9`=g=x7?$r;XDj$7$F$$$k(B.
! 1163: $BFs$DL\$NNc$K$*$1$k(B @code{order} $B$N;XDj$O(B matrix order $B$HF1MM$G$"$k(B.
! 1164: $B$9$J$o$A(B, $B;XDj$5$l$?(B weight vector $B$r:8$+$i=g$K;H$C$F(B weight $B$NHf3S(B
! 1165: $B$r9T$&(B. $B;0$DL\$NNc$bF1MM$G$"$k$,(B, $B$3$3$G$O(B weight vector $B$NMWAG$r(B
! 1166: $BJQ?tKh$K;XDj$7$F$$$k(B. $B;XDj$,$J$$$b$N$O(B 0 $B$H$J$k(B. $B;0$DL\$NNc$G$O(B,
! 1167: @code{order} $B$K$h$k;XDj$G$O9`=g=x$,7hDj$7$J$$(B. $B$3$N>l9g$K$O(B,
! 1168: tie breaker $B$H$7$FA4<!?t5U<-=q<0=g=x$,<+F0E*$K@_Dj$5$l$k(B.
! 1169: $B$3$N;XDjJ}K!$O(B, @code{dp_gr_main}, @code{dp_gr_mod_main} $B$J$I(B
! 1170: $B$NAH$_9~$_4X?t$G$N$_2DG=$G$"$j(B, @code{gr} $B$J$I$N%f!<%6Dj5A4X?t(B
! 1171: $B$G$OL$BP1~$G$"$k(B.
! 1172: \E
! 1173: \BEG
! 1174: In each example, a term ordering is specified as options.
! 1175: In the first example, a variable order, a sugar weight vector
! 1176: and a type of term ordering are specified by options @code{v},
! 1177: @code{sugarweight} and @code{order} respectively.
! 1178: In the second example, an option @code{order} is used
! 1179: to set a matrix ordering. That is, the specified weight vectors
! 1180: are used from left to right for comparing terms.
! 1181: The third example shows a variant of specifying a weight vector,
! 1182: where each component of a weight vector is specified variable by variable,
! 1183: and unspecified components are set to zero. In this example,
! 1184: a term order is not determined only by the specified weight vector.
! 1185: In such a case a tie breaker by the graded reverse lexicographic ordering
! 1186: is set automatically.
! 1187: This type of a term ordering specification can be applied only to builtin
! 1188: functions such as @code{dp_gr_main()}, @code{dp_gr_mod_main()}, not to
! 1189: user defined functions such as @code{gr()}.
1.2 noro 1190: \E
1.1 noro 1191:
1.2 noro 1192: \BJP
1.1 noro 1193: @node $BM-M}<0$r78?t$H$9$k%0%l%V%J4pDl7W;;(B,,, $B%0%l%V%J4pDl$N7W;;(B
1194: @section $BM-M}<0$r78?t$H$9$k%0%l%V%J4pDl7W;;(B
1.2 noro 1195: \E
1196: \BEG
1197: @node Groebner basis computation with rational function coefficients,,, Groebner basis computation
1198: @section Groebner basis computation with rational function coefficients
1199: \E
1.1 noro 1200:
1201: @noindent
1.2 noro 1202: \BJP
1.1 noro 1203: @code{gr()} $B$J$I$N%H%C%W%l%Y%kH!?t$O(B, $B$$$:$l$b(B, $BF~NOB?9`<0%j%9%H$K(B
1204: $B8=$l$kJQ?t(B ($BITDj85(B) $B$H(B, $BJQ?t%j%9%H$K8=$l$kJQ?t$rHf3S$7$F(B, $BJQ?t%j%9%H$K(B
1205: $B$J$$JQ?t$,F~NOB?9`<0$K8=$l$F$$$k>l9g$K$O(B, $B<+F0E*$K(B, $B$=$NJQ?t$r(B, $B78?t(B
1206: $BBN$N85$H$7$F07$&(B.
1.2 noro 1207: \E
1208: \BEG
1209: Such variables that appear within the input polynomials but
1210: not appearing in the input variable list are automatically treated
1211: as elements in the coefficient field
1212: by top level functions, such as @code{gr()}.
1213: \E
1.1 noro 1214:
1215: @example
1216: [64] gr([a*x+b*y-c,d*x+e*y-f],[x,y],2);
1217: [(-e*a+d*b)*x-f*b+e*c,(-e*a+d*b)*y+f*a-d*c]
1218: @end example
1219:
1220: @noindent
1.2 noro 1221: \BJP
1.1 noro 1222: $B$3$NNc$G$O(B, @code{a}, @code{b}, @code{c}, @code{d} $B$,78?tBN$N85$H$7$F(B
1223: $B07$o$l$k(B. $B$9$J$o$A(B, $BM-M}H!?tBN(B
1224: @b{F} = @b{Q}(@code{a},@code{b},@code{c},@code{d}) $B>e$N(B 2 $BJQ?tB?9`<04D(B
1225: @b{F}[@code{x},@code{y}] $B$K$*$1$k%0%l%V%J4pDl$r5a$a$k$3$H$K$J$k(B.
1226: $BCm0U$9$Y$-$3$H$O(B,
1227: $B78?t$,BN$H$7$F07$o$l$F$$$k$3$H$G$"$k(B. $B$9$J$o$A(B, $B78?t$N4V$KB?9`<0(B
1228: $B$H$7$F$N6&DL0x;R$,$"$C$?>l9g$K$O(B, $B7k2L$+$i$=$N0x;R$O=|$+$l$F$$$k(B
1229: $B$?$a(B, $BM-M}?tBN>e$NB?9`<04D>e$NLdBj$H$7$F9M$($?>l9g$N7k2L$H$O0lHL(B
1230: $B$K$O0[$J$k(B. $B$^$?(B, $B<g$H$7$F7W;;8zN(>e$NLdBj$N$?$a(B, $BJ,;6I=8=B?9`<0(B
1231: $B$N78?t$H$7$F<B:]$K5v$5$l$k$N$OB?9`<0$^$G$G$"$k(B. $B$9$J$o$A(B, $BJ,Jl$r(B
1232: $B;}$DM-M}<0$OJ,;6I=8=B?9`<0$N78?t$H$7$F$O5v$5$l$J$$(B.
1.2 noro 1233: \E
1234: \BEG
1235: In this example, variables @code{a}, @code{b}, @code{c}, and @code{d}
1236: are treated as elements in the coefficient field.
1237: In this case, a Groebner basis is computed
1238: on a bi-variate polynomial ring
1239: @b{F}[@code{x},@code{y}]
1240: over rational function field
1241: @b{F} = @b{Q}(@code{a},@code{b},@code{c},@code{d}).
1242: Notice that coefficients are considered as a member in a field.
1243: As a consequence, polynomial factors common to the coefficients
1244: are removed so that the result, in general, is different from
1245: the result that would be obtained when the problem is considered
1246: as a computation of Groebner basis over a polynomial ring
1247: with rational function coefficients.
1248: And note that coefficients of a distributed polynomial are limited
1249: to numbers and polynomials because of efficiency.
1250: \E
1.1 noro 1251:
1.2 noro 1252: \BJP
1.1 noro 1253: @node $B4pDlJQ49(B,,, $B%0%l%V%J4pDl$N7W;;(B
1254: @section $B4pDlJQ49(B
1.2 noro 1255: \E
1256: \BEG
1257: @node Change of ordering,,, Groebner basis computation
1258: @section Change of orderng
1259: \E
1.1 noro 1260:
1261: @noindent
1.2 noro 1262: \BJP
1.1 noro 1263: $B<-=q<0=g=x$N%0%l%V%J4pDl$r5a$a$k>l9g(B, $BD>@\(B @code{gr()} $B$J$I$r5/F0$9$k(B
1264: $B$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,
1265: $B$=$l$rF~NO$H$7$F<-=q<0=g=x$N%0%l%V%J4pDl$r7W;;$9$kJ}$,8zN($,$h$$>l9g(B
1266: $B$,$"$k(B. $B$^$?(B, $BF~NO$,2?$i$+$N=g=x$G$N%0%l%V%J4pDl$K$J$C$F$$$k>l9g(B, $B4pDl(B
1267: $BJQ49$H8F$P$l$kJ}K!$K$h$j(B, Buchberger $B%"%k%4%j%:%`$K$h$i$:$K8zN(NI$/(B
1268: $B<-=q<0=g=x$N%0%l%V%J4pDl$,7W;;$G$-$k>l9g$,$"$k(B. $B$3$N$h$&$JL\E*$N$?$a$N(B
1269: $BH!?t$,(B, $B%f!<%6Dj5AH!?t$H$7$F(B @samp{gr} $B$K$$$/$D$+Dj5A$5$l$F$$$k(B.
1270: $B0J2<$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
1271: $B4{$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
1272: @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 1273: \E
1274: \BEG
1275: When we compute a lex order Groebner basis, it is often efficient to
1276: compute it via Groebner basis with respect to another order such as
1277: degree reverse lex order, rather than to compute it directory by
1278: @code{gr()} etc. If we know that an input is a Groebner basis with
1279: respect to an order, we can apply special methods called change of
1280: ordering for a Groebner basis computation with respect to another
1281: order, without using Buchberger algorithm. The following two functions
1282: are ones for change of ordering such that they convert a Groebner
1283: basis @var{gbase} with respect to the variable order @var{vlist1} and
1284: the order type @var{order} into a lex Groebner basis with respect
1285: to the variable order @var{vlist2}.
1286: \E
1.1 noro 1287:
1288: @table @code
1289: @item tolex(@var{gbase},@var{vlist1},@var{order},@var{vlist2})
1290:
1.2 noro 1291: \BJP
1.1 noro 1292: $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.
1293: $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
1294: $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 1295: \E
1296: \BEG
1297: This function can be used only when @var{gbase} is an ideal over the
1298: rationals. The input @var{gbase} must be a Groebner basis with respect
1299: to the variable order @var{vlist1} and the order type @var{order}. Moreover
1300: the ideal generated by @var{gbase} must be zero-dimensional.
1301: This computes the lex Groebner basis of @var{gbase}
1302: by using the modular change of ordering algorithm. The algorithm first
1303: computes the lex Groebner basis over a finite field. Then each element
1304: in the lex Groebner basis over the rationals is computed with undetermined
1305: coefficient method and linear equation solving by Hensel lifting.
1306: \E
1.1 noro 1307:
1308: @item tolex_tl(@var{gbase},@var{vlist1},@var{order},@var{vlist2},@var{homo})
1309:
1.2 noro 1310: \BJP
1.1 noro 1311: $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
1312: $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
1313: trace-lifting$B$K$*$1$k%0%l%V%J4pDl8uJd$NF,9`(B, $BF,78?t$N@-<A$rMxMQ$7$F(B,
1314: $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
1315: $B$k$?$a(B, $BC1$K(BBuchberger $B%"%k%4%j%:%`$r7+$jJV$9$h$j8zN($h$/7W;;$G$-$k(B.
1316: $B99$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
1317: $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,
1318: $BD>@\<-=q<0=g=x$N7W;;$r9T$&$h$j8zN($,$h$$(B. ($B$b$A$m$sNc30$"$j(B. )
1319: $B0z?t(B @var{homo} $B$,(B 0 $B$G$J$$;~(B, @code{hgr()} $B$HF1MM$K@F<!2=$r7PM3$7$F(B
1320: $B7W;;$r9T$&(B.
1.2 noro 1321: \E
1322: \BEG
1323: This function computes the lex Groebner basis of @var{gbase}. The
1324: input @var{gbase} must be a Groebner basis with respect to the
1325: variable order @var{vlist1} and the order type @var{order}.
1326: Buchberger algorithm with trace lifting is used to compute the lex
1327: Groebner basis, however the Groebner basis check and the ideal
1328: membership check can be omitted by using several properties derived
1329: from the fact that the input is a Groebner basis. So it is more
1330: efficient than simple repetition of Buchberger algorithm. If the input
1331: is zero-dimensional, this function inserts automatically a computation
1332: of Groebner basis with respect to an elimination order, which makes
1333: the whole computation more efficient for many cases. If @var{homo} is
1334: not equal to 0, homogenization is used in each step.
1335: \E
1.1 noro 1336: @end table
1337:
1338: @noindent
1.2 noro 1339: \BJP
1.1 noro 1340: $B$=$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
1341: $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
1342: @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 1343: \E
1344: \BEG
1345: For zero-dimensional systems, there are several fuctions to
1346: compute the minimal polynomial of a polynomial and or a more compact
1347: representation for zeros of the system. They are all defined in @samp{gr}.
1348: Refer to the sections for each functions.
1349: \E
1.1 noro 1350:
1.2 noro 1351: \BJP
1.6 noro 1352: @node Weyl $BBe?t(B,,, $B%0%l%V%J4pDl$N7W;;(B
1353: @section Weyl $BBe?t(B
1354: \E
1355: \BEG
1356: @node Weyl algebra,,, Groebner basis computation
1357: @section Weyl algebra
1358: \E
1359:
1360: @noindent
1361:
1362: \BJP
1363: $B$3$l$^$G$O(B, $BDL>o$N2D49$JB?9`<04D$K$*$1$k%0%l%V%J4pDl7W;;$K$D$$$F(B
1364: $B=R$Y$F$-$?$,(B, $B%0%l%V%J4pDl$NM}O@$O(B, $B$"$k>r7o$rK~$?$9Hs2D49$J(B
1365: $B4D$K$b3HD%$G$-$k(B. $B$3$N$h$&$J4D$NCf$G(B, $B1~MQ>e$b=EMW$J(B,
1366: Weyl $BBe?t(B, $B$9$J$o$AB?9`<04D>e$NHyJ,:nMQAG4D$N1i;;$*$h$S(B
1367: $B%0%l%V%J4pDl7W;;$,(B Risa/Asir $B$K<BAu$5$l$F$$$k(B.
1368:
1369: $BBN(B @code{K} $B>e$N(B @code{n} $B<!85(B Weyl $BBe?t(B
1370: @code{D=K<x1,@dots{},xn,D1,@dots{},Dn>} $B$O(B
1371: \E
1372:
1373: \BEG
1374: So far we have explained Groebner basis computation in
1375: commutative polynomial rings. However Groebner basis can be
1376: considered in more general non-commutative rings.
1377: Weyl algebra is one of such rings and
1378: Risa/Asir implements fundamental operations
1379: in Weyl algebra and Groebner basis computation in Weyl algebra.
1380:
1381: The @code{n} dimensional Weyl algebra over a field @code{K},
1382: @code{D=K<x1,@dots{},xn,D1,@dots{},Dn>} is a non-commutative
1383: algebra which has the following fundamental relations:
1384: \E
1385:
1386: @code{xi*xj-xj*xi=0}, @code{Di*Dj-Dj*Di=0}, @code{Di*xj-xj*Di=0} (@code{i!=j}),
1387: @code{Di*xi-xi*Di=1}
1388:
1389: \BJP
1390: $B$H$$$&4pK\4X78$r;}$D4D$G$"$k(B. @code{D} $B$O(B $BB?9`<04D(B @code{K[x1,@dots{},xn]} $B$r78?t(B
1391: $B$H$9$kHyJ,:nMQAG4D$G(B, @code{Di} $B$O(B @code{xi} $B$K$h$kHyJ,$rI=$9(B. $B8r494X78$K$h$j(B,
1392: @code{D} $B$N85$O(B, @code{x1^i1*@dots{}*xn^in*D1^j1*@dots{}*Dn^jn} $B$J$kC19`(B
1393: $B<0$N(B @code{K} $B@~7A7k9g$H$7$F=q$-I=$9$3$H$,$G$-$k(B.
1394: Risa/Asir $B$K$*$$$F$O(B, $B$3$NC19`<0$r(B, $B2D49$JB?9`<0$HF1MM$K(B
1395: @code{<<i1,@dots{},in,j1,@dots{},jn>>} $B$GI=$9(B. $B$9$J$o$A(B, @code{D} $B$N85$b(B
1396: $BJ,;6I=8=B?9`<0$H$7$FI=$5$l$k(B. $B2C8:;;$O(B, $B2D49$N>l9g$HF1MM$K(B, @code{+}, @code{-}
1397: $B$K$h$j(B
1398: $B<B9T$G$-$k$,(B, $B>h;;$O(B, $BHs2D49@-$r9MN8$7$F(B @code{dp_weyl_mul()} $B$H$$$&4X?t(B
1399: $B$K$h$j<B9T$9$k(B.
1400: \E
1401:
1402: \BEG
1403: @code{D} is the ring of differential operators whose coefficients
1404: are polynomials in @code{K[x1,@dots{},xn]} and
1405: @code{Di} denotes the differentiation with respect to @code{xi}.
1406: According to the commutation relation,
1407: elements of @code{D} can be represented as a @code{K}-linear combination
1408: of monomials @code{x1^i1*@dots{}*xn^in*D1^j1*@dots{}*Dn^jn}.
1409: In Risa/Asir, this type of monomial is represented
1410: by @code{<<i1,@dots{},in,j1,@dots{},jn>>} as in the case of commutative
1411: polynomial.
1412: That is, elements of @code{D} are represented by distributed polynomials.
1413: Addition and subtraction can be done by @code{+}, @code{-},
1414: but multiplication is done by calling @code{dp_weyl_mul()} because of
1415: the non-commutativity of @code{D}.
1416: \E
1417:
1418: @example
1419: [0] A=<<1,2,2,1>>;
1420: (1)*<<1,2,2,1>>
1421: [1] B=<<2,1,1,2>>;
1422: (1)*<<2,1,1,2>>
1423: [2] A*B;
1424: (1)*<<3,3,3,3>>
1425: [3] dp_weyl_mul(A,B);
1426: (1)*<<3,3,3,3>>+(1)*<<3,2,3,2>>+(4)*<<2,3,2,3>>+(4)*<<2,2,2,2>>
1427: +(2)*<<1,3,1,3>>+(2)*<<1,2,1,2>>
1428: @end example
1429:
1430: \BJP
1431: $B%0%l%V%J4pDl7W;;$K$D$$$F$b(B, Weyl $BBe?t@lMQ$N4X?t$H$7$F(B,
1432: $B<!$N4X?t$,MQ0U$7$F$"$k(B.
1433: \E
1434: \BEG
1435: The following functions are avilable for Groebner basis computation
1436: in Weyl algebra:
1437: \E
1438: @code{dp_weyl_gr_main()},
1439: @code{dp_weyl_gr_mod_main()},
1440: @code{dp_weyl_gr_f_main()},
1441: @code{dp_weyl_f4_main()},
1442: @code{dp_weyl_f4_mod_main()}.
1443: \BJP
1444: $B$^$?(B, $B1~MQ$H$7$F(B, global b $B4X?t$N7W;;$,<BAu$5$l$F$$$k(B.
1445: \E
1446: \BEG
1447: Computation of the global b function is implemented as an application.
1448: \E
1449:
1450: \BJP
1.1 noro 1451: @node $B%0%l%V%J4pDl$K4X$9$kH!?t(B,,, $B%0%l%V%J4pDl$N7W;;(B
1452: @section $B%0%l%V%J4pDl$K4X$9$kH!?t(B
1.2 noro 1453: \E
1454: \BEG
1455: @node Functions for Groebner basis computation,,, Groebner basis computation
1456: @section Functions for Groebner basis computation
1457: \E
1.1 noro 1458:
1459: @menu
1460: * gr hgr gr_mod::
1461: * lex_hensel lex_tl tolex tolex_d tolex_tl::
1462: * lex_hensel_gsl tolex_gsl tolex_gsl_d::
1463: * gr_minipoly minipoly::
1464: * tolexm minipolym::
1.6 noro 1465: * 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::
1466: * dp_f4_main dp_f4_mod_main dp_weyl_f4_main dp_weyl_f4_mod_main::
1.1 noro 1467: * dp_gr_flags dp_gr_print::
1468: * dp_ord::
1469: * dp_ptod::
1470: * dp_dtop::
1471: * dp_mod dp_rat::
1472: * dp_homo dp_dehomo::
1473: * dp_ptozp dp_prim::
1474: * dp_nf dp_nf_mod dp_true_nf dp_true_nf_mod::
1475: * dp_hm dp_ht dp_hc dp_rest::
1476: * dp_td dp_sugar::
1477: * dp_lcm::
1478: * dp_redble::
1479: * dp_subd::
1480: * dp_mbase::
1481: * dp_mag::
1482: * dp_red dp_red_mod::
1483: * dp_sp dp_sp_mod::
1484: * p_nf p_nf_mod p_true_nf p_true_nf_mod ::
1485: * p_terms::
1486: * gb_comp::
1487: * katsura hkatsura cyclic hcyclic::
1488: * dp_vtoe dp_etov::
1489: * lex_hensel_gsl tolex_gsl tolex_gsl_d::
1.3 noro 1490: * primadec primedec::
1.5 noro 1491: * primedec_mod::
1.10 noro 1492: * bfunction bfct generic_bfct ann ann0::
1.1 noro 1493: @end menu
1494:
1.2 noro 1495: \JP @node gr hgr gr_mod,,, $B%0%l%V%J4pDl$K4X$9$kH!?t(B
1496: \EG @node gr hgr gr_mod,,, Functions for Groebner basis computation
1.1 noro 1497: @subsection @code{gr}, @code{hgr}, @code{gr_mod}, @code{dgr}
1498: @findex gr
1499: @findex hgr
1500: @findex gr_mod
1501: @findex dgr
1502:
1503: @table @t
1504: @item gr(@var{plist},@var{vlist},@var{order})
1505: @itemx hgr(@var{plist},@var{vlist},@var{order})
1506: @itemx gr_mod(@var{plist},@var{vlist},@var{order},@var{p})
1507: @itemx dgr(@var{plist},@var{vlist},@var{order},@var{procs})
1.2 noro 1508: \JP :: $B%0%l%V%J4pDl$N7W;;(B
1509: \EG :: Groebner basis computation
1.1 noro 1510: @end table
1511:
1512: @table @var
1513: @item return
1.2 noro 1514: \JP $B%j%9%H(B
1515: \EG list
1.4 noro 1516: @item plist vlist procs
1.2 noro 1517: \JP $B%j%9%H(B
1518: \EG list
1.1 noro 1519: @item order
1.2 noro 1520: \JP $B?t(B, $B%j%9%H$^$?$O9TNs(B
1521: \EG number, list or matrix
1.1 noro 1522: @item p
1.2 noro 1523: \JP 2^27 $BL$K~$NAG?t(B
1524: \EG prime less than 2^27
1.1 noro 1525: @end table
1526:
1527: @itemize @bullet
1.2 noro 1528: \BJP
1.1 noro 1529: @item
1530: $BI8=`%i%$%V%i%j$N(B @samp{gr} $B$GDj5A$5$l$F$$$k(B.
