Annotation of OpenXM/src/asir-doc/parts/groebner.texi, Revision 1.4
1.4 ! noro 1: @comment $OpenXM: OpenXM/src/asir-doc/parts/groebner.texi,v 1.3 1999/12/24 04:38:04 noro Exp $
1.2 noro 2: \BJP
1.1 noro 3: @node $B%0%l%V%J4pDl$N7W;;(B,,, Top
4: @chapter $B%0%l%V%J4pDl$N7W;;(B
1.2 noro 5: \E
6: \BEG
7: @node Groebner basis computation,,, Top
8: @chapter Groebner basis computation
9: \E
1.1 noro 10:
11: @menu
1.2 noro 12: \BJP
1.1 noro 13: * $BJ,;6I=8=B?9`<0(B::
14: * $B%U%!%$%k$NFI$_9~$_(B::
15: * $B4pK\E*$JH!?t(B::
16: * $B7W;;$*$h$SI=<($N@)8f(B::
17: * $B9`=g=x$N@_Dj(B::
18: * $BM-M}<0$r78?t$H$9$k%0%l%V%J4pDl7W;;(B::
19: * $B4pDlJQ49(B::
20: * $B%0%l%V%J4pDl$K4X$9$kH!?t(B::
1.2 noro 21: \E
22: \BEG
23: * Distributed polynomial::
24: * Reading files::
25: * Fundamental functions::
26: * Controlling Groebner basis computations::
27: * Setting term orderings::
28: * Groebner basis computation with rational function coefficients::
29: * Change of ordering::
30: * Functions for Groebner basis computation::
31: \E
1.1 noro 32: @end menu
33:
1.2 noro 34: \BJP
1.1 noro 35: @node $BJ,;6I=8=B?9`<0(B,,, $B%0%l%V%J4pDl$N7W;;(B
36: @section $BJ,;6I=8=B?9`<0(B
1.2 noro 37: \E
38: \BEG
39: @node Distributed polynomial,,, Groebner basis computation
40: @section Distributed polynomial
41: \E
1.1 noro 42:
43: @noindent
1.2 noro 44: \BJP
1.1 noro 45: $BJ,;6I=8=B?9`<0$H$O(B, $BB?9`<0$NFbIt7A<0$N0l$D$G$"$k(B. $BDL>o$NB?9`<0(B
46: (@code{type} $B$,(B 2) $B$O(B, $B:F5"I=8=$H8F$P$l$k7A<0$GI=8=$5$l$F$$$k(B. $B$9$J$o(B
47: $B$A(B, $BFCDj$NJQ?t$r<gJQ?t$H$9$k(B 1 $BJQ?tB?9`<0$G(B, $B$=$NB>$NJQ?t$O(B, $B$=$N(B 1 $BJQ(B
48: $B?tB?9`<0$N78?t$K(B, $B<gJQ?t$r4^$^$J$$B?9`<0$H$7$F8=$l$k(B. $B$3$N78?t$,(B, $B$^$?(B,
49: $B$"$kJQ?t$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 50: \E
51: \BEG
52: A distributed polynomial is a polynomial with a special internal
53: representation different from the ordinary one.
54:
55: An ordinary polynomial (having @code{type} 2) is internally represented
56: in a format, called recursive representation.
57: In fact, it is represented as an uni-variate polynomial with respect to
58: a fixed variable, called main variable of that polynomial,
59: where the other variables appear in the coefficients which may again
60: polynomials in such variables other than the previous main variable.
61: A polynomial in the coefficients is again represented as
62: an uni-variate polynomial in a certain fixed variable,
63: the main variable. Thus, by this recursive structure of polynomial
64: representation, it is called the `recursive representation.'
65: \E
1.1 noro 66:
67: @iftex
68: @tex
1.2 noro 69: \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 ))$
70: \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 71: @end tex
72: @end iftex
73: @ifinfo
74: @example
75: (x+y+z)^2 = 1 x^2 + (2 y + (2 z)) x + ((2 z) y + (1 z^2 ))
76: @end example
77: @end ifinfo
78:
79: @noindent
1.2 noro 80: \BJP
1.1 noro 81: $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
82: $BI=8=$H8F$V(B.
1.2 noro 83: \E
84: \BEG
85: On the other hand,
86: we call a representation the distributed representation of a polynomial,
87: if a polynomial is represented, according to its original meaning,
88: as a sum of monomials,
89: where a monomial is the product of power product of variables
90: and a coefficient. We call a polynomial, represented in such an
91: internal format, a distributed polynomial. (This naming may sounds
92: something strange.)
93: \E
1.1 noro 94:
95: @iftex
96: @tex
1.2 noro 97: \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$
98: \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 99: @end tex
100: @end iftex
101: @ifinfo
102: @example
103: (x+y+z)^2 = 1 x^2 + 2 xy + 2 xz + 1 y^2 + 2 yz +1 z^2$
104: @end example
105: @end ifinfo
106:
107: @noindent
1.2 noro 108: \BJP
1.1 noro 109: $B%0%l%V%J4pDl7W;;$K$*$$$F$O(B, $BC19`<0$KCmL\$7$FA`:n$r9T$&$?$aB?9`<0$,J,;6I=8=(B
110: $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,
111: $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$$$k(B.
112: $B$3$3$G(B, $B8e$N@bL@$N$?$a$K(B, $B$$$/$D$+$N8@MU$rDj5A$7$F$*$/(B.
1.2 noro 113: \E
114: \BEG
115: For computation of Groebner basis, efficient operation is expected if
116: polynomials are represented in a distributed representation,
117: because major operations for Groebner basis are performed with respect
118: to monomials.
