Annotation of OpenXM/src/asir-doc/parts/groebner.texi, Revision 1.13
1.13 ! noro 1: @comment $OpenXM: OpenXM/src/asir-doc/parts/groebner.texi,v 1.12 2003/12/27 11:52:07 takayama 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$k(B.
! 1070: \E
! 1071: \BEG
! 1072: Term orders introduced in the previous section can be generalized
! 1073: by setting a weight for each variable.
! 1074: \E
! 1075: @example
! 1076: [0] dp_td(<<1,1,1>>);
! 1077: 3
! 1078: [1] dp_set_weight([1,2,3])$
! 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}.
! 1100: By setting weights, different term orders can be set under a term
! 1101: order type. For example, a polynomial can be made weighted homogeneous
! 1102: by setting an appropriate weight.
! 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.
! 1134:
1.2 noro 1135: \E
1.1 noro 1136:
1.2 noro 1137: \BJP
1.1 noro 1138: @node $BM-M}<0$r78?t$H$9$k%0%l%V%J4pDl7W;;(B,,, $B%0%l%V%J4pDl$N7W;;(B
1139: @section $BM-M}<0$r78?t$H$9$k%0%l%V%J4pDl7W;;(B
1.2 noro 1140: \E
1141: \BEG
1142: @node Groebner basis computation with rational function coefficients,,, Groebner basis computation
1143: @section Groebner basis computation with rational function coefficients
1144: \E
1.1 noro 1145:
1146: @noindent
1.2 noro 1147: \BJP
1.1 noro 1148: @code{gr()} $B$J$I$N%H%C%W%l%Y%kH!?t$O(B, $B$$$:$l$b(B, $BF~NOB?9`<0%j%9%H$K(B
1149: $B8=$l$kJQ?t(B ($BITDj85(B) $B$H(B, $BJQ?t%j%9%H$K8=$l$kJQ?t$rHf3S$7$F(B, $BJQ?t%j%9%H$K(B
1150: $B$J$$JQ?t$,F~NOB?9`<0$K8=$l$F$$$k>l9g$K$O(B, $B<+F0E*$K(B, $B$=$NJQ?t$r(B, $B78?t(B
1151: $BBN$N85$H$7$F07$&(B.
1.2 noro 1152: \E
1153: \BEG
1154: Such variables that appear within the input polynomials but
1155: not appearing in the input variable list are automatically treated
1156: as elements in the coefficient field
1157: by top level functions, such as @code{gr()}.
1158: \E
1.1 noro 1159:
1160: @example
1161: [64] gr([a*x+b*y-c,d*x+e*y-f],[x,y],2);
1162: [(-e*a+d*b)*x-f*b+e*c,(-e*a+d*b)*y+f*a-d*c]
1163: @end example
1164:
1165: @noindent
1.2 noro 1166: \BJP
1.1 noro 1167: $B$3$NNc$G$O(B, @code{a}, @code{b}, @code{c}, @code{d} $B$,78?tBN$N85$H$7$F(B
1168: $B07$o$l$k(B. $B$9$J$o$A(B, $BM-M}H!?tBN(B
1169: @b{F} = @b{Q}(@code{a},@code{b},@code{c},@code{d}) $B>e$N(B 2 $BJQ?tB?9`<04D(B
1170: @b{F}[@code{x},@code{y}] $B$K$*$1$k%0%l%V%J4pDl$r5a$a$k$3$H$K$J$k(B.
1171: $BCm0U$9$Y$-$3$H$O(B,
1172: $B78?t$,BN$H$7$F07$o$l$F$$$k$3$H$G$"$k(B. $B$9$J$o$A(B, $B78?t$N4V$KB?9`<0(B
1173: $B$H$7$F$N6&DL0x;R$,$"$C$?>l9g$K$O(B, $B7k2L$+$i$=$N0x;R$O=|$+$l$F$$$k(B
1174: $B$?$a(B, $BM-M}?tBN>e$NB?9`<04D>e$NLdBj$H$7$F9M$($?>l9g$N7k2L$H$O0lHL(B
1175: $B$K$O0[$J$k(B. $B$^$?(B, $B<g$H$7$F7W;;8zN(>e$NLdBj$N$?$a(B, $BJ,;6I=8=B?9`<0(B
1176: $B$N78?t$H$7$F<B:]$K5v$5$l$k$N$OB?9`<0$^$G$G$"$k(B. $B$9$J$o$A(B, $BJ,Jl$r(B
1177: $B;}$DM-M}<0$OJ,;6I=8=B?9`<0$N78?t$H$7$F$O5v$5$l$J$$(B.
1.2 noro 1178: \E
1179: \BEG
1180: In this example, variables @code{a}, @code{b}, @code{c}, and @code{d}
1181: are treated as elements in the coefficient field.
1182: In this case, a Groebner basis is computed
1183: on a bi-variate polynomial ring
1184: @b{F}[@code{x},@code{y}]
1185: over rational function field
1186: @b{F} = @b{Q}(@code{a},@code{b},@code{c},@code{d}).
1187: Notice that coefficients are considered as a member in a field.
1188: As a consequence, polynomial factors common to the coefficients
1189: are removed so that the result, in general, is different from
1190: the result that would be obtained when the problem is considered
1191: as a computation of Groebner basis over a polynomial ring
1192: with rational function coefficients.
1193: And note that coefficients of a distributed polynomial are limited
1194: to numbers and polynomials because of efficiency.
1195: \E
1.1 noro 1196:
1.2 noro 1197: \BJP
1.1 noro 1198: @node $B4pDlJQ49(B,,, $B%0%l%V%J4pDl$N7W;;(B
1199: @section $B4pDlJQ49(B
1.2 noro 1200: \E
1201: \BEG
1202: @node Change of ordering,,, Groebner basis computation
1203: @section Change of orderng
1204: \E
1.1 noro 1205:
1206: @noindent
1.2 noro 1207: \BJP
1.1 noro 1208: $B<-=q<0=g=x$N%0%l%V%J4pDl$r5a$a$k>l9g(B, $BD>@\(B @code{gr()} $B$J$I$r5/F0$9$k(B
1209: $B$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,
1210: $B$=$l$rF~NO$H$7$F<-=q<0=g=x$N%0%l%V%J4pDl$r7W;;$9$kJ}$,8zN($,$h$$>l9g(B
1211: $B$,$"$k(B. $B$^$?(B, $BF~NO$,2?$i$+$N=g=x$G$N%0%l%V%J4pDl$K$J$C$F$$$k>l9g(B, $B4pDl(B
1212: $BJQ49$H8F$P$l$kJ}K!$K$h$j(B, Buchberger $B%"%k%4%j%:%`$K$h$i$:$K8zN(NI$/(B
1213: $B<-=q<0=g=x$N%0%l%V%J4pDl$,7W;;$G$-$k>l9g$,$"$k(B. $B$3$N$h$&$JL\E*$N$?$a$N(B
1214: $BH!?t$,(B, $B%f!<%6Dj5AH!?t$H$7$F(B @samp{gr} $B$K$$$/$D$+Dj5A$5$l$F$$$k(B.
1215: $B0J2<$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
1216: $B4{$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
1217: @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 1218: \E
1219: \BEG
1220: When we compute a lex order Groebner basis, it is often efficient to
1221: compute it via Groebner basis with respect to another order such as
1222: degree reverse lex order, rather than to compute it directory by
1223: @code{gr()} etc. If we know that an input is a Groebner basis with
1224: respect to an order, we can apply special methods called change of
1225: ordering for a Groebner basis computation with respect to another
1226: order, without using Buchberger algorithm. The following two functions
1227: are ones for change of ordering such that they convert a Groebner
1228: basis @var{gbase} with respect to the variable order @var{vlist1} and
1229: the order type @var{order} into a lex Groebner basis with respect
1230: to the variable order @var{vlist2}.
1231: \E
1.1 noro 1232:
1233: @table @code
1234: @item tolex(@var{gbase},@var{vlist1},@var{order},@var{vlist2})
1235:
1.2 noro 1236: \BJP
1.1 noro 1237: $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.
1238: $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
1239: $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 1240: \E
1241: \BEG
1242: This function can be used only when @var{gbase} is an ideal over the
1243: rationals. The input @var{gbase} must be a Groebner basis with respect
1244: to the variable order @var{vlist1} and the order type @var{order}. Moreover
1245: the ideal generated by @var{gbase} must be zero-dimensional.
1246: This computes the lex Groebner basis of @var{gbase}
1247: by using the modular change of ordering algorithm. The algorithm first
1248: computes the lex Groebner basis over a finite field. Then each element
1249: in the lex Groebner basis over the rationals is computed with undetermined
1250: coefficient method and linear equation solving by Hensel lifting.
1251: \E
1.1 noro 1252:
1253: @item tolex_tl(@var{gbase},@var{vlist1},@var{order},@var{vlist2},@var{homo})
1254:
1.2 noro 1255: \BJP
1.1 noro 1256: $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
1257: $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
1258: trace-lifting$B$K$*$1$k%0%l%V%J4pDl8uJd$NF,9`(B, $BF,78?t$N@-<A$rMxMQ$7$F(B,
1259: $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
1260: $B$k$?$a(B, $BC1$K(BBuchberger $B%"%k%4%j%:%`$r7+$jJV$9$h$j8zN($h$/7W;;$G$-$k(B.
1261: $B99$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
1262: $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,
1263: $BD>@\<-=q<0=g=x$N7W;;$r9T$&$h$j8zN($,$h$$(B. ($B$b$A$m$sNc30$"$j(B. )
1264: $B0z?t(B @var{homo} $B$,(B 0 $B$G$J$$;~(B, @code{hgr()} $B$HF1MM$K@F<!2=$r7PM3$7$F(B
1265: $B7W;;$r9T$&(B.
1.2 noro 1266: \E
1267: \BEG
1268: This function computes the lex Groebner basis of @var{gbase}. The
1269: input @var{gbase} must be a Groebner basis with respect to the
1270: variable order @var{vlist1} and the order type @var{order}.
1271: Buchberger algorithm with trace lifting is used to compute the lex
1272: Groebner basis, however the Groebner basis check and the ideal
1273: membership check can be omitted by using several properties derived
1274: from the fact that the input is a Groebner basis. So it is more
1275: efficient than simple repetition of Buchberger algorithm. If the input
1276: is zero-dimensional, this function inserts automatically a computation
1277: of Groebner basis with respect to an elimination order, which makes
1278: the whole computation more efficient for many cases. If @var{homo} is
1279: not equal to 0, homogenization is used in each step.
1280: \E
1.1 noro 1281: @end table
1282:
1283: @noindent
1.2 noro 1284: \BJP
1.1 noro 1285: $B$=$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
1286: $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
1287: @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 1288: \E
1289: \BEG
1290: For zero-dimensional systems, there are several fuctions to
1291: compute the minimal polynomial of a polynomial and or a more compact
1292: representation for zeros of the system. They are all defined in @samp{gr}.
1293: Refer to the sections for each functions.
1294: \E
1.1 noro 1295:
1.2 noro 1296: \BJP
1.6 noro 1297: @node Weyl $BBe?t(B,,, $B%0%l%V%J4pDl$N7W;;(B
1298: @section Weyl $BBe?t(B
1299: \E
1300: \BEG
1301: @node Weyl algebra,,, Groebner basis computation
1302: @section Weyl algebra
1303: \E
1304:
1305: @noindent
1306:
1307: \BJP
1308: $B$3$l$^$G$O(B, $BDL>o$N2D49$JB?9`<04D$K$*$1$k%0%l%V%J4pDl7W;;$K$D$$$F(B
1309: $B=R$Y$F$-$?$,(B, $B%0%l%V%J4pDl$NM}O@$O(B, $B$"$k>r7o$rK~$?$9Hs2D49$J(B
1310: $B4D$K$b3HD%$G$-$k(B. $B$3$N$h$&$J4D$NCf$G(B, $B1~MQ>e$b=EMW$J(B,
1311: Weyl $BBe?t(B, $B$9$J$o$AB?9`<04D>e$NHyJ,:nMQAG4D$N1i;;$*$h$S(B
1312: $B%0%l%V%J4pDl7W;;$,(B Risa/Asir $B$K<BAu$5$l$F$$$k(B.
1313:
1314: $BBN(B @code{K} $B>e$N(B @code{n} $B<!85(B Weyl $BBe?t(B
1315: @code{D=K<x1,@dots{},xn,D1,@dots{},Dn>} $B$O(B
1316: \E
1317:
1318: \BEG
1319: So far we have explained Groebner basis computation in
1320: commutative polynomial rings. However Groebner basis can be
1321: considered in more general non-commutative rings.
1322: Weyl algebra is one of such rings and
1323: Risa/Asir implements fundamental operations
1324: in Weyl algebra and Groebner basis computation in Weyl algebra.
1325:
1326: The @code{n} dimensional Weyl algebra over a field @code{K},
1327: @code{D=K<x1,@dots{},xn,D1,@dots{},Dn>} is a non-commutative
1328: algebra which has the following fundamental relations:
1329: \E
1330:
1331: @code{xi*xj-xj*xi=0}, @code{Di*Dj-Dj*Di=0}, @code{Di*xj-xj*Di=0} (@code{i!=j}),
1332: @code{Di*xi-xi*Di=1}
1333:
1334: \BJP
1335: $B$H$$$&4pK\4X78$r;}$D4D$G$"$k(B. @code{D} $B$O(B $BB?9`<04D(B @code{K[x1,@dots{},xn]} $B$r78?t(B
1336: $B$H$9$kHyJ,:nMQAG4D$G(B, @code{Di} $B$O(B @code{xi} $B$K$h$kHyJ,$rI=$9(B. $B8r494X78$K$h$j(B,
1337: @code{D} $B$N85$O(B, @code{x1^i1*@dots{}*xn^in*D1^j1*@dots{}*Dn^jn} $B$J$kC19`(B
1338: $B<0$N(B @code{K} $B@~7A7k9g$H$7$F=q$-I=$9$3$H$,$G$-$k(B.
1339: Risa/Asir $B$K$*$$$F$O(B, $B$3$NC19`<0$r(B, $B2D49$JB?9`<0$HF1MM$K(B
1340: @code{<<i1,@dots{},in,j1,@dots{},jn>>} $B$GI=$9(B. $B$9$J$o$A(B, @code{D} $B$N85$b(B
1341: $BJ,;6I=8=B?9`<0$H$7$FI=$5$l$k(B. $B2C8:;;$O(B, $B2D49$N>l9g$HF1MM$K(B, @code{+}, @code{-}
1342: $B$K$h$j(B
1343: $B<B9T$G$-$k$,(B, $B>h;;$O(B, $BHs2D49@-$r9MN8$7$F(B @code{dp_weyl_mul()} $B$H$$$&4X?t(B
1344: $B$K$h$j<B9T$9$k(B.
1345: \E
1346:
1347: \BEG
1348: @code{D} is the ring of differential operators whose coefficients
1349: are polynomials in @code{K[x1,@dots{},xn]} and
1350: @code{Di} denotes the differentiation with respect to @code{xi}.
1351: According to the commutation relation,
1352: elements of @code{D} can be represented as a @code{K}-linear combination
1353: of monomials @code{x1^i1*@dots{}*xn^in*D1^j1*@dots{}*Dn^jn}.
1354: In Risa/Asir, this type of monomial is represented
1355: by @code{<<i1,@dots{},in,j1,@dots{},jn>>} as in the case of commutative
1356: polynomial.
1357: That is, elements of @code{D} are represented by distributed polynomials.
1358: Addition and subtraction can be done by @code{+}, @code{-},
1359: but multiplication is done by calling @code{dp_weyl_mul()} because of
1360: the non-commutativity of @code{D}.
1361: \E
1362:
1363: @example
1364: [0] A=<<1,2,2,1>>;
1365: (1)*<<1,2,2,1>>
1366: [1] B=<<2,1,1,2>>;
1367: (1)*<<2,1,1,2>>
1368: [2] A*B;
1369: (1)*<<3,3,3,3>>
1370: [3] dp_weyl_mul(A,B);
1371: (1)*<<3,3,3,3>>+(1)*<<3,2,3,2>>+(4)*<<2,3,2,3>>+(4)*<<2,2,2,2>>
1372: +(2)*<<1,3,1,3>>+(2)*<<1,2,1,2>>
1373: @end example
1374:
1375: \BJP
1376: $B%0%l%V%J4pDl7W;;$K$D$$$F$b(B, Weyl $BBe?t@lMQ$N4X?t$H$7$F(B,
1377: $B<!$N4X?t$,MQ0U$7$F$"$k(B.
1378: \E
1379: \BEG
1380: The following functions are avilable for Groebner basis computation
1381: in Weyl algebra:
1382: \E
1383: @code{dp_weyl_gr_main()},
1384: @code{dp_weyl_gr_mod_main()},
1385: @code{dp_weyl_gr_f_main()},
1386: @code{dp_weyl_f4_main()},
1387: @code{dp_weyl_f4_mod_main()}.
1388: \BJP
1389: $B$^$?(B, $B1~MQ$H$7$F(B, global b $B4X?t$N7W;;$,<BAu$5$l$F$$$k(B.
1390: \E
1391: \BEG
1392: Computation of the global b function is implemented as an application.
1393: \E
1394:
1395: \BJP
1.1 noro 1396: @node $B%0%l%V%J4pDl$K4X$9$kH!?t(B,,, $B%0%l%V%J4pDl$N7W;;(B
1397: @section $B%0%l%V%J4pDl$K4X$9$kH!?t(B
1.2 noro 1398: \E
1399: \BEG
1400: @node Functions for Groebner basis computation,,, Groebner basis computation
1401: @section Functions for Groebner basis computation
1402: \E
1.1 noro 1403:
1404: @menu
1405: * gr hgr gr_mod::
1406: * lex_hensel lex_tl tolex tolex_d tolex_tl::
1407: * lex_hensel_gsl tolex_gsl tolex_gsl_d::
1408: * gr_minipoly minipoly::
1409: * tolexm minipolym::
1.6 noro 1410: * 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::
1411: * dp_f4_main dp_f4_mod_main dp_weyl_f4_main dp_weyl_f4_mod_main::
1.1 noro 1412: * dp_gr_flags dp_gr_print::
1413: * dp_ord::
1414: * dp_ptod::
1415: * dp_dtop::
1416: * dp_mod dp_rat::
1417: * dp_homo dp_dehomo::
1418: * dp_ptozp dp_prim::
1419: * dp_nf dp_nf_mod dp_true_nf dp_true_nf_mod::
1420: * dp_hm dp_ht dp_hc dp_rest::
1421: * dp_td dp_sugar::
1422: * dp_lcm::
1423: * dp_redble::
1424: * dp_subd::
1425: * dp_mbase::
1426: * dp_mag::
1427: * dp_red dp_red_mod::
1428: * dp_sp dp_sp_mod::
1429: * p_nf p_nf_mod p_true_nf p_true_nf_mod ::
1430: * p_terms::
1431: * gb_comp::
1432: * katsura hkatsura cyclic hcyclic::
1433: * dp_vtoe dp_etov::
1434: * lex_hensel_gsl tolex_gsl tolex_gsl_d::
1.3 noro 1435: * primadec primedec::
1.5 noro 1436: * primedec_mod::
1.10 noro 1437: * bfunction bfct generic_bfct ann ann0::
1.1 noro 1438: @end menu
1439:
1.2 noro 1440: \JP @node gr hgr gr_mod,,, $B%0%l%V%J4pDl$K4X$9$kH!?t(B
1441: \EG @node gr hgr gr_mod,,, Functions for Groebner basis computation
1.1 noro 1442: @subsection @code{gr}, @code{hgr}, @code{gr_mod}, @code{dgr}
1443: @findex gr
1444: @findex hgr
1445: @findex gr_mod
1446: @findex dgr
1447:
1448: @table @t
1449: @item gr(@var{plist},@var{vlist},@var{order})
1450: @itemx hgr(@var{plist},@var{vlist},@var{order})
1451: @itemx gr_mod(@var{plist},@var{vlist},@var{order},@var{p})
1452: @itemx dgr(@var{plist},@var{vlist},@var{order},@var{procs})
1.2 noro 1453: \JP :: $B%0%l%V%J4pDl$N7W;;(B
1454: \EG :: Groebner basis computation
1.1 noro 1455: @end table
1456:
1457: @table @var
1458: @item return
1.2 noro 1459: \JP $B%j%9%H(B
1460: \EG list
1.4 noro 1461: @item plist vlist procs
1.2 noro 1462: \JP $B%j%9%H(B
1463: \EG list
1.1 noro 1464: @item order
1.2 noro 1465: \JP $B?t(B, $B%j%9%H$^$?$O9TNs(B
1466: \EG number, list or matrix
1.1 noro 1467: @item p
1.2 noro 1468: \JP 2^27 $BL$K~$NAG?t(B
1469: \EG prime less than 2^27
1.1 noro 1470: @end table
1471:
1472: @itemize @bullet
1.2 noro 1473: \BJP
1.1 noro 1474: @item
1475: $BI8=`%i%$%V%i%j$N(B @samp{gr} $B$GDj5A$5$l$F$$$k(B.