1531: @item
1532: $B$$$:$l$b(B, $BB?9`<0%j%9%H(B @var{plist} $B$N(B, $BJQ?t=g=x(B @var{vlist}, $B9`=g=x7?(B
1533: @var{order} $B$K4X$9$k%0%l%V%J4pDl$r5a$a$k(B. @code{gr()}, @code{hgr()}
1534: $B$O(B $BM-M}?t78?t(B, @code{gr_mod()} $B$O(B GF(@var{p}) $B78?t$H$7$F7W;;$9$k(B.
1535: @item
1536: @var{vlist} $B$OITDj85$N%j%9%H(B. @var{vlist} $B$K8=$l$J$$ITDj85$O(B,
1537: $B78?tBN$KB0$9$k$H8+$J$5$l$k(B.
1538: @item
1539: @code{gr()}, trace-lifting ($B%b%8%e%i1i;;$rMQ$$$?9bB.2=(B) $B$*$h$S(B sugar
1540: strategy $B$K$h$k7W;;(B, @code{hgr()} $B$O(B trace-lifting $B$*$h$S(B
1541: $B@F<!2=$K$h$k(B $B6:@5$5$l$?(B sugar strategy $B$K$h$k7W;;$r9T$&(B.
1542: @item
1543: @code{dgr()} $B$O(B, @code{gr()}, @code{dgr()} $B$r(B
1544: $B;R%W%m%;%9%j%9%H(B @var{procs} $B$N(B 2 $B$D$N%W%m%;%9$K$h$jF1;~$K7W;;$5$;(B,
1545: $B@h$K7k2L$rJV$7$?J}$N7k2L$rJV$9(B. $B7k2L$OF10l$G$"$k$,(B, $B$I$A$i$NJ}K!$,(B
1546: $B9bB.$+0lHL$K$OITL@$N$?$a(B, $B<B:]$N7P2a;~4V$rC;=L$9$k$N$KM-8z$G$"$k(B.
1547: @item
1548: @code{dgr()} $B$GI=<($5$l$k;~4V$O(B, $B$3$NH!?t$,<B9T$5$l$F$$$k%W%m%;%9$G$N(B
1549: 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 1550: @item
1551: $BB?9`<0%j%9%H(B @var{plist} $B$NMWAG$,J,;6I=8=B?9`<0$N>l9g$O(B
1552: $B7k2L$bJ,;6I=8=B?9`<0$N%j%9%H$G$"$k(B.
1553: $B$3$N>l9g(B, $B0z?t$NJ,;6B?9`<0$OM?$($i$l$?=g=x$K=>$$(B @code{dp_sort} $B$G(B
1554: $B%=!<%H$5$l$F$+$i7W;;$5$l$k(B.
1555: $BB?9`<0%j%9%H$NMWAG$,J,;6I=8=B?9`<0$N>l9g$b(B
1556: $BJQ?t$N?tJ,$NITDj85$N%j%9%H$r(B @var{vlist} $B0z?t$H$7$FM?$($J$$$H$$$1$J$$(B
1557: ($B%@%_!<(B).
1.2 noro 1558: \E
1559: \BEG
1560: @item
1561: These functions are defined in @samp{gr} in the standard library
1562: directory.
1563: @item
1564: They compute a Groebner basis of a polynomial list @var{plist} with
1565: respect to the variable order @var{vlist} and the order type @var{order}.
1566: @code{gr()} and @code{hgr()} compute a Groebner basis over the rationals
1567: and @code{gr_mod} computes over GF(@var{p}).
1568: @item
1569: Variables not included in @var{vlist} are regarded as
1570: included in the ground field.
1571: @item
1572: @code{gr()} uses trace-lifting (an improvement by modular computation)
1573: and sugar strategy.
1574: @code{hgr()} uses trace-lifting and a cured sugar strategy
1575: by using homogenization.
1576: @item
1577: @code{dgr()} executes @code{gr()}, @code{dgr()} simultaneously on
1578: two process in a child process list @var{procs} and returns
1579: the result obtained first. The results returned from both the process
1580: should be equal, but it is not known in advance which method is faster.
1581: Therefore this function is useful to reduce the actual elapsed time.
1582: @item
1583: The CPU time shown after an exection of @code{dgr()} indicates
1584: that of the master process, and most of the time corresponds to the time
1585: for communication.
1.12 takayama 1586: @item
1587: When the elements of @var{plist} are distributed polynomials,
1588: the result is also a list of distributed polynomials.
1589: In this case, firstly the elements of @var{plist} is sorted by @code{dp_sort}
1590: and the Grobner basis computation is started.
1591: Variables must be given in @var{vlist} even in this case
1592: (these variables are dummy).
1.2 noro 1593: \E
1.1 noro 1594: @end itemize
1595:
1596: @example
1597: [0] load("gr")$
1598: [64] load("cyclic")$
1599: [74] G=gr(cyclic(5),[c0,c1,c2,c3,c4],2);
1600: [c4^15+122*c4^10-122*c4^5-1,...]
1601: [75] GM=gr_mod(cyclic(5),[c0,c1,c2,c3,c4],2,31991)$
1602: 24628*c4^15+29453*c4^10+2538*c4^5+7363
1603: [76] (G[0]*24628-GM[0])%31991;
1604: 0
1605: @end example
1606:
1607: @table @t
1.2 noro 1608: \JP @item $B;2>H(B
1609: \EG @item References
1.6 noro 1610: @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 1611: @fref{dp_ord}.
1612: @end table
1613:
1.2 noro 1614: \JP @node lex_hensel lex_tl tolex tolex_d tolex_tl,,, $B%0%l%V%J4pDl$K4X$9$kH!?t(B
1615: \EG @node lex_hensel lex_tl tolex tolex_d tolex_tl,,, Functions for Groebner basis computation
1.1 noro 1616: @subsection @code{lex_hensel}, @code{lex_tl}, @code{tolex}, @code{tolex_d}, @code{tolex_tl}
1617: @findex lex_hensel
1618: @findex lex_tl
1619: @findex tolex
1620: @findex tolex_d
1621: @findex tolex_tl
1622:
1623: @table @t
1624: @item lex_hensel(@var{plist},@var{vlist1},@var{order},@var{vlist2},@var{homo})
1625: @itemx lex_tl(@var{plist},@var{vlist1},@var{order},@var{vlist2},@var{homo})
1.2 noro 1626: \JP :: $B4pDlJQ49$K$h$k<-=q<0=g=x%0%l%V%J4pDl$N7W;;(B
1627: \EG:: Groebner basis computation with respect to a lex order by change of ordering
1.1 noro 1628: @item tolex(@var{plist},@var{vlist1},@var{order},@var{vlist2})
1629: @itemx tolex_d(@var{plist},@var{vlist1},@var{order},@var{vlist2},@var{procs})
1630: @itemx tolex_tl(@var{plist},@var{vlist1},@var{order},@var{vlist2},@var{homo})
1.2 noro 1631: \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
1632: \EG :: Groebner basis computation with respect to a lex order by change of ordering, starting from a Groebner basis
1.1 noro 1633: @end table
1634:
1635: @table @var
1636: @item return
1.2 noro 1637: \JP $B%j%9%H(B
1638: \EG list
1.4 noro 1639: @item plist vlist1 vlist2 procs
1.2 noro 1640: \JP $B%j%9%H(B
1641: \EG list
1.1 noro 1642: @item order
1.2 noro 1643: \JP $B?t(B, $B%j%9%H$^$?$O9TNs(B
1644: \EG number, list or matrix
1.1 noro 1645: @item homo
1.2 noro 1646: \JP $B%U%i%0(B
1647: \EG flag
1.1 noro 1648: @end table
1649:
1650: @itemize @bullet
1.2 noro 1651: \BJP
1.1 noro 1652: @item
1653: $BI8=`%i%$%V%i%j$N(B @samp{gr} $B$GDj5A$5$l$F$$$k(B.
1654: @item
1655: @code{lex_hensel()}, @code{lex_tl()} $B$O(B,
1656: $BB?9`<0%j%9%H(B @var{plist} $B$N(B, $BJQ?t=g=x(B @var{vlist1}, $B9`=g=x7?(B
1657: @var{order} $B$K4X$9$k%0%l%V%J4pDl$r5a$a(B, $B$=$l$r(B, $BJQ?t=g=x(B @var{vlist2}
1658: $B$N<-=q<0=g=x%0%l%V%J4pDl$KJQ49$9$k(B.
1659: @item
1660: @code{tolex()}, @code{tolex_tl()} $B$O(B,
1661: $BJQ?t=g=x(B @var{vlist1}, $B9`=g=x7?(B @var{order} $B$K4X$9$k%0%l%V%J4pDl$G$"$k(B
1662: $BB?9`<0%j%9%H(B @var{plist} $B$rJQ?t=g=x(B @var{vlist2} $B$N<-=q<0=g=x%0%l%V%J(B
1663: $B4pDl$KJQ49$9$k(B.
1664: @code{tolex_d()} $B$O(B, @code{tolex()} $B$K$*$1$k(B, $B3F4pDl$N7W;;$r(B, $B;R%W%m%;%9(B
1665: $B%j%9%H(B @var{procs} $B$N3F%W%m%;%9$KJ,;67W;;$5$;$k(B.
1666: @item
1667: @code{lex_hensel()}, @code{lex_tl()} $B$K$*$$$F$O(B, $B<-=q<0=g=x%0%l%V%J4pDl$N(B
1668: $B7W;;$O<!$N$h$&$K9T$o$l$k(B. (@code{[Noro,Yokoyama]} $B;2>H(B.)
1669: @enumerate
1670: @item
1671: @var{vlist1}, @var{order} $B$K4X$9$k%0%l%V%J4pDl(B @var{G0} $B$r7W;;$9$k(B.
1672: (@code{lex_hensel()} $B$N$_(B. )
1673: @item
1674: @var{G0} $B$N3F85$N(B @var{vlist2} $B$K4X$9$k<-=q<0=g=x$K$*$1$kF,78?t$r3d$i$J$$(B
1675: $B$h$&$JAG?t(B @var{p} $B$rA*$S(B, GF(@var{p}) $B>e$G$N<-=q<0=g=x%0%l%V%J4pDl(B
1676: @var{Gp} $B$r7W;;$9$k(B.
1677: @item
1678: @var{Gp} $B$K8=$l$k$9$Y$F$N9`$N(B, @var{G0} $B$K4X$9$k@55,7A(B @var{NF} $B$r7W;;$9$k(B.
1679: @item
1680: @var{Gp} $B$N3F85(B @var{f} $B$K$D$-(B, @var{f} $B$N78?t$rL$Dj78?t$G(B,
1681: @var{f} $B$N3F9`$rBP1~$9$k(B @var{NF} $B$N85$GCV$-49$((B, $B3F9`$N78?t$r(B 0 $B$HCV$$$?(B,
1682: $BL$Dj78?t$K4X$9$k@~7AJ}Dx<07O(B @var{Lf} $B$r:n$k(B.
1683: @item
1684: @var{Lf} $B$,(B, $BK!(B @var{p} $B$G0l0U2r$r;}$D$3$H$rMQ$$$F(B @var{Lf} $B$N2r$r(B
1685: $BK!(B @var{p}$B$N2r$+$i(B Hensel $B9=@.$K$h$j5a$a$k(B.
1686: @item
1687: $B$9$Y$F$N(B @var{Gp} $B$N85$K$D$-@~7AJ}Dx<0$,2r$1$?$i$=$N2rA4BN$,5a$a$k(B
1688: $B<-=q<0=g=x$G$N%0%l%V%J4pDl(B. $B$b$7$I$l$+$N@~7AJ}Dx<0$N5a2r$K<:GT$7$?$i(B,
1689: @var{p} $B$r$H$jD>$7$F$d$jD>$9(B.
1690: @end enumerate
1691:
1692: @item
1693: @code{lex_tl()}, @code{tolex_tl()} $B$K$*$$$F$O(B, $B<-=q<0=g=x%0%l%V%J4pDl$N(B
1694: $B7W;;$O<!$N$h$&$K9T$o$l$k(B.
1695:
1696: @enumerate
1697: @item
1698: @var{vlist1}, @var{order} $B$K4X$9$k%0%l%V%J4pDl(B @var{G0} $B$r7W;;$9$k(B.
1699: (@code{lex_hensel()} $B$N$_(B. )
1700: @item
1701: @var{G0} $B$,(B 0 $B<!85%7%9%F%`$G$J$$$H$-(B, @var{G0} $B$rF~NO$H$7$F(B,
1702: @var{G0} $B$N3F85$N(B @var{vlist2} $B$K4X$9$k<-=q<0=g=x$K$*$1$kF,78?t$r3d$i$J$$(B
1703: $B$h$&$JAG?t(B @var{p} $B$rA*$S(B, @var{p} $B$rMQ$$$?(B trace-lifting $B$K$h$j<-=q<0(B
1704: $B=g=x$N%0%l%V%J4pDl8uJd$r5a$a(B, $B$b$75a$^$C$?$J$i%A%'%C%/$J$7$K$=$l$,5a$a$k(B
1705: $B%0%l%V%J4pDl$H$J$k(B. $B$b$7<:GT$7$?$i(B, @var{p} $B$r$H$jD>$7$F$d$jD>$9(B.
1706: @item
1707: @var{G0} $B$,(B 0 $B<!85%7%9%F%`$N$H$-(B, @var{G0} $B$rF~NO$H$7$F(B,
1708: $B$^$:(B, @var{vlist2} $B$N:G8e$NJQ?t0J30$r>C5n$9$k>C5n=g=x$K$h$j(B
1709: $B%0%l%V%J4pDl(B @var{G1} $B$r7W;;$7(B, $B$=$l$+$i<-=q<0=g=x$N%0%l%V%J4pDl$r(B
1710: $B7W;;$9$k(B. $B$=$N:](B, $B3F%9%F%C%W$G$O(B, $BF~NO$N3F85$N(B, $B5a$a$k=g=x$K$*$1$k(B
1711: $BF,78?t$r3d$i$J$$AG?t$rMQ$$$?(B trace-lifting $B$G%0%l%V%J4pDl8uJd$r5a$a(B,
1712: $B$b$75a$^$C$?$i%A%'%C%/$J$7$K$=$l$,$=$N=g=x$G$N%0%l%V%J4pDl$H$J$k(B.
1713: @end enumerate
1714:
1715: @item
1716: $BM-M}<078?t$N7W;;$O(B, @code{lex_tl()}, @code{tolex_tl()} $B$N$_<u$1IU$1$k(B.
1717: @item
1718: @code{homo} $B$,(B 0 $B$G$J$$>l9g(B, $BFbIt$G5/F0$5$l$k(B Buchberger $B%"%k%4%j%:%`$K(B
1719: $B$*$$$F(B, $B@F<!2=$,9T$o$l$k(B.
1720: @item
1721: @code{tolex_d()} $B$GI=<($5$l$k;~4V$O(B, $B$3$NH!?t$,<B9T$5$l$F$$$k%W%m%;%9$K(B
1722: $B$*$$$F9T$o$l$?7W;;$KBP1~$7$F$$$F(B, $B;R%W%m%;%9$K$*$1$k;~4V$O4^$^$l$J$$(B.
1.2 noro 1723: \E
1724: \BEG
1725: @item
1726: These functions are defined in @samp{gr} in the standard library
1727: directory.
1728: @item
1729: @code{lex_hensel()} and @code{lex_tl()} first compute a Groebner basis
1730: with respect to the variable order @var{vlist1} and the order type @var{order}.
1731: Then the Groebner basis is converted into a lex order Groebner basis
1732: with respect to the varable order @var{vlist2}.
1733: @item
1734: @code{tolex()} and @code{tolex_tl()} convert a Groebner basis @var{plist}
1735: with respect to the variable order @var{vlist1} and the order type @var{order}
1736: into a lex order Groebner basis
1737: with respect to the varable order @var{vlist2}.
1738: @code{tolex_d()} does computations of basis elements in @code{tolex()}
1739: in parallel on the processes in a child process list @var{procs}.
1740: @item
1741: In @code{lex_hensel()} and @code{tolex_hensel()} a lex order Groebner basis
1742: is computed as follows.(Refer to @code{[Noro,Yokoyama]}.)
1743: @enumerate
1744: @item
1745: Compute a Groebner basis @var{G0} with respect to @var{vlist1} and @var{order}.
1746: (Only in @code{lex_hensel()}. )
1747: @item
1748: Choose a prime which does not divide head coefficients of elements in @var{G0}
1749: with respect to @var{vlist1} and @var{order}. Then compute a lex order
1750: Groebner basis @var{Gp} over GF(@var{p}) with respect to @var{vlist2}.
1751: @item
1752: Compute @var{NF}, the set of all the normal forms with respect to
1753: @var{G0} of terms appearing in @var{Gp}.
1754: @item
1755: For each element @var{f} in @var{Gp}, replace coefficients and terms in @var{f}
1756: with undetermined coefficients and the corresponding polynomials in @var{NF}
1757: respectively, and generate a system of liear equation @var{Lf} by equating
1758: the coefficients of terms in the replaced polynomial with 0.
1759: @item
1760: Solve @var{Lf} by Hensel lifting, starting from the unique mod @var{p}
1761: solution.
1762: @item
1763: If all the linear equations generated from the elements in @var{Gp}
1764: could be solved, then the set of solutions corresponds to a lex order
1765: Groebner basis. Otherwise redo the whole process with another @var{p}.
1766: @end enumerate
1767:
1768: @item
1769: In @code{lex_tl()} and @code{tolex_tl()} a lex order Groebner basis
1770: is computed as follows.(Refer to @code{[Noro,Yokoyama]}.)
1771:
1772: @enumerate
1773: @item
1774: Compute a Groebner basis @var{G0} with respect to @var{vlist1} and @var{order}.
1775: (Only in @code{lex_tl()}. )
1776: @item
1777: If @var{G0} is not zero-dimensional, choose a prime which does not divide
1778: head coefficients of elements in @var{G0} with respect to @var{vlist1} and
1779: @var{order}. Then compute a candidate of a lex order Groebner basis
1780: via trace lifting with @var{p}. If it succeeds the candidate is indeed
1781: a lex order Groebner basis without any check. Otherwise redo the whole
1782: process with another @var{p}.
1783: @item
1784:
1785: If @var{G0} is zero-dimensional, starting from @var{G0},
1786: compute a Groebner basis @var{G1} with respect to an elimination order
1787: to eliminate variables other than the last varibale in @var{vlist2}.
1788: Then compute a lex order Groebner basis stating from @var{G1}. These
1789: computations are done by trace lifting and the selection of a mudulus
1790: @var{p} is the same as in non zero-dimensional cases.
1791: @end enumerate
1792:
1793: @item
1794: Computations with rational function coefficients can be done only by
1795: @code{lex_tl()} and @code{tolex_tl()}.
1796: @item
1797: If @code{homo} is not equal to 0, homogenization is used in Buchberger
1798: algorithm.
1799: @item
1800: The CPU time shown after an execution of @code{tolex_d()} indicates
1801: that of the master process, and it does not include the time in child
1802: processes.