119: From this view point, we provide the object type distributed polynomial
120: with its object identification number 9, and objects having such a type
121: are available by @b{Asir} language.
122:
123: Here, we provide several definitions for the later description.
124: \E
1.1 noro 125:
126: @table @b
1.2 noro 127: \BJP
1.1 noro 128: @item $B9`(B (term)
129: $BJQ?t$NQQ@Q(B. $B$9$J$o$A(B, $B78?t(B 1 $B$NC19`<0$N$3$H(B. @b{Asir} $B$K$*$$$F$O(B,
1.2 noro 130: \E
131: \BEG
132: @item term
133: The power product of variables, i.e., a monomial with coefficient 1.
134: In an @b{Asir} session, it is displayed in the form like
135: \E
1.1 noro 136:
137: @example
138: <<0,1,2,3,4>>
139: @end example
140:
1.2 noro 141: \BJP
1.1 noro 142: $B$H$$$&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
143: $B$r<($9(B. $B3FJQ?t$r(B @code{a}, @code{b}, @code{c}, @code{d}, @code{e} $B$H$9$k$H(B
144: $B$3$N9`$O(B @code{b*c^2*d^3*e^4} $B$rI=$9(B.
1.2 noro 145: \E
146: \BEG
147: and also can be input in such a form.
148: This example shows a term in 5 variables. If we assume the 5 variables
149: as @code{a}, @code{b}, @code{c}, @code{d}, and @code{e},
150: the term represents @code{b*c^2*d^3*e^4} in the ordinary expression.
151: \E
1.1 noro 152:
1.2 noro 153: \BJP
1.1 noro 154: @item $B9`=g=x(B (term order)
155: $BJ,;6I=8=B?9`<0$K$*$1$k9`$O(B, $B<!$N@-<A$rK~$?$9A4=g=x$K$h$j@0Ns$5$l$k(B.
1.2 noro 156: \E
157: \BEG
158: @item term order
159: Terms are ordered according to a total order with the following properties.
160: \E
1.1 noro 161:
162: @enumerate
163: @item
1.2 noro 164: \JP $BG$0U$N9`(B @code{t} $B$KBP$7(B @code{t} > 1
165: \EG For all @code{t} @code{t} > 1.
1.1 noro 166:
167: @item
1.2 noro 168: \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}
169: \EG For all @code{t}, @code{s}, @code{u} @code{t} > @code{s} implies @code{tu} > @code{su}.
1.1 noro 170: @end enumerate
171:
1.2 noro 172: \BJP
1.1 noro 173: $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)
174: $B$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 175: \E
176: \BEG
177: Such a total order is called a term ordering. A term ordering is specified
178: by a variable ordering (a list of variables) and a type of term ordering
179: (an integer, a list or a matrix).
180: \E
1.1 noro 181:
1.2 noro 182: \BJP
1.1 noro 183: @item $BC19`<0(B (monomial)
184: $B9`$H78?t$N@Q(B.
1.2 noro 185: \E
186: \BEG
187: @item monomial
188: The product of a term and a coefficient.
189: In an @b{Asir} session, it is displayed in the form like
190: \E
1.1 noro 191:
192: @example
193: 2*<<0,1,2,3,4>>
194: @end example
195:
1.2 noro 196: \JP $B$H$$$&7A$GI=<($5$l(B, $B$^$?(B, $B$3$N7A$GF~NO2DG=$G$"$k(B.
197: \EG and also can be input in such a form.
1.1 noro 198:
1.2 noro 199: \BJP
1.1 noro 200: @itemx $BF,C19`<0(B (head monomial)
201: @item $BF,9`(B (head term)
202: @itemx $BF,78?t(B (head coefficient)
203: $BJ,;6I=8=B?9`<0$K$*$1$k3FC19`<0$O(B, $B9`=g=x$K$h$j@0Ns$5$l$k(B. $B$3$N;~=g(B
204: $B=x:GBg$NC19`<0$rF,C19`<0(B, $B$=$l$K8=$l$k9`(B, $B78?t$r$=$l$>$lF,9`(B, $BF,78?t(B
205: $B$H8F$V(B.
1.2 noro 206: \E
207: \BEG
208: @itemx head monomial
209: @item head term
210: @itemx head coefficient
211:
212: Monomials in a distributed polynomial is sorted by a total order.
213: In such representation, we call the monomial that is maximum
214: with respect to the order the head monomial, and its term and coefficient
215: the head term and the head coefficient respectively.
216: \E
1.1 noro 217: @end table
218:
1.2 noro 219: \BJP
1.1 noro 220: @node $B%U%!%$%k$NFI$_9~$_(B,,, $B%0%l%V%J4pDl$N7W;;(B
221: @section $B%U%!%$%k$NFI$_9~$_(B
1.2 noro 222: \E
223: \BEG
224: @node Reading files,,, Groebner basis computation
225: @section Reading files
226: \E
1.1 noro 227:
228: @noindent
1.2 noro 229: \BJP
1.1 noro 230: $B%0%l%V%J4pDl$r7W;;$9$k$?$a$N4pK\E*$JH!?t$O(B @code{dp_gr_main()} $B$*$h$S(B
231: @code{dp_gr_mod_main()} $B$J$k(B 2 $B$D$NAH$_9~$_H!?t$G$"$k$,(B, $BDL>o$O(B, $B%Q%i%a%?(B
232: $B@_Dj$J$I$r9T$C$?$N$A$3$l$i$r8F$S=P$9%f!<%6H!?t$rMQ$$$k$N$,JXMx$G$"$k(B.