1476: @item
1477: $B$$$:$l$b(B, $BB?9`<0%j%9%H(B @var{plist} $B$N(B, $BJQ?t=g=x(B @var{vlist}, $B9`=g=x7?(B
1478: @var{order} $B$K4X$9$k%0%l%V%J4pDl$r5a$a$k(B. @code{gr()}, @code{hgr()}
1479: $B$O(B $BM-M}?t78?t(B, @code{gr_mod()} $B$O(B GF(@var{p}) $B78?t$H$7$F7W;;$9$k(B.
1480: @item
1481: @var{vlist} $B$OITDj85$N%j%9%H(B. @var{vlist} $B$K8=$l$J$$ITDj85$O(B,
1482: $B78?tBN$KB0$9$k$H8+$J$5$l$k(B.
1483: @item
1484: @code{gr()}, trace-lifting ($B%b%8%e%i1i;;$rMQ$$$?9bB.2=(B) $B$*$h$S(B sugar
1485: strategy $B$K$h$k7W;;(B, @code{hgr()} $B$O(B trace-lifting $B$*$h$S(B
1486: $B@F<!2=$K$h$k(B $B6:@5$5$l$?(B sugar strategy $B$K$h$k7W;;$r9T$&(B.
1487: @item
1488: @code{dgr()} $B$O(B, @code{gr()}, @code{dgr()} $B$r(B
1489: $B;R%W%m%;%9%j%9%H(B @var{procs} $B$N(B 2 $B$D$N%W%m%;%9$K$h$jF1;~$K7W;;$5$;(B,
1490: $B@h$K7k2L$rJV$7$?J}$N7k2L$rJV$9(B. $B7k2L$OF10l$G$"$k$,(B, $B$I$A$i$NJ}K!$,(B
1491: $B9bB.$+0lHL$K$OITL@$N$?$a(B, $B<B:]$N7P2a;~4V$rC;=L$9$k$N$KM-8z$G$"$k(B.
1492: @item
1493: @code{dgr()} $B$GI=<($5$l$k;~4V$O(B, $B$3$NH!?t$,<B9T$5$l$F$$$k%W%m%;%9$G$N(B
1494: 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 1495: @item
1496: $BB?9`<0%j%9%H(B @var{plist} $B$NMWAG$,J,;6I=8=B?9`<0$N>l9g$O(B
1497: $B7k2L$bJ,;6I=8=B?9`<0$N%j%9%H$G$"$k(B.
1498: $B$3$N>l9g(B, $B0z?t$NJ,;6B?9`<0$OM?$($i$l$?=g=x$K=>$$(B @code{dp_sort} $B$G(B
1499: $B%=!<%H$5$l$F$+$i7W;;$5$l$k(B.
1500: $BB?9`<0%j%9%H$NMWAG$,J,;6I=8=B?9`<0$N>l9g$b(B
1501: $BJQ?t$N?tJ,$NITDj85$N%j%9%H$r(B @var{vlist} $B0z?t$H$7$FM?$($J$$$H$$$1$J$$(B
1502: ($B%@%_!<(B).
1.2 noro 1503: \E
1504: \BEG
1505: @item
1506: These functions are defined in @samp{gr} in the standard library
1507: directory.
1508: @item
1509: They compute a Groebner basis of a polynomial list @var{plist} with
1510: respect to the variable order @var{vlist} and the order type @var{order}.
1511: @code{gr()} and @code{hgr()} compute a Groebner basis over the rationals
1512: and @code{gr_mod} computes over GF(@var{p}).
1513: @item
1514: Variables not included in @var{vlist} are regarded as
1515: included in the ground field.
1516: @item
1517: @code{gr()} uses trace-lifting (an improvement by modular computation)
1518: and sugar strategy.
1519: @code{hgr()} uses trace-lifting and a cured sugar strategy
1520: by using homogenization.
1521: @item
1522: @code{dgr()} executes @code{gr()}, @code{dgr()} simultaneously on
1523: two process in a child process list @var{procs} and returns
1524: the result obtained first. The results returned from both the process
1525: should be equal, but it is not known in advance which method is faster.
1526: Therefore this function is useful to reduce the actual elapsed time.
1527: @item
1528: The CPU time shown after an exection of @code{dgr()} indicates
1529: that of the master process, and most of the time corresponds to the time
1530: for communication.
1.12 takayama 1531: @item
1532: When the elements of @var{plist} are distributed polynomials,
1533: the result is also a list of distributed polynomials.
1534: In this case, firstly the elements of @var{plist} is sorted by @code{dp_sort}
1535: and the Grobner basis computation is started.
1536: Variables must be given in @var{vlist} even in this case
1537: (these variables are dummy).
1.2 noro 1538: \E
1.1 noro 1539: @end itemize
1540:
1541: @example
1542: [0] load("gr")$
1543: [64] load("cyclic")$
1544: [74] G=gr(cyclic(5),[c0,c1,c2,c3,c4],2);
1545: [c4^15+122*c4^10-122*c4^5-1,...]
1546: [75] GM=gr_mod(cyclic(5),[c0,c1,c2,c3,c4],2,31991)$
1547: 24628*c4^15+29453*c4^10+2538*c4^5+7363
1548: [76] (G[0]*24628-GM[0])%31991;
1549: 0
1550: @end example
1551:
1552: @table @t
1.2 noro 1553: \JP @item $B;2>H(B
1554: \EG @item References
1.6 noro 1555: @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 1556: @fref{dp_ord}.
1557: @end table
1558:
1.2 noro 1559: \JP @node lex_hensel lex_tl tolex tolex_d tolex_tl,,, $B%0%l%V%J4pDl$K4X$9$kH!?t(B
1560: \EG @node lex_hensel lex_tl tolex tolex_d tolex_tl,,, Functions for Groebner basis computation
1.1 noro 1561: @subsection @code{lex_hensel}, @code{lex_tl}, @code{tolex}, @code{tolex_d}, @code{tolex_tl}
1562: @findex lex_hensel
1563: @findex lex_tl
1564: @findex tolex
1565: @findex tolex_d
1566: @findex tolex_tl
1567:
1568: @table @t
1569: @item lex_hensel(@var{plist},@var{vlist1},@var{order},@var{vlist2},@var{homo})
1570: @itemx lex_tl(@var{plist},@var{vlist1},@var{order},@var{vlist2},@var{homo})
1.2 noro 1571: \JP :: $B4pDlJQ49$K$h$k<-=q<0=g=x%0%l%V%J4pDl$N7W;;(B
1572: \EG:: Groebner basis computation with respect to a lex order by change of ordering
1.1 noro 1573: @item tolex(@var{plist},@var{vlist1},@var{order},@var{vlist2})
1574: @itemx tolex_d(@var{plist},@var{vlist1},@var{order},@var{vlist2},@var{procs})
1575: @itemx tolex_tl(@var{plist},@var{vlist1},@var{order},@var{vlist2},@var{homo})
1.2 noro 1576: \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
1577: \EG :: Groebner basis computation with respect to a lex order by change of ordering, starting from a Groebner basis
1.1 noro 1578: @end table
1579:
1580: @table @var
1581: @item return
1.2 noro 1582: \JP $B%j%9%H(B
1583: \EG list
1.4 noro 1584: @item plist vlist1 vlist2 procs
1.2 noro 1585: \JP $B%j%9%H(B
1586: \EG list
1.1 noro 1587: @item order
1.2 noro 1588: \JP $B?t(B, $B%j%9%H$^$?$O9TNs(B
1589: \EG number, list or matrix
1.1 noro 1590: @item homo
1.2 noro 1591: \JP $B%U%i%0(B
1592: \EG flag
1.1 noro 1593: @end table
1594:
1595: @itemize @bullet
1.2 noro 1596: \BJP
1.1 noro 1597: @item
1598: $BI8=`%i%$%V%i%j$N(B @samp{gr} $B$GDj5A$5$l$F$$$k(B.
1599: @item
1600: @code{lex_hensel()}, @code{lex_tl()} $B$O(B,
1601: $BB?9`<0%j%9%H(B @var{plist} $B$N(B, $BJQ?t=g=x(B @var{vlist1}, $B9`=g=x7?(B
1602: @var{order} $B$K4X$9$k%0%l%V%J4pDl$r5a$a(B, $B$=$l$r(B, $BJQ?t=g=x(B @var{vlist2}
1603: $B$N<-=q<0=g=x%0%l%V%J4pDl$KJQ49$9$k(B.
1604: @item
1605: @code{tolex()}, @code{tolex_tl()} $B$O(B,
1606: $BJQ?t=g=x(B @var{vlist1}, $B9`=g=x7?(B @var{order} $B$K4X$9$k%0%l%V%J4pDl$G$"$k(B
1607: $BB?9`<0%j%9%H(B @var{plist} $B$rJQ?t=g=x(B @var{vlist2} $B$N<-=q<0=g=x%0%l%V%J(B
1608: $B4pDl$KJQ49$9$k(B.
1609: @code{tolex_d()} $B$O(B, @code{tolex()} $B$K$*$1$k(B, $B3F4pDl$N7W;;$r(B, $B;R%W%m%;%9(B
1610: $B%j%9%H(B @var{procs} $B$N3F%W%m%;%9$KJ,;67W;;$5$;$k(B.
1611: @item
1612: @code{lex_hensel()}, @code{lex_tl()} $B$K$*$$$F$O(B, $B<-=q<0=g=x%0%l%V%J4pDl$N(B
1613: $B7W;;$O<!$N$h$&$K9T$o$l$k(B. (@code{[Noro,Yokoyama]} $B;2>H(B.)
1614: @enumerate
1615: @item
1616: @var{vlist1}, @var{order} $B$K4X$9$k%0%l%V%J4pDl(B @var{G0} $B$r7W;;$9$k(B.
1617: (@code{lex_hensel()} $B$N$_(B. )
1618: @item
1619: @var{G0} $B$N3F85$N(B @var{vlist2} $B$K4X$9$k<-=q<0=g=x$K$*$1$kF,78?t$r3d$i$J$$(B
1620: $B$h$&$JAG?t(B @var{p} $B$rA*$S(B, GF(@var{p}) $B>e$G$N<-=q<0=g=x%0%l%V%J4pDl(B
1621: @var{Gp} $B$r7W;;$9$k(B.
1622: @item
1623: @var{Gp} $B$K8=$l$k$9$Y$F$N9`$N(B, @var{G0} $B$K4X$9$k@55,7A(B @var{NF} $B$r7W;;$9$k(B.
1624: @item
1625: @var{Gp} $B$N3F85(B @var{f} $B$K$D$-(B, @var{f} $B$N78?t$rL$Dj78?t$G(B,
1626: @var{f} $B$N3F9`$rBP1~$9$k(B @var{NF} $B$N85$GCV$-49$((B, $B3F9`$N78?t$r(B 0 $B$HCV$$$?(B,
1627: $BL$Dj78?t$K4X$9$k@~7AJ}Dx<07O(B @var{Lf} $B$r:n$k(B.
1628: @item
1629: @var{Lf} $B$,(B, $BK!(B @var{p} $B$G0l0U2r$r;}$D$3$H$rMQ$$$F(B @var{Lf} $B$N2r$r(B
1630: $BK!(B @var{p}$B$N2r$+$i(B Hensel $B9=@.$K$h$j5a$a$k(B.
1631: @item
1632: $B$9$Y$F$N(B @var{Gp} $B$N85$K$D$-@~7AJ}Dx<0$,2r$1$?$i$=$N2rA4BN$,5a$a$k(B
1633: $B<-=q<0=g=x$G$N%0%l%V%J4pDl(B. $B$b$7$I$l$+$N@~7AJ}Dx<0$N5a2r$K<:GT$7$?$i(B,
1634: @var{p} $B$r$H$jD>$7$F$d$jD>$9(B.
1635: @end enumerate
1636:
1637: @item
1638: @code{lex_tl()}, @code{tolex_tl()} $B$K$*$$$F$O(B, $B<-=q<0=g=x%0%l%V%J4pDl$N(B
1639: $B7W;;$O<!$N$h$&$K9T$o$l$k(B.
1640:
1641: @enumerate
1642: @item
1643: @var{vlist1}, @var{order} $B$K4X$9$k%0%l%V%J4pDl(B @var{G0} $B$r7W;;$9$k(B.
1644: (@code{lex_hensel()} $B$N$_(B. )
1645: @item
1646: @var{G0} $B$,(B 0 $B<!85%7%9%F%`$G$J$$$H$-(B, @var{G0} $B$rF~NO$H$7$F(B,
1647: @var{G0} $B$N3F85$N(B @var{vlist2} $B$K4X$9$k<-=q<0=g=x$K$*$1$kF,78?t$r3d$i$J$$(B
1648: $B$h$&$JAG?t(B @var{p} $B$rA*$S(B, @var{p} $B$rMQ$$$?(B trace-lifting $B$K$h$j<-=q<0(B
1649: $B=g=x$N%0%l%V%J4pDl8uJd$r5a$a(B, $B$b$75a$^$C$?$J$i%A%'%C%/$J$7$K$=$l$,5a$a$k(B
1650: $B%0%l%V%J4pDl$H$J$k(B. $B$b$7<:GT$7$?$i(B, @var{p} $B$r$H$jD>$7$F$d$jD>$9(B.
1651: @item
1652: @var{G0} $B$,(B 0 $B<!85%7%9%F%`$N$H$-(B, @var{G0} $B$rF~NO$H$7$F(B,
1653: $B$^$:(B, @var{vlist2} $B$N:G8e$NJQ?t0J30$r>C5n$9$k>C5n=g=x$K$h$j(B
1654: $B%0%l%V%J4pDl(B @var{G1} $B$r7W;;$7(B, $B$=$l$+$i<-=q<0=g=x$N%0%l%V%J4pDl$r(B
1655: $B7W;;$9$k(B. $B$=$N:](B, $B3F%9%F%C%W$G$O(B, $BF~NO$N3F85$N(B, $B5a$a$k=g=x$K$*$1$k(B
1656: $BF,78?t$r3d$i$J$$AG?t$rMQ$$$?(B trace-lifting $B$G%0%l%V%J4pDl8uJd$r5a$a(B,
1657: $B$b$75a$^$C$?$i%A%'%C%/$J$7$K$=$l$,$=$N=g=x$G$N%0%l%V%J4pDl$H$J$k(B.
1658: @end enumerate
1659:
1660: @item
1661: $BM-M}<078?t$N7W;;$O(B, @code{lex_tl()}, @code{tolex_tl()} $B$N$_<u$1IU$1$k(B.
1662: @item
1663: @code{homo} $B$,(B 0 $B$G$J$$>l9g(B, $BFbIt$G5/F0$5$l$k(B Buchberger $B%"%k%4%j%:%`$K(B
1664: $B$*$$$F(B, $B@F<!2=$,9T$o$l$k(B.
1665: @item
1666: @code{tolex_d()} $B$GI=<($5$l$k;~4V$O(B, $B$3$NH!?t$,<B9T$5$l$F$$$k%W%m%;%9$K(B
1667: $B$*$$$F9T$o$l$?7W;;$KBP1~$7$F$$$F(B, $B;R%W%m%;%9$K$*$1$k;~4V$O4^$^$l$J$$(B.
1.2 noro 1668: \E
1669: \BEG
1670: @item
1671: These functions are defined in @samp{gr} in the standard library
1672: directory.
1673: @item
1674: @code{lex_hensel()} and @code{lex_tl()} first compute a Groebner basis
1675: with respect to the variable order @var{vlist1} and the order type @var{order}.
1676: Then the Groebner basis is converted into a lex order Groebner basis
1677: with respect to the varable order @var{vlist2}.
1678: @item
1679: @code{tolex()} and @code{tolex_tl()} convert a Groebner basis @var{plist}
1680: with respect to the variable order @var{vlist1} and the order type @var{order}
1681: into a lex order Groebner basis
1682: with respect to the varable order @var{vlist2}.
1683: @code{tolex_d()} does computations of basis elements in @code{tolex()}
1684: in parallel on the processes in a child process list @var{procs}.
1685: @item
1686: In @code{lex_hensel()} and @code{tolex_hensel()} a lex order Groebner basis
1687: is computed as follows.(Refer to @code{[Noro,Yokoyama]}.)
1688: @enumerate
1689: @item
1690: Compute a Groebner basis @var{G0} with respect to @var{vlist1} and @var{order}.
1691: (Only in @code{lex_hensel()}. )
1692: @item
1693: Choose a prime which does not divide head coefficients of elements in @var{G0}
1694: with respect to @var{vlist1} and @var{order}. Then compute a lex order
1695: Groebner basis @var{Gp} over GF(@var{p}) with respect to @var{vlist2}.
1696: @item
1697: Compute @var{NF}, the set of all the normal forms with respect to
1698: @var{G0} of terms appearing in @var{Gp}.
1699: @item
1700: For each element @var{f} in @var{Gp}, replace coefficients and terms in @var{f}
1701: with undetermined coefficients and the corresponding polynomials in @var{NF}
1702: respectively, and generate a system of liear equation @var{Lf} by equating
1703: the coefficients of terms in the replaced polynomial with 0.
1704: @item
1705: Solve @var{Lf} by Hensel lifting, starting from the unique mod @var{p}
1706: solution.
1707: @item
1708: If all the linear equations generated from the elements in @var{Gp}
1709: could be solved, then the set of solutions corresponds to a lex order
1710: Groebner basis. Otherwise redo the whole process with another @var{p}.
1711: @end enumerate
1712:
1713: @item
1714: In @code{lex_tl()} and @code{tolex_tl()} a lex order Groebner basis
1715: is computed as follows.(Refer to @code{[Noro,Yokoyama]}.)
1716:
1717: @enumerate
1718: @item
1719: Compute a Groebner basis @var{G0} with respect to @var{vlist1} and @var{order}.
1720: (Only in @code{lex_tl()}. )
1721: @item
1722: If @var{G0} is not zero-dimensional, choose a prime which does not divide
1723: head coefficients of elements in @var{G0} with respect to @var{vlist1} and
1724: @var{order}. Then compute a candidate of a lex order Groebner basis
1725: via trace lifting with @var{p}. If it succeeds the candidate is indeed
1726: a lex order Groebner basis without any check. Otherwise redo the whole
1727: process with another @var{p}.
1728: @item
1729:
1730: If @var{G0} is zero-dimensional, starting from @var{G0},
1731: compute a Groebner basis @var{G1} with respect to an elimination order
1732: to eliminate variables other than the last varibale in @var{vlist2}.
1733: Then compute a lex order Groebner basis stating from @var{G1}. These
1734: computations are done by trace lifting and the selection of a mudulus
1735: @var{p} is the same as in non zero-dimensional cases.
1736: @end enumerate
1737:
1738: @item
1739: Computations with rational function coefficients can be done only by
1740: @code{lex_tl()} and @code{tolex_tl()}.
1741: @item
1742: If @code{homo} is not equal to 0, homogenization is used in Buchberger
1743: algorithm.
1744: @item
1745: The CPU time shown after an execution of @code{tolex_d()} indicates
1746: that of the master process, and it does not include the time in child
1747: processes.
1748: \E
1.1 noro 1749: @end itemize
1750:
1751: @example
1752: [78] K=katsura(5)$
1753: 30msec + gc : 20msec
1754: [79] V=[u5,u4,u3,u2,u1,u0]$
1755: 0msec
1756: [80] G0=hgr(K,V,2)$
1757: 91.558sec + gc : 15.583sec
1758: [81] G1=lex_hensel(K,V,0,V,0)$
1759: 49.049sec + gc : 9.961sec
1760: [82] G2=lex_tl(K,V,0,V,1)$
1761: 31.186sec + gc : 3.500sec
1762: [83] gb_comp(G0,G1);
1763: 1
1764: 10msec
1765: [84] gb_comp(G0,G2);
1766: 1
1767: @end example
1768:
1769: @table @t
1.2 noro 1770: \JP @item $B;2>H(B
1771: \EG @item References
1.6 noro 1772: @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 1773: \JP @fref{dp_ord}, @fref{$BJ,;67W;;(B}
1774: \EG @fref{dp_ord}, @fref{Distributed computation}
1.1 noro 1775: @end table
1776:
1.2 noro 1777: \JP @node lex_hensel_gsl tolex_gsl tolex_gsl_d,,, $B%0%l%V%J4pDl$K4X$9$kH!?t(B
1778: \EG @node lex_hensel_gsl tolex_gsl tolex_gsl_d,,, Functions for Groebner basis computation
1.1 noro 1779: @subsection @code{lex_hensel_gsl}, @code{tolex_gsl}, @code{tolex_gsl_d}
1780: @findex lex_hensel_gsl
1781: @findex tolex_gsl
1782: @findex tolex_gsl_d
1783:
1784: @table @t
1785: @item lex_hensel_gsl(@var{plist},@var{vlist1},@var{order},@var{vlist2},@var{homo})
1.2 noro 1786: \JP :: GSL $B7A<0$N%$%G%"%k4pDl$N7W;;(B
1787: \EG ::Computation of an GSL form ideal basis
1.8 noro 1788: @item tolex_gsl(@var{plist},@var{vlist1},@var{order},@var{vlist2})
1789: @itemx tolex_gsl_d(@var{plist},@var{vlist1},@var{order},@var{vlist2},@var{procs})
1.2 noro 1790: \JP :: $B%0%l%V%J4pDl$rF~NO$H$9$k(B, GSL $B7A<0$N%$%G%"%k4pDl$N7W;;(B
1791: \EG :: Computation of an GSL form ideal basis stating from a Groebner basis
1.1 noro 1792: @end table
1793:
1794: @table @var
1795: @item return
1.2 noro 1796: \JP $B%j%9%H(B
1797: \EG list
1.4 noro 1798: @item plist vlist1 vlist2 procs
1.2 noro 1799: \JP $B%j%9%H(B
1800: \EG list
1.1 noro 1801: @item order
1.2 noro 1802: \JP $B?t(B, $B%j%9%H$^$?$O9TNs(B
1803: \EG number, list or matrix
1.1 noro 1804: @item homo
1.2 noro 1805: \JP $B%U%i%0(B
1806: \EG flag
1.1 noro 1807: @end table
1808:
1809: @itemize @bullet
1.2 noro 1810: \BJP
1.1 noro 1811: @item
1812: @code{lex_hensel_gsl()} $B$O(B @code{lex_hensel()} $B$N(B, @code{tolex_gsl()} $B$O(B
1813: @code{tolex()} $B$NJQ<o$G(B, $B7k2L$N$_$,0[$J$k(B.