1803: \E
1.1 noro 1804: @end itemize
1805:
1806: @example
1807: [78] K=katsura(5)$
1808: 30msec + gc : 20msec
1809: [79] V=[u5,u4,u3,u2,u1,u0]$
1810: 0msec
1811: [80] G0=hgr(K,V,2)$
1812: 91.558sec + gc : 15.583sec
1813: [81] G1=lex_hensel(K,V,0,V,0)$
1814: 49.049sec + gc : 9.961sec
1815: [82] G2=lex_tl(K,V,0,V,1)$
1816: 31.186sec + gc : 3.500sec
1817: [83] gb_comp(G0,G1);
1818: 1
1819: 10msec
1820: [84] gb_comp(G0,G2);
1821: 1
1822: @end example
1823:
1824: @table @t
1.2 noro 1825: \JP @item $B;2>H(B
1826: \EG @item References
1.6 noro 1827: @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 1828: \JP @fref{dp_ord}, @fref{$BJ,;67W;;(B}
1829: \EG @fref{dp_ord}, @fref{Distributed computation}
1.1 noro 1830: @end table
1831:
1.2 noro 1832: \JP @node lex_hensel_gsl tolex_gsl tolex_gsl_d,,, $B%0%l%V%J4pDl$K4X$9$kH!?t(B
1833: \EG @node lex_hensel_gsl tolex_gsl tolex_gsl_d,,, Functions for Groebner basis computation
1.1 noro 1834: @subsection @code{lex_hensel_gsl}, @code{tolex_gsl}, @code{tolex_gsl_d}
1835: @findex lex_hensel_gsl
1836: @findex tolex_gsl
1837: @findex tolex_gsl_d
1838:
1839: @table @t
1840: @item lex_hensel_gsl(@var{plist},@var{vlist1},@var{order},@var{vlist2},@var{homo})
1.2 noro 1841: \JP :: GSL $B7A<0$N%$%G%"%k4pDl$N7W;;(B
1842: \EG ::Computation of an GSL form ideal basis
1.8 noro 1843: @item tolex_gsl(@var{plist},@var{vlist1},@var{order},@var{vlist2})
1844: @itemx tolex_gsl_d(@var{plist},@var{vlist1},@var{order},@var{vlist2},@var{procs})
1.2 noro 1845: \JP :: $B%0%l%V%J4pDl$rF~NO$H$9$k(B, GSL $B7A<0$N%$%G%"%k4pDl$N7W;;(B
1846: \EG :: Computation of an GSL form ideal basis stating from a Groebner basis
1.1 noro 1847: @end table
1848:
1849: @table @var
1850: @item return
1.2 noro 1851: \JP $B%j%9%H(B
1852: \EG list
1.4 noro 1853: @item plist vlist1 vlist2 procs
1.2 noro 1854: \JP $B%j%9%H(B
1855: \EG list
1.1 noro 1856: @item order
1.2 noro 1857: \JP $B?t(B, $B%j%9%H$^$?$O9TNs(B
1858: \EG number, list or matrix
1.1 noro 1859: @item homo
1.2 noro 1860: \JP $B%U%i%0(B
1861: \EG flag
1.1 noro 1862: @end table
1863:
1864: @itemize @bullet
1.2 noro 1865: \BJP
1.1 noro 1866: @item
1867: @code{lex_hensel_gsl()} $B$O(B @code{lex_hensel()} $B$N(B, @code{tolex_gsl()} $B$O(B
1868: @code{tolex()} $B$NJQ<o$G(B, $B7k2L$N$_$,0[$J$k(B.
1869: @code{tolex_gsl_d()} $B$O(B, $B4pDl7W;;$r(B, @code{procs} $B$G;XDj$5$l$k;R%W%m%;%9$K(B
1870: $BJ,;67W;;$5$;$k(B.
1871: @item
1872: $BF~NO$,(B 0 $B<!85%7%9%F%`$G(B, $B$=$N<-=q<0=g=x%0%l%V%J4pDl$,(B
1873: @code{[f0,x1-f1,...,xn-fn]} (@code{f0},...,@code{fn} $B$O(B
1874: @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,
1875: @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)
1876: $B$rJV$9(B.
1.2 noro 1877: $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 1878: @code{x0} $B$N(B1 $BJQ?tB?9`<0$G(B,
1879: $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')]}
1880: $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
1881: $B$h$kDL>o$N%0%l%V%J4pDl$rJV$9(B.
1882: @item
1883: GSL $B7A<0$K$h$jI=$5$l$k4pDl$O%0%l%V%J4pDl$G$O$J$$$,(B, $B0lHL$K78?t$,(B SL $B7A<0(B
1884: $B$N%0%l%V%J4pDl$h$jHs>o$K>.$5$$$?$a7W;;$bB.$/(B, $B2r$b5a$a$d$9$$(B.
1885: @code{tolex_gsl_d()} $B$GI=<($5$l$k;~4V$O(B, $B$3$NH!?t$,<B9T$5$l$F$$$k%W%m%;%9$K(B
1886: $B$*$$$F9T$o$l$?7W;;$KBP1~$7$F$$$F(B, $B;R%W%m%;%9$K$*$1$k;~4V$O4^$^$l$J$$(B.
1.2 noro 1887: \E
1888: \BEG
1889: @item
1890: @code{lex_hensel_gsl()} and @code{lex_hensel()} are variants of
1891: @code{tolex_gsl()} and @code{tolex()} respectively. The results are
1892: Groebner basis or a kind of ideal basis, called GSL form.
1893: @code{tolex_gsl_d()} does basis computations in parallel on child
1894: processes specified in @code{procs}.
1895:
1896: @item
1897: If the input is zero-dimensional and a lex order Groebner basis has
1898: the form @code{[f0,x1-f1,...,xn-fn]} (@code{f0},...,@code{fn} are
1899: univariate polynomials of @code{x0}; SL form), then this these
1900: functions return a list such as
1901: @code{[[x1,g1,d1],...,[xn,gn,dn],[x0,f0,f0']]} (GSL form). In this list
1902: @code{gi} is a univariate polynomial of @code{x0} such that
1903: @code{di*f0'*fi-gi} divides @code{f0} and the roots of the input ideal is
1904: @code{[x1=g1/(d1*f0'),...,xn=gn/(dn*f0')]} for @code{x0}
1905: such that @code{f0(x0)=0}.
1906: If the lex order Groebner basis does not have the above form,
1907: these functions return
1908: a lex order Groebner basis computed by @code{tolex()}.
1909: @item
1910: Though an ideal basis represented as GSL form is not a Groebner basis
1911: we can expect that the coefficients are much smaller than those in a Groebner
1912: basis and that the computation is efficient.
1913: The CPU time shown after an execution of @code{tolex_gsl_d()} indicates
1914: that of the master process, and it does not include the time in child
1915: processes.
1916: \E
1.1 noro 1917: @end itemize
1918:
1919: @example
1920: [103] K=katsura(5)$
1921: [104] V=[u5,u4,u3,u2,u1,u0]$
1922: [105] G0=gr(K,V,0)$
1923: [106] GSL=tolex_gsl(G0,V,0,V)$
1924: [107] GSL[0];
1925: [u1,8635837421130477667200000000*u0^31-...]
1926: [108] GSL[1];
1927: [u2,10352277157007342793600000000*u0^31-...]
1928: [109] GSL[5];
1.5 noro 1929: [u0,11771021876193064124640000000*u0^32-...,
1930: 376672700038178051988480000000*u0^31-...]
1.1 noro 1931: @end example
1932:
1933: @table @t
1.2 noro 1934: \JP @item $B;2>H(B
1935: \EG @item References
1.1 noro 1936: @fref{lex_hensel lex_tl tolex tolex_d tolex_tl},
1.2 noro 1937: \JP @fref{$BJ,;67W;;(B}
1938: \EG @fref{Distributed computation}
1.1 noro 1939: @end table
1940:
1.2 noro 1941: \JP @node gr_minipoly minipoly,,, $B%0%l%V%J4pDl$K4X$9$kH!?t(B
1942: \EG @node gr_minipoly minipoly,,, Functions for Groebner basis computation
1.1 noro 1943: @subsection @code{gr_minipoly}, @code{minipoly}
1944: @findex gr_minipoly
1945: @findex minipoly
1946:
1947: @table @t
1948: @item gr_minipoly(@var{plist},@var{vlist},@var{order},@var{poly},@var{v},@var{homo})
1.2 noro 1949: \JP :: $BB?9`<0$N(B, $B%$%G%"%k$rK!$H$7$?:G>.B?9`<0$N7W;;(B
1950: \EG :: Computation of the minimal polynomial of a polynomial modulo an ideal
1.1 noro 1951: @item minipoly(@var{plist},@var{vlist},@var{order},@var{poly},@var{v})
1.2 noro 1952: \JP :: $B%0%l%V%J4pDl$rF~NO$H$9$k(B, $BB?9`<0$N:G>.B?9`<0$N7W;;(B
1953: \EG :: Computation of the minimal polynomial of a polynomial modulo an ideal
1.1 noro 1954: @end table
1955:
1956: @table @var
1957: @item return
1.2 noro 1958: \JP $BB?9`<0(B
1959: \EG polynomial
1.4 noro 1960: @item plist vlist
1.2 noro 1961: \JP $B%j%9%H(B
1962: \EG list
1.1 noro 1963: @item order
1.2 noro 1964: \JP $B?t(B, $B%j%9%H$^$?$O9TNs(B
1965: \EG number, list or matrix
1.1 noro 1966: @item poly
1.2 noro 1967: \JP $BB?9`<0(B
1968: \EG polynomial
1.1 noro 1969: @item v
1.2 noro 1970: \JP $BITDj85(B
1971: \EG indeterminate
1.1 noro 1972: @item homo
1.2 noro 1973: \JP $B%U%i%0(B
1974: \EG flag
1.1 noro 1975: @end table
1976:
1977: @itemize @bullet
1.2 noro 1978: \BJP
1.1 noro 1979: @item
1980: @code{gr_minipoly()} $B$O%0%l%V%J4pDl$N7W;;$+$i9T$$(B, @code{minipoly()} $B$O(B
1981: $BF~NO$r%0%l%V%J4pDl$H$_$J$9(B.
1982: @item
1983: $B%$%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,
1984: K[@var{v}] $B$N85(B f(@var{v}) $B$K(B f(@var{p}) mod I $B$rBP1~$5$;$k(B
1985: $B4D=`F17?$N3K$O(B 0 $B$G$J$$B?9`<0$K$h$j@8@.$5$l$k(B. $B$3$N@8@.85$r(B @var{p}
1986: $B$N(B, $BK!(B @var{I} $B$G$N:G>.B?9`<0$H8F$V(B.
1987: @item
1988: @code{gr_minipoly()}, @code{minipoly()} $B$O(B, $BB?9`<0(B @var{p} $B$N:G>.B?9`<0(B
1989: $B$r5a$a(B, @var{v} $B$rJQ?t$H$9$kB?9`<0$H$7$FJV$9(B.
1990: @item
1991: $B:G>.B?9`<0$O(B, $B%0%l%V%J4pDl$N(B 1 $B$D$N85$H$7$F7W;;$9$k$3$H$b$G$-$k$,(B,
1992: $B:G>.B?9`<0$N$_$r5a$a$?$$>l9g(B, @code{minipoly()}, @code{gr_minipoly()} $B$O(B
1993: $B%0%l%V%J4pDl$rMQ$$$kJ}K!$KHf$Y$F8zN($,$h$$(B.
1994: @item
1995: @code{gr_minipoly()} $B$K;XDj$9$k9`=g=x$H$7$F$O(B, $BDL>oA4<!?t5U<-=q<0=g=x$r(B
1996: $BMQ$$$k(B.
1.2 noro 1997: \E
1998: \BEG
1999: @item
2000: @code{gr_minipoly()} begins by computing a Groebner basis.
2001: @code{minipoly()} regards an input as a Groebner basis with respect to
2002: the variable order @var{vlist} and the order type @var{order}.
2003: @item
2004: Let K be a field. If an ideal @var{I} in K[X] is zero-dimensional, then, for
2005: a polynomial @var{p} in K[X], the kernel of a homomorphism from
2006: K[@var{v}] to K[X]/@var{I} which maps f(@var{v}) to f(@var{p}) mod @var{I}
2007: is generated by a polynomial. The generator is called the minimal polynomial
2008: of @var{p} modulo @var{I}.
2009: @item
2010: @code{gr_minipoly()} and @code{minipoly()} computes the minimal polynomial
2011: of a polynomial @var{p} and returns it as a polynomial of @var{v}.
2012: @item
2013: The minimal polynomial can be computed as an element of a Groebner basis.
2014: But if we are only interested in the minimal polynomial,
2015: @code{minipoly()} and @code{gr_minipoly()} can compute it more efficiently
2016: than methods using Groebner basis computation.
2017: @item
2018: It is recommended to use a degree reverse lex order as a term order
2019: for @code{gr_minipoly()}.
2020: \E
1.1 noro 2021: @end itemize
2022:
2023: @example
2024: [117] G=tolex(G0,V,0,V)$
2025: 43.818sec + gc : 11.202sec
2026: [118] GSL=tolex_gsl(G0,V,0,V)$
2027: 17.123sec + gc : 2.590sec
2028: [119] MP=minipoly(G0,V,0,u0,z)$
2029: 4.370sec + gc : 780msec
2030: @end example
2031:
2032: @table @t
1.2 noro 2033: \JP @item $B;2>H(B
2034: \EG @item References
1.1 noro 2035: @fref{lex_hensel lex_tl tolex tolex_d tolex_tl}.
2036: @end table
2037:
1.2 noro 2038: \JP @node tolexm minipolym,,, $B%0%l%V%J4pDl$K4X$9$kH!?t(B
2039: \EG @node tolexm minipolym,,, Functions for Groebner basis computation
1.1 noro 2040: @subsection @code{tolexm}, @code{minipolym}
2041: @findex tolexm
2042: @findex minipolym
2043:
2044: @table @t
2045: @item tolexm(@var{plist},@var{vlist1},@var{order},@var{vlist2},@var{mod})
1.2 noro 2046: \JP :: $BK!(B @var{mod} $B$G$N4pDlJQ49$K$h$k%0%l%V%J4pDl7W;;(B
2047: \EG :: Groebner basis computation modulo @var{mod} by change of ordering.
1.1 noro 2048: @item minipolym(@var{plist},@var{vlist1},@var{order},@var{poly},@var{v},@var{mod})
1.2 noro 2049: \JP :: $BK!(B @var{mod} $B$G$N%0%l%V%J4pDl$K$h$kB?9`<0$N:G>.B?9`<0$N7W;;(B
2050: \EG :: Minimal polynomial computation modulo @var{mod} the same method as
1.1 noro 2051: @end table
2052:
2053: @table @var
2054: @item return
1.2 noro 2055: \JP @code{tolexm()} : $B%j%9%H(B, @code{minipolym()} : $BB?9`<0(B
2056: \EG @code{tolexm()} : list, @code{minipolym()} : polynomial
1.4 noro 2057: @item plist vlist1 vlist2
1.2 noro 2058: \JP $B%j%9%H(B
2059: \EG list
1.1 noro 2060: @item order
1.2 noro 2061: \JP $B?t(B, $B%j%9%H$^$?$O9TNs(B
2062: \EG number, list or matrix
1.1 noro 2063: @item mod
1.2 noro 2064: \JP $BAG?t(B
2065: \EG prime
1.1 noro 2066: @end table
2067:
2068: @itemize @bullet
1.2 noro 2069: \BJP
1.1 noro 2070: @item
2071: $BF~NO(B @var{plist} $B$O$$$:$l$b(B $BJQ?t=g=x(B @var{vlist1}, $B9`=g=x7?(B @var{order},
2072: $BK!(B @var{mod} $B$K$*$1$k%0%l%V%J4pDl$G$J$1$l$P$J$i$J$$(B.
2073: @item
2074: @code{minipolym()} $B$O(B @code{minipoly} $B$KBP1~$9$k7W;;$rK!(B @var{mod}$B$G9T$&(B.
2075: @item
2076: @code{tolexm()} $B$O(B FGLM $BK!$K$h$k4pDlJQ49$K$h$j(B @var{vlist2},
2077: $B<-=q<0=g=x$K$h$k%0%l%V%J4pDl$r7W;;$9$k(B.
1.2 noro 2078: \E
2079: \BEG
2080: @item
2081: An input @var{plist} must be a Groebner basis modulo @var{mod}
2082: with respect to the variable order @var{vlist1} and the order type @var{order}.
2083: @item
2084: @code{minipolym()} executes the same computation as in @code{minipoly}.
2085: @item
2086: @code{tolexm()} computes a lex order Groebner basis modulo @var{mod}
2087: with respect to the variable order @var{vlist2}, by using FGLM algorithm.
2088: \E
1.1 noro 2089: @end itemize
2090:
2091: @example
2092: [197] tolexm(G0,V,0,V,31991);
2093: [8271*u0^31+10435*u0^30+816*u0^29+26809*u0^28+...,...]
2094: [198] minipolym(G0,V,0,u0,z,31991);
2095: z^32+11405*z^31+20868*z^30+21602*z^29+...
2096: @end example
2097:
2098: @table @t
1.2 noro 2099: \JP @item $B;2>H(B
2100: \EG @item References
1.1 noro 2101: @fref{lex_hensel lex_tl tolex tolex_d tolex_tl},
2102: @fref{gr_minipoly minipoly}.
2103: @end table
2104:
1.6 noro 2105: \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
2106: \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
2107: @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 2108: @findex dp_gr_main
2109: @findex dp_gr_mod_main
1.5 noro 2110: @findex dp_gr_f_main
1.6 noro 2111: @findex dp_weyl_gr_main
2112: @findex dp_weyl_gr_mod_main
2113: @findex dp_weyl_gr_f_main
1.1 noro 2114:
2115: @table @t
2116: @item dp_gr_main(@var{plist},@var{vlist},@var{homo},@var{modular},@var{order})
2117: @itemx dp_gr_mod_main(@var{plist},@var{vlist},@var{homo},@var{modular},@var{order})
1.5 noro 2118: @itemx dp_gr_f_main(@var{plist},@var{vlist},@var{homo},@var{order})
1.6 noro 2119: @itemx dp_weyl_gr_main(@var{plist},@var{vlist},@var{homo},@var{modular},@var{order})
2120: @itemx dp_weyl_gr_mod_main(@var{plist},@var{vlist},@var{homo},@var{modular},@var{order})
2121: @itemx dp_weyl_gr_f_main(@var{plist},@var{vlist},@var{homo},@var{order})
1.2 noro 2122: \JP :: $B%0%l%V%J4pDl$N7W;;(B ($BAH$_9~$_H!?t(B)
2123: \EG :: Groebner basis computation (built-in functions)
1.1 noro 2124: @end table
2125:
2126: @table @var
2127: @item return
1.2 noro 2128: \JP $B%j%9%H(B
2129: \EG list
1.4 noro 2130: @item plist vlist
1.2 noro 2131: \JP $B%j%9%H(B
2132: \EG list
1.1 noro 2133: @item order
1.2 noro 2134: \JP $B?t(B, $B%j%9%H$^$?$O9TNs(B
2135: \EG number, list or matrix
1.1 noro 2136: @item homo
1.2 noro 2137: \JP $B%U%i%0(B
2138: \EG flag
1.1 noro 2139: @item modular
1.2 noro 2140: \JP $B%U%i%0$^$?$OAG?t(B
2141: \EG flag or prime
1.1 noro 2142: @end table
2143:
2144: @itemize @bullet
1.2 noro 2145: \BJP
1.1 noro 2146: @item
2147: $B$3$l$i$NH!?t$O(B, $B%0%l%V%J4pDl7W;;$N4pK\E*AH$_9~$_H!?t$G$"$j(B, @code{gr()},
2148: @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 2149: $B$r9T$C$F$$$k(B. $B4X?tL>$K(B weyl $B$,F~$C$F$$$k$b$N$O(B, Weyl $BBe?t>e$N7W;;(B
2150: $B$N$?$a$N4X?t$G$"$k(B.
1.1 noro 2151: @item
1.6 noro 2152: @code{dp_gr_f_main()}, @code{dp_weyl_f_main()} $B$O(B, $B<o!9$NM-8BBN>e$N%0%l%V%J4pDl$r7W;;$9$k(B
1.5 noro 2153: $B>l9g$KMQ$$$k(B. $BF~NO$O(B, $B$"$i$+$8$a(B, @code{simp_ff()} $B$J$I$G(B,
2154: $B9M$($kM-8BBN>e$K<M1F$5$l$F$$$kI,MW$,$"$k(B.
2155: @item
1.1 noro 2156: $B%U%i%0(B @var{homo} $B$,(B 0 $B$G$J$$;~(B, $BF~NO$r@F<!2=$7$F$+$i(B Buchberger $B%"%k%4%j%:%`(B
2157: $B$r<B9T$9$k(B.
2158: @item
2159: @code{dp_gr_mod_main()} $B$KBP$7$F$O(B, @var{modular} $B$O(B, GF(@var{modular}) $B>e(B
2160: $B$G$N7W;;$r0UL#$9$k(B.
2161: @code{dp_gr_main()} $B$KBP$7$F$O(B, @var{modular} $B$O<!$N$h$&$J0UL#$r;}$D(B.
2162: @enumerate
2163: @item
2164: @var{modular} $B$,(B 1 $B$N;~(B, trace-lifting $B$K$h$k7W;;$r9T$&(B. $BAG?t$O(B
2165: @code{lprime(0)} $B$+$i=g$K@.8y$9$k$^$G(B @code{lprime()} $B$r8F$S=P$7$F@8@.$9$k(B.
2166: @item
2167: @var{modular} $B$,(B 2 $B0J>e$N<+A3?t$N;~(B, $B$=$NCM$rAG?t$H$_$J$7$F(B trace-lifting
2168: $B$r9T$&(B. $B$=$NAG?t$G<:GT$7$?>l9g(B, 0 $B$rJV$9(B.
2169: @item
2170: @var{modular} $B$,Ii$N>l9g(B,
2171: @var{-modular} $B$KBP$7$F>e=R$N5,B'$,E,MQ$5$l$k$,(B, trace-lifting $B$N:G=*(B
2172: $BCJ3,$N%0%l%V%J4pDl%A%'%C%/$H%$%G%"%k%a%s%P%7%C%W%A%'%C%/$,>JN,$5$l$k(B.