233: $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
234: $B$_9~$`$3$H$K$h$j;HMQ2DG=$H$J$k(B. @samp{gr} $B$O(B, @b{Asir} $B$NI8=`(B
235: $B%i%$%V%i%j%G%#%l%/%H%j$KCV$+$l$F$$$k(B. $B$h$C$F(B, $B4D6-JQ?t(B @code{ASIR_LIBDIR}
236: $B$rFC$K0[$J$k%Q%9$K@_Dj$7$J$$8B$j(B, $B%U%!%$%kL>$N$_$GFI$_9~$`$3$H$,$G$-$k(B.
1.2 noro 237: \E
238: \BEG
239: Facilities for computing Groebner bases are provided not by built-in
240: functions but by a set of user functions written in @b{Asir}.
241: The set of functions is provided as a file (sometimes called package),
242: named @samp{gr}.
243: The facilities will be ready to use after you load the package by
244: @code{load()}. The package @samp{gr} is placed in the standard library
245: directory of @b{Asir}. Therefore, it is loaded simply by specifying
246: its file name, unless the environment variable @code{ASIR_LIBDIR}
247: is set to a non-standard one.
248: \E
1.1 noro 249:
250: @example
251: [0] load("gr")$
252: @end example
253:
1.2 noro 254: \BJP
1.1 noro 255: @node $B4pK\E*$JH!?t(B,,, $B%0%l%V%J4pDl$N7W;;(B
256: @section $B4pK\E*$JH!?t(B
1.2 noro 257: \E
258: \BEG
259: @node Fundamental functions,,, Groebner basis computation
260: @section Fundamental functions
261: \E
1.1 noro 262:
263: @noindent
1.2 noro 264: \BJP
1.1 noro 265: @samp{gr} $B$G$O?tB?$/$NH!?t$,Dj5A$5$l$F$$$k$,(B, $BD>@\(B
266: $B%0%l%V%J4pDl$r7W;;$9$k$?$a$N%H%C%W%l%Y%k$O<!$N(B 3 $B$D$G$"$k(B.
267: $B0J2<$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,
268: @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 269: \E
270: \BEG
271: There are many functions and options defined in the package @samp{gr}.
272: Usually not so many of them are used. Top level functions for Groebner
273: basis computation are the following three functions.
274:
275: In the following description, @var{plist}, @var{vlist}, @var{order}
276: and @var{p} stand for a list of polynomials, a list of variables
277: (indeterminates), a type of term ordering and a prime less than
278: @code{2^27} respectively.
279: \E
1.1 noro 280:
281: @table @code
282: @item gr(@var{plist},@var{vlist},@var{order})
283:
1.2 noro 284: \BJP
1.1 noro 285: Gebauer-Moeller $B$K$h$k(B useless pair elimination criteria, sugar
286: strategy $B$*$h$S(B Traverso $B$K$h$k(B trace-lifting $B$rMQ$$$?(B Buchberger $B%"%k(B
287: $B%4%j%:%`$K$h$kM-M}?t78?t%0%l%V%J4pDl7W;;H!?t(B. $B0lHL$K$O$3$NH!?t$rMQ$$$k(B.
1.2 noro 288: \E
289: \BEG
290: Function that computes Groebner bases over the rationals. The
291: algorithm is Buchberger algorithm with useless pair elimination
292: criteria by Gebauer-Moeller, sugar strategy and trace-lifting by
293: Traverso. For ordinary computation, this function is used.
294: \E
1.1 noro 295:
296: @item hgr(@var{plist},@var{vlist},@var{order})
297:
1.2 noro 298: \BJP
1.1 noro 299: $BF~NOB?9`<0$r@F<!2=$7$?8e(B @code{gr()} $B$N%0%l%V%J4pDl8uJd@8@.It$K$h$j8u(B
300: $BJd@8@.$7(B, $BHs@F<!2=(B, interreduce $B$7$?$b$N$r(B @code{gr()} $B$N%0%l%V%J4pDl(B
301: $B%A%'%C%/It$G%A%'%C%/$9$k(B. 0 $B<!85%7%9%F%`(B ($B2r$N8D?t$,M-8B8D$NJ}Dx<07O(B)
302: $B$N>l9g(B, sugar strategy $B$,78?tKDD%$r0z$-5/$3$9>l9g$,$"$k(B. $B$3$N$h$&$J>l(B
303: $B9g(B, strategy $B$r@F<!2=$K$h$k(B strategy $B$KCV$-49$($k$3$H$K$h$j78?tKDD%$r(B
304: $BM^@)$9$k$3$H$,$G$-$k>l9g$,B?$$(B.
1.2 noro 305: \E
306: \BEG
307: After homogenizing the input polynomials a candidate of the \gr basis
308: is computed by trace-lifting. Then the candidate is dehomogenized and
309: checked whether it is indeed a Groebner basis of the input. Sugar
310: strategy often causes intermediate coefficient swells. It is
311: empirically known that the combination of homogenization and supresses
312: the swells for such cases.
313: \E
1.1 noro 314:
315: @item gr_mod(@var{plist},@var{vlist},@var{order},@var{p})
316:
1.2 noro 317: \BJP
1.1 noro 318: Gebauer-Moeller $B$K$h$k(B useless pair elimination criteria, sugar
319: strategy $B$*$h$S(B Buchberger $B%"%k%4%j%:%`$K$h$k(B GF(p) $B78?t%0%l%V%J4pDl7W(B
320: $B;;H!?t(B.
1.2 noro 321: \E
322: \BEG
323: Function that computes Groebner bases over GF(@var{p}). The same
324: algorithm as @code{gr()} is used.