1814: @code{tolex_gsl_d()} $B$O(B, $B4pDl7W;;$r(B, @code{procs} $B$G;XDj$5$l$k;R%W%m%;%9$K(B
1815: $BJ,;67W;;$5$;$k(B.
1816: @item
1817: $BF~NO$,(B 0 $B<!85%7%9%F%`$G(B, $B$=$N<-=q<0=g=x%0%l%V%J4pDl$,(B
1818: @code{[f0,x1-f1,...,xn-fn]} (@code{f0},...,@code{fn} $B$O(B
1819: @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,
1820: @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)
1821: $B$rJV$9(B.
1.2 noro 1822: $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 1823: @code{x0} $B$N(B1 $BJQ?tB?9`<0$G(B,
1824: $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')]}
1825: $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
1826: $B$h$kDL>o$N%0%l%V%J4pDl$rJV$9(B.
1827: @item
1828: GSL $B7A<0$K$h$jI=$5$l$k4pDl$O%0%l%V%J4pDl$G$O$J$$$,(B, $B0lHL$K78?t$,(B SL $B7A<0(B
1829: $B$N%0%l%V%J4pDl$h$jHs>o$K>.$5$$$?$a7W;;$bB.$/(B, $B2r$b5a$a$d$9$$(B.
1830: @code{tolex_gsl_d()} $B$GI=<($5$l$k;~4V$O(B, $B$3$NH!?t$,<B9T$5$l$F$$$k%W%m%;%9$K(B
1831: $B$*$$$F9T$o$l$?7W;;$KBP1~$7$F$$$F(B, $B;R%W%m%;%9$K$*$1$k;~4V$O4^$^$l$J$$(B.
1.2 noro 1832: \E
1833: \BEG
1834: @item
1835: @code{lex_hensel_gsl()} and @code{lex_hensel()} are variants of
1836: @code{tolex_gsl()} and @code{tolex()} respectively. The results are
1837: Groebner basis or a kind of ideal basis, called GSL form.
1838: @code{tolex_gsl_d()} does basis computations in parallel on child
1839: processes specified in @code{procs}.
1840:
1841: @item
1842: If the input is zero-dimensional and a lex order Groebner basis has
1843: the form @code{[f0,x1-f1,...,xn-fn]} (@code{f0},...,@code{fn} are
1844: univariate polynomials of @code{x0}; SL form), then this these
1845: functions return a list such as
1846: @code{[[x1,g1,d1],...,[xn,gn,dn],[x0,f0,f0']]} (GSL form). In this list
1847: @code{gi} is a univariate polynomial of @code{x0} such that
1848: @code{di*f0'*fi-gi} divides @code{f0} and the roots of the input ideal is
1849: @code{[x1=g1/(d1*f0'),...,xn=gn/(dn*f0')]} for @code{x0}
1850: such that @code{f0(x0)=0}.
1851: If the lex order Groebner basis does not have the above form,
1852: these functions return
1853: a lex order Groebner basis computed by @code{tolex()}.
1854: @item
1855: Though an ideal basis represented as GSL form is not a Groebner basis
1856: we can expect that the coefficients are much smaller than those in a Groebner
1857: basis and that the computation is efficient.
1858: The CPU time shown after an execution of @code{tolex_gsl_d()} indicates
1859: that of the master process, and it does not include the time in child
1860: processes.
1861: \E
1.1 noro 1862: @end itemize
1863:
1864: @example
1865: [103] K=katsura(5)$
1866: [104] V=[u5,u4,u3,u2,u1,u0]$
1867: [105] G0=gr(K,V,0)$
1868: [106] GSL=tolex_gsl(G0,V,0,V)$
1869: [107] GSL[0];
1870: [u1,8635837421130477667200000000*u0^31-...]
1871: [108] GSL[1];
1872: [u2,10352277157007342793600000000*u0^31-...]
1873: [109] GSL[5];
1.5 noro 1874: [u0,11771021876193064124640000000*u0^32-...,
1875: 376672700038178051988480000000*u0^31-...]
1.1 noro 1876: @end example
1877:
1878: @table @t
1.2 noro 1879: \JP @item $B;2>H(B
1880: \EG @item References
1.1 noro 1881: @fref{lex_hensel lex_tl tolex tolex_d tolex_tl},
1.2 noro 1882: \JP @fref{$BJ,;67W;;(B}
1883: \EG @fref{Distributed computation}
1.1 noro 1884: @end table
1885:
1.2 noro 1886: \JP @node gr_minipoly minipoly,,, $B%0%l%V%J4pDl$K4X$9$kH!?t(B
1887: \EG @node gr_minipoly minipoly,,, Functions for Groebner basis computation
1.1 noro 1888: @subsection @code{gr_minipoly}, @code{minipoly}
1889: @findex gr_minipoly
1890: @findex minipoly
1891:
1892: @table @t
1893: @item gr_minipoly(@var{plist},@var{vlist},@var{order},@var{poly},@var{v},@var{homo})
1.2 noro 1894: \JP :: $BB?9`<0$N(B, $B%$%G%"%k$rK!$H$7$?:G>.B?9`<0$N7W;;(B
1895: \EG :: Computation of the minimal polynomial of a polynomial modulo an ideal
1.1 noro 1896: @item minipoly(@var{plist},@var{vlist},@var{order},@var{poly},@var{v})
1.2 noro 1897: \JP :: $B%0%l%V%J4pDl$rF~NO$H$9$k(B, $BB?9`<0$N:G>.B?9`<0$N7W;;(B
1898: \EG :: Computation of the minimal polynomial of a polynomial modulo an ideal
1.1 noro 1899: @end table
1900:
1901: @table @var
1902: @item return
1.2 noro 1903: \JP $BB?9`<0(B
1904: \EG polynomial
1.4 noro 1905: @item plist vlist
1.2 noro 1906: \JP $B%j%9%H(B
1907: \EG list
1.1 noro 1908: @item order
1.2 noro 1909: \JP $B?t(B, $B%j%9%H$^$?$O9TNs(B
1910: \EG number, list or matrix
1.1 noro 1911: @item poly
1.2 noro 1912: \JP $BB?9`<0(B
1913: \EG polynomial
1.1 noro 1914: @item v
1.2 noro 1915: \JP $BITDj85(B
1916: \EG indeterminate
1.1 noro 1917: @item homo
1.2 noro 1918: \JP $B%U%i%0(B
1919: \EG flag
1.1 noro 1920: @end table
1921:
1922: @itemize @bullet
1.2 noro 1923: \BJP
1.1 noro 1924: @item
1925: @code{gr_minipoly()} $B$O%0%l%V%J4pDl$N7W;;$+$i9T$$(B, @code{minipoly()} $B$O(B
1926: $BF~NO$r%0%l%V%J4pDl$H$_$J$9(B.
1927: @item
1928: $B%$%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,
1929: K[@var{v}] $B$N85(B f(@var{v}) $B$K(B f(@var{p}) mod I $B$rBP1~$5$;$k(B
1930: $B4D=`F17?$N3K$O(B 0 $B$G$J$$B?9`<0$K$h$j@8@.$5$l$k(B. $B$3$N@8@.85$r(B @var{p}
1931: $B$N(B, $BK!(B @var{I} $B$G$N:G>.B?9`<0$H8F$V(B.
1932: @item
1933: @code{gr_minipoly()}, @code{minipoly()} $B$O(B, $BB?9`<0(B @var{p} $B$N:G>.B?9`<0(B
1934: $B$r5a$a(B, @var{v} $B$rJQ?t$H$9$kB?9`<0$H$7$FJV$9(B.
1935: @item
1936: $B:G>.B?9`<0$O(B, $B%0%l%V%J4pDl$N(B 1 $B$D$N85$H$7$F7W;;$9$k$3$H$b$G$-$k$,(B,
1937: $B:G>.B?9`<0$N$_$r5a$a$?$$>l9g(B, @code{minipoly()}, @code{gr_minipoly()} $B$O(B
1938: $B%0%l%V%J4pDl$rMQ$$$kJ}K!$KHf$Y$F8zN($,$h$$(B.
1939: @item
1940: @code{gr_minipoly()} $B$K;XDj$9$k9`=g=x$H$7$F$O(B, $BDL>oA4<!?t5U<-=q<0=g=x$r(B
1941: $BMQ$$$k(B.
1.2 noro 1942: \E
1943: \BEG
1944: @item
1945: @code{gr_minipoly()} begins by computing a Groebner basis.
1946: @code{minipoly()} regards an input as a Groebner basis with respect to
1947: the variable order @var{vlist} and the order type @var{order}.
1948: @item
1949: Let K be a field. If an ideal @var{I} in K[X] is zero-dimensional, then, for
1950: a polynomial @var{p} in K[X], the kernel of a homomorphism from
1951: K[@var{v}] to K[X]/@var{I} which maps f(@var{v}) to f(@var{p}) mod @var{I}
1952: is generated by a polynomial. The generator is called the minimal polynomial
1953: of @var{p} modulo @var{I}.
1954: @item
1955: @code{gr_minipoly()} and @code{minipoly()} computes the minimal polynomial
1956: of a polynomial @var{p} and returns it as a polynomial of @var{v}.
1957: @item
1958: The minimal polynomial can be computed as an element of a Groebner basis.
1959: But if we are only interested in the minimal polynomial,
1960: @code{minipoly()} and @code{gr_minipoly()} can compute it more efficiently
1961: than methods using Groebner basis computation.
1962: @item
1963: It is recommended to use a degree reverse lex order as a term order
1964: for @code{gr_minipoly()}.
1965: \E
1.1 noro 1966: @end itemize
1967:
1968: @example
1969: [117] G=tolex(G0,V,0,V)$
1970: 43.818sec + gc : 11.202sec
1971: [118] GSL=tolex_gsl(G0,V,0,V)$
1972: 17.123sec + gc : 2.590sec
1973: [119] MP=minipoly(G0,V,0,u0,z)$
1974: 4.370sec + gc : 780msec
1975: @end example
1976:
1977: @table @t
1.2 noro 1978: \JP @item $B;2>H(B
1979: \EG @item References
1.1 noro 1980: @fref{lex_hensel lex_tl tolex tolex_d tolex_tl}.
1981: @end table
1982:
1.2 noro 1983: \JP @node tolexm minipolym,,, $B%0%l%V%J4pDl$K4X$9$kH!?t(B
1984: \EG @node tolexm minipolym,,, Functions for Groebner basis computation
1.1 noro 1985: @subsection @code{tolexm}, @code{minipolym}
1986: @findex tolexm
1987: @findex minipolym
1988:
1989: @table @t
1990: @item tolexm(@var{plist},@var{vlist1},@var{order},@var{vlist2},@var{mod})
1.2 noro 1991: \JP :: $BK!(B @var{mod} $B$G$N4pDlJQ49$K$h$k%0%l%V%J4pDl7W;;(B
1992: \EG :: Groebner basis computation modulo @var{mod} by change of ordering.
1.1 noro 1993: @item minipolym(@var{plist},@var{vlist1},@var{order},@var{poly},@var{v},@var{mod})
1.2 noro 1994: \JP :: $BK!(B @var{mod} $B$G$N%0%l%V%J4pDl$K$h$kB?9`<0$N:G>.B?9`<0$N7W;;(B
1995: \EG :: Minimal polynomial computation modulo @var{mod} the same method as
1.1 noro 1996: @end table
1997:
1998: @table @var
1999: @item return
1.2 noro 2000: \JP @code{tolexm()} : $B%j%9%H(B, @code{minipolym()} : $BB?9`<0(B
2001: \EG @code{tolexm()} : list, @code{minipolym()} : polynomial
1.4 noro 2002: @item plist vlist1 vlist2
1.2 noro 2003: \JP $B%j%9%H(B
2004: \EG list
1.1 noro 2005: @item order
1.2 noro 2006: \JP $B?t(B, $B%j%9%H$^$?$O9TNs(B
2007: \EG number, list or matrix
1.1 noro 2008: @item mod
1.2 noro 2009: \JP $BAG?t(B
2010: \EG prime
1.1 noro 2011: @end table
2012:
2013: @itemize @bullet
1.2 noro 2014: \BJP
1.1 noro 2015: @item
2016: $BF~NO(B @var{plist} $B$O$$$:$l$b(B $BJQ?t=g=x(B @var{vlist1}, $B9`=g=x7?(B @var{order},
2017: $BK!(B @var{mod} $B$K$*$1$k%0%l%V%J4pDl$G$J$1$l$P$J$i$J$$(B.
2018: @item
2019: @code{minipolym()} $B$O(B @code{minipoly} $B$KBP1~$9$k7W;;$rK!(B @var{mod}$B$G9T$&(B.
2020: @item
2021: @code{tolexm()} $B$O(B FGLM $BK!$K$h$k4pDlJQ49$K$h$j(B @var{vlist2},
2022: $B<-=q<0=g=x$K$h$k%0%l%V%J4pDl$r7W;;$9$k(B.
1.2 noro 2023: \E
2024: \BEG
2025: @item
2026: An input @var{plist} must be a Groebner basis modulo @var{mod}
2027: with respect to the variable order @var{vlist1} and the order type @var{order}.
2028: @item
2029: @code{minipolym()} executes the same computation as in @code{minipoly}.
2030: @item
2031: @code{tolexm()} computes a lex order Groebner basis modulo @var{mod}
2032: with respect to the variable order @var{vlist2}, by using FGLM algorithm.
2033: \E
1.1 noro 2034: @end itemize
2035:
2036: @example
2037: [197] tolexm(G0,V,0,V,31991);
2038: [8271*u0^31+10435*u0^30+816*u0^29+26809*u0^28+...,...]
2039: [198] minipolym(G0,V,0,u0,z,31991);
2040: z^32+11405*z^31+20868*z^30+21602*z^29+...
2041: @end example
2042:
2043: @table @t
1.2 noro 2044: \JP @item $B;2>H(B
2045: \EG @item References
1.1 noro 2046: @fref{lex_hensel lex_tl tolex tolex_d tolex_tl},
2047: @fref{gr_minipoly minipoly}.
2048: @end table
2049:
1.6 noro 2050: \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
2051: \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
2052: @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 2053: @findex dp_gr_main
2054: @findex dp_gr_mod_main
1.5 noro 2055: @findex dp_gr_f_main
1.6 noro 2056: @findex dp_weyl_gr_main
2057: @findex dp_weyl_gr_mod_main
2058: @findex dp_weyl_gr_f_main
1.1 noro 2059:
2060: @table @t
2061: @item dp_gr_main(@var{plist},@var{vlist},@var{homo},@var{modular},@var{order})
2062: @itemx dp_gr_mod_main(@var{plist},@var{vlist},@var{homo},@var{modular},@var{order})
1.5 noro 2063: @itemx dp_gr_f_main(@var{plist},@var{vlist},@var{homo},@var{order})
1.6 noro 2064: @itemx dp_weyl_gr_main(@var{plist},@var{vlist},@var{homo},@var{modular},@var{order})
2065: @itemx dp_weyl_gr_mod_main(@var{plist},@var{vlist},@var{homo},@var{modular},@var{order})
2066: @itemx dp_weyl_gr_f_main(@var{plist},@var{vlist},@var{homo},@var{order})
1.2 noro 2067: \JP :: $B%0%l%V%J4pDl$N7W;;(B ($BAH$_9~$_H!?t(B)
2068: \EG :: Groebner basis computation (built-in functions)
1.1 noro 2069: @end table
2070:
2071: @table @var
2072: @item return
1.2 noro 2073: \JP $B%j%9%H(B
2074: \EG list
1.4 noro 2075: @item plist vlist
1.2 noro 2076: \JP $B%j%9%H(B
2077: \EG list
1.1 noro 2078: @item order
1.2 noro 2079: \JP $B?t(B, $B%j%9%H$^$?$O9TNs(B
2080: \EG number, list or matrix
1.1 noro 2081: @item homo
1.2 noro 2082: \JP $B%U%i%0(B
2083: \EG flag
1.1 noro 2084: @item modular
1.2 noro 2085: \JP $B%U%i%0$^$?$OAG?t(B
2086: \EG flag or prime
1.1 noro 2087: @end table
2088:
2089: @itemize @bullet
1.2 noro 2090: \BJP
1.1 noro 2091: @item
2092: $B$3$l$i$NH!?t$O(B, $B%0%l%V%J4pDl7W;;$N4pK\E*AH$_9~$_H!?t$G$"$j(B, @code{gr()},
2093: @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 2094: $B$r9T$C$F$$$k(B. $B4X?tL>$K(B weyl $B$,F~$C$F$$$k$b$N$O(B, Weyl $BBe?t>e$N7W;;(B
2095: $B$N$?$a$N4X?t$G$"$k(B.
1.1 noro 2096: @item
1.6 noro 2097: @code{dp_gr_f_main()}, @code{dp_weyl_f_main()} $B$O(B, $B<o!9$NM-8BBN>e$N%0%l%V%J4pDl$r7W;;$9$k(B
1.5 noro 2098: $B>l9g$KMQ$$$k(B. $BF~NO$O(B, $B$"$i$+$8$a(B, @code{simp_ff()} $B$J$I$G(B,
2099: $B9M$($kM-8BBN>e$K<M1F$5$l$F$$$kI,MW$,$"$k(B.
2100: @item
1.1 noro 2101: $B%U%i%0(B @var{homo} $B$,(B 0 $B$G$J$$;~(B, $BF~NO$r@F<!2=$7$F$+$i(B Buchberger $B%"%k%4%j%:%`(B
2102: $B$r<B9T$9$k(B.
2103: @item
2104: @code{dp_gr_mod_main()} $B$KBP$7$F$O(B, @var{modular} $B$O(B, GF(@var{modular}) $B>e(B
2105: $B$G$N7W;;$r0UL#$9$k(B.
2106: @code{dp_gr_main()} $B$KBP$7$F$O(B, @var{modular} $B$O<!$N$h$&$J0UL#$r;}$D(B.
2107: @enumerate
2108: @item
2109: @var{modular} $B$,(B 1 $B$N;~(B, trace-lifting $B$K$h$k7W;;$r9T$&(B. $BAG?t$O(B
2110: @code{lprime(0)} $B$+$i=g$K@.8y$9$k$^$G(B @code{lprime()} $B$r8F$S=P$7$F@8@.$9$k(B.
2111: @item
2112: @var{modular} $B$,(B 2 $B0J>e$N<+A3?t$N;~(B, $B$=$NCM$rAG?t$H$_$J$7$F(B trace-lifting
2113: $B$r9T$&(B. $B$=$NAG?t$G<:GT$7$?>l9g(B, 0 $B$rJV$9(B.
2114: @item
2115: @var{modular} $B$,Ii$N>l9g(B,
2116: @var{-modular} $B$KBP$7$F>e=R$N5,B'$,E,MQ$5$l$k$,(B, trace-lifting $B$N:G=*(B
2117: $BCJ3,$N%0%l%V%J4pDl%A%'%C%/$H%$%G%"%k%a%s%P%7%C%W%A%'%C%/$,>JN,$5$l$k(B.
2118: @end enumerate
2119:
2120: @item
2121: @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
2122: @code{dp_gr_main(P,V,1,1,O)}, @code{gr_mod(P,V,O,M)} $B$O(B
2123: @code{dp_gr_mod_main(P,V,0,M,O)} $B$r$=$l$>$l<B9T$9$k(B.
2124: @item
2125: @var{homo}, @var{modular} $B$NB>$K(B, @code{dp_gr_flags()} $B$G@_Dj$5$l$k(B
2126: $B$5$^$6$^$J%U%i%0$K$h$j7W;;$,@)8f$5$l$k(B.
1.2 noro 2127: \E
2128: \BEG
2129: @item
2130: These functions are fundamental built-in functions for Groebner basis
2131: computation and @code{gr()},@code{hgr()} and @code{gr_mod()}
1.6 noro 2132: are all interfaces to these functions. Functions whose names
2133: contain weyl are those for computation in Weyl algebra.