2173: @end enumerate
2174:
2175: @item
2176: @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
2177: @code{dp_gr_main(P,V,1,1,O)}, @code{gr_mod(P,V,O,M)} $B$O(B
2178: @code{dp_gr_mod_main(P,V,0,M,O)} $B$r$=$l$>$l<B9T$9$k(B.
2179: @item
2180: @var{homo}, @var{modular} $B$NB>$K(B, @code{dp_gr_flags()} $B$G@_Dj$5$l$k(B
2181: $B$5$^$6$^$J%U%i%0$K$h$j7W;;$,@)8f$5$l$k(B.
1.2 noro 2182: \E
2183: \BEG
2184: @item
2185: These functions are fundamental built-in functions for Groebner basis
2186: computation and @code{gr()},@code{hgr()} and @code{gr_mod()}
1.6 noro 2187: are all interfaces to these functions. Functions whose names
2188: contain weyl are those for computation in Weyl algebra.
1.2 noro 2189: @item
1.6 noro 2190: @code{dp_gr_f_main()} and @code{dp_weyl_gr_f_main()}
2191: are functions for Groebner basis computation
1.5 noro 2192: over various finite fields. Coefficients of input polynomials
2193: must be converted to elements of a finite field
2194: currently specified by @code{setmod_ff()}.
2195: @item
1.2 noro 2196: If @var{homo} is not equal to 0, homogenization is applied before entering
2197: Buchberger algorithm
2198: @item
2199: For @code{dp_gr_mod_main()}, @var{modular} means a computation over
2200: GF(@var{modular}).
2201: For @code{dp_gr_main()}, @var{modular} has the following mean.
2202: @enumerate
2203: @item
2204: If @var{modular} is 1 , trace lifting is used. Primes for trace lifting
2205: are generated by @code{lprime()}, starting from @code{lprime(0)}, until
2206: the computation succeeds.
2207: @item
2208: If @var{modular} is an integer greater than 1, the integer is regarded as a
2209: prime and trace lifting is executed by using the prime. If the computation
2210: fails then 0 is returned.
2211: @item
2212: If @var{modular} is negative, the above rule is applied for @var{-modular}
2213: but the Groebner basis check and ideal-membership check are omitted in
2214: the last stage of trace lifting.
2215: @end enumerate
2216:
2217: @item
2218: @code{gr(P,V,O)}, @code{hgr(P,V,O)} and @code{gr_mod(P,V,O,M)} execute
2219: @code{dp_gr_main(P,V,0,1,O)}, @code{dp_gr_main(P,V,1,1,O)}
2220: and @code{dp_gr_mod_main(P,V,0,M,O)} respectively.
2221: @item
2222: Actual computation is controlled by various parameters set by
2223: @code{dp_gr_flags()}, other then by @var{homo} and @var{modular}.
2224: \E
1.1 noro 2225: @end itemize
2226:
2227: @table @t
1.2 noro 2228: \JP @item $B;2>H(B
2229: \EG @item References
1.1 noro 2230: @fref{dp_ord},
2231: @fref{dp_gr_flags dp_gr_print},
2232: @fref{gr hgr gr_mod},
1.5 noro 2233: @fref{setmod_ff},
1.2 noro 2234: \JP @fref{$B7W;;$*$h$SI=<($N@)8f(B}.
2235: \EG @fref{Controlling Groebner basis computations}
1.1 noro 2236: @end table
2237:
1.6 noro 2238: \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
2239: \EG @node dp_f4_main dp_f4_mod_main dp_weyl_f4_main dp_weyl_f4_mod_main,,, Functions for Groebner basis computation
2240: @subsection @code{dp_f4_main}, @code{dp_f4_mod_main}, @code{dp_weyl_f4_main}, @code{dp_weyl_f4_mod_main}
1.1 noro 2241: @findex dp_f4_main
2242: @findex dp_f4_mod_main
1.6 noro 2243: @findex dp_weyl_f4_main
2244: @findex dp_weyl_f4_mod_main
1.1 noro 2245:
2246: @table @t
2247: @item dp_f4_main(@var{plist},@var{vlist},@var{order})
2248: @itemx dp_f4_mod_main(@var{plist},@var{vlist},@var{order})
1.6 noro 2249: @itemx dp_weyl_f4_main(@var{plist},@var{vlist},@var{order})
2250: @itemx dp_weyl_f4_mod_main(@var{plist},@var{vlist},@var{order})
1.2 noro 2251: \JP :: F4 $B%"%k%4%j%:%`$K$h$k%0%l%V%J4pDl$N7W;;(B ($BAH$_9~$_H!?t(B)
2252: \EG :: Groebner basis computation by F4 algorithm (built-in functions)
1.1 noro 2253: @end table
2254:
2255: @table @var
2256: @item return
1.2 noro 2257: \JP $B%j%9%H(B
2258: \EG list
1.4 noro 2259: @item plist vlist
1.2 noro 2260: \JP $B%j%9%H(B
2261: \EG list
1.1 noro 2262: @item order
1.2 noro 2263: \JP $B?t(B, $B%j%9%H$^$?$O9TNs(B
2264: \EG number, list or matrix
1.1 noro 2265: @end table
2266:
2267: @itemize @bullet
1.2 noro 2268: \BJP
1.1 noro 2269: @item
2270: F4 $B%"%k%4%j%:%`$K$h$j%0%l%V%J4pDl$N7W;;$r9T$&(B.
2271: @item
2272: F4 $B%"%k%4%j%:%`$O(B, J.C. Faugere $B$K$h$jDs>'$5$l$??7@$Be%0%l%V%J4pDl(B
2273: $B;;K!$G$"$j(B, $BK\<BAu$O(B, $BCf9q>jM>DjM}$K$h$k@~7AJ}Dx<05a2r$rMQ$$$?(B
2274: $B;n83E*$J<BAu$G$"$k(B.
2275: @item
1.6 noro 2276: $B@F<!2=$N0z?t$,$J$$$3$H$r=|$1$P(B, $B0z?t$*$h$SF0:n$O$=$l$>$l(B
2277: @code{dp_gr_main()}, @code{dp_gr_mod_main()},
2278: @code{dp_weyl_gr_main()}, @code{dp_weyl_gr_mod_main()}
1.1 noro 2279: $B$HF1MM$G$"$k(B.
1.2 noro 2280: \E
2281: \BEG
2282: @item
2283: These functions compute Groebner bases by F4 algorithm.
2284: @item
2285: F4 is a new generation algorithm for Groebner basis computation
2286: invented by J.C. Faugere. The current implementation of @code{dp_f4_main()}
2287: uses Chinese Remainder theorem and not highly optimized.
2288: @item
2289: Arguments and actions are the same as those of
1.6 noro 2290: @code{dp_gr_main()}, @code{dp_gr_mod_main()},
2291: @code{dp_weyl_gr_main()}, @code{dp_weyl_gr_mod_main()},
2292: except for lack of the argument for controlling homogenization.
1.2 noro 2293: \E
1.1 noro 2294: @end itemize
2295:
2296: @table @t
1.2 noro 2297: \JP @item $B;2>H(B
2298: \EG @item References
1.1 noro 2299: @fref{dp_ord},
2300: @fref{dp_gr_flags dp_gr_print},
2301: @fref{gr hgr gr_mod},
1.2 noro 2302: \JP @fref{$B7W;;$*$h$SI=<($N@)8f(B}.
2303: \EG @fref{Controlling Groebner basis computations}
1.1 noro 2304: @end table
2305:
1.2 noro 2306: \JP @node dp_gr_flags dp_gr_print,,, $B%0%l%V%J4pDl$K4X$9$kH!?t(B
2307: \EG @node dp_gr_flags dp_gr_print,,, Functions for Groebner basis computation
1.1 noro 2308: @subsection @code{dp_gr_flags}, @code{dp_gr_print}
2309: @findex dp_gr_flags
2310: @findex dp_gr_print
2311:
2312: @table @t
2313: @item dp_gr_flags([@var{list}])
1.7 noro 2314: @itemx dp_gr_print([@var{i}])
1.2 noro 2315: \JP :: $B7W;;$*$h$SI=<(MQ%Q%i%a%?$N@_Dj(B, $B;2>H(B
2316: \BEG :: Set and show various parameters for cotrolling computations
2317: and showing informations.
2318: \E
1.1 noro 2319: @end table
2320:
2321: @table @var
2322: @item return
1.2 noro 2323: \JP $B@_DjCM(B
2324: \EG value currently set
1.1 noro 2325: @item list
1.2 noro 2326: \JP $B%j%9%H(B
2327: \EG list
1.7 noro 2328: @item i
2329: \JP $B@0?t(B
2330: \EG integer
1.1 noro 2331: @end table
2332:
2333: @itemize @bullet
1.2 noro 2334: \BJP
1.1 noro 2335: @item
1.5 noro 2336: @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 2337: $B$J%Q%i%a%?$r@_Dj(B, $B;2>H$9$k(B.
2338: @item
2339: $B0z?t$,$J$$>l9g(B, $B8=:_$N@_Dj$,JV$5$l$k(B.
2340: @item
2341: $B0z?t$O(B, @code{["Print",1,"NoSugar",1,...]} $B$J$k7A$N%j%9%H$G(B, $B:8$+$i=g$K(B
2342: $B@_Dj$5$l$k(B. $B%Q%i%a%?L>$OJ8;zNs$GM?$($kI,MW$,$"$k(B.
2343: @item
1.7 noro 2344: @code{dp_gr_print()} $B$O(B, $BFC$K%Q%i%a%?(B @code{Print}, @code{PrintShort} $B$NCM$rD>@\@_Dj(B, $B;2>H(B
2345: $B$G$-$k(B. $B@_Dj$5$l$kCM$O<!$NDL$j$G$"$k!#(B
2346: @table @var
2347: @item i=0
2348: @code{Print=0}, @code{PrintShort=0}
2349: @item i=1
2350: @code{Print=1}, @code{PrintShort=0}
2351: @item i=2
2352: @code{Print=0}, @code{PrintShort=1}
2353: @end table
2354: $B$3$l$O(B, @code{dp_gr_main()} $B$J$I$r%5%V%k!<%A%s$H$7$FMQ$$$k%f!<%6(B
2355: $BH!?t$K$*$$$F(B, $B$=$N%5%V%k!<%A%s$,Cf4V>pJs$NI=<((B
1.1 noro 2356: $B$r9T$&:]$K(B, $B?WB.$K%U%i%0$r8+$k$3$H$,$G$-$k$h$&$KMQ0U$5$l$F$$$k(B.
1.2 noro 2357: \E
2358: \BEG
2359: @item
2360: @code{dp_gr_flags()} sets and shows various parameters for Groebner basis
2361: computation.
2362: @item
2363: If no argument is specified the current settings are returned.
2364: @item
2365: Arguments must be specified as a list such as
2366: @code{["Print",1,"NoSugar",1,...]}. Names of parameters must be character
2367: strings.
2368: @item
2369: @code{dp_gr_print()} is used to set and show the value of a parameter
1.7 noro 2370: @code{Print} and @code{PrintShort}.
2371: @table @var
2372: @item i=0
2373: @code{Print=0}, @code{PrintShort=0}
2374: @item i=1
2375: @code{Print=1}, @code{PrintShort=0}
2376: @item i=2
2377: @code{Print=0}, @code{PrintShort=1}
2378: @end table
2379: This functions is prepared to get quickly the value
2380: when a user defined function calling @code{dp_gr_main()} etc.
1.2 noro 2381: uses the value as a flag for showing intermediate informations.
2382: \E
1.1 noro 2383: @end itemize
2384:
2385: @table @t
1.2 noro 2386: \JP @item $B;2>H(B
2387: \EG @item References
2388: \JP @fref{$B7W;;$*$h$SI=<($N@)8f(B}
2389: \EG @fref{Controlling Groebner basis computations}
1.1 noro 2390: @end table
2391:
1.2 noro 2392: \JP @node dp_ord,,, $B%0%l%V%J4pDl$K4X$9$kH!?t(B
2393: \EG @node dp_ord,,, Functions for Groebner basis computation
1.1 noro 2394: @subsection @code{dp_ord}
2395: @findex dp_ord
2396:
2397: @table @t
2398: @item dp_ord([@var{order}])
1.2 noro 2399: \JP :: $BJQ?t=g=x7?$N@_Dj(B, $B;2>H(B
2400: \EG :: Set and show the ordering type.
1.1 noro 2401: @end table
2402:
2403: @table @var
2404: @item return
1.2 noro 2405: \JP $BJQ?t=g=x7?(B ($B?t(B, $B%j%9%H$^$?$O9TNs(B)
2406: \EG ordering type (number, list or matrix)
1.1 noro 2407: @item order
1.2 noro 2408: \JP $B?t(B, $B%j%9%H$^$?$O9TNs(B
2409: \EG number, list or matrix
1.1 noro 2410: @end table
2411:
2412: @itemize @bullet
1.2 noro 2413: \BJP
1.1 noro 2414: @item
2415: $B0z?t$,$"$k;~(B, $BJQ?t=g=x7?$r(B @var{order} $B$K@_Dj$9$k(B. $B0z?t$,$J$$;~(B,
2416: $B8=:_@_Dj$5$l$F$$$kJQ?t=g=x7?$rJV$9(B.
2417:
2418: @item
2419: $BJ,;6I=8=B?9`<0$K4X$9$kH!?t(B, $B1i;;$O0z?t$H$7$FJQ?t=g=x7?$r$H$k$b$N$H$H$i$J$$$b$N(B
2420: $B$,$"$j(B, $B$H$i$J$$$b$N$K4X$7$F$O(B, $B$=$N;~E@$G@_Dj$5$l$F$$$kCM$rMQ$$$F7W;;$,(B
2421: $B9T$o$l$k(B.
2422:
2423: @item
2424: @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()}
2425: $B$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.
2426:
2427: @item
2428: $BJ,;6I=8=B?9`<0$N;MB'1i;;$b(B, $B@_Dj$5$l$F$$$kCM$rMQ$$$F7W;;$5$l$k(B. $B=>$C$F(B,
2429: $B$=$NB?9`<0$,@8@.$5$l$?;~E@$K$*$1$kJQ?t=g=x7?$,(B, $B;MB'1i;;;~$K@5$7$/@_Dj(B
2430: $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
2431: $B7?$K4p$E$$$F@8@.$5$l$?$b$N$G$J$1$l$P$J$i$J$$(B.
2432:
2433: @item
2434: $B%H%C%W%l%Y%kH!?t0J30$NH!?t$rD>@\8F$S=P$9>l9g$K$O(B, $B$3$NH!?t$K$h$j(B
2435: $BJQ?t=g=x7?$r@5$7$/@_Dj$7$J$1$l$P$J$i$J$$(B.
1.2 noro 2436: \E
2437: \BEG
2438: @item
2439: If an argument is specified, the function
2440: sets the current ordering type to @var{order}.
2441: If no argument is specified, the function returns the ordering
2442: type currently set.
2443:
2444: @item
2445: There are two types of functions concerning distributed polynomial,
2446: functions which take a ordering type and those which don't take it.
2447: The latter ones use the current setting.
2448:
2449: @item
2450: Functions such as @code{gr()}, which need a ordering type as an argument,
2451: call @code{dp_ord()} internally during the execution.
2452: The setting remains after the execution.
2453:
2454: Fundamental arithmetics for distributed polynomial also use the current
2455: setting. Therefore, when such arithmetics for distributed polynomials
2456: are done, the current setting must coincide with the ordering type
2457: which was used upon the creation of the polynomials. It is assumed
2458: that such polynomials were generated under the same ordering type.
2459:
2460: @item
2461: Type of term ordering must be correctly set by this function
2462: when functions other than top level functions are called directly.
2463: \E
1.1 noro 2464: @end itemize
2465:
2466: @example
2467: [19] dp_ord(0)$
2468: [20] <<1,2,3>>+<<3,1,1>>;
2469: (1)*<<1,2,3>>+(1)*<<3,1,1>>
2470: [21] dp_ord(2)$
2471: [22] <<1,2,3>>+<<3,1,1>>;
2472: (1)*<<3,1,1>>+(1)*<<1,2,3>>
2473: @end example
2474:
2475: @table @t
1.2 noro 2476: \JP @item $B;2>H(B
2477: \EG @item References
2478: \JP @fref{$B9`=g=x$N@_Dj(B}
2479: \EG @fref{Setting term orderings}
1.1 noro 2480: @end table
2481:
1.2 noro 2482: \JP @node dp_ptod,,, $B%0%l%V%J4pDl$K4X$9$kH!?t(B
2483: \EG @node dp_ptod,,, Functions for Groebner basis computation
1.1 noro 2484: @subsection @code{dp_ptod}
2485: @findex dp_ptod
2486:
2487: @table @t
2488: @item dp_ptod(@var{poly},@var{vlist})
1.2 noro 2489: \JP :: $BB?9`<0$rJ,;6I=8=B?9`<0$KJQ49$9$k(B.
2490: \EG :: Converts an ordinary polynomial into a distributed polynomial.
1.1 noro 2491: @end table
2492:
2493: @table @var
2494: @item return
1.2 noro 2495: \JP $BJ,;6I=8=B?9`<0(B
2496: \EG distributed polynomial
1.1 noro 2497: @item poly
1.2 noro 2498: \JP $BB?9`<0(B
2499: \EG polynomial
1.1 noro 2500: @item vlist
1.2 noro 2501: \JP $B%j%9%H(B
2502: \EG list
1.1 noro 2503: @end table
2504:
2505: @itemize @bullet
1.2 noro 2506: \BJP
1.1 noro 2507: @item
2508: $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.
2509: @item
2510: @var{vlist} $B$K4^$^$l$J$$ITDj85$O(B, $B78?tBN$KB0$9$k$H$7$FJQ49$5$l$k(B.
1.2 noro 2511: \E
2512: \BEG
2513: @item
2514: According to the variable ordering @var{vlist} and current
2515: type of term ordering, this function converts an ordinary
2516: polynomial into a distributed polynomial.
2517: @item
2518: Indeterminates not included in @var{vlist} are regarded to belong to
2519: the coefficient field.
2520: \E
1.1 noro 2521: @end itemize
2522:
2523: @example
2524: [50] dp_ord(0);
2525: 1
2526: [51] dp_ptod((x+y+z)^2,[x,y,z]);
2527: (1)*<<2,0,0>>+(2)*<<1,1,0>>+(1)*<<0,2,0>>+(2)*<<1,0,1>>+(2)*<<0,1,1>>
2528: +(1)*<<0,0,2>>
2529: [52] dp_ptod((x+y+z)^2,[x,y]);
1.5 noro 2530: (1)*<<2,0>>+(2)*<<1,1>>+(1)*<<0,2>>+(2*z)*<<1,0>>+(2*z)*<<0,1>>
2531: +(z^2)*<<0,0>>
1.1 noro 2532: @end example
2533:
2534: @table @t
1.2 noro 2535: \JP @item $B;2>H(B
2536: \EG @item References
1.1 noro 2537: @fref{dp_dtop},
2538: @fref{dp_ord}.
2539: @end table
2540:
1.2 noro 2541: \JP @node dp_dtop,,, $B%0%l%V%J4pDl$K4X$9$kH!?t(B
2542: \EG @node dp_dtop,,, Functions for Groebner basis computation
1.1 noro 2543: @subsection @code{dp_dtop}
2544: @findex dp_dtop
2545:
2546: @table @t
2547: @item dp_dtop(@var{dpoly},@var{vlist})
1.2 noro 2548: \JP :: $BJ,;6I=8=B?9`<0$rB?9`<0$KJQ49$9$k(B.
2549: \EG :: Converts a distributed polynomial into an ordinary polynomial.
1.1 noro 2550: @end table
2551:
2552: @table @var
2553: @item return
1.2 noro 2554: \JP $BB?9`<0(B
2555: \EG polynomial
1.1 noro 2556: @item dpoly
1.2 noro 2557: \JP $BJ,;6I=8=B?9`<0(B
2558: \EG distributed polynomial
1.1 noro 2559: @item vlist
1.2 noro 2560: \JP $B%j%9%H(B
2561: \EG list
1.1 noro 2562: @end table
2563:
2564: @itemize @bullet
1.2 noro 2565: \BJP
1.1 noro 2566: @item
2567: $BJ,;6I=8=B?9`<0$r(B, $BM?$($i$l$?ITDj85%j%9%H$rMQ$$$FB?9`<0$KJQ49$9$k(B.
2568: @item
2569: $BITDj85%j%9%H$O(B, $BD9$5J,;6I=8=B?9`<0$NJQ?t$N8D?t$H0lCW$7$F$$$l$P2?$G$b$h$$(B.
1.2 noro 2570: \E
2571: \BEG
2572: @item
2573: This function converts a distributed polynomial into an ordinary polynomial
2574: according to a list of indeterminates @var{vlist}.
2575: @item
2576: @var{vlist} is such a list that its length coincides with the number of
2577: variables of @var{dpoly}.