325: \E
1.1 noro 326:
327: @end table
328:
1.2 noro 329: \BJP
1.1 noro 330: @node $B7W;;$*$h$SI=<($N@)8f(B,,, $B%0%l%V%J4pDl$N7W;;(B
331: @section $B7W;;$*$h$SI=<($N@)8f(B
1.2 noro 332: \E
333: \BEG
334: @node Controlling Groebner basis computations,,, Groebner basis computation
335: @section Controlling Groebner basis computations
336: \E
1.1 noro 337:
338: @noindent
1.2 noro 339: \BJP
1.1 noro 340: $B%0%l%V%J4pDl$N7W;;$K$*$$$F(B, $B$5$^$6$^$J%Q%i%a%?@_Dj$r9T$&$3$H$K$h$j7W;;(B,
341: $BI=<($r@)8f$9$k$3$H$,$G$-$k(B. $B$3$l$i$O(B, $BAH$_9~$_H!?t(B @code{dp_gr_flags()}
342: $B$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
343: $B$H(B, $B8=:_@_Dj$5$l$F$$$k%Q%i%a%?$,(B, $BL>A0$HCM$N%j%9%H$GJV$5$l$k(B.
1.2 noro 344: \E
345: \BEG
346: One can cotrol a Groebner basis computation by setting various parameters.
347: These parameters can be set and examined by a built-in function
348: @code{dp_gr_flags()}. Without argument it returns the current settings.
349: \E
1.1 noro 350:
351: @example
352: [100] dp_gr_flags();
353: [Demand,0,NoSugar,0,NoCriB,0,NoGC,0,NoMC,0,NoRA,0,NoGCD,0,Top,0,ShowMag,1,
354: Print,1,Stat,0,Reverse,0,InterReduce,0,Multiple,0]
355: [101]
356: @end example
357:
1.2 noro 358: \BJP
1.1 noro 359: $B0J2<$G(B, $B3F%Q%i%a%?$N0UL#$r@bL@$9$k(B. on $B$N>l9g$H$O(B, $B%Q%i%a%?$,(B 0 $B$G$J$$>l9g$r(B
360: $B$$$&(B. $B$3$l$i$N%Q%i%a%?$N=i4|CM$OA4$F(B 0 (off) $B$G$"$k(B.
1.2 noro 361: \E
362: \BEG
363: The return value is a list which contains the names of parameters and their
364: values. The meaning of the parameters are as follows. `on' means that the
365: parameter is not zero.
366: \E
1.1 noro 367:
368: @table @code
369: @item NoSugar
1.2 noro 370: \BJP
1.1 noro 371: on $B$N>l9g(B, sugar strategy $B$NBe$o$j$K(B Buchberger$B$N(B normal strategy $B$,MQ(B
372: $B$$$i$l$k(B.
1.2 noro 373: \E
374: \BEG
375: If `on', Buchberger's normal strategy is used instead of sugar strategy.
376: \E
1.1 noro 377:
378: @item NoCriB
1.2 noro 379: \JP on $B$N>l9g(B, $BITI,MWBP8!=P5,=`$N$&$A(B, $B5,=`(B B $B$rE,MQ$7$J$$(B.
380: \EG If `on', criterion B among the Gebauer-Moeller's criteria is not applied.
1.1 noro 381:
382: @item NoGC
1.2 noro 383: \JP on $B$N>l9g(B, $B7k2L$,%0%l%V%J4pDl$K$J$C$F$$$k$+$I$&$+$N%A%'%C%/$r9T$o$J$$(B.
384: \BEG
385: If `on', the check that a Groebner basis candidate is indeed a Groebner basis,
386: is not executed.
387: \E
1.1 noro 388:
389: @item NoMC
1.2 noro 390: \BJP
1.1 noro 391: on $B$N>l9g(B, $B7k2L$,F~NO%$%G%"%k$HF1Ey$N%$%G%"%k$G$"$k$+$I$&$+$N%A%'%C%/(B
392: $B$r9T$o$J$$(B.
1.2 noro 393: \E
394: \BEG
395: If `on', the check that the resulting polynomials generates the same ideal as
396: the ideal generated by the input, is not executed.
397: \E
1.1 noro 398:
399: @item NoRA
1.2 noro 400: \BJP
1.1 noro 401: on $B$N>l9g(B, $B7k2L$r(B reduced $B%0%l%V%J4pDl$K$9$k$?$a$N(B
402: interreduce $B$r9T$o$J$$(B.
1.2 noro 403: \E
404: \BEG
405: If `on', the interreduction, which makes the Groebner basis reduced, is not
406: executed.
407: \E
1.1 noro 408:
409: @item NoGCD
1.2 noro 410: \BJP
1.1 noro 411: on $B$N>l9g(B, $BM-M}<078?t$N%0%l%V%J4pDl7W;;$K$*$$$F(B, $B@8@.$5$l$?B?9`<0$N(B,
412: $B78?t$N(B content $B$r$H$i$J$$(B.
1.2 noro 413: \E
414: \BEG
415: If `on', content removals are not executed during a Groebner basis computation
416: over a rational function field.
417: \E
1.1 noro 418:
419: @item Top
1.2 noro 420: \JP on $B$N>l9g(B, normal form $B7W;;$K$*$$$FF,9`>C5n$N$_$r9T$&(B.
421: \EG If `on', Only the head term of the polynomial being reduced is reduced.
1.1 noro 422:
1.2 noro 423: @comment @item Interreduce
424: @comment \BJP
425: @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
426: @comment $B$h$k(B normal form $B$GCV$-49$($k(B.