1.2 noro 2134: @item
1.6 noro 2135: @code{dp_gr_f_main()} and @code{dp_weyl_gr_f_main()}
2136: are functions for Groebner basis computation
1.5 noro 2137: over various finite fields. Coefficients of input polynomials
2138: must be converted to elements of a finite field
2139: currently specified by @code{setmod_ff()}.
2140: @item
1.2 noro 2141: If @var{homo} is not equal to 0, homogenization is applied before entering
2142: Buchberger algorithm
2143: @item
2144: For @code{dp_gr_mod_main()}, @var{modular} means a computation over
2145: GF(@var{modular}).
2146: For @code{dp_gr_main()}, @var{modular} has the following mean.
2147: @enumerate
2148: @item
2149: If @var{modular} is 1 , trace lifting is used. Primes for trace lifting
2150: are generated by @code{lprime()}, starting from @code{lprime(0)}, until
2151: the computation succeeds.
2152: @item
2153: If @var{modular} is an integer greater than 1, the integer is regarded as a
2154: prime and trace lifting is executed by using the prime. If the computation
2155: fails then 0 is returned.
2156: @item
2157: If @var{modular} is negative, the above rule is applied for @var{-modular}
2158: but the Groebner basis check and ideal-membership check are omitted in
2159: the last stage of trace lifting.
2160: @end enumerate
2161:
2162: @item
2163: @code{gr(P,V,O)}, @code{hgr(P,V,O)} and @code{gr_mod(P,V,O,M)} execute
2164: @code{dp_gr_main(P,V,0,1,O)}, @code{dp_gr_main(P,V,1,1,O)}
2165: and @code{dp_gr_mod_main(P,V,0,M,O)} respectively.
2166: @item
2167: Actual computation is controlled by various parameters set by
2168: @code{dp_gr_flags()}, other then by @var{homo} and @var{modular}.
2169: \E
1.1 noro 2170: @end itemize
2171:
2172: @table @t
1.2 noro 2173: \JP @item $B;2>H(B
2174: \EG @item References
1.1 noro 2175: @fref{dp_ord},
2176: @fref{dp_gr_flags dp_gr_print},
2177: @fref{gr hgr gr_mod},
1.5 noro 2178: @fref{setmod_ff},
1.2 noro 2179: \JP @fref{$B7W;;$*$h$SI=<($N@)8f(B}.
2180: \EG @fref{Controlling Groebner basis computations}
1.1 noro 2181: @end table
2182:
1.6 noro 2183: \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
2184: \EG @node dp_f4_main dp_f4_mod_main dp_weyl_f4_main dp_weyl_f4_mod_main,,, Functions for Groebner basis computation
2185: @subsection @code{dp_f4_main}, @code{dp_f4_mod_main}, @code{dp_weyl_f4_main}, @code{dp_weyl_f4_mod_main}
1.1 noro 2186: @findex dp_f4_main
2187: @findex dp_f4_mod_main
1.6 noro 2188: @findex dp_weyl_f4_main
2189: @findex dp_weyl_f4_mod_main
1.1 noro 2190:
2191: @table @t
2192: @item dp_f4_main(@var{plist},@var{vlist},@var{order})
2193: @itemx dp_f4_mod_main(@var{plist},@var{vlist},@var{order})
1.6 noro 2194: @itemx dp_weyl_f4_main(@var{plist},@var{vlist},@var{order})
2195: @itemx dp_weyl_f4_mod_main(@var{plist},@var{vlist},@var{order})
1.2 noro 2196: \JP :: F4 $B%"%k%4%j%:%`$K$h$k%0%l%V%J4pDl$N7W;;(B ($BAH$_9~$_H!?t(B)
2197: \EG :: Groebner basis computation by F4 algorithm (built-in functions)
1.1 noro 2198: @end table
2199:
2200: @table @var
2201: @item return
1.2 noro 2202: \JP $B%j%9%H(B
2203: \EG list
1.4 noro 2204: @item plist vlist
1.2 noro 2205: \JP $B%j%9%H(B
2206: \EG list
1.1 noro 2207: @item order
1.2 noro 2208: \JP $B?t(B, $B%j%9%H$^$?$O9TNs(B
2209: \EG number, list or matrix
1.1 noro 2210: @end table
2211:
2212: @itemize @bullet
1.2 noro 2213: \BJP
1.1 noro 2214: @item
2215: F4 $B%"%k%4%j%:%`$K$h$j%0%l%V%J4pDl$N7W;;$r9T$&(B.
2216: @item
2217: F4 $B%"%k%4%j%:%`$O(B, J.C. Faugere $B$K$h$jDs>'$5$l$??7@$Be%0%l%V%J4pDl(B
2218: $B;;K!$G$"$j(B, $BK\<BAu$O(B, $BCf9q>jM>DjM}$K$h$k@~7AJ}Dx<05a2r$rMQ$$$?(B
2219: $B;n83E*$J<BAu$G$"$k(B.
2220: @item
1.6 noro 2221: $B@F<!2=$N0z?t$,$J$$$3$H$r=|$1$P(B, $B0z?t$*$h$SF0:n$O$=$l$>$l(B
2222: @code{dp_gr_main()}, @code{dp_gr_mod_main()},
2223: @code{dp_weyl_gr_main()}, @code{dp_weyl_gr_mod_main()}
1.1 noro 2224: $B$HF1MM$G$"$k(B.
1.2 noro 2225: \E
2226: \BEG
2227: @item
2228: These functions compute Groebner bases by F4 algorithm.
2229: @item
2230: F4 is a new generation algorithm for Groebner basis computation
2231: invented by J.C. Faugere. The current implementation of @code{dp_f4_main()}
2232: uses Chinese Remainder theorem and not highly optimized.
2233: @item
2234: Arguments and actions are the same as those of
1.6 noro 2235: @code{dp_gr_main()}, @code{dp_gr_mod_main()},
2236: @code{dp_weyl_gr_main()}, @code{dp_weyl_gr_mod_main()},
2237: except for lack of the argument for controlling homogenization.
1.2 noro 2238: \E
1.1 noro 2239: @end itemize
2240:
2241: @table @t
1.2 noro 2242: \JP @item $B;2>H(B
2243: \EG @item References
1.1 noro 2244: @fref{dp_ord},
2245: @fref{dp_gr_flags dp_gr_print},
2246: @fref{gr hgr gr_mod},
1.2 noro 2247: \JP @fref{$B7W;;$*$h$SI=<($N@)8f(B}.
2248: \EG @fref{Controlling Groebner basis computations}
1.1 noro 2249: @end table
2250:
1.2 noro 2251: \JP @node dp_gr_flags dp_gr_print,,, $B%0%l%V%J4pDl$K4X$9$kH!?t(B
2252: \EG @node dp_gr_flags dp_gr_print,,, Functions for Groebner basis computation
1.1 noro 2253: @subsection @code{dp_gr_flags}, @code{dp_gr_print}
2254: @findex dp_gr_flags
2255: @findex dp_gr_print
2256:
2257: @table @t
2258: @item dp_gr_flags([@var{list}])
1.7 noro 2259: @itemx dp_gr_print([@var{i}])
1.2 noro 2260: \JP :: $B7W;;$*$h$SI=<(MQ%Q%i%a%?$N@_Dj(B, $B;2>H(B
2261: \BEG :: Set and show various parameters for cotrolling computations
2262: and showing informations.
2263: \E
1.1 noro 2264: @end table
2265:
2266: @table @var
2267: @item return
1.2 noro 2268: \JP $B@_DjCM(B
2269: \EG value currently set
1.1 noro 2270: @item list
1.2 noro 2271: \JP $B%j%9%H(B
2272: \EG list
1.7 noro 2273: @item i
2274: \JP $B@0?t(B
2275: \EG integer
1.1 noro 2276: @end table
2277:
2278: @itemize @bullet
1.2 noro 2279: \BJP
1.1 noro 2280: @item
1.5 noro 2281: @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 2282: $B$J%Q%i%a%?$r@_Dj(B, $B;2>H$9$k(B.
2283: @item
2284: $B0z?t$,$J$$>l9g(B, $B8=:_$N@_Dj$,JV$5$l$k(B.
2285: @item
2286: $B0z?t$O(B, @code{["Print",1,"NoSugar",1,...]} $B$J$k7A$N%j%9%H$G(B, $B:8$+$i=g$K(B
2287: $B@_Dj$5$l$k(B. $B%Q%i%a%?L>$OJ8;zNs$GM?$($kI,MW$,$"$k(B.
2288: @item
1.7 noro 2289: @code{dp_gr_print()} $B$O(B, $BFC$K%Q%i%a%?(B @code{Print}, @code{PrintShort} $B$NCM$rD>@\@_Dj(B, $B;2>H(B
2290: $B$G$-$k(B. $B@_Dj$5$l$kCM$O<!$NDL$j$G$"$k!#(B
2291: @table @var
2292: @item i=0
2293: @code{Print=0}, @code{PrintShort=0}
2294: @item i=1
2295: @code{Print=1}, @code{PrintShort=0}
2296: @item i=2
2297: @code{Print=0}, @code{PrintShort=1}
2298: @end table
2299: $B$3$l$O(B, @code{dp_gr_main()} $B$J$I$r%5%V%k!<%A%s$H$7$FMQ$$$k%f!<%6(B
2300: $BH!?t$K$*$$$F(B, $B$=$N%5%V%k!<%A%s$,Cf4V>pJs$NI=<((B
1.1 noro 2301: $B$r9T$&:]$K(B, $B?WB.$K%U%i%0$r8+$k$3$H$,$G$-$k$h$&$KMQ0U$5$l$F$$$k(B.
1.2 noro 2302: \E
2303: \BEG
2304: @item
2305: @code{dp_gr_flags()} sets and shows various parameters for Groebner basis
2306: computation.
2307: @item
2308: If no argument is specified the current settings are returned.
2309: @item
2310: Arguments must be specified as a list such as
2311: @code{["Print",1,"NoSugar",1,...]}. Names of parameters must be character
2312: strings.
2313: @item
2314: @code{dp_gr_print()} is used to set and show the value of a parameter
1.7 noro 2315: @code{Print} and @code{PrintShort}.
2316: @table @var
2317: @item i=0
2318: @code{Print=0}, @code{PrintShort=0}
2319: @item i=1
2320: @code{Print=1}, @code{PrintShort=0}
2321: @item i=2
2322: @code{Print=0}, @code{PrintShort=1}
2323: @end table
2324: This functions is prepared to get quickly the value
2325: when a user defined function calling @code{dp_gr_main()} etc.
1.2 noro 2326: uses the value as a flag for showing intermediate informations.
2327: \E
1.1 noro 2328: @end itemize
2329:
2330: @table @t
1.2 noro 2331: \JP @item $B;2>H(B
2332: \EG @item References
2333: \JP @fref{$B7W;;$*$h$SI=<($N@)8f(B}
2334: \EG @fref{Controlling Groebner basis computations}
1.1 noro 2335: @end table
2336:
1.2 noro 2337: \JP @node dp_ord,,, $B%0%l%V%J4pDl$K4X$9$kH!?t(B
2338: \EG @node dp_ord,,, Functions for Groebner basis computation
1.1 noro 2339: @subsection @code{dp_ord}
2340: @findex dp_ord
2341:
2342: @table @t
2343: @item dp_ord([@var{order}])
1.2 noro 2344: \JP :: $BJQ?t=g=x7?$N@_Dj(B, $B;2>H(B
2345: \EG :: Set and show the ordering type.
1.1 noro 2346: @end table
2347:
2348: @table @var
2349: @item return
1.2 noro 2350: \JP $BJQ?t=g=x7?(B ($B?t(B, $B%j%9%H$^$?$O9TNs(B)
2351: \EG ordering type (number, list or matrix)
1.1 noro 2352: @item order
1.2 noro 2353: \JP $B?t(B, $B%j%9%H$^$?$O9TNs(B
2354: \EG number, list or matrix
1.1 noro 2355: @end table
2356:
2357: @itemize @bullet
1.2 noro 2358: \BJP
1.1 noro 2359: @item
2360: $B0z?t$,$"$k;~(B, $BJQ?t=g=x7?$r(B @var{order} $B$K@_Dj$9$k(B. $B0z?t$,$J$$;~(B,
2361: $B8=:_@_Dj$5$l$F$$$kJQ?t=g=x7?$rJV$9(B.
2362:
2363: @item
2364: $BJ,;6I=8=B?9`<0$K4X$9$kH!?t(B, $B1i;;$O0z?t$H$7$FJQ?t=g=x7?$r$H$k$b$N$H$H$i$J$$$b$N(B
2365: $B$,$"$j(B, $B$H$i$J$$$b$N$K4X$7$F$O(B, $B$=$N;~E@$G@_Dj$5$l$F$$$kCM$rMQ$$$F7W;;$,(B
2366: $B9T$o$l$k(B.
2367:
2368: @item
2369: @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()}
2370: $B$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.
2371:
2372: @item
2373: $BJ,;6I=8=B?9`<0$N;MB'1i;;$b(B, $B@_Dj$5$l$F$$$kCM$rMQ$$$F7W;;$5$l$k(B. $B=>$C$F(B,
2374: $B$=$NB?9`<0$,@8@.$5$l$?;~E@$K$*$1$kJQ?t=g=x7?$,(B, $B;MB'1i;;;~$K@5$7$/@_Dj(B
2375: $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
2376: $B7?$K4p$E$$$F@8@.$5$l$?$b$N$G$J$1$l$P$J$i$J$$(B.
2377:
2378: @item
2379: $B%H%C%W%l%Y%kH!?t0J30$NH!?t$rD>@\8F$S=P$9>l9g$K$O(B, $B$3$NH!?t$K$h$j(B
2380: $BJQ?t=g=x7?$r@5$7$/@_Dj$7$J$1$l$P$J$i$J$$(B.
1.2 noro 2381: \E
2382: \BEG
2383: @item
2384: If an argument is specified, the function
2385: sets the current ordering type to @var{order}.
2386: If no argument is specified, the function returns the ordering
2387: type currently set.
2388:
2389: @item
2390: There are two types of functions concerning distributed polynomial,
2391: functions which take a ordering type and those which don't take it.
2392: The latter ones use the current setting.
2393:
2394: @item
2395: Functions such as @code{gr()}, which need a ordering type as an argument,
2396: call @code{dp_ord()} internally during the execution.
2397: The setting remains after the execution.
2398:
2399: Fundamental arithmetics for distributed polynomial also use the current
2400: setting. Therefore, when such arithmetics for distributed polynomials
2401: are done, the current setting must coincide with the ordering type
2402: which was used upon the creation of the polynomials. It is assumed
2403: that such polynomials were generated under the same ordering type.
2404:
2405: @item
2406: Type of term ordering must be correctly set by this function
2407: when functions other than top level functions are called directly.
2408: \E
1.1 noro 2409: @end itemize
2410:
2411: @example
2412: [19] dp_ord(0)$
2413: [20] <<1,2,3>>+<<3,1,1>>;
2414: (1)*<<1,2,3>>+(1)*<<3,1,1>>
2415: [21] dp_ord(2)$
2416: [22] <<1,2,3>>+<<3,1,1>>;
2417: (1)*<<3,1,1>>+(1)*<<1,2,3>>
2418: @end example
2419:
2420: @table @t
1.2 noro 2421: \JP @item $B;2>H(B
2422: \EG @item References
2423: \JP @fref{$B9`=g=x$N@_Dj(B}
2424: \EG @fref{Setting term orderings}
1.1 noro 2425: @end table
2426:
1.2 noro 2427: \JP @node dp_ptod,,, $B%0%l%V%J4pDl$K4X$9$kH!?t(B
2428: \EG @node dp_ptod,,, Functions for Groebner basis computation
1.1 noro 2429: @subsection @code{dp_ptod}
2430: @findex dp_ptod
2431:
2432: @table @t
2433: @item dp_ptod(@var{poly},@var{vlist})
1.2 noro 2434: \JP :: $BB?9`<0$rJ,;6I=8=B?9`<0$KJQ49$9$k(B.
2435: \EG :: Converts an ordinary polynomial into a distributed polynomial.
1.1 noro 2436: @end table
2437:
2438: @table @var
2439: @item return
1.2 noro 2440: \JP $BJ,;6I=8=B?9`<0(B
2441: \EG distributed polynomial
1.1 noro 2442: @item poly
1.2 noro 2443: \JP $BB?9`<0(B
2444: \EG polynomial
1.1 noro 2445: @item vlist
1.2 noro 2446: \JP $B%j%9%H(B
2447: \EG list
1.1 noro 2448: @end table
2449:
2450: @itemize @bullet
1.2 noro 2451: \BJP
1.1 noro 2452: @item
2453: $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.
2454: @item
2455: @var{vlist} $B$K4^$^$l$J$$ITDj85$O(B, $B78?tBN$KB0$9$k$H$7$FJQ49$5$l$k(B.
1.2 noro 2456: \E
2457: \BEG
2458: @item
2459: According to the variable ordering @var{vlist} and current
2460: type of term ordering, this function converts an ordinary
2461: polynomial into a distributed polynomial.
2462: @item
2463: Indeterminates not included in @var{vlist} are regarded to belong to
2464: the coefficient field.
2465: \E
1.1 noro 2466: @end itemize
2467:
2468: @example
2469: [50] dp_ord(0);
2470: 1
2471: [51] dp_ptod((x+y+z)^2,[x,y,z]);
2472: (1)*<<2,0,0>>+(2)*<<1,1,0>>+(1)*<<0,2,0>>+(2)*<<1,0,1>>+(2)*<<0,1,1>>
2473: +(1)*<<0,0,2>>
2474: [52] dp_ptod((x+y+z)^2,[x,y]);
1.5 noro 2475: (1)*<<2,0>>+(2)*<<1,1>>+(1)*<<0,2>>+(2*z)*<<1,0>>+(2*z)*<<0,1>>
2476: +(z^2)*<<0,0>>
1.1 noro 2477: @end example
2478:
2479: @table @t
1.2 noro 2480: \JP @item $B;2>H(B
2481: \EG @item References
1.1 noro 2482: @fref{dp_dtop},
2483: @fref{dp_ord}.
2484: @end table
2485:
1.2 noro 2486: \JP @node dp_dtop,,, $B%0%l%V%J4pDl$K4X$9$kH!?t(B
2487: \EG @node dp_dtop,,, Functions for Groebner basis computation
1.1 noro 2488: @subsection @code{dp_dtop}
2489: @findex dp_dtop
2490:
2491: @table @t
2492: @item dp_dtop(@var{dpoly},@var{vlist})
1.2 noro 2493: \JP :: $BJ,;6I=8=B?9`<0$rB?9`<0$KJQ49$9$k(B.
2494: \EG :: Converts a distributed polynomial into an ordinary polynomial.
1.1 noro 2495: @end table
2496:
2497: @table @var
2498: @item return
1.2 noro 2499: \JP $BB?9`<0(B
2500: \EG polynomial
1.1 noro 2501: @item dpoly
1.2 noro 2502: \JP $BJ,;6I=8=B?9`<0(B
2503: \EG distributed polynomial
1.1 noro 2504: @item vlist
1.2 noro 2505: \JP $B%j%9%H(B
2506: \EG list
1.1 noro 2507: @end table
2508:
2509: @itemize @bullet
1.2 noro 2510: \BJP
1.1 noro 2511: @item
2512: $BJ,;6I=8=B?9`<0$r(B, $BM?$($i$l$?ITDj85%j%9%H$rMQ$$$FB?9`<0$KJQ49$9$k(B.
2513: @item
2514: $BITDj85%j%9%H$O(B, $BD9$5J,;6I=8=B?9`<0$NJQ?t$N8D?t$H0lCW$7$F$$$l$P2?$G$b$h$$(B.
1.2 noro 2515: \E
2516: \BEG
2517: @item
2518: This function converts a distributed polynomial into an ordinary polynomial
2519: according to a list of indeterminates @var{vlist}.
2520: @item
2521: @var{vlist} is such a list that its length coincides with the number of
2522: variables of @var{dpoly}.
2523: \E
1.1 noro 2524: @end itemize
2525:
2526: @example
2527: [53] T=dp_ptod((x+y+z)^2,[x,y]);
1.5 noro 2528: (1)*<<2,0>>+(2)*<<1,1>>+(1)*<<0,2>>+(2*z)*<<1,0>>+(2*z)*<<0,1>>
2529: +(z^2)*<<0,0>>
1.1 noro 2530: [54] P=dp_dtop(T,[a,b]);
2531: z^2+(2*a+2*b)*z+a^2+2*b*a+b^2
2532: @end example
2533:
1.2 noro 2534: \JP @node dp_mod dp_rat,,, $B%0%l%V%J4pDl$K4X$9$kH!?t(B
2535: \EG @node dp_mod dp_rat,,, Functions for Groebner basis computation
1.1 noro 2536: @subsection @code{dp_mod}, @code{dp_rat}
2537: @findex dp_mod
2538: @findex dp_rat
2539:
2540: @table @t
2541: @item dp_mod(@var{p},@var{mod},@var{subst})
1.2 noro 2542: \JP :: $BM-M}?t78?tJ,;6I=8=B?9`<0$NM-8BBN78?t$X$NJQ49(B
2543: \EG :: Converts a disributed polynomial into one with coefficients in a finite field.