2578: \E
1.1 noro 2579: @end itemize
2580:
2581: @example
2582: [53] T=dp_ptod((x+y+z)^2,[x,y]);
1.5 noro 2583: (1)*<<2,0>>+(2)*<<1,1>>+(1)*<<0,2>>+(2*z)*<<1,0>>+(2*z)*<<0,1>>
2584: +(z^2)*<<0,0>>
1.1 noro 2585: [54] P=dp_dtop(T,[a,b]);
2586: z^2+(2*a+2*b)*z+a^2+2*b*a+b^2
2587: @end example
2588:
1.2 noro 2589: \JP @node dp_mod dp_rat,,, $B%0%l%V%J4pDl$K4X$9$kH!?t(B
2590: \EG @node dp_mod dp_rat,,, Functions for Groebner basis computation
1.1 noro 2591: @subsection @code{dp_mod}, @code{dp_rat}
2592: @findex dp_mod
2593: @findex dp_rat
2594:
2595: @table @t
2596: @item dp_mod(@var{p},@var{mod},@var{subst})
1.2 noro 2597: \JP :: $BM-M}?t78?tJ,;6I=8=B?9`<0$NM-8BBN78?t$X$NJQ49(B
2598: \EG :: Converts a disributed polynomial into one with coefficients in a finite field.
1.1 noro 2599: @item dp_rat(@var{p})
1.2 noro 2600: \JP :: $BM-8BBN78?tJ,;6I=8=B?9`<0$NM-M}?t78?t$X$NJQ49(B
2601: \BEG
2602: :: Converts a distributed polynomial with coefficients in a finite field into
2603: one with coefficients in the rationals.
2604: \E
1.1 noro 2605: @end table
2606:
2607: @table @var
2608: @item return
1.2 noro 2609: \JP $BJ,;6I=8=B?9`<0(B
2610: \EG distributed polynomial
1.1 noro 2611: @item p
1.2 noro 2612: \JP $BJ,;6I=8=B?9`<0(B
2613: \EG distributed polynomial
1.1 noro 2614: @item mod
1.2 noro 2615: \JP $BAG?t(B
2616: \EG prime
1.1 noro 2617: @item subst
1.2 noro 2618: \JP $B%j%9%H(B
2619: \EG list
1.1 noro 2620: @end table
2621:
2622: @itemize @bullet
1.2 noro 2623: \BJP
1.1 noro 2624: @item
2625: @code{dp_nf_mod()}, @code{dp_true_nf_mod()} $B$O(B, $BF~NO$H$7$FM-8BBN78?t$N(B
2626: $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
2627: $BM-M}?t78?tJ,;6I=8=B?9`<0$rJQ49$7$FMQ$$$k$3$H$,$G$-$k(B. $B$^$?(B, $BF@$i$l$?(B
2628: $B7k2L$O(B, $BM-8BBN78?tB?9`<0$H$O1i;;$G$-$k$,(B, $BM-M}?t78?tB?9`<0$H$O1i;;$G$-$J$$(B
2629: $B$?$a(B, @code{dp_rat()} $B$K$h$jJQ49$9$kI,MW$,$"$k(B.
2630: @item
2631: $BM-8BBN78?t$N1i;;$K$*$$$F$O(B, $B$"$i$+$8$a(B @code{setmod()} $B$K$h$jM-8BBN$N85$N(B
2632: $B8D?t$r;XDj$7$F$*$/I,MW$,$"$k(B.
2633: @item
2634: @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
2635: $B$7$?8eM-8BBN78?t$KJQ49$9$k$H$$$&A`:n$r9T$&:]$N(B, $BBeF~CM$r;XDj$9$k$b$N$G(B,
2636: @code{[[@var{var},@var{value}],...]} $B$N7A$N%j%9%H$G$"$k(B.
1.2 noro 2637: \E
2638: \BEG
2639: @item
2640: @code{dp_nf_mod()} and @code{dp_true_nf_mod()} require
2641: distributed polynomials with coefficients in a finite field as arguments.
2642: @code{dp_mod()} is used to convert distributed polynomials with rational
2643: number coefficients into appropriate ones.
2644: Polynomials with coefficients in a finite field
2645: cannot be used as inputs of operations with polynomials
2646: with rational number coefficients. @code{dp_rat()} is used for such cases.
2647: @item
2648: The ground finite field must be set in advance by using @code{setmod()}.
2649: @item
2650: @var{subst} is such a list as @code{[[@var{var},@var{value}],...]}.
2651: This is valid when the ground field of the input polynomial is a
2652: rational function field. @var{var}'s are variables in the ground field and
2653: the list means that @var{value} is substituted for @var{var} before
2654: converting the coefficients into elements of a finite field.
2655: \E
1.1 noro 2656: @end itemize
2657:
2658: @example
2659: @end example
2660:
2661: @table @t
1.2 noro 2662: \JP @item $B;2>H(B
2663: \EG @item References
1.1 noro 2664: @fref{dp_nf dp_nf_mod dp_true_nf dp_true_nf_mod},
2665: @fref{subst psubst},
2666: @fref{setmod}.
2667: @end table
2668:
1.2 noro 2669: \JP @node dp_homo dp_dehomo,,, $B%0%l%V%J4pDl$K4X$9$kH!?t(B
2670: \EG @node dp_homo dp_dehomo,,, Functions for Groebner basis computation
1.1 noro 2671: @subsection @code{dp_homo}, @code{dp_dehomo}
2672: @findex dp_homo
2673: @findex dp_dehomo
2674:
2675: @table @t
2676: @item dp_homo(@var{dpoly})
1.2 noro 2677: \JP :: $BJ,;6I=8=B?9`<0$N@F<!2=(B
2678: \EG :: Homogenize a distributed polynomial
1.1 noro 2679: @item dp_dehomo(@var{dpoly})
1.2 noro 2680: \JP :: $B@F<!J,;6I=8=B?9`<0$NHs@F<!2=(B
2681: \EG :: Dehomogenize a homogenious distributed polynomial
1.1 noro 2682: @end table
2683:
2684: @table @var
2685: @item return
1.2 noro 2686: \JP $BJ,;6I=8=B?9`<0(B
2687: \EG distributed polynomial
1.1 noro 2688: @item dpoly
1.2 noro 2689: \JP $BJ,;6I=8=B?9`<0(B
2690: \EG distributed polynomial
1.1 noro 2691: @end table
2692:
2693: @itemize @bullet
1.2 noro 2694: \BJP
1.1 noro 2695: @item
2696: @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
2697: 1 $B?-$P$7(B, $B:G8e$N@.J,$NCM$r(B @var{d}-@code{deg(@var{t})}
2698: (@var{d} $B$O(B @var{dpoly} $B$NA4<!?t(B) $B$H$7$?J,;6I=8=B?9`<0$rJV$9(B.
2699: @item
2700: @code{dp_dehomo()} $B$O(B, @var{dpoly} $B$N3F9`$K$D$$$F(B, $B;X?t%Y%/%H%k$N:G8e$N@.J,(B
2701: $B$r<h$j=|$$$?J,;6B?9`<0$rJV$9(B.
2702: @item
2703: $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
2704: $B@5$7$/@_Dj$9$kI,MW$,$"$k(B.
2705: @item
2706: @code{hgr()} $B$J$I$K$*$$$F(B, $BFbItE*$KMQ$$$i$l$F$$$k(B.
1.2 noro 2707: \E
2708: \BEG
2709: @item
2710: @code{dp_homo()} makes a copy of @var{dpoly}, extends
2711: the length of the exponent vector of each term @var{t} in the copy by 1,
2712: and sets the value of the newly appended
2713: component to @var{d}-@code{deg(@var{t})}, where @var{d} is the total
2714: degree of @var{dpoly}.
2715: @item
2716: @code{dp_dehomo()} make a copy of @var{dpoly} and removes the last component
2717: of each terms in the copy.
2718: @item
2719: Appropriate term orderings must be set when the results are used as inputs
2720: of some operations.
2721: @item
2722: These are used internally in @code{hgr()} etc.
2723: \E
1.1 noro 2724: @end itemize
2725:
2726: @example
2727: [202] X=<<1,2,3>>+3*<<1,2,1>>;
2728: (1)*<<1,2,3>>+(3)*<<1,2,1>>
2729: [203] dp_homo(X);
2730: (1)*<<1,2,3,0>>+(3)*<<1,2,1,2>>
2731: [204] dp_dehomo(@@);
2732: (1)*<<1,2,3>>+(3)*<<1,2,1>>
2733: @end example
2734:
2735: @table @t
1.2 noro 2736: \JP @item $B;2>H(B
2737: \EG @item References
1.1 noro 2738: @fref{gr hgr gr_mod}.
2739: @end table
2740:
1.2 noro 2741: \JP @node dp_ptozp dp_prim,,, $B%0%l%V%J4pDl$K4X$9$kH!?t(B
2742: \EG @node dp_ptozp dp_prim,,, Functions for Groebner basis computation
1.1 noro 2743: @subsection @code{dp_ptozp}, @code{dp_prim}
2744: @findex dp_ptozp
2745: @findex dp_prim
2746:
2747: @table @t
2748: @item dp_ptozp(@var{dpoly})
1.2 noro 2749: \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.
2750: \BEG
2751: :: Converts a distributed polynomial @var{poly} with rational coefficients
2752: into an integral distributed polynomial such that GCD of all its coefficients
2753: is 1.
2754: \E
1.1 noro 2755: @itemx dp_prim(@var{dpoly})
1.2 noro 2756: \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.
2757: \BEG
2758: :: Converts a distributed polynomial @var{poly} with rational function
2759: coefficients into an integral distributed polynomial such that polynomial
2760: GCD of all its coefficients is 1.
2761: \E
1.1 noro 2762: @end table
2763:
2764: @table @var
2765: @item return
1.2 noro 2766: \JP $BJ,;6I=8=B?9`<0(B
2767: \EG distributed polynomial
1.1 noro 2768: @item dpoly
1.2 noro 2769: \JP $BJ,;6I=8=B?9`<0(B
2770: \EG distributed polynomial
1.1 noro 2771: @end table
2772:
2773: @itemize @bullet
1.2 noro 2774: \BJP
1.1 noro 2775: @item
2776: @code{dp_ptozp()} $B$O(B, @code{ptozp()} $B$KAjEv$9$kA`:n$rJ,;6I=8=B?9`<0$K(B
2777: $BBP$7$F9T$&(B. $B78?t$,B?9`<0$r4^$`>l9g(B, $B78?t$K4^$^$l$kB?9`<06&DL0x;R$O(B
2778: $B<h$j=|$+$J$$(B.
2779: @item
2780: @code{dp_prim()} $B$O(B, $B78?t$,B?9`<0$r4^$`>l9g(B, $B78?t$K4^$^$l$kB?9`<06&DL0x;R(B
2781: $B$r<h$j=|$/(B.
1.2 noro 2782: \E
2783: \BEG
2784: @item
2785: @code{dp_ptozp()} executes the same operation as @code{ptozp()} for
2786: a distributed polynomial. If the coefficients include polynomials,
2787: polynomial contents included in the coefficients are not removed.
2788: @item
2789: @code{dp_prim()} removes polynomial contents.
2790: \E
1.1 noro 2791: @end itemize
2792:
2793: @example
2794: [208] X=dp_ptod(3*(x-y)*(y-z)*(z-x),[x]);
2795: (-3*y+3*z)*<<2>>+(3*y^2-3*z^2)*<<1>>+(-3*z*y^2+3*z^2*y)*<<0>>
2796: [209] dp_ptozp(X);
2797: (-y+z)*<<2>>+(y^2-z^2)*<<1>>+(-z*y^2+z^2*y)*<<0>>
2798: [210] dp_prim(X);
2799: (1)*<<2>>+(-y-z)*<<1>>+(z*y)*<<0>>
2800: @end example
2801:
2802: @table @t
1.2 noro 2803: \JP @item $B;2>H(B
2804: \EG @item References
1.1 noro 2805: @fref{ptozp}.
2806: @end table
2807:
1.2 noro 2808: \JP @node dp_nf dp_nf_mod dp_true_nf dp_true_nf_mod,,, $B%0%l%V%J4pDl$K4X$9$kH!?t(B
2809: \EG @node dp_nf dp_nf_mod dp_true_nf dp_true_nf_mod,,, Functions for Groebner basis computation
1.1 noro 2810: @subsection @code{dp_nf}, @code{dp_nf_mod}, @code{dp_true_nf}, @code{dp_true_nf_mod}
2811: @findex dp_nf
2812: @findex dp_true_nf
2813: @findex dp_nf_mod
2814: @findex dp_true_nf_mod
2815:
2816: @table @t
2817: @item dp_nf(@var{indexlist},@var{dpoly},@var{dpolyarray},@var{fullreduce})
2818: @item dp_nf_mod(@var{indexlist},@var{dpoly},@var{dpolyarray},@var{fullreduce},@var{mod})
1.2 noro 2819: \JP :: $BJ,;6I=8=B?9`<0$N@55,7A$r5a$a$k(B. ($B7k2L$ODj?tG\$5$l$F$$$k2DG=@-$"$j(B)
1.1 noro 2820:
1.2 noro 2821: \BEG
2822: :: Computes the normal form of a distributed polynomial.
2823: (The result may be multiplied by a constant in the ground field.)
2824: \E
1.1 noro 2825: @item dp_true_nf(@var{indexlist},@var{dpoly},@var{dpolyarray},@var{fullreduce})
2826: @item dp_true_nf_mod(@var{indexlist},@var{dpoly},@var{dpolyarray},@var{fullreduce},@var{mod})
1.2 noro 2827: \JP :: $BJ,;6I=8=B?9`<0$N@55,7A$r5a$a$k(B. ($B??$N7k2L$r(B @code{[$BJ,;R(B, $BJ,Jl(B]} $B$N7A$GJV$9(B)
2828: \BEG
2829: :: Computes the normal form of a distributed polynomial. (The true result
2830: is returned in such a list as @code{[numerator, denominator]})
2831: \E
1.1 noro 2832: @end table
2833:
2834: @table @var
2835: @item return
1.2 noro 2836: \JP @code{dp_nf()} : $BJ,;6I=8=B?9`<0(B, @code{dp_true_nf()} : $B%j%9%H(B
2837: \EG @code{dp_nf()} : distributed polynomial, @code{dp_true_nf()} : list
1.1 noro 2838: @item indexlist
1.2 noro 2839: \JP $B%j%9%H(B
2840: \EG list
1.1 noro 2841: @item dpoly
1.2 noro 2842: \JP $BJ,;6I=8=B?9`<0(B
2843: \EG distributed polynomial
1.1 noro 2844: @item dpolyarray
1.2 noro 2845: \JP $BG[Ns(B
2846: \EG array of distributed polynomial
1.1 noro 2847: @item fullreduce
1.2 noro 2848: \JP $B%U%i%0(B
2849: \EG flag
1.1 noro 2850: @item mod
1.2 noro 2851: \JP $BAG?t(B
2852: \EG prime
1.1 noro 2853: @end table
2854:
2855: @itemize @bullet
1.2 noro 2856: \BJP
1.1 noro 2857: @item
2858: $BJ,;6I=8=B?9`<0(B @var{dpoly} $B$N@55,7A$r5a$a$k(B.
2859: @item
2860: @code{dp_nf_mod()}, @code{dp_true_nf_mod()} $B$NF~NO$O(B, @code{dp_mod()} $B$J$I(B
2861: $B$K$h$j(B, $BM-8BBN>e$NJ,;6I=8=B?9`<0$K$J$C$F$$$J$1$l$P$J$i$J$$(B.
2862: @item
2863: $B7k2L$KM-M}?t(B, $BM-M}<0$,4^$^$l$k$N$rHr$1$k$?$a(B, @code{dp_nf()} $B$O(B
2864: $B??$NCM$NDj?tG\$NCM$rJV$9(B. $BM-M}<078?t$N>l9g$N(B @code{dp_nf_mod()} $B$bF1MM(B
2865: $B$G$"$k$,(B, $B78?tBN$,M-8BBN$N>l9g(B @code{dp_nf_mod()} $B$O??$NCM$rJV$9(B.
2866: @item
2867: @code{dp_true_nf()}, @code{dp_true_nf_mod()} $B$O(B,
2868: @code{[@var{nm},@var{dn}]} $B$J$k7A$N%j%9%H$rJV$9(B.
2869: $B$?$@$7(B, @var{nm} $B$O78?t$KJ,?t(B, $BM-M}<0$r4^$^$J$$J,;6I=8=B?9`<0(B, @var{dn} $B$O(B
2870: $B?t$^$?$OB?9`<0$G(B @var{nm}/@var{dn} $B$,??$NCM$H$J$k(B.
2871: @item
2872: @var{dpolyarray} $B$OJ,;6I=8=B?9`<0$rMWAG$H$9$k%Y%/%H%k(B,
2873: @var{indexlist} $B$O@55,2=7W;;$KMQ$$$k(B @var{dpolyarray} $B$NMWAG$N%$%s%G%C%/%9(B
2874: $B$N%j%9%H(B.
2875: @item
2876: @var{fullreduce} $B$,(B 0 $B$G$J$$$H$-A4$F$N9`$KBP$7$F4JLs$r9T$&(B. @var{fullreduce}
2877: $B$,(B 0 $B$N$H$-F,9`$N$_$KBP$7$F4JLs$r9T$&(B.
2878: @item
2879: @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.
2880: @item
2881: $B0lHL$K$O(B @var{indexlist} $B$NM?$(J}$K$h$jH!?t$NCM$O0[$J$k2DG=@-$,$"$k$,(B,
2882: $B%0%l%V%J4pDl$KBP$7$F$O0l0UE*$KDj$^$k(B.
2883: @item
2884: $BJ,;6I=8=$G$J$$8GDj$5$l$?B?9`<0=89g$K$h$k@55,7A$rB??t5a$a$kI,MW$,$"$k>l9g(B
2885: $B$KJXMx$G$"$k(B. $BC10l$N1i;;$K4X$7$F$O(B, @code{p_nf}, @code{p_true_nf} $B$r(B
2886: $BMQ$$$k$H$h$$(B.
1.2 noro 2887: \E
2888: \BEG
2889: @item
2890: Computes the normal form of a distributed polynomial.
2891: @item
2892: @code{dp_nf_mod()} and @code{dp_true_nf_mod()} require
2893: distributed polynomials with coefficients in a finite field as arguments.
2894: @item
2895: The result of @code{dp_nf()} may be multiplied by a constant in the
2896: ground field in order to make the result integral. The same is true
2897: for @code{dp_nf_mod()}, but it returns the true normal form if
2898: the ground field is a finite field.
2899: @item
2900: @code{dp_true_nf()} and @code{dp_true_nf_mod()} return
2901: such a list as @code{[@var{nm},@var{dn}]}.
2902: Here @var{nm} is a distributed polynomial whose coefficients are integral
2903: in the ground field, @var{dn} is an integral element in the ground
2904: field and @var{nm}/@var{dn} is the true normal form.
2905: @item
2906: @var{dpolyarray} is a vector whose components are distributed polynomials
2907: and @var{indexlist} is a list of indices which is used for the normal form
2908: computation.
2909: @item
2910: When argument @var{fullreduce} has non-zero value,
2911: all terms are reduced. When it has value 0,
2912: only the head term is reduced.
2913: @item
2914: As for the polynomials specified by @var{indexlist}, one specified by
2915: an index placed at the preceding position has priority to be selected.
2916: @item
2917: In general, the result of the function may be different depending on
2918: @var{indexlist}. However, the result is unique for Groebner bases.
2919: @item
2920: These functions are useful when a fixed non-distributed polynomial set
2921: is used as a set of reducers to compute normal forms of many polynomials.
2922: For single computation @code{p_nf} and @code{p_true_nf} are sufficient.
2923: \E
1.1 noro 2924: @end itemize
2925:
2926: @example
2927: [0] load("gr")$
2928: [64] load("katsura")$
2929: [69] K=katsura(4)$
2930: [70] dp_ord(2)$
2931: [71] V=[u0,u1,u2,u3,u4]$
2932: [72] DP1=newvect(length(K),map(dp_ptod,K,V))$
2933: [73] G=gr(K,V,2)$
2934: [74] DP2=newvect(length(G),map(dp_ptod,G,V))$
2935: [75] T=dp_ptod((u0-u1+u2-u3+u4)^2,V)$
2936: [76] dp_dtop(dp_nf([0,1,2,3,4],T,DP1,1),V);
1.5 noro 2937: u4^2+(6*u3+2*u2+6*u1-2)*u4+9*u3^2+(6*u2+18*u1-6)*u3+u2^2
2938: +(6*u1-2)*u2+9*u1^2-6*u1+1
1.1 noro 2939: [77] dp_dtop(dp_nf([4,3,2,1,0],T,DP1,1),V);
2940: -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
2941: [78] dp_dtop(dp_nf([0,1,2,3,4],T,DP2,1),V);
1.5 noro 2942: -11380879768451657780886122972730785203470970010204714556333530492210
2943: 456775930005716505560062087150928400876150217079820311439477560587583
2944: 488*u4^15+...
1.1 noro 2945: [79] dp_dtop(dp_nf([4,3,2,1,0],T,DP2,1),V);
1.5 noro 2946: -11380879768451657780886122972730785203470970010204714556333530492210
2947: 456775930005716505560062087150928400876150217079820311439477560587583
2948: 488*u4^15+...
1.1 noro 2949: [80] @@78==@@79;
2950: 1
2951: @end example
2952:
2953: @table @t
1.2 noro 2954: \JP @item $B;2>H(B
2955: \EG @item References
1.1 noro 2956: @fref{dp_dtop},
2957: @fref{dp_ord},
2958: @fref{dp_mod dp_rat},
2959: @fref{p_nf p_nf_mod p_true_nf p_true_nf_mod}.