427: @comment \E
428: @comment \BEG
429: @comment If `on', intermediate basis elements are reduced by using a newly generated
430: @comment basis element.
431: @comment \E
1.1 noro 432:
433: @item Reverse
1.2 noro 434: \BJP
1.1 noro 435: 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
436: $B@h$7$FA*$V(B.
1.2 noro 437: \E
438: \BEG
439: If `on', the selection strategy of reducer in a normal form computation
440: is such that a newer reducer is used first.
441: \E
1.1 noro 442:
443: @item Print
1.2 noro 444: \JP on $B$N>l9g(B, $B%0%l%V%J4pDl7W;;$NESCf$K$*$1$k$5$^$6$^$J>pJs$rI=<($9$k(B.
445: \BEG
446: If `on', various informations during a Groebner basis computation is
447: displayed.
448: \E
1.1 noro 449:
450: @item Stat
1.2 noro 451: \BJP
1.1 noro 452: 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
453: $B$l$k%G!<%?$NFb(B, $B=87W%G!<%?$N$_$,I=<($5$l$k(B.
1.2 noro 454: \E
455: \BEG
456: If `on', a summary of informations is shown after a Groebner basis
457: computation. Note that the summary is always shown if @code{Print} is `on'.
458: \E
1.1 noro 459:
460: @item ShowMag
1.2 noro 461: \BJP
1.1 noro 462: 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
463: $B78?t$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 464: \E
465: \BEG
466: If `on' and @code{Print} is `on', the sum of bit length of
467: coefficients of a generated basis element, which we call @var{magnitude},
468: is shown after every normal computation. After comleting the
469: computation the maximal value among the sums is shown.
470: \E
1.1 noro 471:
472: @item Multiple
1.2 noro 473: \BJP
1.1 noro 474: 0 $B$G$J$$@0?t$N;~(B, $BM-M}?t>e$N@55,7A7W;;$K$*$$$F(B, $B78?t$N%S%C%HD9$NOB$,(B
475: @code{Multiple} $BG\$K$J$k$4$H$K78?tA4BN$N(B GCD $B$,7W;;$5$l(B, $B$=$N(B GCD $B$G(B
476: $B3d$C$?B?9`<0$r4JLs$9$k(B. @code{Multiple} $B$,(B 1 $B$J$i$P(B, $B4JLs$9$k$4$H$K(B
477: GCD $B7W;;$,9T$o$l0lHL$K$O8zN($,0-$/$J$k$,(B, @code{Multiple} $B$r(B 2 $BDxEY(B
478: $B$H$9$k$H(B, $B5pBg$J@0?t$,78?t$K8=$l$k>l9g(B, $B8zN($,NI$/$J$k>l9g$,$"$k(B.
1.2 noro 479: \E
480: \BEG
481: If a non-zero integer, in a normal form computation
482: over the rationals, the integer content of the polynomial being
483: reduced is removed when its magnitude becomes @code{Multiple} times
484: larger than a registered value, which is set to the magnitude of the
485: input polynomial. After each content removal the registered value is
486: set to the magnitude of the resulting polynomial. @code{Multiple} is
487: equal to 1, the simiplification is done after every normal form computation.
488: It is empirically known that it is often efficient to set @code{Multiple} to 2
489: for the case where large integers appear during the computation.
490: \E
1.1 noro 491:
492: @item Demand
1.2 noro 493:
494: \BJP
1.1 noro 495: $B@5Ev$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
496: $BCf$K$*$+$l$:(B, $B$=$N%G%#%l%/%H%jCf$K%P%$%J%j%G!<%?$H$7$FCV$+$l(B, $B$=$NB?9`(B
497: $B<0$rMQ$$$k(B normal form $B7W;;$N:](B, $B<+F0E*$K%a%b%jCf$K%m!<%I$5$l$k(B. $B3FB?(B
498: $B9`<0$O(B, $BFbIt$G$N%$%s%G%C%/%9$r%U%!%$%kL>$K;}$D%U%!%$%k$K3JG<$5$l$k(B.
499: $B$3$3$G;XDj$5$l$?%G%#%l%/%H%j$K=q$+$l$?%U%!%$%k$O<+F0E*$K$O>C5n$5$l$J$$(B
500: $B$?$a(B, $B%f!<%6$,@UG$$r;}$C$F>C5n$9$kI,MW$,$"$k(B.
1.2 noro 501: \E
502: \BEG
503: If the value (a character string) is a valid directory name, then
504: generated basis elements are put in the directory and are loaded on
505: demand during normal form computations. Each elements is saved in the
506: binary form and its name coincides with the index internally used in
507: the computation. These binary files are not removed automatically
508: and one should remove them by hand.
509: \E
1.1 noro 510: @end table
511:
512: @noindent
1.2 noro 513: \JP @code{Print} $B$,(B 0 $B$G$J$$>l9g<!$N$h$&$J%G!<%?$,I=<($5$l$k(B.
514: \EG If @code{Print} is `on', the following informations are shown.
1.1 noro 515:
516: @example
517: [93] gr(cyclic(4),[c0,c1,c2,c3],0)$
518: mod= 99999989, eval = []
519: (0)(0)<<0,2,0,0>>(2,3),nb=2,nab=5,rp=2,sugar=2,mag=4
520: (0)(0)<<0,1,2,0>>(1,2),nb=3,nab=6,rp=2,sugar=3,mag=4
521: (0)(0)<<0,1,1,2>>(0,1),nb=4,nab=7,rp=3,sugar=4,mag=6
522: .