1.1 noro 2544: @item dp_rat(@var{p})
1.2 noro 2545: \JP :: $BM-8BBN78?tJ,;6I=8=B?9`<0$NM-M}?t78?t$X$NJQ49(B
2546: \BEG
2547: :: Converts a distributed polynomial with coefficients in a finite field into
2548: one with coefficients in the rationals.
2549: \E
1.1 noro 2550: @end table
2551:
2552: @table @var
2553: @item return
1.2 noro 2554: \JP $BJ,;6I=8=B?9`<0(B
2555: \EG distributed polynomial
1.1 noro 2556: @item p
1.2 noro 2557: \JP $BJ,;6I=8=B?9`<0(B
2558: \EG distributed polynomial
1.1 noro 2559: @item mod
1.2 noro 2560: \JP $BAG?t(B
2561: \EG prime
1.1 noro 2562: @item subst
1.2 noro 2563: \JP $B%j%9%H(B
2564: \EG list
1.1 noro 2565: @end table
2566:
2567: @itemize @bullet
1.2 noro 2568: \BJP
1.1 noro 2569: @item
2570: @code{dp_nf_mod()}, @code{dp_true_nf_mod()} $B$O(B, $BF~NO$H$7$FM-8BBN78?t$N(B
2571: $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
2572: $BM-M}?t78?tJ,;6I=8=B?9`<0$rJQ49$7$FMQ$$$k$3$H$,$G$-$k(B. $B$^$?(B, $BF@$i$l$?(B
2573: $B7k2L$O(B, $BM-8BBN78?tB?9`<0$H$O1i;;$G$-$k$,(B, $BM-M}?t78?tB?9`<0$H$O1i;;$G$-$J$$(B
2574: $B$?$a(B, @code{dp_rat()} $B$K$h$jJQ49$9$kI,MW$,$"$k(B.
2575: @item
2576: $BM-8BBN78?t$N1i;;$K$*$$$F$O(B, $B$"$i$+$8$a(B @code{setmod()} $B$K$h$jM-8BBN$N85$N(B
2577: $B8D?t$r;XDj$7$F$*$/I,MW$,$"$k(B.
2578: @item
2579: @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
2580: $B$7$?8eM-8BBN78?t$KJQ49$9$k$H$$$&A`:n$r9T$&:]$N(B, $BBeF~CM$r;XDj$9$k$b$N$G(B,
2581: @code{[[@var{var},@var{value}],...]} $B$N7A$N%j%9%H$G$"$k(B.
1.2 noro 2582: \E
2583: \BEG
2584: @item
2585: @code{dp_nf_mod()} and @code{dp_true_nf_mod()} require
2586: distributed polynomials with coefficients in a finite field as arguments.
2587: @code{dp_mod()} is used to convert distributed polynomials with rational
2588: number coefficients into appropriate ones.
2589: Polynomials with coefficients in a finite field
2590: cannot be used as inputs of operations with polynomials
2591: with rational number coefficients. @code{dp_rat()} is used for such cases.
2592: @item
2593: The ground finite field must be set in advance by using @code{setmod()}.
2594: @item
2595: @var{subst} is such a list as @code{[[@var{var},@var{value}],...]}.
2596: This is valid when the ground field of the input polynomial is a
2597: rational function field. @var{var}'s are variables in the ground field and
2598: the list means that @var{value} is substituted for @var{var} before
2599: converting the coefficients into elements of a finite field.
2600: \E
1.1 noro 2601: @end itemize
2602:
2603: @example
2604: @end example
2605:
2606: @table @t
1.2 noro 2607: \JP @item $B;2>H(B
2608: \EG @item References
1.1 noro 2609: @fref{dp_nf dp_nf_mod dp_true_nf dp_true_nf_mod},
2610: @fref{subst psubst},
2611: @fref{setmod}.
2612: @end table
2613:
1.2 noro 2614: \JP @node dp_homo dp_dehomo,,, $B%0%l%V%J4pDl$K4X$9$kH!?t(B
2615: \EG @node dp_homo dp_dehomo,,, Functions for Groebner basis computation
1.1 noro 2616: @subsection @code{dp_homo}, @code{dp_dehomo}
2617: @findex dp_homo
2618: @findex dp_dehomo
2619:
2620: @table @t
2621: @item dp_homo(@var{dpoly})
1.2 noro 2622: \JP :: $BJ,;6I=8=B?9`<0$N@F<!2=(B
2623: \EG :: Homogenize a distributed polynomial
1.1 noro 2624: @item dp_dehomo(@var{dpoly})
1.2 noro 2625: \JP :: $B@F<!J,;6I=8=B?9`<0$NHs@F<!2=(B
2626: \EG :: Dehomogenize a homogenious distributed polynomial
1.1 noro 2627: @end table
2628:
2629: @table @var
2630: @item return
1.2 noro 2631: \JP $BJ,;6I=8=B?9`<0(B
2632: \EG distributed polynomial
1.1 noro 2633: @item dpoly
1.2 noro 2634: \JP $BJ,;6I=8=B?9`<0(B
2635: \EG distributed polynomial
1.1 noro 2636: @end table
2637:
2638: @itemize @bullet
1.2 noro 2639: \BJP
1.1 noro 2640: @item
2641: @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
2642: 1 $B?-$P$7(B, $B:G8e$N@.J,$NCM$r(B @var{d}-@code{deg(@var{t})}
2643: (@var{d} $B$O(B @var{dpoly} $B$NA4<!?t(B) $B$H$7$?J,;6I=8=B?9`<0$rJV$9(B.
2644: @item
2645: @code{dp_dehomo()} $B$O(B, @var{dpoly} $B$N3F9`$K$D$$$F(B, $B;X?t%Y%/%H%k$N:G8e$N@.J,(B
2646: $B$r<h$j=|$$$?J,;6B?9`<0$rJV$9(B.
2647: @item
2648: $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
2649: $B@5$7$/@_Dj$9$kI,MW$,$"$k(B.
2650: @item
2651: @code{hgr()} $B$J$I$K$*$$$F(B, $BFbItE*$KMQ$$$i$l$F$$$k(B.
1.2 noro 2652: \E
2653: \BEG
2654: @item
2655: @code{dp_homo()} makes a copy of @var{dpoly}, extends
2656: the length of the exponent vector of each term @var{t} in the copy by 1,
2657: and sets the value of the newly appended
2658: component to @var{d}-@code{deg(@var{t})}, where @var{d} is the total
2659: degree of @var{dpoly}.
2660: @item
2661: @code{dp_dehomo()} make a copy of @var{dpoly} and removes the last component
2662: of each terms in the copy.
2663: @item
2664: Appropriate term orderings must be set when the results are used as inputs
2665: of some operations.
2666: @item
2667: These are used internally in @code{hgr()} etc.
2668: \E
1.1 noro 2669: @end itemize
2670:
2671: @example
2672: [202] X=<<1,2,3>>+3*<<1,2,1>>;
2673: (1)*<<1,2,3>>+(3)*<<1,2,1>>
2674: [203] dp_homo(X);
2675: (1)*<<1,2,3,0>>+(3)*<<1,2,1,2>>
2676: [204] dp_dehomo(@@);
2677: (1)*<<1,2,3>>+(3)*<<1,2,1>>
2678: @end example
2679:
2680: @table @t
1.2 noro 2681: \JP @item $B;2>H(B
2682: \EG @item References
1.1 noro 2683: @fref{gr hgr gr_mod}.
2684: @end table
2685:
1.2 noro 2686: \JP @node dp_ptozp dp_prim,,, $B%0%l%V%J4pDl$K4X$9$kH!?t(B
2687: \EG @node dp_ptozp dp_prim,,, Functions for Groebner basis computation
1.1 noro 2688: @subsection @code{dp_ptozp}, @code{dp_prim}
2689: @findex dp_ptozp
2690: @findex dp_prim
2691:
2692: @table @t
2693: @item dp_ptozp(@var{dpoly})
1.2 noro 2694: \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.
2695: \BEG
2696: :: Converts a distributed polynomial @var{poly} with rational coefficients
2697: into an integral distributed polynomial such that GCD of all its coefficients
2698: is 1.
2699: \E
1.1 noro 2700: @itemx dp_prim(@var{dpoly})
1.2 noro 2701: \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.
2702: \BEG
2703: :: Converts a distributed polynomial @var{poly} with rational function
2704: coefficients into an integral distributed polynomial such that polynomial
2705: GCD of all its coefficients is 1.
2706: \E
1.1 noro 2707: @end table
2708:
2709: @table @var
2710: @item return
1.2 noro 2711: \JP $BJ,;6I=8=B?9`<0(B
2712: \EG distributed polynomial
1.1 noro 2713: @item dpoly
1.2 noro 2714: \JP $BJ,;6I=8=B?9`<0(B
2715: \EG distributed polynomial
1.1 noro 2716: @end table
2717:
2718: @itemize @bullet
1.2 noro 2719: \BJP
1.1 noro 2720: @item
2721: @code{dp_ptozp()} $B$O(B, @code{ptozp()} $B$KAjEv$9$kA`:n$rJ,;6I=8=B?9`<0$K(B
2722: $BBP$7$F9T$&(B. $B78?t$,B?9`<0$r4^$`>l9g(B, $B78?t$K4^$^$l$kB?9`<06&DL0x;R$O(B
2723: $B<h$j=|$+$J$$(B.
2724: @item
2725: @code{dp_prim()} $B$O(B, $B78?t$,B?9`<0$r4^$`>l9g(B, $B78?t$K4^$^$l$kB?9`<06&DL0x;R(B
2726: $B$r<h$j=|$/(B.
1.2 noro 2727: \E
2728: \BEG
2729: @item
2730: @code{dp_ptozp()} executes the same operation as @code{ptozp()} for
2731: a distributed polynomial. If the coefficients include polynomials,
2732: polynomial contents included in the coefficients are not removed.
2733: @item
2734: @code{dp_prim()} removes polynomial contents.
2735: \E
1.1 noro 2736: @end itemize
2737:
2738: @example
2739: [208] X=dp_ptod(3*(x-y)*(y-z)*(z-x),[x]);
2740: (-3*y+3*z)*<<2>>+(3*y^2-3*z^2)*<<1>>+(-3*z*y^2+3*z^2*y)*<<0>>
2741: [209] dp_ptozp(X);
2742: (-y+z)*<<2>>+(y^2-z^2)*<<1>>+(-z*y^2+z^2*y)*<<0>>
2743: [210] dp_prim(X);
2744: (1)*<<2>>+(-y-z)*<<1>>+(z*y)*<<0>>
2745: @end example
2746:
2747: @table @t
1.2 noro 2748: \JP @item $B;2>H(B
2749: \EG @item References
1.1 noro 2750: @fref{ptozp}.
2751: @end table
2752:
1.2 noro 2753: \JP @node dp_nf dp_nf_mod dp_true_nf dp_true_nf_mod,,, $B%0%l%V%J4pDl$K4X$9$kH!?t(B
2754: \EG @node dp_nf dp_nf_mod dp_true_nf dp_true_nf_mod,,, Functions for Groebner basis computation
1.1 noro 2755: @subsection @code{dp_nf}, @code{dp_nf_mod}, @code{dp_true_nf}, @code{dp_true_nf_mod}
2756: @findex dp_nf
2757: @findex dp_true_nf
2758: @findex dp_nf_mod
2759: @findex dp_true_nf_mod
2760:
2761: @table @t
2762: @item dp_nf(@var{indexlist},@var{dpoly},@var{dpolyarray},@var{fullreduce})
2763: @item dp_nf_mod(@var{indexlist},@var{dpoly},@var{dpolyarray},@var{fullreduce},@var{mod})
1.2 noro 2764: \JP :: $BJ,;6I=8=B?9`<0$N@55,7A$r5a$a$k(B. ($B7k2L$ODj?tG\$5$l$F$$$k2DG=@-$"$j(B)
1.1 noro 2765:
1.2 noro 2766: \BEG
2767: :: Computes the normal form of a distributed polynomial.
2768: (The result may be multiplied by a constant in the ground field.)
2769: \E
1.1 noro 2770: @item dp_true_nf(@var{indexlist},@var{dpoly},@var{dpolyarray},@var{fullreduce})
2771: @item dp_true_nf_mod(@var{indexlist},@var{dpoly},@var{dpolyarray},@var{fullreduce},@var{mod})
1.2 noro 2772: \JP :: $BJ,;6I=8=B?9`<0$N@55,7A$r5a$a$k(B. ($B??$N7k2L$r(B @code{[$BJ,;R(B, $BJ,Jl(B]} $B$N7A$GJV$9(B)
2773: \BEG
2774: :: Computes the normal form of a distributed polynomial. (The true result
2775: is returned in such a list as @code{[numerator, denominator]})
2776: \E
1.1 noro 2777: @end table
2778:
2779: @table @var
2780: @item return
1.2 noro 2781: \JP @code{dp_nf()} : $BJ,;6I=8=B?9`<0(B, @code{dp_true_nf()} : $B%j%9%H(B
2782: \EG @code{dp_nf()} : distributed polynomial, @code{dp_true_nf()} : list
1.1 noro 2783: @item indexlist
1.2 noro 2784: \JP $B%j%9%H(B
2785: \EG list
1.1 noro 2786: @item dpoly
1.2 noro 2787: \JP $BJ,;6I=8=B?9`<0(B
2788: \EG distributed polynomial
1.1 noro 2789: @item dpolyarray
1.2 noro 2790: \JP $BG[Ns(B
2791: \EG array of distributed polynomial
1.1 noro 2792: @item fullreduce
1.2 noro 2793: \JP $B%U%i%0(B
2794: \EG flag
1.1 noro 2795: @item mod
1.2 noro 2796: \JP $BAG?t(B
2797: \EG prime
1.1 noro 2798: @end table
2799:
2800: @itemize @bullet
1.2 noro 2801: \BJP
1.1 noro 2802: @item
2803: $BJ,;6I=8=B?9`<0(B @var{dpoly} $B$N@55,7A$r5a$a$k(B.
2804: @item
2805: @code{dp_nf_mod()}, @code{dp_true_nf_mod()} $B$NF~NO$O(B, @code{dp_mod()} $B$J$I(B
2806: $B$K$h$j(B, $BM-8BBN>e$NJ,;6I=8=B?9`<0$K$J$C$F$$$J$1$l$P$J$i$J$$(B.
2807: @item
2808: $B7k2L$KM-M}?t(B, $BM-M}<0$,4^$^$l$k$N$rHr$1$k$?$a(B, @code{dp_nf()} $B$O(B
2809: $B??$NCM$NDj?tG\$NCM$rJV$9(B. $BM-M}<078?t$N>l9g$N(B @code{dp_nf_mod()} $B$bF1MM(B
2810: $B$G$"$k$,(B, $B78?tBN$,M-8BBN$N>l9g(B @code{dp_nf_mod()} $B$O??$NCM$rJV$9(B.
2811: @item
2812: @code{dp_true_nf()}, @code{dp_true_nf_mod()} $B$O(B,
2813: @code{[@var{nm},@var{dn}]} $B$J$k7A$N%j%9%H$rJV$9(B.
2814: $B$?$@$7(B, @var{nm} $B$O78?t$KJ,?t(B, $BM-M}<0$r4^$^$J$$J,;6I=8=B?9`<0(B, @var{dn} $B$O(B
2815: $B?t$^$?$OB?9`<0$G(B @var{nm}/@var{dn} $B$,??$NCM$H$J$k(B.
2816: @item
2817: @var{dpolyarray} $B$OJ,;6I=8=B?9`<0$rMWAG$H$9$k%Y%/%H%k(B,
2818: @var{indexlist} $B$O@55,2=7W;;$KMQ$$$k(B @var{dpolyarray} $B$NMWAG$N%$%s%G%C%/%9(B
2819: $B$N%j%9%H(B.
2820: @item
2821: @var{fullreduce} $B$,(B 0 $B$G$J$$$H$-A4$F$N9`$KBP$7$F4JLs$r9T$&(B. @var{fullreduce}
2822: $B$,(B 0 $B$N$H$-F,9`$N$_$KBP$7$F4JLs$r9T$&(B.
2823: @item
2824: @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.
2825: @item
2826: $B0lHL$K$O(B @var{indexlist} $B$NM?$(J}$K$h$jH!?t$NCM$O0[$J$k2DG=@-$,$"$k$,(B,
2827: $B%0%l%V%J4pDl$KBP$7$F$O0l0UE*$KDj$^$k(B.
2828: @item
2829: $BJ,;6I=8=$G$J$$8GDj$5$l$?B?9`<0=89g$K$h$k@55,7A$rB??t5a$a$kI,MW$,$"$k>l9g(B
2830: $B$KJXMx$G$"$k(B. $BC10l$N1i;;$K4X$7$F$O(B, @code{p_nf}, @code{p_true_nf} $B$r(B
2831: $BMQ$$$k$H$h$$(B.
1.2 noro 2832: \E
2833: \BEG
2834: @item
2835: Computes the normal form of a distributed polynomial.
2836: @item
2837: @code{dp_nf_mod()} and @code{dp_true_nf_mod()} require
2838: distributed polynomials with coefficients in a finite field as arguments.
2839: @item
2840: The result of @code{dp_nf()} may be multiplied by a constant in the
2841: ground field in order to make the result integral. The same is true
2842: for @code{dp_nf_mod()}, but it returns the true normal form if
2843: the ground field is a finite field.
2844: @item
2845: @code{dp_true_nf()} and @code{dp_true_nf_mod()} return
2846: such a list as @code{[@var{nm},@var{dn}]}.
2847: Here @var{nm} is a distributed polynomial whose coefficients are integral
2848: in the ground field, @var{dn} is an integral element in the ground
2849: field and @var{nm}/@var{dn} is the true normal form.
2850: @item
2851: @var{dpolyarray} is a vector whose components are distributed polynomials
2852: and @var{indexlist} is a list of indices which is used for the normal form
2853: computation.
2854: @item
2855: When argument @var{fullreduce} has non-zero value,
2856: all terms are reduced. When it has value 0,
2857: only the head term is reduced.
2858: @item
2859: As for the polynomials specified by @var{indexlist}, one specified by
2860: an index placed at the preceding position has priority to be selected.
2861: @item
2862: In general, the result of the function may be different depending on
2863: @var{indexlist}. However, the result is unique for Groebner bases.
2864: @item
2865: These functions are useful when a fixed non-distributed polynomial set
2866: is used as a set of reducers to compute normal forms of many polynomials.
2867: For single computation @code{p_nf} and @code{p_true_nf} are sufficient.
2868: \E
1.1 noro 2869: @end itemize
2870:
2871: @example
2872: [0] load("gr")$
2873: [64] load("katsura")$
2874: [69] K=katsura(4)$
2875: [70] dp_ord(2)$
2876: [71] V=[u0,u1,u2,u3,u4]$
2877: [72] DP1=newvect(length(K),map(dp_ptod,K,V))$
2878: [73] G=gr(K,V,2)$
2879: [74] DP2=newvect(length(G),map(dp_ptod,G,V))$
2880: [75] T=dp_ptod((u0-u1+u2-u3+u4)^2,V)$
2881: [76] dp_dtop(dp_nf([0,1,2,3,4],T,DP1,1),V);
1.5 noro 2882: u4^2+(6*u3+2*u2+6*u1-2)*u4+9*u3^2+(6*u2+18*u1-6)*u3+u2^2
2883: +(6*u1-2)*u2+9*u1^2-6*u1+1
1.1 noro 2884: [77] dp_dtop(dp_nf([4,3,2,1,0],T,DP1,1),V);
2885: -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
2886: [78] dp_dtop(dp_nf([0,1,2,3,4],T,DP2,1),V);
1.5 noro 2887: -11380879768451657780886122972730785203470970010204714556333530492210
2888: 456775930005716505560062087150928400876150217079820311439477560587583
2889: 488*u4^15+...
1.1 noro 2890: [79] dp_dtop(dp_nf([4,3,2,1,0],T,DP2,1),V);
1.5 noro 2891: -11380879768451657780886122972730785203470970010204714556333530492210
2892: 456775930005716505560062087150928400876150217079820311439477560587583
2893: 488*u4^15+...
1.1 noro 2894: [80] @@78==@@79;
2895: 1
2896: @end example
2897:
2898: @table @t
1.2 noro 2899: \JP @item $B;2>H(B
2900: \EG @item References
1.1 noro 2901: @fref{dp_dtop},
2902: @fref{dp_ord},
2903: @fref{dp_mod dp_rat},
2904: @fref{p_nf p_nf_mod p_true_nf p_true_nf_mod}.
2905: @end table
2906:
1.2 noro 2907: \JP @node dp_hm dp_ht dp_hc dp_rest,,, $B%0%l%V%J4pDl$K4X$9$kH!?t(B
2908: \EG @node dp_hm dp_ht dp_hc dp_rest,,, Functions for Groebner basis computation
1.1 noro 2909: @subsection @code{dp_hm}, @code{dp_ht}, @code{dp_hc}, @code{dp_rest}
2910: @findex dp_hm
2911: @findex dp_ht
2912: @findex dp_hc
2913: @findex dp_rest
2914:
2915: @table @t
2916: @item dp_hm(@var{dpoly})
1.2 noro 2917: \JP :: $BF,C19`<0$r<h$j=P$9(B.