2960: @end table
2961:
1.2 noro 2962: \JP @node dp_hm dp_ht dp_hc dp_rest,,, $B%0%l%V%J4pDl$K4X$9$kH!?t(B
2963: \EG @node dp_hm dp_ht dp_hc dp_rest,,, Functions for Groebner basis computation
1.1 noro 2964: @subsection @code{dp_hm}, @code{dp_ht}, @code{dp_hc}, @code{dp_rest}
2965: @findex dp_hm
2966: @findex dp_ht
2967: @findex dp_hc
2968: @findex dp_rest
2969:
2970: @table @t
2971: @item dp_hm(@var{dpoly})
1.2 noro 2972: \JP :: $BF,C19`<0$r<h$j=P$9(B.
2973: \EG :: Gets the head monomial.
1.1 noro 2974: @item dp_ht(@var{dpoly})
1.2 noro 2975: \JP :: $BF,9`$r<h$j=P$9(B.
2976: \EG :: Gets the head term.
1.1 noro 2977: @item dp_hc(@var{dpoly})
1.2 noro 2978: \JP :: $BF,78?t$r<h$j=P$9(B.
2979: \EG :: Gets the head coefficient.
1.1 noro 2980: @item dp_rest(@var{dpoly})
1.2 noro 2981: \JP :: $BF,C19`<0$r<h$j=|$$$?;D$j$rJV$9(B.
2982: \EG :: Gets the remainder of the polynomial where the head monomial is removed.
1.1 noro 2983: @end table
2984:
2985: @table @var
1.2 noro 2986: \BJP
1.1 noro 2987: @item return
2988: @code{dp_hm()}, @code{dp_ht()}, @code{dp_rest()} : $BJ,;6I=8=B?9`<0(B,
2989: @code{dp_hc()} : $B?t$^$?$OB?9`<0(B
2990: @item dpoly
2991: $BJ,;6I=8=B?9`<0(B
1.2 noro 2992: \E
2993: \BEG
2994: @item return
2995: @code{dp_hm()}, @code{dp_ht()}, @code{dp_rest()} : distributed polynomial
2996: @code{dp_hc()} : number or polynomial
2997: @item dpoly
2998: distributed polynomial
2999: \E
1.1 noro 3000: @end table
3001:
3002: @itemize @bullet
1.2 noro 3003: \BJP
1.1 noro 3004: @item
3005: $B$3$l$i$O(B, $BJ,;6I=8=B?9`<0$N3FItJ,$r<h$j=P$9$?$a$NH!?t$G$"$k(B.
3006: @item
3007: $BJ,;6I=8=B?9`<0(B @var{p} $B$KBP$7<!$,@.$jN)$D(B.
1.2 noro 3008: \E
3009: \BEG
3010: @item
3011: These are used to get various parts of a distributed polynomial.
3012: @item
3013: The next equations hold for a distributed polynomial @var{p}.
3014: \E
1.1 noro 3015: @table @code
3016: @item @var{p} = dp_hm(@var{p}) + dp_rest(@var{p})
3017: @item dp_hm(@var{p}) = dp_hc(@var{p}) dp_ht(@var{p})
3018: @end table
3019: @end itemize
3020:
3021: @example
3022: [87] dp_ord(0)$
3023: [88] X=ptozp((a46^2+7/10*a46+7/48)*u3^4-50/27*a46^2-35/27*a46-49/216)$
3024: [89] T=dp_ptod(X,[u3,u4,a46])$
3025: [90] dp_hm(T);
3026: (2160)*<<4,0,2>>
3027: [91] dp_ht(T);
3028: (1)*<<4,0,2>>
3029: [92] dp_hc(T);
3030: 2160
3031: [93] dp_rest(T);
3032: (1512)*<<4,0,1>>+(315)*<<4,0,0>>+(-4000)*<<0,0,2>>+(-2800)*<<0,0,1>>
3033: +(-490)*<<0,0,0>>
3034: @end example
3035:
1.2 noro 3036: \JP @node dp_td dp_sugar,,, $B%0%l%V%J4pDl$K4X$9$kH!?t(B
3037: \EG @node dp_td dp_sugar,,, Functions for Groebner basis computation
1.1 noro 3038: @subsection @code{dp_td}, @code{dp_sugar}
3039: @findex dp_td
3040: @findex dp_sugar
3041:
3042: @table @t
3043: @item dp_td(@var{dpoly})
1.2 noro 3044: \JP :: $BF,9`$NA4<!?t$rJV$9(B.
3045: \EG :: Gets the total degree of the head term.
1.1 noro 3046: @item dp_sugar(@var{dpoly})
1.2 noro 3047: \JP :: $BB?9`<0$N(B @code{sugar} $B$rJV$9(B.
3048: \EG :: Gets the @code{sugar} of a polynomial.
1.1 noro 3049: @end table
3050:
3051: @table @var
3052: @item return
1.2 noro 3053: \JP $B<+A3?t(B
3054: \EG non-negative integer
1.1 noro 3055: @item dpoly
1.2 noro 3056: \JP $BJ,;6I=8=B?9`<0(B
3057: \EG distributed polynomial
1.1 noro 3058: @item onoff
1.2 noro 3059: \JP $B%U%i%0(B
3060: \EG flag
1.1 noro 3061: @end table
3062:
3063: @itemize @bullet
1.2 noro 3064: \BJP
1.1 noro 3065: @item
3066: @code{dp_td()} $B$O(B, $BF,9`$NA4<!?t(B, $B$9$J$o$A3FJQ?t$N;X?t$NOB$rJV$9(B.
3067: @item
3068: $BJ,;6I=8=B?9`<0$,@8@.$5$l$k$H(B, @code{sugar} $B$H8F$P$l$k$"$k@0?t$,IUM?(B
3069: $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.
3070: @item
3071: @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
3072: $B7hDj$9$k$?$a$N=EMW$J;X?K$H$J$k(B.
1.2 noro 3073: \E
3074: \BEG
3075: @item
3076: Function @code{dp_td()} returns the total degree of the head term,
3077: i.e., the sum of all exponent of variables in that term.
3078: @item
3079: Upon creation of a distributed polynomial, an integer called @code{sugar}
3080: is associated. This value is
3081: the total degree of the virtually homogenized one of the original
3082: polynomial.
3083: @item
3084: The quantity @code{sugar} is an important guide to determine the
3085: selection strategy of critical pairs in Groebner basis computation.
3086: \E
1.1 noro 3087: @end itemize
3088:
3089: @example
3090: [74] dp_ord(0)$
3091: [75] X=<<1,2>>+<<0,1>>$
3092: [76] Y=<<1,2>>+<<1,0>>$
3093: [77] Z=X-Y;
3094: (-1)*<<1,0>>+(1)*<<0,1>>
3095: [78] dp_sugar(T);
3096: 3
3097: @end example
3098:
1.2 noro 3099: \JP @node dp_lcm,,, $B%0%l%V%J4pDl$K4X$9$kH!?t(B
3100: \EG @node dp_lcm,,, Functions for Groebner basis computation
1.1 noro 3101: @subsection @code{dp_lcm}
3102: @findex dp_lcm
3103:
3104: @table @t
3105: @item dp_lcm(@var{dpoly1},@var{dpoly2})
1.2 noro 3106: \JP :: $B:G>.8xG\9`$rJV$9(B.
3107: \EG :: Returns the least common multiple of the head terms of the given two polynomials.
1.1 noro 3108: @end table
3109:
3110: @table @var
3111: @item return
1.2 noro 3112: \JP $BJ,;6I=8=B?9`<0(B
3113: \EG distributed polynomial
1.4 noro 3114: @item dpoly1 dpoly2
1.2 noro 3115: \JP $BJ,;6I=8=B?9`<0(B
3116: \EG distributed polynomial
1.1 noro 3117: @end table
3118:
3119: @itemize @bullet
1.2 noro 3120: \BJP
1.1 noro 3121: @item
3122: $B$=$l$>$l$N0z?t$NF,9`$N:G>.8xG\9`$rJV$9(B. $B78?t$O(B 1 $B$G$"$k(B.
1.2 noro 3123: \E
3124: \BEG
3125: @item
3126: Returns the least common multiple of the head terms of the given
3127: two polynomials, where coefficient is always set to 1.
3128: \E
1.1 noro 3129: @end itemize
3130:
3131: @example
3132: [100] dp_lcm(<<1,2,3,4,5>>,<<5,4,3,2,1>>);
3133: (1)*<<5,4,3,4,5>>
3134: @end example
3135:
3136: @table @t
1.2 noro 3137: \JP @item $B;2>H(B
3138: \EG @item References
1.1 noro 3139: @fref{p_nf p_nf_mod p_true_nf p_true_nf_mod}.
3140: @end table
3141:
1.2 noro 3142: \JP @node dp_redble,,, $B%0%l%V%J4pDl$K4X$9$kH!?t(B
3143: \EG @node dp_redble,,, Functions for Groebner basis computation
1.1 noro 3144: @subsection @code{dp_redble}
3145: @findex dp_redble
3146:
3147: @table @t
3148: @item dp_redble(@var{dpoly1},@var{dpoly2})
1.2 noro 3149: \JP :: $BF,9`$I$&$7$,@0=|2DG=$+$I$&$+D4$Y$k(B.
3150: \EG :: Checks whether one head term is divisible by the other head term.
1.1 noro 3151: @end table
3152:
3153: @table @var
3154: @item return
1.2 noro 3155: \JP $B@0?t(B
3156: \EG integer
1.4 noro 3157: @item dpoly1 dpoly2
1.2 noro 3158: \JP $BJ,;6I=8=B?9`<0(B
3159: \EG distributed polynomial
1.1 noro 3160: @end table
3161:
3162: @itemize @bullet
1.2 noro 3163: \BJP
1.1 noro 3164: @item
3165: @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
3166: 0 $B$rJV$9(B.
3167: @item
3168: $BB?9`<0$N4JLs$r9T$&:](B, $B$I$N9`$r4JLs$G$-$k$+$rC5$9$N$KMQ$$$k(B.
1.2 noro 3169: \E
3170: \BEG
3171: @item
3172: Returns 1 if the head term of @var{dpoly2} divides the head term of
3173: @var{dpoly1}; otherwise 0.
3174: @item
3175: Used for finding candidate terms at reduction of polynomials.
3176: \E
1.1 noro 3177: @end itemize
3178:
3179: @example
3180: [148] C;
3181: (1)*<<1,1,1,0,0>>+(1)*<<0,1,1,1,0>>+(1)*<<1,1,0,0,1>>+(1)*<<1,0,0,1,1>>
3182: [149] T;
3183: (3)*<<2,1,0,0,0>>+(3)*<<1,2,0,0,0>>+(1)*<<0,3,0,0,0>>+(6)*<<1,1,1,0,0>>
3184: [150] for ( ; T; T = dp_rest(T)) print(dp_redble(T,C));
3185: 0
3186: 0
3187: 0
3188: 1
3189: @end example
3190:
3191: @table @t
1.2 noro 3192: \JP @item $B;2>H(B
3193: \EG @item References
1.1 noro 3194: @fref{dp_red dp_red_mod}.
3195: @end table
3196:
1.2 noro 3197: \JP @node dp_subd,,, $B%0%l%V%J4pDl$K4X$9$kH!?t(B
3198: \EG @node dp_subd,,, Functions for Groebner basis computation
1.1 noro 3199: @subsection @code{dp_subd}
3200: @findex dp_subd
3201:
3202: @table @t
3203: @item dp_subd(@var{dpoly1},@var{dpoly2})
1.2 noro 3204: \JP :: $BF,9`$N>&C19`<0$rJV$9(B.
3205: \EG :: Returns the quotient monomial of the head terms.
1.1 noro 3206: @end table
3207:
3208: @table @var
3209: @item return
1.2 noro 3210: \JP $BJ,;6I=8=B?9`<0(B
3211: \EG distributed polynomial
1.4 noro 3212: @item dpoly1 dpoly2
1.2 noro 3213: \JP $BJ,;6I=8=B?9`<0(B
3214: \EG distributed polynomial
1.1 noro 3215: @end table
3216:
3217: @itemize @bullet
1.2 noro 3218: \BJP
1.1 noro 3219: @item
3220: @code{dp_ht(@var{dpoly1})/dp_ht(@var{dpoly2})} $B$r5a$a$k(B. $B7k2L$N78?t$O(B 1
3221: $B$G$"$k(B.
3222: @item
3223: $B3d$j@Z$l$k$3$H$,$"$i$+$8$a$o$+$C$F$$$kI,MW$,$"$k(B.
1.2 noro 3224: \E
3225: \BEG
3226: @item
3227: Gets @code{dp_ht(@var{dpoly1})/dp_ht(@var{dpoly2})}.
3228: The coefficient of the result is always set to 1.
3229: @item
3230: Divisibility assumed.
3231: \E
1.1 noro 3232: @end itemize
3233:
3234: @example
3235: [162] dp_subd(<<1,2,3,4,5>>,<<1,1,2,3,4>>);
3236: (1)*<<0,1,1,1,1>>
3237: @end example
3238:
3239: @table @t
1.2 noro 3240: \JP @item $B;2>H(B
3241: \EG @item References
1.1 noro 3242: @fref{dp_red dp_red_mod}.
3243: @end table
3244:
1.2 noro 3245: \JP @node dp_vtoe dp_etov,,, $B%0%l%V%J4pDl$K4X$9$kH!?t(B
3246: \EG @node dp_vtoe dp_etov,,, Functions for Groebner basis computation
1.1 noro 3247: @subsection @code{dp_vtoe}, @code{dp_etov}
3248: @findex dp_vtoe
3249: @findex dp_etov
3250:
3251: @table @t
3252: @item dp_vtoe(@var{vect})
1.2 noro 3253: \JP :: $B;X?t%Y%/%H%k$r9`$KJQ49(B
3254: \EG :: Converts an exponent vector into a term.
1.1 noro 3255: @item dp_etov(@var{dpoly})
1.2 noro 3256: \JP :: $BF,9`$r;X?t%Y%/%H%k$KJQ49(B
3257: \EG :: Convert the head term of a distributed polynomial into an exponent vector.
1.1 noro 3258: @end table
3259:
3260: @table @var
3261: @item return
1.2 noro 3262: \JP @code{dp_vtoe} : $BJ,;6I=8=B?9`<0(B, @code{dp_etov} : $B%Y%/%H%k(B
3263: \EG @code{dp_vtoe} : distributed polynomial, @code{dp_etov} : vector
1.1 noro 3264: @item vect
1.2 noro 3265: \JP $B%Y%/%H%k(B
3266: \EG vector
1.1 noro 3267: @item dpoly
1.2 noro 3268: \JP $BJ,;6I=8=B?9`<0(B
3269: \EG distributed polynomial
1.1 noro 3270: @end table
3271:
3272: @itemize @bullet
1.2 noro 3273: \BJP
1.1 noro 3274: @item
3275: @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.
3276: @item
3277: @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
3278: $B%Y%/%H%k$KJQ49$9$k(B.
1.2 noro 3279: \E
3280: \BEG
3281: @item
3282: @code{dp_vtoe()} generates a term whose exponent vector is @var{vect}.
3283: @item
3284: @code{dp_etov()} generates a vector which is the exponent vector of the
3285: head term of @code{dpoly}.
3286: \E
1.1 noro 3287: @end itemize
3288:
3289: @example
3290: [211] X=<<1,2,3>>;
3291: (1)*<<1,2,3>>
3292: [212] V=dp_etov(X);
3293: [ 1 2 3 ]
3294: [213] V[2]++$
3295: [214] Y=dp_vtoe(V);
3296: (1)*<<1,2,4>>
3297: @end example
3298:
1.2 noro 3299: \JP @node dp_mbase,,, $B%0%l%V%J4pDl$K4X$9$kH!?t(B
3300: \EG @node dp_mbase,,, Functions for Groebner basis computation
1.1 noro 3301: @subsection @code{dp_mbase}
3302: @findex dp_mbase
3303:
3304: @table @t
3305: @item dp_mbase(@var{dplist})
1.2 noro 3306: \JP :: monomial $B4pDl$N7W;;(B
3307: \EG :: Computes the monomial basis
1.1 noro 3308: @end table
3309:
3310: @table @var
3311: @item return
1.2 noro 3312: \JP $BJ,;6I=8=B?9`<0$N%j%9%H(B
3313: \EG list of distributed polynomial
1.1 noro 3314: @item dplist
1.2 noro 3315: \JP $BJ,;6I=8=B?9`<0$N%j%9%H(B
3316: \EG list of distributed polynomial
1.1 noro 3317: @end table
3318:
3319: @itemize @bullet
1.2 noro 3320: \BJP
1.1 noro 3321: @item
3322: $B$"$k=g=x$G%0%l%V%J4pDl$H$J$C$F$$$kB?9`<0=89g$N(B, $B$=$N=g=x$K4X$9$kJ,;6I=8=(B
3323: $B$G$"$k(B @var{dplist} $B$K$D$$$F(B,
3324: @var{dplist} $B$,(B K[X] $BCf$G@8@.$9$k%$%G%"%k(B I $B$,(B 0 $B<!85$N;~(B,
3325: K $B>eM-8B<!85@~7A6u4V$G$"$k(B K[X]/I $B$N(B monomial $B$K$h$k4pDl$r5a$a$k(B.
3326: @item
3327: $BF@$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 3328: \E
3329: \BEG
3330: @item
3331: Assuming that @var{dplist} is a list of distributed polynomials which
3332: is a Groebner basis with respect to the current ordering type and
3333: that the ideal @var{I} generated by @var{dplist} in K[X] is zero-dimensional,
3334: this function computes the monomial basis of a finite dimenstional K-vector
3335: space K[X]/I.
3336: @item
3337: The number of elements in the monomial basis is equal to the
3338: K-dimenstion of K[X]/I.
3339: \E
1.1 noro 3340: @end itemize
3341:
3342: @example
3343: [215] K=katsura(5)$
3344: [216] V=[u5,u4,u3,u2,u1,u0]$
3345: [217] G0=gr(K,V,0)$
3346: [218] H=map(dp_ptod,G0,V)$
3347: [219] map(dp_ptod,dp_mbase(H),V)$
3348: [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,
3349: 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,
3350: u1*u2,u1^2,u4*u0,u3*u0,u2*u0,u1*u0,u0^2,u4,u3,u2,u1,u0,1]
3351: @end example
3352:
3353: @table @t
1.2 noro 3354: \JP @item $B;2>H(B
3355: \EG @item References
1.1 noro 3356: @fref{gr hgr gr_mod}.
3357: @end table
3358:
1.2 noro 3359: \JP @node dp_mag,,, $B%0%l%V%J4pDl$K4X$9$kH!?t(B
3360: \EG @node dp_mag,,, Functions for Groebner basis computation
1.1 noro 3361: @subsection @code{dp_mag}
3362: @findex dp_mag
3363:
3364: @table @t
3365: @item dp_mag(@var{p})
1.2 noro 3366: \JP :: $B78?t$N%S%C%HD9$NOB$rJV$9(B
3367: \EG :: Computes the sum of bit lengths of coefficients of a distributed polynomial.
1.1 noro 3368: @end table
3369:
3370: @table @var
3371: @item return
1.2 noro 3372: \JP $B?t(B
3373: \EG integer
1.1 noro 3374: @item p
1.2 noro 3375: \JP $BJ,;6I=8=B?9`<0(B
3376: \EG distributed polynomial
1.1 noro 3377: @end table
3378:
3379: @itemize @bullet
1.2 noro 3380: \BJP
1.1 noro 3381: @item
3382: $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)
3383: $B$N%S%C%HD9$NAmOB$rJV$9(B.
3384: @item
3385: $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
3386: $B78?tKDD%$,LdBj$H$J$j(B, $BESCf@8@.$5$l$kB?9`<0$,78?tKDD%$r5/$3$7$F$$$k$+$I$&$+(B
3387: $B$NH=Dj$KLrN)$D(B.
3388: @item
3389: @code{dp_gr_flags()} $B$G(B, @code{ShowMag}, @code{Print} $B$r(B on $B$K$9$k$3$H$K$h$j(B
3390: $BESCf@8@.$5$l$kB?9`<0$K$?$$$9$k(B @code{dp_mag()} $B$NCM$r8+$k$3$H$,$G$-$k(B.
1.2 noro 3391: \E
3392: \BEG
3393: @item
3394: This function computes the sum of bit lengths of coefficients of a
3395: distributed polynomial @var{p}. If a coefficient is non integral,
3396: the sum of bit lengths of the numerator and the denominator is taken.
3397: @item
3398: This is a measure of the size of a polynomial. Especially for
3399: zero-dimensional system coefficient swells are often serious and
3400: the returned value is useful to detect such swells.
3401: @item
3402: If @code{ShowMag} and @code{Print} for @code{dp_gr_flags()} are on,
3403: values of @code{dp_mag()} for intermediate basis elements are shown.
3404: \E
1.1 noro 3405: @end itemize
3406:
3407: @example
3408: [221] X=dp_ptod((x+2*y)^10,[x,y])$
3409: [222] dp_mag(X);
3410: 115
3411: @end example
3412:
3413: @table @t
1.2 noro 3414: \JP @item $B;2>H(B
3415: \EG @item References
1.1 noro 3416: @fref{dp_gr_flags dp_gr_print}.