523: (0)(0)<<0,0,3,2>>(5,6),nb=5,nab=8,rp=2,sugar=5,mag=4
524: (0)(0)<<0,1,0,4>>(4,6),nb=6,nab=9,rp=3,sugar=5,mag=4
525: (0)(0)<<0,0,2,4>>(6,8),nb=7,nab=10,rp=4,sugar=6,mag=6
526: ....gb done
527: reduceall
528: .......
529: membercheck
530: (0,0)(0,0)(0,0)(0,0)
531: gbcheck total 8 pairs
532: ........
533: UP=(0,0)SP=(0,0)SPM=(0,0)NF=(0,0)NFM=(0.010002,0)ZNFM=(0.010002,0)PZ=(0,0)
534: NP=(0,0)MP=(0,0)RA=(0,0)MC=(0,0)GC=(0,0)T=40,B=0 M=8 F=6 D=12 ZR=5 NZR=6
535: Max_mag=6
536: [94]
537: @end example
538:
539: @noindent
1.2 noro 540: \BJP
1.1 noro 541: $B:G=i$KI=<($5$l$k(B @code{mod}, @code{eval} $B$O(B, trace-lifting $B$GMQ$$$i$l$kK!(B
542: $B$G$"$k(B. @code{mod} $B$OAG?t(B, @code{eval} $B$OM-M}<078?t$N>l9g$KMQ$$$i$l$k(B
543: $B?t$N%j%9%H$G$"$k(B.
1.2 noro 544: \E
545: \BEG
546: In this example @code{mod} and @code{eval} indicate moduli used in
547: trace-lifting. @code{mod} is a prime and @code{eval} is a list of integers
548: used for evaluation when the ground field is a field of rational functions.
549: \E
1.1 noro 550:
551: @noindent
1.2 noro 552: \JP $B7W;;ESCf$GB?9`<0$,@8@.$5$l$kKh$K<!$N7A$N%G!<%?$,I=<($5$l$k(B.
553: \EG The following information is shown after every normal form computation.
1.1 noro 554:
555: @example
556: (TNF)(TCONT)HT(INDEX),nb=NB,nab=NAB,rp=RP,sugar=S,mag=M
557: @end example
558:
559: @noindent
1.2 noro 560: \JP $B$=$l$i$N0UL#$O<!$NDL$j(B.
561: \EG Meaning of each component is as follows.
1.1 noro 562:
563: @table @code
1.2 noro 564: \BJP
1.1 noro 565: @item TNF
1.2 noro 566:
1.1 noro 567: normal form $B7W;;;~4V(B ($BIC(B)
568:
569: @item TCONT
1.2 noro 570:
1.1 noro 571: content $B7W;;;~4V(B ($BIC(B)
572:
573: @item HT
1.2 noro 574:
1.1 noro 575: $B@8@.$5$l$?B?9`<0$NF,9`(B
576:
577: @item INDEX
1.2 noro 578:
1.1 noro 579: S-$BB?9`<0$r9=@.$9$kB?9`<0$N%$%s%G%C%/%9$N%Z%"(B
580:
581: @item NB
1.2 noro 582:
1.1 noro 583: $B8=:_$N(B, $B>iD9@-$r=|$$$?4pDl$N?t(B
584:
585: @item NAB
1.2 noro 586:
1.1 noro 587: $B8=:_$^$G$K@8@.$5$l$?4pDl$N?t(B
588:
589: @item RP
1.2 noro 590:
1.1 noro 591: $B;D$j$N%Z%"$N?t(B
592:
593: @item S
1.2 noro 594:
1.1 noro 595: $B@8@.$5$l$?B?9`<0$N(B sugar $B$NCM(B
596:
597: @item M
1.2 noro 598:
1.1 noro 599: $B@8@.$5$l$?B?9`<0$N78?t$N%S%C%HD9$NOB(B (@code{ShowMag} $B$,(B on $B$N;~$KI=<($5$l$k(B. )
1.2 noro 600: \E
601: \BEG
602: @item TNF
603:
604: CPU time for normal form computation (second)
605:
606: @item TCONT
607:
608: CPU time for content removal(second)
609:
610: @item HT
611:
612: Head term of the generated basis element
613:
614: @item INDEX
615:
616: Pair of indices which corresponds to the reduced S-polynomial
617:
618: @item NB
619:
620: Number of basis elements after removing redundancy
621:
622: @item NAB
623:
624: Number of all the basis elements
625:
626: @item RP
627:
628: Number of remaining pairs
629:
630: @item S
631:
632: Sugar of the generated basis element
633:
634: @item M
635:
636: Magnitude of the genrated basis element (shown if @code{ShowMag} is `on'.)
637: \E
1.1 noro 638: @end table
639:
640: @noindent
1.2 noro 641: \BJP
1.1 noro 642: $B:G8e$K(B, $B=87W%G!<%?$,I=<($5$l$k(B. $B0UL#$O<!$NDL$j(B.
643: ($B;~4V$NI=<($K$*$$$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 644: \E
645: \BEG
646: The summary of the informations shown after a Groebner basis
647: computation is as follows. If a component shows timings and it
648: contains two numbers, they are a pair of time for computation and time
649: for garbage colection.