2918: \EG :: Gets the head monomial.
1.1 noro 2919: @item dp_ht(@var{dpoly})
1.2 noro 2920: \JP :: $BF,9`$r<h$j=P$9(B.
2921: \EG :: Gets the head term.
1.1 noro 2922: @item dp_hc(@var{dpoly})
1.2 noro 2923: \JP :: $BF,78?t$r<h$j=P$9(B.
2924: \EG :: Gets the head coefficient.
1.1 noro 2925: @item dp_rest(@var{dpoly})
1.2 noro 2926: \JP :: $BF,C19`<0$r<h$j=|$$$?;D$j$rJV$9(B.
2927: \EG :: Gets the remainder of the polynomial where the head monomial is removed.
1.1 noro 2928: @end table
2929:
2930: @table @var
1.2 noro 2931: \BJP
1.1 noro 2932: @item return
2933: @code{dp_hm()}, @code{dp_ht()}, @code{dp_rest()} : $BJ,;6I=8=B?9`<0(B,
2934: @code{dp_hc()} : $B?t$^$?$OB?9`<0(B
2935: @item dpoly
2936: $BJ,;6I=8=B?9`<0(B
1.2 noro 2937: \E
2938: \BEG
2939: @item return
2940: @code{dp_hm()}, @code{dp_ht()}, @code{dp_rest()} : distributed polynomial
2941: @code{dp_hc()} : number or polynomial
2942: @item dpoly
2943: distributed polynomial
2944: \E
1.1 noro 2945: @end table
2946:
2947: @itemize @bullet
1.2 noro 2948: \BJP
1.1 noro 2949: @item
2950: $B$3$l$i$O(B, $BJ,;6I=8=B?9`<0$N3FItJ,$r<h$j=P$9$?$a$NH!?t$G$"$k(B.
2951: @item
2952: $BJ,;6I=8=B?9`<0(B @var{p} $B$KBP$7<!$,@.$jN)$D(B.
1.2 noro 2953: \E
2954: \BEG
2955: @item
2956: These are used to get various parts of a distributed polynomial.
2957: @item
2958: The next equations hold for a distributed polynomial @var{p}.
2959: \E
1.1 noro 2960: @table @code
2961: @item @var{p} = dp_hm(@var{p}) + dp_rest(@var{p})
2962: @item dp_hm(@var{p}) = dp_hc(@var{p}) dp_ht(@var{p})
2963: @end table
2964: @end itemize
2965:
2966: @example
2967: [87] dp_ord(0)$
2968: [88] X=ptozp((a46^2+7/10*a46+7/48)*u3^4-50/27*a46^2-35/27*a46-49/216)$
2969: [89] T=dp_ptod(X,[u3,u4,a46])$
2970: [90] dp_hm(T);
2971: (2160)*<<4,0,2>>
2972: [91] dp_ht(T);
2973: (1)*<<4,0,2>>
2974: [92] dp_hc(T);
2975: 2160
2976: [93] dp_rest(T);
2977: (1512)*<<4,0,1>>+(315)*<<4,0,0>>+(-4000)*<<0,0,2>>+(-2800)*<<0,0,1>>
2978: +(-490)*<<0,0,0>>
2979: @end example
2980:
1.2 noro 2981: \JP @node dp_td dp_sugar,,, $B%0%l%V%J4pDl$K4X$9$kH!?t(B
2982: \EG @node dp_td dp_sugar,,, Functions for Groebner basis computation
1.1 noro 2983: @subsection @code{dp_td}, @code{dp_sugar}
2984: @findex dp_td
2985: @findex dp_sugar
2986:
2987: @table @t
2988: @item dp_td(@var{dpoly})
1.2 noro 2989: \JP :: $BF,9`$NA4<!?t$rJV$9(B.
2990: \EG :: Gets the total degree of the head term.
1.1 noro 2991: @item dp_sugar(@var{dpoly})
1.2 noro 2992: \JP :: $BB?9`<0$N(B @code{sugar} $B$rJV$9(B.
2993: \EG :: Gets the @code{sugar} of a polynomial.
1.1 noro 2994: @end table
2995:
2996: @table @var
2997: @item return
1.2 noro 2998: \JP $B<+A3?t(B
2999: \EG non-negative integer
1.1 noro 3000: @item dpoly
1.2 noro 3001: \JP $BJ,;6I=8=B?9`<0(B
3002: \EG distributed polynomial
1.1 noro 3003: @item onoff
1.2 noro 3004: \JP $B%U%i%0(B
3005: \EG flag
1.1 noro 3006: @end table
3007:
3008: @itemize @bullet
1.2 noro 3009: \BJP
1.1 noro 3010: @item
3011: @code{dp_td()} $B$O(B, $BF,9`$NA4<!?t(B, $B$9$J$o$A3FJQ?t$N;X?t$NOB$rJV$9(B.
3012: @item
3013: $BJ,;6I=8=B?9`<0$,@8@.$5$l$k$H(B, @code{sugar} $B$H8F$P$l$k$"$k@0?t$,IUM?(B
3014: $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.
3015: @item
3016: @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
3017: $B7hDj$9$k$?$a$N=EMW$J;X?K$H$J$k(B.
1.2 noro 3018: \E
3019: \BEG
3020: @item
3021: Function @code{dp_td()} returns the total degree of the head term,
3022: i.e., the sum of all exponent of variables in that term.
3023: @item
3024: Upon creation of a distributed polynomial, an integer called @code{sugar}
3025: is associated. This value is
3026: the total degree of the virtually homogenized one of the original
3027: polynomial.
3028: @item
3029: The quantity @code{sugar} is an important guide to determine the
3030: selection strategy of critical pairs in Groebner basis computation.
3031: \E
1.1 noro 3032: @end itemize
3033:
3034: @example
3035: [74] dp_ord(0)$
3036: [75] X=<<1,2>>+<<0,1>>$
3037: [76] Y=<<1,2>>+<<1,0>>$
3038: [77] Z=X-Y;
3039: (-1)*<<1,0>>+(1)*<<0,1>>
3040: [78] dp_sugar(T);
3041: 3
3042: @end example
3043:
1.2 noro 3044: \JP @node dp_lcm,,, $B%0%l%V%J4pDl$K4X$9$kH!?t(B
3045: \EG @node dp_lcm,,, Functions for Groebner basis computation
1.1 noro 3046: @subsection @code{dp_lcm}
3047: @findex dp_lcm
3048:
3049: @table @t
3050: @item dp_lcm(@var{dpoly1},@var{dpoly2})
1.2 noro 3051: \JP :: $B:G>.8xG\9`$rJV$9(B.
3052: \EG :: Returns the least common multiple of the head terms of the given two polynomials.
1.1 noro 3053: @end table
3054:
3055: @table @var
3056: @item return
1.2 noro 3057: \JP $BJ,;6I=8=B?9`<0(B
3058: \EG distributed polynomial
1.4 noro 3059: @item dpoly1 dpoly2
1.2 noro 3060: \JP $BJ,;6I=8=B?9`<0(B
3061: \EG distributed polynomial
1.1 noro 3062: @end table
3063:
3064: @itemize @bullet
1.2 noro 3065: \BJP
1.1 noro 3066: @item
3067: $B$=$l$>$l$N0z?t$NF,9`$N:G>.8xG\9`$rJV$9(B. $B78?t$O(B 1 $B$G$"$k(B.
1.2 noro 3068: \E
3069: \BEG
3070: @item
3071: Returns the least common multiple of the head terms of the given
3072: two polynomials, where coefficient is always set to 1.
3073: \E
1.1 noro 3074: @end itemize
3075:
3076: @example
3077: [100] dp_lcm(<<1,2,3,4,5>>,<<5,4,3,2,1>>);
3078: (1)*<<5,4,3,4,5>>
3079: @end example
3080:
3081: @table @t
1.2 noro 3082: \JP @item $B;2>H(B
3083: \EG @item References
1.1 noro 3084: @fref{p_nf p_nf_mod p_true_nf p_true_nf_mod}.
3085: @end table
3086:
1.2 noro 3087: \JP @node dp_redble,,, $B%0%l%V%J4pDl$K4X$9$kH!?t(B
3088: \EG @node dp_redble,,, Functions for Groebner basis computation
1.1 noro 3089: @subsection @code{dp_redble}
3090: @findex dp_redble
3091:
3092: @table @t
3093: @item dp_redble(@var{dpoly1},@var{dpoly2})
1.2 noro 3094: \JP :: $BF,9`$I$&$7$,@0=|2DG=$+$I$&$+D4$Y$k(B.
3095: \EG :: Checks whether one head term is divisible by the other head term.
1.1 noro 3096: @end table
3097:
3098: @table @var
3099: @item return
1.2 noro 3100: \JP $B@0?t(B
3101: \EG integer
1.4 noro 3102: @item dpoly1 dpoly2
1.2 noro 3103: \JP $BJ,;6I=8=B?9`<0(B
3104: \EG distributed polynomial
1.1 noro 3105: @end table
3106:
3107: @itemize @bullet
1.2 noro 3108: \BJP
1.1 noro 3109: @item
3110: @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
3111: 0 $B$rJV$9(B.
3112: @item
3113: $BB?9`<0$N4JLs$r9T$&:](B, $B$I$N9`$r4JLs$G$-$k$+$rC5$9$N$KMQ$$$k(B.
1.2 noro 3114: \E
3115: \BEG
3116: @item
3117: Returns 1 if the head term of @var{dpoly2} divides the head term of
3118: @var{dpoly1}; otherwise 0.
3119: @item
3120: Used for finding candidate terms at reduction of polynomials.
3121: \E
1.1 noro 3122: @end itemize
3123:
3124: @example
3125: [148] C;
3126: (1)*<<1,1,1,0,0>>+(1)*<<0,1,1,1,0>>+(1)*<<1,1,0,0,1>>+(1)*<<1,0,0,1,1>>
3127: [149] T;
3128: (3)*<<2,1,0,0,0>>+(3)*<<1,2,0,0,0>>+(1)*<<0,3,0,0,0>>+(6)*<<1,1,1,0,0>>
3129: [150] for ( ; T; T = dp_rest(T)) print(dp_redble(T,C));
3130: 0
3131: 0
3132: 0
3133: 1
3134: @end example
3135:
3136: @table @t
1.2 noro 3137: \JP @item $B;2>H(B
3138: \EG @item References
1.1 noro 3139: @fref{dp_red dp_red_mod}.
3140: @end table
3141:
1.2 noro 3142: \JP @node dp_subd,,, $B%0%l%V%J4pDl$K4X$9$kH!?t(B
3143: \EG @node dp_subd,,, Functions for Groebner basis computation
1.1 noro 3144: @subsection @code{dp_subd}
3145: @findex dp_subd
3146:
3147: @table @t
3148: @item dp_subd(@var{dpoly1},@var{dpoly2})
1.2 noro 3149: \JP :: $BF,9`$N>&C19`<0$rJV$9(B.
3150: \EG :: Returns the quotient monomial of the head terms.
1.1 noro 3151: @end table
3152:
3153: @table @var
3154: @item return
1.2 noro 3155: \JP $BJ,;6I=8=B?9`<0(B
3156: \EG distributed polynomial
1.4 noro 3157: @item dpoly1 dpoly2
1.2 noro 3158: \JP $BJ,;6I=8=B?9`<0(B
3159: \EG distributed polynomial
1.1 noro 3160: @end table
3161:
3162: @itemize @bullet
1.2 noro 3163: \BJP
1.1 noro 3164: @item
3165: @code{dp_ht(@var{dpoly1})/dp_ht(@var{dpoly2})} $B$r5a$a$k(B. $B7k2L$N78?t$O(B 1
3166: $B$G$"$k(B.
3167: @item
3168: $B3d$j@Z$l$k$3$H$,$"$i$+$8$a$o$+$C$F$$$kI,MW$,$"$k(B.
1.2 noro 3169: \E
3170: \BEG
3171: @item
3172: Gets @code{dp_ht(@var{dpoly1})/dp_ht(@var{dpoly2})}.
3173: The coefficient of the result is always set to 1.
3174: @item
3175: Divisibility assumed.
3176: \E
1.1 noro 3177: @end itemize
3178:
3179: @example
3180: [162] dp_subd(<<1,2,3,4,5>>,<<1,1,2,3,4>>);
3181: (1)*<<0,1,1,1,1>>
3182: @end example
3183:
3184: @table @t
1.2 noro 3185: \JP @item $B;2>H(B
3186: \EG @item References
1.1 noro 3187: @fref{dp_red dp_red_mod}.
3188: @end table
3189:
1.2 noro 3190: \JP @node dp_vtoe dp_etov,,, $B%0%l%V%J4pDl$K4X$9$kH!?t(B
3191: \EG @node dp_vtoe dp_etov,,, Functions for Groebner basis computation
1.1 noro 3192: @subsection @code{dp_vtoe}, @code{dp_etov}
3193: @findex dp_vtoe
3194: @findex dp_etov
3195:
3196: @table @t
3197: @item dp_vtoe(@var{vect})
1.2 noro 3198: \JP :: $B;X?t%Y%/%H%k$r9`$KJQ49(B
3199: \EG :: Converts an exponent vector into a term.
1.1 noro 3200: @item dp_etov(@var{dpoly})
1.2 noro 3201: \JP :: $BF,9`$r;X?t%Y%/%H%k$KJQ49(B
3202: \EG :: Convert the head term of a distributed polynomial into an exponent vector.
1.1 noro 3203: @end table
3204:
3205: @table @var
3206: @item return
1.2 noro 3207: \JP @code{dp_vtoe} : $BJ,;6I=8=B?9`<0(B, @code{dp_etov} : $B%Y%/%H%k(B
3208: \EG @code{dp_vtoe} : distributed polynomial, @code{dp_etov} : vector
1.1 noro 3209: @item vect
1.2 noro 3210: \JP $B%Y%/%H%k(B
3211: \EG vector
1.1 noro 3212: @item dpoly
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_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.
3221: @item
3222: @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
3223: $B%Y%/%H%k$KJQ49$9$k(B.
1.2 noro 3224: \E
3225: \BEG
3226: @item
3227: @code{dp_vtoe()} generates a term whose exponent vector is @var{vect}.
3228: @item
3229: @code{dp_etov()} generates a vector which is the exponent vector of the
3230: head term of @code{dpoly}.
3231: \E
1.1 noro 3232: @end itemize
3233:
3234: @example
3235: [211] X=<<1,2,3>>;
3236: (1)*<<1,2,3>>
3237: [212] V=dp_etov(X);
3238: [ 1 2 3 ]
3239: [213] V[2]++$
3240: [214] Y=dp_vtoe(V);
3241: (1)*<<1,2,4>>
3242: @end example
3243:
1.2 noro 3244: \JP @node dp_mbase,,, $B%0%l%V%J4pDl$K4X$9$kH!?t(B
3245: \EG @node dp_mbase,,, Functions for Groebner basis computation
1.1 noro 3246: @subsection @code{dp_mbase}
3247: @findex dp_mbase
3248:
3249: @table @t
3250: @item dp_mbase(@var{dplist})
1.2 noro 3251: \JP :: monomial $B4pDl$N7W;;(B
3252: \EG :: Computes the monomial basis
1.1 noro 3253: @end table
3254:
3255: @table @var
3256: @item return
1.2 noro 3257: \JP $BJ,;6I=8=B?9`<0$N%j%9%H(B
3258: \EG list of distributed polynomial
1.1 noro 3259: @item dplist
1.2 noro 3260: \JP $BJ,;6I=8=B?9`<0$N%j%9%H(B
3261: \EG list of distributed polynomial
1.1 noro 3262: @end table
3263:
3264: @itemize @bullet
1.2 noro 3265: \BJP
1.1 noro 3266: @item
3267: $B$"$k=g=x$G%0%l%V%J4pDl$H$J$C$F$$$kB?9`<0=89g$N(B, $B$=$N=g=x$K4X$9$kJ,;6I=8=(B
3268: $B$G$"$k(B @var{dplist} $B$K$D$$$F(B,
3269: @var{dplist} $B$,(B K[X] $BCf$G@8@.$9$k%$%G%"%k(B I $B$,(B 0 $B<!85$N;~(B,
3270: K $B>eM-8B<!85@~7A6u4V$G$"$k(B K[X]/I $B$N(B monomial $B$K$h$k4pDl$r5a$a$k(B.
3271: @item
3272: $BF@$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 3273: \E
3274: \BEG
3275: @item
3276: Assuming that @var{dplist} is a list of distributed polynomials which
3277: is a Groebner basis with respect to the current ordering type and
3278: that the ideal @var{I} generated by @var{dplist} in K[X] is zero-dimensional,
3279: this function computes the monomial basis of a finite dimenstional K-vector
3280: space K[X]/I.
3281: @item
3282: The number of elements in the monomial basis is equal to the
3283: K-dimenstion of K[X]/I.
3284: \E
1.1 noro 3285: @end itemize
3286:
3287: @example
3288: [215] K=katsura(5)$
3289: [216] V=[u5,u4,u3,u2,u1,u0]$
3290: [217] G0=gr(K,V,0)$
3291: [218] H=map(dp_ptod,G0,V)$
3292: [219] map(dp_ptod,dp_mbase(H),V)$
3293: [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,
3294: 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,
3295: u1*u2,u1^2,u4*u0,u3*u0,u2*u0,u1*u0,u0^2,u4,u3,u2,u1,u0,1]
3296: @end example
3297:
3298: @table @t
1.2 noro 3299: \JP @item $B;2>H(B
3300: \EG @item References
1.1 noro 3301: @fref{gr hgr gr_mod}.
3302: @end table
3303:
1.2 noro 3304: \JP @node dp_mag,,, $B%0%l%V%J4pDl$K4X$9$kH!?t(B
3305: \EG @node dp_mag,,, Functions for Groebner basis computation
1.1 noro 3306: @subsection @code{dp_mag}
3307: @findex dp_mag
3308:
3309: @table @t
3310: @item dp_mag(@var{p})
1.2 noro 3311: \JP :: $B78?t$N%S%C%HD9$NOB$rJV$9(B
3312: \EG :: Computes the sum of bit lengths of coefficients of a distributed polynomial.
1.1 noro 3313: @end table
3314:
3315: @table @var
3316: @item return
1.2 noro 3317: \JP $B?t(B
3318: \EG integer
1.1 noro 3319: @item p
1.2 noro 3320: \JP $BJ,;6I=8=B?9`<0(B
3321: \EG distributed polynomial
1.1 noro 3322: @end table
3323:
3324: @itemize @bullet
1.2 noro 3325: \BJP
1.1 noro 3326: @item
3327: $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)
3328: $B$N%S%C%HD9$NAmOB$rJV$9(B.
3329: @item
3330: $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
3331: $B78?tKDD%$,LdBj$H$J$j(B, $BESCf@8@.$5$l$kB?9`<0$,78?tKDD%$r5/$3$7$F$$$k$+$I$&$+(B
3332: $B$NH=Dj$KLrN)$D(B.
3333: @item
3334: @code{dp_gr_flags()} $B$G(B, @code{ShowMag}, @code{Print} $B$r(B on $B$K$9$k$3$H$K$h$j(B
3335: $BESCf@8@.$5$l$kB?9`<0$K$?$$$9$k(B @code{dp_mag()} $B$NCM$r8+$k$3$H$,$G$-$k(B.
1.2 noro 3336: \E
3337: \BEG
3338: @item
3339: This function computes the sum of bit lengths of coefficients of a
3340: distributed polynomial @var{p}. If a coefficient is non integral,
3341: the sum of bit lengths of the numerator and the denominator is taken.
3342: @item
3343: This is a measure of the size of a polynomial. Especially for
3344: zero-dimensional system coefficient swells are often serious and
3345: the returned value is useful to detect such swells.
3346: @item
3347: If @code{ShowMag} and @code{Print} for @code{dp_gr_flags()} are on,
3348: values of @code{dp_mag()} for intermediate basis elements are shown.
3349: \E
1.1 noro 3350: @end itemize
3351:
3352: @example
3353: [221] X=dp_ptod((x+2*y)^10,[x,y])$
3354: [222] dp_mag(X);
3355: 115
3356: @end example
3357:
3358: @table @t
1.2 noro 3359: \JP @item $B;2>H(B
3360: \EG @item References
1.1 noro 3361: @fref{dp_gr_flags dp_gr_print}.