3417: @end table
3418:
1.2 noro 3419: \JP @node dp_red dp_red_mod,,, $B%0%l%V%J4pDl$K4X$9$kH!?t(B
3420: \EG @node dp_red dp_red_mod,,, Functions for Groebner basis computation
1.1 noro 3421: @subsection @code{dp_red}, @code{dp_red_mod}
3422: @findex dp_red
3423: @findex dp_red_mod
3424:
3425: @table @t
3426: @item dp_red(@var{dpoly1},@var{dpoly2},@var{dpoly3})
3427: @item dp_red_mod(@var{dpoly1},@var{dpoly2},@var{dpoly3},@var{mod})
1.2 noro 3428: \JP :: $B0l2s$N4JLsA`:n(B
3429: \EG :: Single reduction operation
1.1 noro 3430: @end table
3431:
3432: @table @var
3433: @item return
1.2 noro 3434: \JP $B%j%9%H(B
3435: \EG list
1.4 noro 3436: @item dpoly1 dpoly2 dpoly3
1.2 noro 3437: \JP $BJ,;6I=8=B?9`<0(B
3438: \EG distributed polynomial
1.1 noro 3439: @item vlist
1.2 noro 3440: \JP $B%j%9%H(B
3441: \EG list
1.1 noro 3442: @item mod
1.2 noro 3443: \JP $BAG?t(B
3444: \EG prime
1.1 noro 3445: @end table
3446:
3447: @itemize @bullet
1.2 noro 3448: \BJP
1.1 noro 3449: @item
3450: @var{dpoly1} + @var{dpoly2} $B$J$kJ,;6I=8=B?9`<0$r(B @var{dpoly3} $B$G(B
3451: 1 $B2s4JLs$9$k(B.
3452: @item
3453: @code{dp_red_mod()} $B$NF~NO$O(B, $BA4$FM-8BBN78?t$KJQ49$5$l$F$$$kI,MW$,$"$k(B.
3454: @item
3455: $B4JLs$5$l$k9`$O(B @var{dpoly2} $B$NF,9`$G$"$k(B. $B=>$C$F(B, @var{dpoly2} $B$N(B
3456: $BF,9`$,(B @var{dpoly3} $B$NF,9`$G3d$j@Z$l$k$3$H$,$"$i$+$8$a$o$+$C$F$$$J$1$l$P(B
3457: $B$J$i$J$$(B.
3458: @item
3459: $B0z?t$,@0?t78?t$N;~(B, $B4JLs$O(B, $BJ,?t$,8=$l$J$$$h$&(B, $B@0?t(B @var{a}, @var{b},
1.4 noro 3460: $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 3461: @item
3462: $B7k2L$O(B, @code{[@var{a dpoly1},@var{a dpoly2 - bt dpoly3}]} $B$J$k%j%9%H$G$"$k(B.
1.2 noro 3463: \E
3464: \BEG
3465: @item
3466: Reduces a distributed polynomial, @var{dpoly1} + @var{dpoly2},
3467: by @var{dpoly3} for single time.
3468: @item
3469: An input for @code{dp_red_mod()} must be converted into a distributed
3470: polynomial with coefficients in a finite field.
3471: @item
3472: This implies that
3473: the divisibility of the head term of @var{dpoly2} by the head term of
3474: @var{dpoly3} is assumed.
3475: @item
3476: When integral coefficients, computation is so carefully performed that
3477: no rational operations appear in the reduction procedure.
3478: It is computed for integers @var{a} and @var{b}, and a term @var{t} as:
1.4 noro 3479: @var{a}(@var{dpoly1} + @var{dpoly2})-@var{bt} @var{dpoly3}.
1.2 noro 3480: @item
3481: The result is a list @code{[@var{a dpoly1},@var{a dpoly2 - bt dpoly3}]}.
3482: \E
1.1 noro 3483: @end itemize
3484:
3485: @example
3486: [157] D=(3)*<<2,1,0,0,0>>+(3)*<<1,2,0,0,0>>+(1)*<<0,3,0,0,0>>;
3487: (3)*<<2,1,0,0,0>>+(3)*<<1,2,0,0,0>>+(1)*<<0,3,0,0,0>>
3488: [158] R=(6)*<<1,1,1,0,0>>;
3489: (6)*<<1,1,1,0,0>>
3490: [159] C=12*<<1,1,1,0,0>>+(1)*<<0,1,1,1,0>>+(1)*<<1,1,0,0,1>>;
3491: (12)*<<1,1,1,0,0>>+(1)*<<0,1,1,1,0>>+(1)*<<1,1,0,0,1>>
3492: [160] dp_red(D,R,C);
1.5 noro 3493: [(6)*<<2,1,0,0,0>>+(6)*<<1,2,0,0,0>>+(2)*<<0,3,0,0,0>>,
3494: (-1)*<<0,1,1,1,0>>+(-1)*<<1,1,0,0,1>>]
1.1 noro 3495: @end example
3496:
3497: @table @t
1.2 noro 3498: \JP @item $B;2>H(B
3499: \EG @item References
1.1 noro 3500: @fref{dp_mod dp_rat}.
3501: @end table
3502:
1.2 noro 3503: \JP @node dp_sp dp_sp_mod,,, $B%0%l%V%J4pDl$K4X$9$kH!?t(B
3504: \EG @node dp_sp dp_sp_mod,,, Functions for Groebner basis computation
1.1 noro 3505: @subsection @code{dp_sp}, @code{dp_sp_mod}
3506: @findex dp_sp
3507: @findex dp_sp_mod
3508:
3509: @table @t
3510: @item dp_sp(@var{dpoly1},@var{dpoly2})
3511: @item dp_sp_mod(@var{dpoly1},@var{dpoly2},@var{mod})
1.2 noro 3512: \JP :: S-$BB?9`<0$N7W;;(B
3513: \EG :: Computation of an S-polynomial
1.1 noro 3514: @end table
3515:
3516: @table @var
3517: @item return
1.2 noro 3518: \JP $BJ,;6I=8=B?9`<0(B
3519: \EG distributed polynomial
1.4 noro 3520: @item dpoly1 dpoly2
1.2 noro 3521: \JP $BJ,;6I=8=B?9`<0(B
3522: \EG distributed polynomial
1.1 noro 3523: @item mod
1.2 noro 3524: \JP $BAG?t(B
3525: \EG prime
1.1 noro 3526: @end table
3527:
3528: @itemize @bullet
1.2 noro 3529: \BJP
1.1 noro 3530: @item
3531: @var{dpoly1}, @var{dpoly2} $B$N(B S-$BB?9`<0$r7W;;$9$k(B.
3532: @item
3533: @code{dp_sp_mod()} $B$NF~NO$O(B, $BA4$FM-8BBN78?t$KJQ49$5$l$F$$$kI,MW$,$"$k(B.
3534: @item
3535: $B7k2L$KM-M}?t(B, $BM-M}<0$,F~$k$N$rHr$1$k$?$a(B, $B7k2L$,Dj?tG\(B, $B$"$k$$$OB?9`<0(B
3536: $BG\$5$l$F$$$k2DG=@-$,$"$k(B.
1.2 noro 3537: \E
3538: \BEG
3539: @item
3540: This function computes the S-polynomial of @var{dpoly1} and @var{dpoly2}.
3541: @item
3542: Inputs of @code{dp_sp_mod()} must be polynomials with coefficients in a
3543: finite field.
3544: @item
3545: The result may be multiplied by a constant in the ground field in order to
3546: make the result integral.
3547: \E
1.1 noro 3548: @end itemize
3549:
3550: @example
3551: [227] X=dp_ptod(x^2*y+x*y,[x,y]);
3552: (1)*<<2,1>>+(1)*<<1,1>>
3553: [228] Y=dp_ptod(x*y^2+x*y,[x,y]);
3554: (1)*<<1,2>>+(1)*<<1,1>>
3555: [229] dp_sp(X,Y);
3556: (-1)*<<2,1>>+(1)*<<1,2>>
3557: @end example
3558:
3559: @table @t
1.2 noro 3560: \JP @item $B;2>H(B
3561: \EG @item References
1.1 noro 3562: @fref{dp_mod dp_rat}.
3563: @end table
1.2 noro 3564: \JP @node p_nf p_nf_mod p_true_nf p_true_nf_mod,,, $B%0%l%V%J4pDl$K4X$9$kH!?t(B
3565: \EG @node p_nf p_nf_mod p_true_nf p_true_nf_mod,,, Functions for Groebner basis computation
1.1 noro 3566: @subsection @code{p_nf}, @code{p_nf_mod}, @code{p_true_nf}, @code{p_true_nf_mod}
3567: @findex p_nf
3568: @findex p_nf_mod
3569: @findex p_true_nf
3570: @findex p_true_nf_mod
3571:
3572: @table @t
3573: @item p_nf(@var{poly},@var{plist},@var{vlist},@var{order})
3574: @itemx p_nf_mod(@var{poly},@var{plist},@var{vlist},@var{order},@var{mod})
1.2 noro 3575: \JP :: $BI=8=B?9`<0$N@55,7A$r5a$a$k(B. ($B7k2L$ODj?tG\$5$l$F$$$k2DG=@-$"$j(B)
3576: \BEG
3577: :: Computes the normal form of the given polynomial.
3578: (The result may be multiplied by a constant.)
3579: \E
1.1 noro 3580: @item p_true_nf(@var{poly},@var{plist},@var{vlist},@var{order})
3581: @itemx p_true_nf_mod(@var{poly},@var{plist},@var{vlist},@var{order},@var{mod})
1.2 noro 3582: \JP :: $BI=8=B?9`<0$N@55,7A$r5a$a$k(B. ($B??$N7k2L$r(B @code{[$BJ,;R(B, $BJ,Jl(B]} $B$N7A$GJV$9(B)
3583: \BEG
3584: :: Computes the normal form of the given polynomial. (The result is returned
3585: as a form of @code{[numerator, denominator]})
3586: \E
1.1 noro 3587: @end table
3588:
3589: @table @var
3590: @item return
1.2 noro 3591: \JP @code{p_nf} : $BB?9`<0(B, @code{p_true_nf} : $B%j%9%H(B
3592: \EG @code{p_nf} : polynomial, @code{p_true_nf} : list
1.1 noro 3593: @item poly
1.2 noro 3594: \JP $BB?9`<0(B
3595: \EG polynomial
1.4 noro 3596: @item plist vlist
1.2 noro 3597: \JP $B%j%9%H(B
3598: \EG list
1.1 noro 3599: @item order
1.2 noro 3600: \JP $B?t(B, $B%j%9%H$^$?$O9TNs(B
3601: \EG number, list or matrix
1.1 noro 3602: @item mod
1.2 noro 3603: \JP $BAG?t(B
3604: \EG prime
1.1 noro 3605: @end table
3606:
3607: @itemize @bullet
1.2 noro 3608: \BJP
1.1 noro 3609: @item
3610: @samp{gr} $B$GDj5A$5$l$F$$$k(B.
3611: @item
3612: $BB?9`<0$N(B, $BB?9`<0%j%9%H$K$h$k@55,7A$r5a$a$k(B.
3613: @item
3614: @code{dp_nf()}, @code{dp_true_nf()}, @code{dp_nf_mod()}, @code{dp_true_nf_mod}
3615: $B$KBP$9$k%$%s%?%U%'!<%9$G$"$k(B.
3616: @item
3617: @var{poly} $B$*$h$S(B @var{plist} $B$O(B, $BJQ?t=g=x(B @var{vlist} $B$*$h$S(B
3618: $BJQ?t=g=x7?(B @var{otype} $B$K=>$C$FJ,;6I=8=B?9`<0$KJQ49$5$l(B,
3619: @code{dp_nf()}, @code{dp_true_nf()}, @code{dp_nf_mod()},
3620: @code{dp_true_nf_mod()} $B$KEO$5$l$k(B.
3621: @item
3622: @code{dp_nf()}, @code{dp_true_nf()}, @code{dp_nf_mod()},
3623: @code{dp_true_nf_mod()} $B$O(B @var{fullreduce} $B$,(B 1 $B$G8F$S=P$5$l$k(B.
3624: @item
3625: $B7k2L$OB?9`<0$KJQ49$5$l$F=PNO$5$l$k(B.
3626: @item
3627: @code{p_true_nf()}, @code{p_true_nf_mod()} $B$N=PNO$K4X$7$F$O(B,
3628: @code{dp_true_nf()}, @code{dp_true_nf_mod()} $B$N9`$r;2>H(B.
1.2 noro 3629: \E
3630: \BEG
3631: @item
3632: Defined in the package @samp{gr}.
3633: @item
3634: Obtains the normal form of a polynomial by a polynomial list.
3635: @item
3636: These are interfaces to @code{dp_nf()}, @code{dp_true_nf()}, @code{dp_nf_mod()},
3637: @code{dp_true_nf_mod}
3638: @item
3639: The polynomial @var{poly} and the polynomials in @var{plist} is
3640: converted, according to the variable ordering @var{vlist} and
3641: type of term ordering @var{otype}, into their distributed polynomial
3642: counterparts and passed to @code{dp_nf()}.
3643: @item
3644: @code{dp_nf()}, @code{dp_true_nf()}, @code{dp_nf_mod()} and
3645: @code{dp_true_nf_mod()}
3646: is called with value 1 for @var{fullreduce}.
3647: @item
3648: The result is converted back into an ordinary polynomial.
3649: @item
3650: As for @code{p_true_nf()}, @code{p_true_nf_mod()}
3651: refer to @code{dp_true_nf()} and @code{dp_true_nf_mod()}.
3652: \E
1.1 noro 3653: @end itemize
3654:
3655: @example
3656: [79] K = katsura(5)$
3657: [80] V = [u5,u4,u3,u2,u1,u0]$
3658: [81] G = hgr(K,V,2)$
3659: [82] p_nf(K[1],G,V,2);
3660: 0
3661: [83] L = p_true_nf(K[1]+1,G,V,2);
3662: [-1503...,-1503...]
3663: [84] L[0]/L[1];
3664: 1
3665: @end example
3666:
3667: @table @t
1.2 noro 3668: \JP @item $B;2>H(B
3669: \EG @item References
1.1 noro 3670: @fref{dp_ptod},
3671: @fref{dp_dtop},
3672: @fref{dp_ord},
3673: @fref{dp_nf dp_nf_mod dp_true_nf dp_true_nf_mod}.
3674: @end table
3675:
1.2 noro 3676: \JP @node p_terms,,, $B%0%l%V%J4pDl$K4X$9$kH!?t(B
3677: \EG @node p_terms,,, Functions for Groebner basis computation
1.1 noro 3678: @subsection @code{p_terms}
3679: @findex p_terms
3680:
3681: @table @t
3682: @item p_terms(@var{poly},@var{vlist},@var{order})
1.2 noro 3683: \JP :: $BB?9`<0$K$"$i$o$l$kC19`$r%j%9%H$K$9$k(B.
3684: \EG :: Monomials appearing in the given polynomial is collected into a list.
1.1 noro 3685: @end table
3686:
3687: @table @var
3688: @item return
1.2 noro 3689: \JP $B%j%9%H(B
3690: \EG list
1.1 noro 3691: @item poly
1.2 noro 3692: \JP $BB?9`<0(B
3693: \EG polynomial
1.1 noro 3694: @item vlist
1.2 noro 3695: \JP $B%j%9%H(B
3696: \EG list
1.1 noro 3697: @item order
1.2 noro 3698: \JP $B?t(B, $B%j%9%H$^$?$O9TNs(B
3699: \EG number, list or matrix
1.1 noro 3700: @end table
3701:
3702: @itemize @bullet
1.2 noro 3703: \BJP
1.1 noro 3704: @item
3705: @samp{gr} $B$GDj5A$5$l$F$$$k(B.
3706: @item
3707: $BB?9`<0$rC19`$KE83+$7$?;~$K8=$l$k9`$r%j%9%H$K$7$FJV$9(B.
3708: @var{vlist} $B$*$h$S(B @var{order} $B$K$h$jDj$^$k9`=g=x$K$h$j(B, $B=g=x$N9b$$$b$N(B
3709: $B$,%j%9%H$N@hF,$KMh$k$h$&$K%=!<%H$5$l$k(B.
3710: @item
3711: $B%0%l%V%J4pDl$O$7$P$7$P78?t$,5pBg$K$J$k$?$a(B, $B<B:]$K$I$N9`$,8=$l$F(B
3712: $B$$$k$N$+$r8+$k$?$a$J$I$KMQ$$$k(B.
1.2 noro 3713: \E
3714: \BEG
3715: @item
3716: Defined in the package @samp{gr}.
3717: @item
3718: This returns a list which contains all non-zero monomials in the given
3719: polynomial. The monomials are ordered according to the current
3720: type of term ordering and @var{vlist}.
3721: @item
3722: Since polynomials in a Groebner base often have very large coefficients,
3723: examining a polynomial as it is may sometimes be difficult to perform.
3724: For such a case, this function enables to examine which term is really
3725: exists.
3726: \E
1.1 noro 3727: @end itemize
3728:
3729: @example
3730: [233] G=gr(katsura(5),[u5,u4,u3,u2,u1,u0],2)$
3731: [234] p_terms(G[0],[u5,u4,u3,u2,u1,u0],2);
1.5 noro 3732: [u5,u0^31,u0^30,u0^29,u0^28,u0^27,u0^26,u0^25,u0^24,u0^23,u0^22,
3733: u0^21,u0^20,u0^19,u0^18,u0^17,u0^16,u0^15,u0^14,u0^13,u0^12,u0^11,
3734: u0^10,u0^9,u0^8,u0^7,u0^6,u0^5,u0^4,u0^3,u0^2,u0,1]
1.1 noro 3735: @end example
3736:
1.2 noro 3737: \JP @node gb_comp,,, $B%0%l%V%J4pDl$K4X$9$kH!?t(B
3738: \EG @node gb_comp,,, Functions for Groebner basis computation
1.1 noro 3739: @subsection @code{gb_comp}
3740: @findex gb_comp
3741:
3742: @table @t
3743: @item gb_comp(@var{plist1}, @var{plist2})
1.2 noro 3744: \JP :: $BB?9`<0%j%9%H$,(B, $BId9f$r=|$$$F=89g$H$7$FEy$7$$$+$I$&$+D4$Y$k(B.
3745: \EG :: Checks whether two polynomial lists are equal or not as a set
1.1 noro 3746: @end table
3747:
3748: @table @var
1.2 noro 3749: \JP @item return 0 $B$^$?$O(B 1
3750: \EG @item return 0 or 1
1.4 noro 3751: @item plist1 plist2
1.1 noro 3752: @end table
3753:
3754: @itemize @bullet
1.2 noro 3755: \BJP
1.1 noro 3756: @item
3757: @var{plist1}, @var{plist2} $B$K$D$$$F(B, $BId9f$r=|$$$F=89g$H$7$FEy$7$$$+$I$&$+(B
3758: $BD4$Y$k(B.
3759: @item
3760: $B0[$J$kJ}K!$G5a$a$?%0%l%V%J4pDl$O(B, $B4pDl$N=g=x(B, $BId9f$,0[$J$k>l9g$,$"$j(B,
3761: $B$=$l$i$,Ey$7$$$+$I$&$+$rD4$Y$k$?$a$KMQ$$$k(B.
1.2 noro 3762: \E
3763: \BEG
3764: @item
3765: This function checks whether @var{plist1} and @var{plist2} are equal or
3766: not as a set .
3767: @item
3768: For the same input and the same term ordering different
3769: functions for Groebner basis computations may produce different outputs
3770: as lists. This function compares such lists whether they are equal
3771: as a generating set of an ideal.
3772: \E
1.1 noro 3773: @end itemize
3774:
3775: @example
3776: [243] C=cyclic(6)$
3777: [244] V=[c0,c1,c2,c3,c4,c5]$
3778: [245] G0=gr(C,V,0)$
3779: [246] G=tolex(G0,V,0,V)$
3780: [247] GG=lex_tl(C,V,0,V,0)$
3781: [248] gb_comp(G,GG);
3782: 1
3783: @end example
3784:
1.2 noro 3785: \JP @node katsura hkatsura cyclic hcyclic,,, $B%0%l%V%J4pDl$K4X$9$kH!?t(B
3786: \EG @node katsura hkatsura cyclic hcyclic,,, Functions for Groebner basis computation
1.1 noro 3787: @subsection @code{katsura}, @code{hkatsura}, @code{cyclic}, @code{hcyclic}
3788: @findex katsura
3789: @findex hkatsura
3790: @findex cyclic
3791: @findex hcyclic
3792:
3793: @table @t
3794: @item katsura(@var{n})
3795: @item hkatsura(@var{n})
3796: @item cyclic(@var{n})
3797: @item hcyclic(@var{n})
1.2 noro 3798: \JP :: $BB?9`<0%j%9%H$N@8@.(B
3799: \EG :: Generates a polynomial list of standard benchmark.
1.1 noro 3800: @end table
3801:
3802: @table @var
3803: @item return
1.2 noro 3804: \JP $B%j%9%H(B
3805: \EG list
1.1 noro 3806: @item n
1.2 noro 3807: \JP $B@0?t(B
3808: \EG integer
1.1 noro 3809: @end table
3810:
3811: @itemize @bullet
1.2 noro 3812: \BJP
1.1 noro 3813: @item
3814: @code{katsura()} $B$O(B @samp{katsura}, @code{cyclic()} $B$O(B @samp{cyclic}
3815: $B$GDj5A$5$l$F$$$k(B.
3816: @item
3817: $B%0%l%V%J4pDl7W;;$G$7$P$7$P%F%9%H(B, $B%Y%s%A%^!<%/$KMQ$$$i$l$k(B @code{katsura},
3818: @code{cyclic} $B$*$h$S$=$N@F<!2=$r@8@.$9$k(B.