650: \E
1.1 noro 651:
652: @table @code
1.2 noro 653: \BJP
1.1 noro 654: @item UP
1.2 noro 655:
1.1 noro 656: $B%Z%"$N%j%9%H$NA`:n$K$+$+$C$?;~4V(B
657:
658: @item SP
1.2 noro 659:
1.1 noro 660: $BM-M}?t>e$N(B S-$BB?9`<07W;;;~4V(B
661:
662: @item SPM
1.2 noro 663:
1.1 noro 664: $BM-8BBN>e$N(B S-$BB?9`<07W;;;~4V(B
665:
666: @item NF
1.2 noro 667:
1.1 noro 668: $BM-M}?t>e$N(B normal form $B7W;;;~4V(B
669:
670: @item NFM
1.2 noro 671:
1.1 noro 672: $BM-8BBN>e$N(B normal form $B7W;;;~4V(B
673:
674: @item ZNFM
1.2 noro 675:
1.1 noro 676: @code{NFM} $B$NFb(B, 0 $B$X$N(B reduction $B$K$+$+$C$?;~4V(B
677:
678: @item PZ
1.2 noro 679:
1.1 noro 680: content $B7W;;;~4V(B
681:
682: @item NP
1.2 noro 683:
1.1 noro 684: $BM-M}?t78?tB?9`<0$N78?t$KBP$9$k>jM>1i;;$N7W;;;~4V(B
685:
686: @item MP
1.2 noro 687:
1.1 noro 688: S-$BB?9`<0$r@8@.$9$k%Z%"$NA*Br$K$+$+$C$?;~4V(B
689:
690: @item RA
1.2 noro 691:
1.1 noro 692: interreduce $B7W;;;~4V(B
693:
694: @item MC
1.2 noro 695:
1.1 noro 696: trace-lifting $B$K$*$1$k(B, $BF~NOB?9`<0$N%a%s%P%7%C%W7W;;;~4V(B
697:
698: @item GC
1.2 noro 699:
1.1 noro 700: $B7k2L$N%0%l%V%J4pDl8uJd$N%0%l%V%J4pDl%A%'%C%/;~4V(B
701:
702: @item T
1.2 noro 703:
1.1 noro 704: $B@8@.$5$l$?%Z%"$N?t(B
705:
706: @item B, M, F, D
1.2 noro 707:
1.1 noro 708: $B3F(B criterion $B$K$h$j=|$+$l$?%Z%"$N?t(B
709:
710: @item ZR
1.2 noro 711:
1.1 noro 712: 0 $B$K(B reduce $B$5$l$?%Z%"$N?t(B
713:
714: @item NZR
1.2 noro 715:
1.1 noro 716: 0 $B$G$J$$B?9`<0$K(B reduce $B$5$l$?%Z%"$N?t(B
717:
718: @item Max_mag
1.2 noro 719:
1.1 noro 720: $B@8@.$5$l$?B?9`<0$N(B, $B78?t$N%S%C%HD9$NOB$N:GBgCM(B
1.2 noro 721: \E
722: \BEG
723: @item UP
724:
725: Time to manipulate the list of critical pairs
726:
727: @item SP
728:
729: Time to compute S-polynomials over the rationals
730:
731: @item SPM
732:
733: Time to compute S-polynomials over a finite field
734:
735: @item NF
736:
737: Time to compute normal forms over the rationals
738:
739: @item NFM
740:
741: Time to compute normal forms over a finite field
742:
743: @item ZNFM
744:
745: Time for zero reductions in @code{NFM}
746:
747: @item PZ
748:
749: Time to remove integer contets
750:
751: @item NP
752:
753: Time to compute remainders for coefficients of polynomials with coeffieints
754: in the rationals
755:
756: @item MP
757:
758: Time to select pairs from which S-polynomials are computed
759:
760: @item RA
761:
762: Time to interreduce the Groebner basis candidate
763:
764: @item MC
1.1 noro 765:
1.2 noro 766: Time to check that each input polynomial is a member of the ideal
767: generated by the Groebner basis candidate.
768:
769: @item GC
770:
771: Time to check that the Groebner basis candidate is a Groebner basis
772:
773: @item T
774:
775: Number of critical pairs generated
776:
777: @item B, M, F, D
778:
779: Number of critical pairs removed by using each criterion
780:
781: @item ZR
782:
783: Number of S-polynomials reduced to 0
784:
785: @item NZR
786:
787: Number of S-polynomials reduced to non-zero results
788:
789: @item Max_mag
790:
791: Maximal magnitude among all the generated polynomials
792: \E
1.1 noro 793: @end table
794:
1.2 noro 795: \BJP
1.1 noro 796: @node $B9`=g=x$N@_Dj(B,,, $B%0%l%V%J4pDl$N7W;;(B
797: @section $B9`=g=x$N@_Dj(B
1.2 noro 798: \E
799: \BEG
800: @node Setting term orderings,,, Groebner basis computation
801: @section Setting term orderings
802: \E
1.1 noro 803:
804: @noindent
1.2 noro 805: \BJP
1.1 noro 806: $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
807: $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
808: $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
809: $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
810: $B9TNs$GI=8=$5$l$k(B.
1.2 noro 811: \E
812: \BEG
813: A term is internally represented as an integer vector whose components
814: are exponents with respect to variables. A variable list specifies the
815: correspondences between variables and components. A type of term ordering
816: specifies a total order for integer vectors. A type of term ordering is
817: represented by an integer, a list of integer or matrices.
818: \E
1.1 noro 819:
820: @noindent
1.2 noro 821: \JP $B4pK\E*$J9`=g=x7?$H$7$F<!$N(B 3 $B$D$,$"$k(B.
822: \EG There are following three fundamental types.