3362: @end table
3363:
1.2 noro 3364: \JP @node dp_red dp_red_mod,,, $B%0%l%V%J4pDl$K4X$9$kH!?t(B
3365: \EG @node dp_red dp_red_mod,,, Functions for Groebner basis computation
1.1 noro 3366: @subsection @code{dp_red}, @code{dp_red_mod}
3367: @findex dp_red
3368: @findex dp_red_mod
3369:
3370: @table @t
3371: @item dp_red(@var{dpoly1},@var{dpoly2},@var{dpoly3})
3372: @item dp_red_mod(@var{dpoly1},@var{dpoly2},@var{dpoly3},@var{mod})
1.2 noro 3373: \JP :: $B0l2s$N4JLsA`:n(B
3374: \EG :: Single reduction operation
1.1 noro 3375: @end table
3376:
3377: @table @var
3378: @item return
1.2 noro 3379: \JP $B%j%9%H(B
3380: \EG list
1.4 noro 3381: @item dpoly1 dpoly2 dpoly3
1.2 noro 3382: \JP $BJ,;6I=8=B?9`<0(B
3383: \EG distributed polynomial
1.1 noro 3384: @item vlist
1.2 noro 3385: \JP $B%j%9%H(B
3386: \EG list
1.1 noro 3387: @item mod
1.2 noro 3388: \JP $BAG?t(B
3389: \EG prime
1.1 noro 3390: @end table
3391:
3392: @itemize @bullet
1.2 noro 3393: \BJP
1.1 noro 3394: @item
3395: @var{dpoly1} + @var{dpoly2} $B$J$kJ,;6I=8=B?9`<0$r(B @var{dpoly3} $B$G(B
3396: 1 $B2s4JLs$9$k(B.
3397: @item
3398: @code{dp_red_mod()} $B$NF~NO$O(B, $BA4$FM-8BBN78?t$KJQ49$5$l$F$$$kI,MW$,$"$k(B.
3399: @item
3400: $B4JLs$5$l$k9`$O(B @var{dpoly2} $B$NF,9`$G$"$k(B. $B=>$C$F(B, @var{dpoly2} $B$N(B
3401: $BF,9`$,(B @var{dpoly3} $B$NF,9`$G3d$j@Z$l$k$3$H$,$"$i$+$8$a$o$+$C$F$$$J$1$l$P(B
3402: $B$J$i$J$$(B.
3403: @item
3404: $B0z?t$,@0?t78?t$N;~(B, $B4JLs$O(B, $BJ,?t$,8=$l$J$$$h$&(B, $B@0?t(B @var{a}, @var{b},
1.4 noro 3405: $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 3406: @item
3407: $B7k2L$O(B, @code{[@var{a dpoly1},@var{a dpoly2 - bt dpoly3}]} $B$J$k%j%9%H$G$"$k(B.
1.2 noro 3408: \E
3409: \BEG
3410: @item
3411: Reduces a distributed polynomial, @var{dpoly1} + @var{dpoly2},
3412: by @var{dpoly3} for single time.
3413: @item
3414: An input for @code{dp_red_mod()} must be converted into a distributed
3415: polynomial with coefficients in a finite field.
3416: @item
3417: This implies that
3418: the divisibility of the head term of @var{dpoly2} by the head term of
3419: @var{dpoly3} is assumed.
3420: @item
3421: When integral coefficients, computation is so carefully performed that
3422: no rational operations appear in the reduction procedure.
3423: It is computed for integers @var{a} and @var{b}, and a term @var{t} as:
1.4 noro 3424: @var{a}(@var{dpoly1} + @var{dpoly2})-@var{bt} @var{dpoly3}.
1.2 noro 3425: @item
3426: The result is a list @code{[@var{a dpoly1},@var{a dpoly2 - bt dpoly3}]}.
3427: \E
1.1 noro 3428: @end itemize
3429:
3430: @example
3431: [157] D=(3)*<<2,1,0,0,0>>+(3)*<<1,2,0,0,0>>+(1)*<<0,3,0,0,0>>;
3432: (3)*<<2,1,0,0,0>>+(3)*<<1,2,0,0,0>>+(1)*<<0,3,0,0,0>>
3433: [158] R=(6)*<<1,1,1,0,0>>;
3434: (6)*<<1,1,1,0,0>>
3435: [159] C=12*<<1,1,1,0,0>>+(1)*<<0,1,1,1,0>>+(1)*<<1,1,0,0,1>>;
3436: (12)*<<1,1,1,0,0>>+(1)*<<0,1,1,1,0>>+(1)*<<1,1,0,0,1>>
3437: [160] dp_red(D,R,C);
1.5 noro 3438: [(6)*<<2,1,0,0,0>>+(6)*<<1,2,0,0,0>>+(2)*<<0,3,0,0,0>>,
3439: (-1)*<<0,1,1,1,0>>+(-1)*<<1,1,0,0,1>>]
1.1 noro 3440: @end example
3441:
3442: @table @t
1.2 noro 3443: \JP @item $B;2>H(B
3444: \EG @item References
1.1 noro 3445: @fref{dp_mod dp_rat}.
3446: @end table
3447:
1.2 noro 3448: \JP @node dp_sp dp_sp_mod,,, $B%0%l%V%J4pDl$K4X$9$kH!?t(B
3449: \EG @node dp_sp dp_sp_mod,,, Functions for Groebner basis computation
1.1 noro 3450: @subsection @code{dp_sp}, @code{dp_sp_mod}
3451: @findex dp_sp
3452: @findex dp_sp_mod
3453:
3454: @table @t
3455: @item dp_sp(@var{dpoly1},@var{dpoly2})
3456: @item dp_sp_mod(@var{dpoly1},@var{dpoly2},@var{mod})
1.2 noro 3457: \JP :: S-$BB?9`<0$N7W;;(B
3458: \EG :: Computation of an S-polynomial
1.1 noro 3459: @end table
3460:
3461: @table @var
3462: @item return
1.2 noro 3463: \JP $BJ,;6I=8=B?9`<0(B
3464: \EG distributed polynomial
1.4 noro 3465: @item dpoly1 dpoly2
1.2 noro 3466: \JP $BJ,;6I=8=B?9`<0(B
3467: \EG distributed polynomial
1.1 noro 3468: @item mod
1.2 noro 3469: \JP $BAG?t(B
3470: \EG prime
1.1 noro 3471: @end table
3472:
3473: @itemize @bullet
1.2 noro 3474: \BJP
1.1 noro 3475: @item
3476: @var{dpoly1}, @var{dpoly2} $B$N(B S-$BB?9`<0$r7W;;$9$k(B.
3477: @item
3478: @code{dp_sp_mod()} $B$NF~NO$O(B, $BA4$FM-8BBN78?t$KJQ49$5$l$F$$$kI,MW$,$"$k(B.
3479: @item
3480: $B7k2L$KM-M}?t(B, $BM-M}<0$,F~$k$N$rHr$1$k$?$a(B, $B7k2L$,Dj?tG\(B, $B$"$k$$$OB?9`<0(B
3481: $BG\$5$l$F$$$k2DG=@-$,$"$k(B.
1.2 noro 3482: \E
3483: \BEG
3484: @item
3485: This function computes the S-polynomial of @var{dpoly1} and @var{dpoly2}.
3486: @item
3487: Inputs of @code{dp_sp_mod()} must be polynomials with coefficients in a
3488: finite field.
3489: @item
3490: The result may be multiplied by a constant in the ground field in order to
3491: make the result integral.
3492: \E
1.1 noro 3493: @end itemize
3494:
3495: @example
3496: [227] X=dp_ptod(x^2*y+x*y,[x,y]);
3497: (1)*<<2,1>>+(1)*<<1,1>>
3498: [228] Y=dp_ptod(x*y^2+x*y,[x,y]);
3499: (1)*<<1,2>>+(1)*<<1,1>>
3500: [229] dp_sp(X,Y);
3501: (-1)*<<2,1>>+(1)*<<1,2>>
3502: @end example
3503:
3504: @table @t
1.2 noro 3505: \JP @item $B;2>H(B
3506: \EG @item References
1.1 noro 3507: @fref{dp_mod dp_rat}.
3508: @end table
1.2 noro 3509: \JP @node p_nf p_nf_mod p_true_nf p_true_nf_mod,,, $B%0%l%V%J4pDl$K4X$9$kH!?t(B
3510: \EG @node p_nf p_nf_mod p_true_nf p_true_nf_mod,,, Functions for Groebner basis computation
1.1 noro 3511: @subsection @code{p_nf}, @code{p_nf_mod}, @code{p_true_nf}, @code{p_true_nf_mod}
3512: @findex p_nf
3513: @findex p_nf_mod
3514: @findex p_true_nf
3515: @findex p_true_nf_mod
3516:
3517: @table @t
3518: @item p_nf(@var{poly},@var{plist},@var{vlist},@var{order})
3519: @itemx p_nf_mod(@var{poly},@var{plist},@var{vlist},@var{order},@var{mod})
1.2 noro 3520: \JP :: $BI=8=B?9`<0$N@55,7A$r5a$a$k(B. ($B7k2L$ODj?tG\$5$l$F$$$k2DG=@-$"$j(B)
3521: \BEG
3522: :: Computes the normal form of the given polynomial.
3523: (The result may be multiplied by a constant.)
3524: \E
1.1 noro 3525: @item p_true_nf(@var{poly},@var{plist},@var{vlist},@var{order})
3526: @itemx p_true_nf_mod(@var{poly},@var{plist},@var{vlist},@var{order},@var{mod})
1.2 noro 3527: \JP :: $BI=8=B?9`<0$N@55,7A$r5a$a$k(B. ($B??$N7k2L$r(B @code{[$BJ,;R(B, $BJ,Jl(B]} $B$N7A$GJV$9(B)
3528: \BEG
3529: :: Computes the normal form of the given polynomial. (The result is returned
3530: as a form of @code{[numerator, denominator]})
3531: \E
1.1 noro 3532: @end table
3533:
3534: @table @var
3535: @item return
1.2 noro 3536: \JP @code{p_nf} : $BB?9`<0(B, @code{p_true_nf} : $B%j%9%H(B
3537: \EG @code{p_nf} : polynomial, @code{p_true_nf} : list
1.1 noro 3538: @item poly
1.2 noro 3539: \JP $BB?9`<0(B
3540: \EG polynomial
1.4 noro 3541: @item plist vlist
1.2 noro 3542: \JP $B%j%9%H(B
3543: \EG list
1.1 noro 3544: @item order
1.2 noro 3545: \JP $B?t(B, $B%j%9%H$^$?$O9TNs(B
3546: \EG number, list or matrix
1.1 noro 3547: @item mod
1.2 noro 3548: \JP $BAG?t(B
3549: \EG prime
1.1 noro 3550: @end table
3551:
3552: @itemize @bullet
1.2 noro 3553: \BJP
1.1 noro 3554: @item
3555: @samp{gr} $B$GDj5A$5$l$F$$$k(B.
3556: @item
3557: $BB?9`<0$N(B, $BB?9`<0%j%9%H$K$h$k@55,7A$r5a$a$k(B.
3558: @item
3559: @code{dp_nf()}, @code{dp_true_nf()}, @code{dp_nf_mod()}, @code{dp_true_nf_mod}
3560: $B$KBP$9$k%$%s%?%U%'!<%9$G$"$k(B.
3561: @item
3562: @var{poly} $B$*$h$S(B @var{plist} $B$O(B, $BJQ?t=g=x(B @var{vlist} $B$*$h$S(B
3563: $BJQ?t=g=x7?(B @var{otype} $B$K=>$C$FJ,;6I=8=B?9`<0$KJQ49$5$l(B,
3564: @code{dp_nf()}, @code{dp_true_nf()}, @code{dp_nf_mod()},
3565: @code{dp_true_nf_mod()} $B$KEO$5$l$k(B.
3566: @item
3567: @code{dp_nf()}, @code{dp_true_nf()}, @code{dp_nf_mod()},
3568: @code{dp_true_nf_mod()} $B$O(B @var{fullreduce} $B$,(B 1 $B$G8F$S=P$5$l$k(B.
3569: @item
3570: $B7k2L$OB?9`<0$KJQ49$5$l$F=PNO$5$l$k(B.
3571: @item
3572: @code{p_true_nf()}, @code{p_true_nf_mod()} $B$N=PNO$K4X$7$F$O(B,
3573: @code{dp_true_nf()}, @code{dp_true_nf_mod()} $B$N9`$r;2>H(B.
1.2 noro 3574: \E
3575: \BEG
3576: @item
3577: Defined in the package @samp{gr}.
3578: @item
3579: Obtains the normal form of a polynomial by a polynomial list.
3580: @item
3581: These are interfaces to @code{dp_nf()}, @code{dp_true_nf()}, @code{dp_nf_mod()},
3582: @code{dp_true_nf_mod}
3583: @item
3584: The polynomial @var{poly} and the polynomials in @var{plist} is
3585: converted, according to the variable ordering @var{vlist} and
3586: type of term ordering @var{otype}, into their distributed polynomial
3587: counterparts and passed to @code{dp_nf()}.
3588: @item
3589: @code{dp_nf()}, @code{dp_true_nf()}, @code{dp_nf_mod()} and
3590: @code{dp_true_nf_mod()}
3591: is called with value 1 for @var{fullreduce}.
3592: @item
3593: The result is converted back into an ordinary polynomial.
3594: @item
3595: As for @code{p_true_nf()}, @code{p_true_nf_mod()}
3596: refer to @code{dp_true_nf()} and @code{dp_true_nf_mod()}.
3597: \E
1.1 noro 3598: @end itemize
3599:
3600: @example
3601: [79] K = katsura(5)$
3602: [80] V = [u5,u4,u3,u2,u1,u0]$
3603: [81] G = hgr(K,V,2)$
3604: [82] p_nf(K[1],G,V,2);
3605: 0
3606: [83] L = p_true_nf(K[1]+1,G,V,2);
3607: [-1503...,-1503...]
3608: [84] L[0]/L[1];
3609: 1
3610: @end example
3611:
3612: @table @t
1.2 noro 3613: \JP @item $B;2>H(B
3614: \EG @item References
1.1 noro 3615: @fref{dp_ptod},
3616: @fref{dp_dtop},
3617: @fref{dp_ord},
3618: @fref{dp_nf dp_nf_mod dp_true_nf dp_true_nf_mod}.
3619: @end table
3620:
1.2 noro 3621: \JP @node p_terms,,, $B%0%l%V%J4pDl$K4X$9$kH!?t(B
3622: \EG @node p_terms,,, Functions for Groebner basis computation
1.1 noro 3623: @subsection @code{p_terms}
3624: @findex p_terms
3625:
3626: @table @t
3627: @item p_terms(@var{poly},@var{vlist},@var{order})
1.2 noro 3628: \JP :: $BB?9`<0$K$"$i$o$l$kC19`$r%j%9%H$K$9$k(B.
3629: \EG :: Monomials appearing in the given polynomial is collected into a list.
1.1 noro 3630: @end table
3631:
3632: @table @var
3633: @item return
1.2 noro 3634: \JP $B%j%9%H(B
3635: \EG list
1.1 noro 3636: @item poly
1.2 noro 3637: \JP $BB?9`<0(B
3638: \EG polynomial
1.1 noro 3639: @item vlist
1.2 noro 3640: \JP $B%j%9%H(B
3641: \EG list
1.1 noro 3642: @item order
1.2 noro 3643: \JP $B?t(B, $B%j%9%H$^$?$O9TNs(B
3644: \EG number, list or matrix
1.1 noro 3645: @end table
3646:
3647: @itemize @bullet
1.2 noro 3648: \BJP
1.1 noro 3649: @item
3650: @samp{gr} $B$GDj5A$5$l$F$$$k(B.
3651: @item
3652: $BB?9`<0$rC19`$KE83+$7$?;~$K8=$l$k9`$r%j%9%H$K$7$FJV$9(B.
3653: @var{vlist} $B$*$h$S(B @var{order} $B$K$h$jDj$^$k9`=g=x$K$h$j(B, $B=g=x$N9b$$$b$N(B
3654: $B$,%j%9%H$N@hF,$KMh$k$h$&$K%=!<%H$5$l$k(B.
3655: @item
3656: $B%0%l%V%J4pDl$O$7$P$7$P78?t$,5pBg$K$J$k$?$a(B, $B<B:]$K$I$N9`$,8=$l$F(B
3657: $B$$$k$N$+$r8+$k$?$a$J$I$KMQ$$$k(B.
1.2 noro 3658: \E
3659: \BEG
3660: @item
3661: Defined in the package @samp{gr}.
3662: @item
3663: This returns a list which contains all non-zero monomials in the given
3664: polynomial. The monomials are ordered according to the current
3665: type of term ordering and @var{vlist}.
3666: @item
3667: Since polynomials in a Groebner base often have very large coefficients,
3668: examining a polynomial as it is may sometimes be difficult to perform.
3669: For such a case, this function enables to examine which term is really
3670: exists.
3671: \E
1.1 noro 3672: @end itemize
3673:
3674: @example
3675: [233] G=gr(katsura(5),[u5,u4,u3,u2,u1,u0],2)$
3676: [234] p_terms(G[0],[u5,u4,u3,u2,u1,u0],2);
1.5 noro 3677: [u5,u0^31,u0^30,u0^29,u0^28,u0^27,u0^26,u0^25,u0^24,u0^23,u0^22,
3678: u0^21,u0^20,u0^19,u0^18,u0^17,u0^16,u0^15,u0^14,u0^13,u0^12,u0^11,
3679: u0^10,u0^9,u0^8,u0^7,u0^6,u0^5,u0^4,u0^3,u0^2,u0,1]
1.1 noro 3680: @end example
3681:
1.2 noro 3682: \JP @node gb_comp,,, $B%0%l%V%J4pDl$K4X$9$kH!?t(B
3683: \EG @node gb_comp,,, Functions for Groebner basis computation
1.1 noro 3684: @subsection @code{gb_comp}
3685: @findex gb_comp
3686:
3687: @table @t
3688: @item gb_comp(@var{plist1}, @var{plist2})
1.2 noro 3689: \JP :: $BB?9`<0%j%9%H$,(B, $BId9f$r=|$$$F=89g$H$7$FEy$7$$$+$I$&$+D4$Y$k(B.
3690: \EG :: Checks whether two polynomial lists are equal or not as a set
1.1 noro 3691: @end table
3692:
3693: @table @var
1.2 noro 3694: \JP @item return 0 $B$^$?$O(B 1
3695: \EG @item return 0 or 1
1.4 noro 3696: @item plist1 plist2
1.1 noro 3697: @end table
3698:
3699: @itemize @bullet
1.2 noro 3700: \BJP
1.1 noro 3701: @item
3702: @var{plist1}, @var{plist2} $B$K$D$$$F(B, $BId9f$r=|$$$F=89g$H$7$FEy$7$$$+$I$&$+(B
3703: $BD4$Y$k(B.
3704: @item
3705: $B0[$J$kJ}K!$G5a$a$?%0%l%V%J4pDl$O(B, $B4pDl$N=g=x(B, $BId9f$,0[$J$k>l9g$,$"$j(B,
3706: $B$=$l$i$,Ey$7$$$+$I$&$+$rD4$Y$k$?$a$KMQ$$$k(B.
1.2 noro 3707: \E
3708: \BEG
3709: @item
3710: This function checks whether @var{plist1} and @var{plist2} are equal or
3711: not as a set .
3712: @item
3713: For the same input and the same term ordering different
3714: functions for Groebner basis computations may produce different outputs
3715: as lists. This function compares such lists whether they are equal
3716: as a generating set of an ideal.
3717: \E
1.1 noro 3718: @end itemize
3719:
3720: @example
3721: [243] C=cyclic(6)$
3722: [244] V=[c0,c1,c2,c3,c4,c5]$
3723: [245] G0=gr(C,V,0)$
3724: [246] G=tolex(G0,V,0,V)$
3725: [247] GG=lex_tl(C,V,0,V,0)$
3726: [248] gb_comp(G,GG);
3727: 1
3728: @end example
3729:
1.2 noro 3730: \JP @node katsura hkatsura cyclic hcyclic,,, $B%0%l%V%J4pDl$K4X$9$kH!?t(B
3731: \EG @node katsura hkatsura cyclic hcyclic,,, Functions for Groebner basis computation
1.1 noro 3732: @subsection @code{katsura}, @code{hkatsura}, @code{cyclic}, @code{hcyclic}
3733: @findex katsura
3734: @findex hkatsura
3735: @findex cyclic
3736: @findex hcyclic
3737:
3738: @table @t
3739: @item katsura(@var{n})
3740: @item hkatsura(@var{n})
3741: @item cyclic(@var{n})
3742: @item hcyclic(@var{n})
1.2 noro 3743: \JP :: $BB?9`<0%j%9%H$N@8@.(B
3744: \EG :: Generates a polynomial list of standard benchmark.
1.1 noro 3745: @end table
3746:
3747: @table @var
3748: @item return
1.2 noro 3749: \JP $B%j%9%H(B
3750: \EG list
1.1 noro 3751: @item n
1.2 noro 3752: \JP $B@0?t(B
3753: \EG integer
1.1 noro 3754: @end table
3755:
3756: @itemize @bullet
1.2 noro 3757: \BJP
1.1 noro 3758: @item
3759: @code{katsura()} $B$O(B @samp{katsura}, @code{cyclic()} $B$O(B @samp{cyclic}
3760: $B$GDj5A$5$l$F$$$k(B.
3761: @item
3762: $B%0%l%V%J4pDl7W;;$G$7$P$7$P%F%9%H(B, $B%Y%s%A%^!<%/$KMQ$$$i$l$k(B @code{katsura},
3763: @code{cyclic} $B$*$h$S$=$N@F<!2=$r@8@.$9$k(B.
3764: @item
3765: @code{cyclic} $B$O(B @code{Arnborg}, @code{Lazard}, @code{Davenport} $B$J$I$N(B
3766: $BL>$G8F$P$l$k$3$H$b$"$k(B.