3819: @item
3820: @code{cyclic} $B$O(B @code{Arnborg}, @code{Lazard}, @code{Davenport} $B$J$I$N(B
3821: $BL>$G8F$P$l$k$3$H$b$"$k(B.
1.2 noro 3822: \E
3823: \BEG
3824: @item
3825: Function @code{katsura()} is defined in @samp{katsura}, and
3826: function @code{cyclic()} in @samp{cyclic}.
3827: @item
3828: These functions generate a series of polynomial sets, respectively,
3829: which are often used for testing and bench marking:
3830: @code{katsura}, @code{cyclic} and their homogenized versions.
3831: @item
3832: Polynomial set @code{cyclic} is sometimes called by other name:
3833: @code{Arnborg}, @code{Lazard}, and @code{Davenport}.
3834: \E
1.1 noro 3835: @end itemize
3836:
3837: @example
3838: [74] load("katsura")$
3839: [79] load("cyclic")$
3840: [89] katsura(5);
3841: [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 3842: 2*u3*u0+2*u1*u4-u3+(2*u1+2*u5)*u2,2*u2*u0+2*u2*u4+(2*u1+2*u5)*u3
3843: -u2+u1^2,2*u1*u0+(2*u3+2*u5)*u4+2*u2*u3+2*u1*u2-u1,
1.1 noro 3844: u0^2-u0+2*u4^2+2*u3^2+2*u2^2+2*u1^2+2*u5^2]
3845: [90] hkatsura(5);
3846: [-t+u0+2*u4+2*u3+2*u2+2*u1+2*u5,
3847: -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,
3848: -u2*t+2*u2*u0+2*u2*u4+(2*u1+2*u5)*u3+u1^2,
3849: -u1*t+2*u1*u0+(2*u3+2*u5)*u4+2*u2*u3+2*u1*u2,
3850: -u0*t+u0^2+2*u4^2+2*u3^2+2*u2^2+2*u1^2+2*u5^2]
3851: [91] cyclic(6);
3852: [c5*c4*c3*c2*c1*c0-1,
3853: ((((c4+c5)*c3+c5*c4)*c2+c5*c4*c3)*c1+c5*c4*c3*c2)*c0+c5*c4*c3*c2*c1,
3854: (((c3+c5)*c2+c5*c4)*c1+c5*c4*c3)*c0+c4*c3*c2*c1+c5*c4*c3*c2,
3855: ((c2+c5)*c1+c5*c4)*c0+c3*c2*c1+c4*c3*c2+c5*c4*c3,
3856: (c1+c5)*c0+c2*c1+c3*c2+c4*c3+c5*c4,c0+c1+c2+c3+c4+c5]
3857: [92] hcyclic(6);
3858: [-c^6+c5*c4*c3*c2*c1*c0,
3859: ((((c4+c5)*c3+c5*c4)*c2+c5*c4*c3)*c1+c5*c4*c3*c2)*c0+c5*c4*c3*c2*c1,
3860: (((c3+c5)*c2+c5*c4)*c1+c5*c4*c3)*c0+c4*c3*c2*c1+c5*c4*c3*c2,
3861: ((c2+c5)*c1+c5*c4)*c0+c3*c2*c1+c4*c3*c2+c5*c4*c3,
3862: (c1+c5)*c0+c2*c1+c3*c2+c4*c3+c5*c4,c0+c1+c2+c3+c4+c5]
3863: @end example
3864:
3865: @table @t
1.2 noro 3866: \JP @item $B;2>H(B
3867: \EG @item References
1.1 noro 3868: @fref{dp_dtop}.
3869: @end table
3870:
1.3 noro 3871: \JP @node primadec primedec,,, $B%0%l%V%J4pDl$K4X$9$kH!?t(B
3872: \EG @node primadec primedec,,, Functions for Groebner basis computation
3873: @subsection @code{primadec}, @code{primedec}
3874: @findex primadec
3875: @findex primedec
3876:
3877: @table @t
3878: @item primadec(@var{plist},@var{vlist})
3879: @item primedec(@var{plist},@var{vlist})
3880: \JP :: $B%$%G%"%k$NJ,2r(B
3881: \EG :: Computes decompositions of ideals.
3882: @end table
3883:
3884: @table @var
3885: @item return
3886: @itemx plist
3887: \JP $BB?9`<0%j%9%H(B
3888: \EG list of polynomials
3889: @item vlist
3890: \JP $BJQ?t%j%9%H(B
3891: \EG list of variables
3892: @end table
3893:
3894: @itemize @bullet
3895: \BJP
3896: @item
3897: @code{primadec()}, @code{primedec} $B$O(B @samp{primdec} $B$GDj5A$5$l$F$$$k(B.
3898: @item
3899: @code{primadec()}, @code{primedec()} $B$O$=$l$>$lM-M}?tBN>e$G$N%$%G%"%k$N(B
3900: $B=`AGJ,2r(B, $B:,4p$NAG%$%G%"%kJ,2r$r9T$&(B.
3901: @item
3902: $B0z?t$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.
3903: @item
3904: @code{primadec} $B$O(B @code{[$B=`AG@.J,(B, $BIUB0AG%$%G%"%k(B]} $B$N%j%9%H$rJV$9(B.
3905: @item
3906: @code{primadec} $B$O(B $BAG0x;R$N%j%9%H$rJV$9(B.
3907: @item
3908: $B7k2L$K$*$$$F(B, $BB?9`<0%j%9%H$H$7$FI=<($5$l$F$$$k3F%$%G%"%k$OA4$F(B
3909: $B%0%l%V%J4pDl$G$"$k(B. $BBP1~$9$k9`=g=x$O(B, $B$=$l$>$l(B
3910: $BJQ?t(B @code{PRIMAORD}, @code{PRIMEORD} $B$K3JG<$5$l$F$$$k(B.
3911: @item
3912: @code{primadec} $B$O(B @code{[Shimoyama,Yokoyama]} $B$N=`AGJ,2r%"%k%4%j%:%`(B
3913: $B$r<BAu$7$F$$$k(B.
3914: @item
3915: $B$b$7AG0x;R$N$_$r5a$a$?$$$J$i(B, @code{primedec} $B$r;H$&J}$,$h$$(B.
3916: $B$3$l$O(B, $BF~NO%$%G%"%k$,:,4p%$%G%"%k$G$J$$>l9g$K(B, @code{primadec}
3917: $B$N7W;;$KM>J,$J%3%9%H$,I,MW$H$J$k>l9g$,$"$k$+$i$G$"$k(B.
3918: \E
3919: \BEG
3920: @item
3921: Function @code{primadec()} and @code{primedec} are defined in @samp{primdec}.
3922: @item
3923: @code{primadec()}, @code{primedec()} are the function for primary
3924: ideal decomposition and prime decomposition of the radical over the
3925: rationals respectively.
3926: @item
3927: The arguments are a list of polynomials and a list of variables.
3928: These functions accept ideals with rational function coefficients only.
3929: @item
3930: @code{primadec} returns the list of pair lists consisting a primary component
3931: and its associated prime.
3932: @item
3933: @code{primedec} returns the list of prime components.
3934: @item
3935: Each component is a Groebner basis and the corresponding term order
3936: is indicated by the global variables @code{PRIMAORD}, @code{PRIMEORD}
3937: respectively.
3938: @item
3939: @code{primadec} implements the primary decompostion algorithm
3940: in @code{[Shimoyama,Yokoyama]}.
3941: @item
3942: If one only wants to know the prime components of an ideal, then
3943: use @code{primedec} because @code{primadec} may need additional costs
3944: if an input ideal is not radical.
3945: \E
3946: @end itemize
3947:
3948: @example
3949: [84] load("primdec")$
3950: [102] primedec([p*q*x-q^2*y^2+q^2*y,-p^2*x^2+p^2*x+p*q*y,
3951: (q^3*y^4-2*q^3*y^3+q^3*y^2)*x-q^3*y^4+q^3*y^3,
3952: -q^3*y^4+2*q^3*y^3+(-q^3+p*q^2)*y^2],[p,q,x,y]);
3953: [[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]]
3954: [103] primadec([x,z*y,w*y^2,w^2*y-z^3,y^3],[x,y,z,w]);
3955: [[[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]]]
3956: @end example
3957:
3958: @table @t
3959: \JP @item $B;2>H(B
3960: \EG @item References
3961: @fref{fctr sqfr},
3962: \JP @fref{$B9`=g=x$N@_Dj(B}.
3963: \EG @fref{Setting term orderings}.
3964: @end table
1.5 noro 3965:
3966: \JP @node primedec_mod,,, $B%0%l%V%J4pDl$K4X$9$kH!?t(B
3967: \EG @node primedec_mod,,, Functions for Groebner basis computation
3968: @subsection @code{primedec_mod}
3969: @findex primedec_mod
3970:
3971: @table @t
3972: @item primedec_mod(@var{plist},@var{vlist},@var{ord},@var{mod},@var{strategy})
3973: \JP :: $B%$%G%"%k$NJ,2r(B
3974: \EG :: Computes decompositions of ideals over small finite fields.
3975: @end table
3976:
3977: @table @var
3978: @item return
3979: @itemx plist
3980: \JP $BB?9`<0%j%9%H(B
3981: \EG list of polynomials
3982: @item vlist
3983: \JP $BJQ?t%j%9%H(B
3984: \EG list of variables
3985: @item ord
3986: \JP $B?t(B, $B%j%9%H$^$?$O9TNs(B
3987: \EG number, list or matrix
3988: @item mod
3989: \JP $B@5@0?t(B
3990: \EG positive integer
3991: @item strategy
3992: \JP $B@0?t(B
3993: \EG integer
3994: @end table
3995:
3996: @itemize @bullet
3997: \BJP
3998: @item
3999: @code{primedec_mod()} $B$O(B @samp{primdec_mod}
4000: $B$GDj5A$5$l$F$$$k(B. @code{[Yokoyama]} $B$NAG%$%G%"%kJ,2r%"%k%4%j%:%`(B
4001: $B$r<BAu$7$F$$$k(B.
4002: @item
4003: @code{primedec_mod()} $B$OM-8BBN>e$G$N%$%G%"%k$N(B
4004: $B:,4p$NAG%$%G%"%kJ,2r$r9T$$(B, $BAG%$%G%"%k$N%j%9%H$rJV$9(B.
4005: @item
4006: @code{primedec_mod()} $B$O(B, GF(@var{mod}) $B>e$G$NJ,2r$rM?$($k(B.
4007: $B7k2L$N3F@.J,$N@8@.85$O(B, $B@0?t78?tB?9`<0$G$"$k(B.
4008: @item
4009: $B7k2L$K$*$$$F(B, $BB?9`<0%j%9%H$H$7$FI=<($5$l$F$$$k3F%$%G%"%k$OA4$F(B
4010: [@var{vlist},@var{ord}] $B$G;XDj$5$l$k9`=g=x$K4X$9$k%0%l%V%J4pDl$G$"$k(B.
4011: @item
4012: @var{strategy} $B$,(B 0 $B$G$J$$$H$-(B, incremental $B$K(B component $B$N6&DL(B
4013: $BItJ,$r7W;;$9$k$3$H$K$h$k(B early termination $B$r9T$&(B. $B0lHL$K(B,
4014: $B%$%G%"%k$N<!85$,9b$$>l9g$KM-8z$@$,(B, 0 $B<!85$N>l9g$J$I(B, $B<!85$,>.$5$$(B
4015: $B>l9g$K$O(B overhead $B$,Bg$-$$>l9g$,$"$k(B.
1.7 noro 4016: @item
4017: $B7W;;ESCf$GFbIt>pJs$r8+$?$$>l9g$K$O!"(B
4018: $BA0$b$C$F(B @code{dp_gr_print(2)} $B$r<B9T$7$F$*$1$P$h$$(B.
1.5 noro 4019: \E
4020: \BEG
4021: @item
4022: Function @code{primedec_mod()}
4023: is defined in @samp{primdec_mod} and implements the prime decomposition
4024: algorithm in @code{[Yokoyama]}.
4025: @item
4026: @code{primedec_mod()}
4027: is the function for prime ideal decomposition
4028: of the radical of a polynomial ideal over small finite field,
4029: and they return a list of prime ideals, which are associated primes
4030: of the input ideal.
4031: @item
4032: @code{primedec_mod()} gives the decomposition over GF(@var{mod}).
4033: The generators of each resulting component consists of integral polynomials.
4034: @item
4035: Each resulting component is a Groebner basis with respect to
4036: a term order specified by [@var{vlist},@var{ord}].
4037: @item
4038: If @var{strategy} is non zero, then the early termination strategy
4039: is tried by computing the intersection of obtained components
4040: incrementally. In general, this strategy is useful when the krull
4041: dimension of the ideal is high, but it may add some overhead
4042: if the dimension is small.
1.7 noro 4043: @item
4044: If you want to see internal information during the computation,
4045: execute @code{dp_gr_print(2)} in advance.
1.5 noro 4046: \E
4047: @end itemize
4048:
4049: @example
4050: [0] load("primdec_mod")$
4051: [246] PP444=[x^8+x^2+t,y^8+y^2+t,z^8+z^2+t]$
4052: [247] primedec_mod(PP444,[x,y,z,t],0,2,1);
4053: [[y+z,x+z,z^8+z^2+t],[x+y,y^2+y+z^2+z+1,z^8+z^2+t],
4054: [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],
4055: [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],
4056: [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],
4057: [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]]
4058: [248]
4059: @end example
4060:
4061: @table @t
4062: \JP @item $B;2>H(B
4063: \EG @item References
4064: @fref{modfctr},
1.6 noro 4065: @fref{dp_gr_main dp_gr_mod_main dp_gr_f_main dp_weyl_gr_main dp_weyl_gr_mod_main dp_weyl_gr_f_main},
1.5 noro 4066: \JP @fref{$B9`=g=x$N@_Dj(B}.
1.7 noro 4067: \EG @fref{Setting term orderings},
4068: @fref{dp_gr_flags dp_gr_print}.
1.5 noro 4069: @end table
4070:
1.10 noro 4071: \JP @node bfunction bfct generic_bfct ann ann0,,, $B%0%l%V%J4pDl$K4X$9$kH!?t(B
4072: \EG @node bfunction bfct generic_bfct ann ann0,,, Functions for Groebner basis computation
4073: @subsection @code{bfunction}, @code{bfct}, @code{generic_bfct}, @code{ann}, @code{ann0}
1.6 noro 4074: @findex bfunction
1.9 noro 4075: @findex bfct
1.6 noro 4076: @findex generic_bfct
1.10 noro 4077: @findex ann
4078: @findex ann0
1.5 noro 4079:
1.6 noro 4080: @table @t
4081: @item bfunction(@var{f})
1.10 noro 4082: @itemx bfct(@var{f})
4083: @itemx generic_bfct(@var{plist},@var{vlist},@var{dvlist},@var{weight})
4084: \JP :: @var{b} $B4X?t$N7W;;(B
4085: \EG :: Computes the global @var{b} function of a polynomial or an ideal
4086: @item ann(@var{f})
4087: @itemx ann0(@var{f})
4088: \JP :: $BB?9`<0$N%Y%-$N(B annihilator $B$N7W;;(B
4089: \EG :: Computes the annihilator of a power of polynomial
1.6 noro 4090: @end table
1.10 noro 4091:
1.6 noro 4092: @table @var
4093: @item return
1.10 noro 4094: \JP $BB?9`<0$^$?$O%j%9%H(B
4095: \EG polynomial or list
4096: @item f
1.6 noro 4097: \JP $BB?9`<0(B
4098: \EG polynomial
4099: @item plist
4100: \JP $BB?9`<0%j%9%H(B
4101: \EG list of polynomials
4102: @item vlist dvlist
4103: \JP $BJQ?t%j%9%H(B
4104: \EG list of variables
4105: @end table
1.5 noro 4106:
1.6 noro 4107: @itemize @bullet
4108: \BJP
4109: @item @samp{bfct} $B$GDj5A$5$l$F$$$k(B.
1.10 noro 4110: @item @code{bfunction(@var{f})}, @code{bfct(@var{f})} $B$OB?9`<0(B @var{f} $B$N(B global @var{b} $B4X?t(B @code{b(s)} $B$r(B
1.6 noro 4111: $B7W;;$9$k(B. @code{b(s)} $B$O(B, Weyl $BBe?t(B @code{D} $B>e$N0lJQ?tB?9`<04D(B @code{D[s]}
4112: $B$N85(B @code{P(x,s)} $B$,B8:_$7$F(B, @code{P(x,s)f^(s+1)=b(s)f^s} $B$rK~$?$9$h$&$J(B
4113: $BB?9`<0(B @code{b(s)} $B$NCf$G(B, $B<!?t$,:G$bDc$$$b$N$G$"$k(B.
4114: @item @code{generic_bfct(@var{f},@var{vlist},@var{dvlist},@var{weight})}
4115: $B$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 4116: $B%&%'%$%H(B @var{weight} $B$K4X$9$k(B global @var{b} $B4X?t$r7W;;$9$k(B.
1.6 noro 4117: @var{vlist} $B$O(B @code{x}-$BJQ?t(B, @var{vlist} $B$OBP1~$9$k(B @code{D}-$BJQ?t(B
4118: $B$r=g$KJB$Y$k(B.
1.9 noro 4119: @item @code{bfunction} $B$H(B @code{bfct} $B$G$OMQ$$$F$$$k%"%k%4%j%:%`$,(B
1.11 noro 4120: $B0[$J$k(B. $B$I$A$i$,9bB.$+$OF~NO$K$h$k(B.
1.10 noro 4121: @item @code{ann(@var{f})} $B$O(B, @code{@var{f}^s} $B$N(B annihilator ideal
4122: $B$N@8@.7O$rJV$9(B. @code{ann(@var{f})} $B$O(B, @code{[@var{a},@var{list}]}
4123: $B$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,
4124: @var{list} $B$O(B @code{ann(@var{f})} $B$N7k2L$N(B @code{s}$ $B$K(B, @var{a} $B$r(B
4125: $BBeF~$7$?$b$N$G$"$k(B.
1.7 noro 4126: @item $B>\:Y$K$D$$$F$O(B, [Saito,Sturmfels,Takayama] $B$r8+$h(B.
1.6 noro 4127: \E
4128: \BEG
4129: @item These functions are defined in @samp{bfct}.
1.10 noro 4130: @item @code{bfunction(@var{f})} and @code{bfct(@var{f})} compute the global @var{b}-function @code{b(s)} of
1.6 noro 4131: a polynomial @var{f}.
4132: @code{b(s)} is a polynomial of the minimal degree
4133: such that there exists @code{P(x,s)} in D[s], which is a polynomial
4134: ring over Weyl algebra @code{D}, and @code{P(x,s)f^(s+1)=b(s)f^s} holds.
4135: @item @code{generic_bfct(@var{f},@var{vlist},@var{dvlist},@var{weight})}
1.10 noro 4136: computes the global @var{b}-function of a left ideal @code{I} in @code{D}
1.6 noro 4137: generated by @var{plist}, with respect to @var{weight}.
4138: @var{vlist} is the list of @code{x}-variables,
4139: @var{vlist} is the list of corresponding @code{D}-variables.
1.9 noro 4140: @item @code{bfunction(@var{f})} and @code{bfct(@var{f})} implement
4141: different algorithms and the efficiency depends on inputs.
1.10 noro 4142: @item @code{ann(@var{f})} returns the generator set of the annihilator
4143: ideal of @code{@var{f}^s}.
4144: @code{ann(@var{f})} returns a list @code{[@var{a},@var{list}]},
4145: where @var{a} is the minimal integral root of the global @var{b}-function
4146: of @var{f}, and @var{list} is a list of polynomials obtained by
4147: substituting @code{s} in @code{ann(@var{f})} with @var{a}.
1.7 noro 4148: @item See [Saito,Sturmfels,Takayama] for the details.
1.6 noro 4149: \E
4150: @end itemize
4151:
4152: @example
4153: [0] load("bfct")$
4154: [216] bfunction(x^3+y^3+z^3+x^2*y^2*z^2+x*y*z);
4155: -9*s^5-63*s^4-173*s^3-233*s^2-154*s-40
4156: [217] fctr(@@);
4157: [[-1,1],[s+2,1],[3*s+4,1],[3*s+5,1],[s+1,2]]
4158: [218] F = [4*x^3*dt+y*z*dt+dx,x*z*dt+4*y^3*dt+dy,
4159: x*y*dt+5*z^4*dt+dz,-x^4-z*y*x-y^4-z^5+t]$
4160: [219] generic_bfct(F,[t,z,y,x],[dt,dz,dy,dx],[1,0,0,0]);
4161: 20000*s^10-70000*s^9+101750*s^8-79375*s^7+35768*s^6-9277*s^5
4162: +1278*s^4-72*s^3
1.10 noro 4163: [220] P=x^3-y^2$
4164: [221] ann(P);
4165: [2*dy*x+3*dx*y^2,-3*dx*x-2*dy*y+6*s]
4166: [222] ann0(P);
4167: [-1,[2*dy*x+3*dx*y^2,-3*dx*x-2*dy*y-6]]
1.6 noro 4168: @end example
4169:
4170: @table @t
4171: \JP @item $B;2>H(B
4172: \EG @item References
4173: \JP @fref{Weyl $BBe?t(B}.
4174: \EG @fref{Weyl algebra}.
4175: @end table
1.5 noro 4176:
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