1.2 noro 3767: \E
3768: \BEG
3769: @item
3770: Function @code{katsura()} is defined in @samp{katsura}, and
3771: function @code{cyclic()} in @samp{cyclic}.
3772: @item
3773: These functions generate a series of polynomial sets, respectively,
3774: which are often used for testing and bench marking:
3775: @code{katsura}, @code{cyclic} and their homogenized versions.
3776: @item
3777: Polynomial set @code{cyclic} is sometimes called by other name:
3778: @code{Arnborg}, @code{Lazard}, and @code{Davenport}.
3779: \E
1.1 noro 3780: @end itemize
3781:
3782: @example
3783: [74] load("katsura")$
3784: [79] load("cyclic")$
3785: [89] katsura(5);
3786: [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 3787: 2*u3*u0+2*u1*u4-u3+(2*u1+2*u5)*u2,2*u2*u0+2*u2*u4+(2*u1+2*u5)*u3
3788: -u2+u1^2,2*u1*u0+(2*u3+2*u5)*u4+2*u2*u3+2*u1*u2-u1,
1.1 noro 3789: u0^2-u0+2*u4^2+2*u3^2+2*u2^2+2*u1^2+2*u5^2]
3790: [90] hkatsura(5);
3791: [-t+u0+2*u4+2*u3+2*u2+2*u1+2*u5,
3792: -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,
3793: -u2*t+2*u2*u0+2*u2*u4+(2*u1+2*u5)*u3+u1^2,
3794: -u1*t+2*u1*u0+(2*u3+2*u5)*u4+2*u2*u3+2*u1*u2,
3795: -u0*t+u0^2+2*u4^2+2*u3^2+2*u2^2+2*u1^2+2*u5^2]
3796: [91] cyclic(6);
3797: [c5*c4*c3*c2*c1*c0-1,
3798: ((((c4+c5)*c3+c5*c4)*c2+c5*c4*c3)*c1+c5*c4*c3*c2)*c0+c5*c4*c3*c2*c1,
3799: (((c3+c5)*c2+c5*c4)*c1+c5*c4*c3)*c0+c4*c3*c2*c1+c5*c4*c3*c2,
3800: ((c2+c5)*c1+c5*c4)*c0+c3*c2*c1+c4*c3*c2+c5*c4*c3,
3801: (c1+c5)*c0+c2*c1+c3*c2+c4*c3+c5*c4,c0+c1+c2+c3+c4+c5]
3802: [92] hcyclic(6);
3803: [-c^6+c5*c4*c3*c2*c1*c0,
3804: ((((c4+c5)*c3+c5*c4)*c2+c5*c4*c3)*c1+c5*c4*c3*c2)*c0+c5*c4*c3*c2*c1,
3805: (((c3+c5)*c2+c5*c4)*c1+c5*c4*c3)*c0+c4*c3*c2*c1+c5*c4*c3*c2,
3806: ((c2+c5)*c1+c5*c4)*c0+c3*c2*c1+c4*c3*c2+c5*c4*c3,
3807: (c1+c5)*c0+c2*c1+c3*c2+c4*c3+c5*c4,c0+c1+c2+c3+c4+c5]
3808: @end example
3809:
3810: @table @t
1.2 noro 3811: \JP @item $B;2>H(B
3812: \EG @item References
1.1 noro 3813: @fref{dp_dtop}.
3814: @end table
3815:
1.3 noro 3816: \JP @node primadec primedec,,, $B%0%l%V%J4pDl$K4X$9$kH!?t(B
3817: \EG @node primadec primedec,,, Functions for Groebner basis computation
3818: @subsection @code{primadec}, @code{primedec}
3819: @findex primadec
3820: @findex primedec
3821:
3822: @table @t
3823: @item primadec(@var{plist},@var{vlist})
3824: @item primedec(@var{plist},@var{vlist})
3825: \JP :: $B%$%G%"%k$NJ,2r(B
3826: \EG :: Computes decompositions of ideals.
3827: @end table
3828:
3829: @table @var
3830: @item return
3831: @itemx plist
3832: \JP $BB?9`<0%j%9%H(B
3833: \EG list of polynomials
3834: @item vlist
3835: \JP $BJQ?t%j%9%H(B
3836: \EG list of variables
3837: @end table
3838:
3839: @itemize @bullet
3840: \BJP
3841: @item
3842: @code{primadec()}, @code{primedec} $B$O(B @samp{primdec} $B$GDj5A$5$l$F$$$k(B.
3843: @item
3844: @code{primadec()}, @code{primedec()} $B$O$=$l$>$lM-M}?tBN>e$G$N%$%G%"%k$N(B
3845: $B=`AGJ,2r(B, $B:,4p$NAG%$%G%"%kJ,2r$r9T$&(B.
3846: @item
3847: $B0z?t$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.
3848: @item
3849: @code{primadec} $B$O(B @code{[$B=`AG@.J,(B, $BIUB0AG%$%G%"%k(B]} $B$N%j%9%H$rJV$9(B.
3850: @item
3851: @code{primadec} $B$O(B $BAG0x;R$N%j%9%H$rJV$9(B.
3852: @item
3853: $B7k2L$K$*$$$F(B, $BB?9`<0%j%9%H$H$7$FI=<($5$l$F$$$k3F%$%G%"%k$OA4$F(B
3854: $B%0%l%V%J4pDl$G$"$k(B. $BBP1~$9$k9`=g=x$O(B, $B$=$l$>$l(B
3855: $BJQ?t(B @code{PRIMAORD}, @code{PRIMEORD} $B$K3JG<$5$l$F$$$k(B.
3856: @item
3857: @code{primadec} $B$O(B @code{[Shimoyama,Yokoyama]} $B$N=`AGJ,2r%"%k%4%j%:%`(B
3858: $B$r<BAu$7$F$$$k(B.
3859: @item
3860: $B$b$7AG0x;R$N$_$r5a$a$?$$$J$i(B, @code{primedec} $B$r;H$&J}$,$h$$(B.
3861: $B$3$l$O(B, $BF~NO%$%G%"%k$,:,4p%$%G%"%k$G$J$$>l9g$K(B, @code{primadec}
3862: $B$N7W;;$KM>J,$J%3%9%H$,I,MW$H$J$k>l9g$,$"$k$+$i$G$"$k(B.
3863: \E
3864: \BEG
3865: @item
3866: Function @code{primadec()} and @code{primedec} are defined in @samp{primdec}.
3867: @item
3868: @code{primadec()}, @code{primedec()} are the function for primary
3869: ideal decomposition and prime decomposition of the radical over the
3870: rationals respectively.
3871: @item
3872: The arguments are a list of polynomials and a list of variables.
3873: These functions accept ideals with rational function coefficients only.
3874: @item
3875: @code{primadec} returns the list of pair lists consisting a primary component
3876: and its associated prime.
3877: @item
3878: @code{primedec} returns the list of prime components.
3879: @item
3880: Each component is a Groebner basis and the corresponding term order
3881: is indicated by the global variables @code{PRIMAORD}, @code{PRIMEORD}
3882: respectively.
3883: @item
3884: @code{primadec} implements the primary decompostion algorithm
3885: in @code{[Shimoyama,Yokoyama]}.
3886: @item
3887: If one only wants to know the prime components of an ideal, then
3888: use @code{primedec} because @code{primadec} may need additional costs
3889: if an input ideal is not radical.
3890: \E
3891: @end itemize
3892:
3893: @example
3894: [84] load("primdec")$
3895: [102] primedec([p*q*x-q^2*y^2+q^2*y,-p^2*x^2+p^2*x+p*q*y,
3896: (q^3*y^4-2*q^3*y^3+q^3*y^2)*x-q^3*y^4+q^3*y^3,
3897: -q^3*y^4+2*q^3*y^3+(-q^3+p*q^2)*y^2],[p,q,x,y]);
3898: [[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]]
3899: [103] primadec([x,z*y,w*y^2,w^2*y-z^3,y^3],[x,y,z,w]);
3900: [[[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]]]
3901: @end example
3902:
3903: @table @t
3904: \JP @item $B;2>H(B
3905: \EG @item References
3906: @fref{fctr sqfr},
3907: \JP @fref{$B9`=g=x$N@_Dj(B}.
3908: \EG @fref{Setting term orderings}.
3909: @end table
1.5 noro 3910:
3911: \JP @node primedec_mod,,, $B%0%l%V%J4pDl$K4X$9$kH!?t(B
3912: \EG @node primedec_mod,,, Functions for Groebner basis computation
3913: @subsection @code{primedec_mod}
3914: @findex primedec_mod
3915:
3916: @table @t
3917: @item primedec_mod(@var{plist},@var{vlist},@var{ord},@var{mod},@var{strategy})
3918: \JP :: $B%$%G%"%k$NJ,2r(B
3919: \EG :: Computes decompositions of ideals over small finite fields.
3920: @end table
3921:
3922: @table @var
3923: @item return
3924: @itemx plist
3925: \JP $BB?9`<0%j%9%H(B
3926: \EG list of polynomials
3927: @item vlist
3928: \JP $BJQ?t%j%9%H(B
3929: \EG list of variables
3930: @item ord
3931: \JP $B?t(B, $B%j%9%H$^$?$O9TNs(B
3932: \EG number, list or matrix
3933: @item mod
3934: \JP $B@5@0?t(B
3935: \EG positive integer
3936: @item strategy
3937: \JP $B@0?t(B
3938: \EG integer
3939: @end table
3940:
3941: @itemize @bullet
3942: \BJP
3943: @item
3944: @code{primedec_mod()} $B$O(B @samp{primdec_mod}
3945: $B$GDj5A$5$l$F$$$k(B. @code{[Yokoyama]} $B$NAG%$%G%"%kJ,2r%"%k%4%j%:%`(B
3946: $B$r<BAu$7$F$$$k(B.
3947: @item
3948: @code{primedec_mod()} $B$OM-8BBN>e$G$N%$%G%"%k$N(B
3949: $B:,4p$NAG%$%G%"%kJ,2r$r9T$$(B, $BAG%$%G%"%k$N%j%9%H$rJV$9(B.
3950: @item
3951: @code{primedec_mod()} $B$O(B, GF(@var{mod}) $B>e$G$NJ,2r$rM?$($k(B.
3952: $B7k2L$N3F@.J,$N@8@.85$O(B, $B@0?t78?tB?9`<0$G$"$k(B.
3953: @item
3954: $B7k2L$K$*$$$F(B, $BB?9`<0%j%9%H$H$7$FI=<($5$l$F$$$k3F%$%G%"%k$OA4$F(B
3955: [@var{vlist},@var{ord}] $B$G;XDj$5$l$k9`=g=x$K4X$9$k%0%l%V%J4pDl$G$"$k(B.
3956: @item
3957: @var{strategy} $B$,(B 0 $B$G$J$$$H$-(B, incremental $B$K(B component $B$N6&DL(B
3958: $BItJ,$r7W;;$9$k$3$H$K$h$k(B early termination $B$r9T$&(B. $B0lHL$K(B,
3959: $B%$%G%"%k$N<!85$,9b$$>l9g$KM-8z$@$,(B, 0 $B<!85$N>l9g$J$I(B, $B<!85$,>.$5$$(B
3960: $B>l9g$K$O(B overhead $B$,Bg$-$$>l9g$,$"$k(B.
1.7 noro 3961: @item
3962: $B7W;;ESCf$GFbIt>pJs$r8+$?$$>l9g$K$O!"(B
3963: $BA0$b$C$F(B @code{dp_gr_print(2)} $B$r<B9T$7$F$*$1$P$h$$(B.
1.5 noro 3964: \E
3965: \BEG
3966: @item
3967: Function @code{primedec_mod()}
3968: is defined in @samp{primdec_mod} and implements the prime decomposition
3969: algorithm in @code{[Yokoyama]}.
3970: @item
3971: @code{primedec_mod()}
3972: is the function for prime ideal decomposition
3973: of the radical of a polynomial ideal over small finite field,
3974: and they return a list of prime ideals, which are associated primes
3975: of the input ideal.
3976: @item
3977: @code{primedec_mod()} gives the decomposition over GF(@var{mod}).
3978: The generators of each resulting component consists of integral polynomials.
3979: @item
3980: Each resulting component is a Groebner basis with respect to
3981: a term order specified by [@var{vlist},@var{ord}].
3982: @item
3983: If @var{strategy} is non zero, then the early termination strategy
3984: is tried by computing the intersection of obtained components
3985: incrementally. In general, this strategy is useful when the krull
3986: dimension of the ideal is high, but it may add some overhead
3987: if the dimension is small.
1.7 noro 3988: @item
3989: If you want to see internal information during the computation,
3990: execute @code{dp_gr_print(2)} in advance.
1.5 noro 3991: \E
3992: @end itemize
3993:
3994: @example
3995: [0] load("primdec_mod")$
3996: [246] PP444=[x^8+x^2+t,y^8+y^2+t,z^8+z^2+t]$
3997: [247] primedec_mod(PP444,[x,y,z,t],0,2,1);
3998: [[y+z,x+z,z^8+z^2+t],[x+y,y^2+y+z^2+z+1,z^8+z^2+t],
3999: [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],
4000: [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],
4001: [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],
4002: [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]]
4003: [248]
4004: @end example
4005:
4006: @table @t
4007: \JP @item $B;2>H(B
4008: \EG @item References
4009: @fref{modfctr},
1.6 noro 4010: @fref{dp_gr_main dp_gr_mod_main dp_gr_f_main dp_weyl_gr_main dp_weyl_gr_mod_main dp_weyl_gr_f_main},
1.5 noro 4011: \JP @fref{$B9`=g=x$N@_Dj(B}.
1.7 noro 4012: \EG @fref{Setting term orderings},
4013: @fref{dp_gr_flags dp_gr_print}.
1.5 noro 4014: @end table
4015:
1.10 noro 4016: \JP @node bfunction bfct generic_bfct ann ann0,,, $B%0%l%V%J4pDl$K4X$9$kH!?t(B
4017: \EG @node bfunction bfct generic_bfct ann ann0,,, Functions for Groebner basis computation
4018: @subsection @code{bfunction}, @code{bfct}, @code{generic_bfct}, @code{ann}, @code{ann0}
1.6 noro 4019: @findex bfunction
1.9 noro 4020: @findex bfct
1.6 noro 4021: @findex generic_bfct
1.10 noro 4022: @findex ann
4023: @findex ann0
1.5 noro 4024:
1.6 noro 4025: @table @t
4026: @item bfunction(@var{f})
1.10 noro 4027: @itemx bfct(@var{f})
4028: @itemx generic_bfct(@var{plist},@var{vlist},@var{dvlist},@var{weight})
4029: \JP :: @var{b} $B4X?t$N7W;;(B
4030: \EG :: Computes the global @var{b} function of a polynomial or an ideal
4031: @item ann(@var{f})
4032: @itemx ann0(@var{f})
4033: \JP :: $BB?9`<0$N%Y%-$N(B annihilator $B$N7W;;(B
4034: \EG :: Computes the annihilator of a power of polynomial
1.6 noro 4035: @end table
1.10 noro 4036:
1.6 noro 4037: @table @var
4038: @item return
1.10 noro 4039: \JP $BB?9`<0$^$?$O%j%9%H(B
4040: \EG polynomial or list
4041: @item f
1.6 noro 4042: \JP $BB?9`<0(B
4043: \EG polynomial
4044: @item plist
4045: \JP $BB?9`<0%j%9%H(B
4046: \EG list of polynomials
4047: @item vlist dvlist
4048: \JP $BJQ?t%j%9%H(B
4049: \EG list of variables
4050: @end table
1.5 noro 4051:
1.6 noro 4052: @itemize @bullet
4053: \BJP
4054: @item @samp{bfct} $B$GDj5A$5$l$F$$$k(B.
1.10 noro 4055: @item @code{bfunction(@var{f})}, @code{bfct(@var{f})} $B$OB?9`<0(B @var{f} $B$N(B global @var{b} $B4X?t(B @code{b(s)} $B$r(B
1.6 noro 4056: $B7W;;$9$k(B. @code{b(s)} $B$O(B, Weyl $BBe?t(B @code{D} $B>e$N0lJQ?tB?9`<04D(B @code{D[s]}
4057: $B$N85(B @code{P(x,s)} $B$,B8:_$7$F(B, @code{P(x,s)f^(s+1)=b(s)f^s} $B$rK~$?$9$h$&$J(B
4058: $BB?9`<0(B @code{b(s)} $B$NCf$G(B, $B<!?t$,:G$bDc$$$b$N$G$"$k(B.
4059: @item @code{generic_bfct(@var{f},@var{vlist},@var{dvlist},@var{weight})}
4060: $B$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 4061: $B%&%'%$%H(B @var{weight} $B$K4X$9$k(B global @var{b} $B4X?t$r7W;;$9$k(B.
1.6 noro 4062: @var{vlist} $B$O(B @code{x}-$BJQ?t(B, @var{vlist} $B$OBP1~$9$k(B @code{D}-$BJQ?t(B
4063: $B$r=g$KJB$Y$k(B.
1.9 noro 4064: @item @code{bfunction} $B$H(B @code{bfct} $B$G$OMQ$$$F$$$k%"%k%4%j%:%`$,(B
1.11 noro 4065: $B0[$J$k(B. $B$I$A$i$,9bB.$+$OF~NO$K$h$k(B.
1.10 noro 4066: @item @code{ann(@var{f})} $B$O(B, @code{@var{f}^s} $B$N(B annihilator ideal
4067: $B$N@8@.7O$rJV$9(B. @code{ann(@var{f})} $B$O(B, @code{[@var{a},@var{list}]}
4068: $B$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,
4069: @var{list} $B$O(B @code{ann(@var{f})} $B$N7k2L$N(B @code{s}$ $B$K(B, @var{a} $B$r(B
4070: $BBeF~$7$?$b$N$G$"$k(B.
1.7 noro 4071: @item $B>\:Y$K$D$$$F$O(B, [Saito,Sturmfels,Takayama] $B$r8+$h(B.
1.6 noro 4072: \E
4073: \BEG
4074: @item These functions are defined in @samp{bfct}.
1.10 noro 4075: @item @code{bfunction(@var{f})} and @code{bfct(@var{f})} compute the global @var{b}-function @code{b(s)} of
1.6 noro 4076: a polynomial @var{f}.
4077: @code{b(s)} is a polynomial of the minimal degree
4078: such that there exists @code{P(x,s)} in D[s], which is a polynomial
4079: ring over Weyl algebra @code{D}, and @code{P(x,s)f^(s+1)=b(s)f^s} holds.
4080: @item @code{generic_bfct(@var{f},@var{vlist},@var{dvlist},@var{weight})}
1.10 noro 4081: computes the global @var{b}-function of a left ideal @code{I} in @code{D}
1.6 noro 4082: generated by @var{plist}, with respect to @var{weight}.
4083: @var{vlist} is the list of @code{x}-variables,
4084: @var{vlist} is the list of corresponding @code{D}-variables.
1.9 noro 4085: @item @code{bfunction(@var{f})} and @code{bfct(@var{f})} implement
4086: different algorithms and the efficiency depends on inputs.
1.10 noro 4087: @item @code{ann(@var{f})} returns the generator set of the annihilator
4088: ideal of @code{@var{f}^s}.
4089: @code{ann(@var{f})} returns a list @code{[@var{a},@var{list}]},
4090: where @var{a} is the minimal integral root of the global @var{b}-function
4091: of @var{f}, and @var{list} is a list of polynomials obtained by
4092: substituting @code{s} in @code{ann(@var{f})} with @var{a}.
1.7 noro 4093: @item See [Saito,Sturmfels,Takayama] for the details.
1.6 noro 4094: \E
4095: @end itemize
4096:
4097: @example
4098: [0] load("bfct")$
4099: [216] bfunction(x^3+y^3+z^3+x^2*y^2*z^2+x*y*z);
4100: -9*s^5-63*s^4-173*s^3-233*s^2-154*s-40
4101: [217] fctr(@@);
4102: [[-1,1],[s+2,1],[3*s+4,1],[3*s+5,1],[s+1,2]]
4103: [218] F = [4*x^3*dt+y*z*dt+dx,x*z*dt+4*y^3*dt+dy,
4104: x*y*dt+5*z^4*dt+dz,-x^4-z*y*x-y^4-z^5+t]$
4105: [219] generic_bfct(F,[t,z,y,x],[dt,dz,dy,dx],[1,0,0,0]);
4106: 20000*s^10-70000*s^9+101750*s^8-79375*s^7+35768*s^6-9277*s^5
4107: +1278*s^4-72*s^3
1.10 noro 4108: [220] P=x^3-y^2$
4109: [221] ann(P);
4110: [2*dy*x+3*dx*y^2,-3*dx*x-2*dy*y+6*s]
4111: [222] ann0(P);
4112: [-1,[2*dy*x+3*dx*y^2,-3*dx*x-2*dy*y-6]]
1.6 noro 4113: @end example
4114:
4115: @table @t
4116: \JP @item $B;2>H(B
4117: \EG @item References
4118: \JP @fref{Weyl $BBe?t(B}.
4119: \EG @fref{Weyl algebra}.
4120: @end table
1.5 noro 4121:
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