1.1 noro 823:
824: @table @code
1.2 noro 825: \JP @item 0 (DegRevLex; @b{$BA4<!?t5U<-=q<0=g=x(B})
826: \EG @item 0 (DegRevLex; @b{total degree reverse lexicographic ordering})
1.1 noro 827:
1.2 noro 828: \BJP
1.1 noro 829: $B0lHL$K(B, $B$3$N=g=x$K$h$k%0%l%V%J4pDl7W;;$,:G$b9bB.$G$"$k(B. $B$?$@$7(B,
830: $BJ}Dx<0$r2r$/$H$$$&L\E*$KMQ$$$k$3$H$O(B, $B0lHL$K$O$G$-$J$$(B. $B$3$N(B
831: $B=g=x$K$h$k%0%l%V%J4pDl$O(B, $B2r$N8D?t$N7W;;(B, $B%$%G%"%k$N%a%s%P%7%C%W$d(B,
832: $BB>$NJQ?t=g=x$X$N4pDlJQ49$N$?$a$N%=!<%9$H$7$FMQ$$$i$l$k(B.
1.2 noro 833: \E
834: \BEG
835: In general, computation by this ordering shows the fastest speed
836: in most Groebner basis computations.
837: However, for the purpose to solve polynomial equations, this type
838: of ordering is, in general, not so suitable.
839: The Groebner bases obtained by this ordering is used for computing
840: the number of solutions, solving ideal membership problem and seeds
841: for conversion to other Groebner bases under different ordering.
842: \E
1.1 noro 843:
1.2 noro 844: \JP @item 1 (DegLex; @b{$BA4<!?t<-=q<0=g=x(B})
845: \EG @item 1 (DegLex; @b{total degree lexicographic ordering})
1.1 noro 846:
1.2 noro 847: \BJP
1.1 noro 848: $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,
849: @code{DegRevLex} $B$HF1MMD>@\$=$N7k2L$rMQ$$$k$3$H$O:$Fq$G$"$k(B. $B$7$+$7(B,
850: $B<-=q<0=g=x$N%0%l%V%J4pDl$r5a$a$k:]$K(B, $B@F<!2=8e$K$3$N=g=x$G%0%l%V%J4pDl(B
851: $B$r5a$a$F$$$k(B.
1.2 noro 852: \E
853: \BEG
854: By this type term ordering, Groebner bases are obtained fairly faster
855: than Lex (lexicographic) ordering, too.
856: Alike the @code{DegRevLex} ordering, the result, in general, cannot directly
857: be used for solving polynomial equations.
858: It is used, however, in such a way
859: that a Groebner basis is computed in this ordering after homogenization
860: to obtain the final lexicographic Groebner basis.
861: \E
1.1 noro 862:
1.2 noro 863: \JP @item 2 (Lex; @b{$B<-=q<0=g=x(B})
864: \EG @item 2 (Lex; @b{lexicographic ordering})
1.1 noro 865:
1.2 noro 866: \BJP
1.1 noro 867: $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
868: $B7W;;;~4V$,$+$+$j2a$.$k$N$,FqE@$G$"$k(B. $BFC$K(B, $B2r$,M-8B8D$N>l9g(B, $B7k2L$N(B
869: $B78?t$,6K$a$FD9Bg$JB?G\D9?t$K$J$k>l9g$,B?$$(B. $B$3$N>l9g(B, @code{gr()},
870: @code{hgr()} $B$K$h$k7W;;$,6K$a$FM-8z$K$J$k>l9g$,B?$$(B.
1.2 noro 871: \E
872: \BEG
873: Groebner bases computed by this ordering give the most convenient
874: Groebner bases for solving the polynomial equations.
875: The only and serious shortcoming is the enormously long computation
876: time.
877: It is often observed that the number coefficients of the result becomes
878: very very long integers, especially if the ideal is 0-dimensional.
879: For such a case, it is empirically true for many cases
880: that i.e., computation by
881: @code{gr()} and/or @code{hgr()} may be quite effective.
882: \E
1.1 noro 883: @end table
884:
885: @noindent
1.2 noro 886: \BJP
1.1 noro 887: $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.
888: $B$3$l$O(B,
1.2 noro 889: \E
890: \BEG
891: By combining these fundamental orderingl into a list, one can make
892: various term ordering called elimination orderings.
893: \E
1.1 noro 894:
895: @code{[[O1,L1],[O2,L2],...]}
896:
897: @noindent
1.2 noro 898: \BJP
1.1 noro 899: $B$G;XDj$5$l$k(B. @code{Oi} $B$O(B 0, 1, 2 $B$N$$$:$l$+$G(B, @code{Li} $B$OJQ?t$N8D(B
900: $B?t$rI=$9(B. $B$3$N;XDj$O(B, $BJQ?t$r@hF,$+$i(B @code{L1}, @code{L2} , ...$B8D(B
901: $B$:$D$NAH$KJ,$1(B, $B$=$l$>$l$NJQ?t$K4X$7(B, $B=g$K(B @code{O1}, @code{O2},
902: ...$B$N9`=g=x7?$GBg>.$,7hDj$9$k$^$GHf3S$9$k$3$H$r0UL#$9$k(B. $B$3$N7?$N(B
903: $B=g=x$O0lHL$K>C5n=g=x$H8F$P$l$k(B.
1.2 noro 904: \E
905: \BEG
906: In this example @code{Oi} indicates 0, 1 or 2 and @code{Li} indicates
907: the number of variables subject to the correspoinding orderings.
908: This specification means the following.
909:
910: The variable list is separated into sub lists from left to right where
911: the @code{i}-th list contains @code{Li} members and it corresponds to
912: the ordering of type @code{Oi}. The result of a comparison is equal
913: to that for the leftmost different sub components. This type of ordering
914: is called an elimination ordering.
915: \E
1.1 noro 916:
917: @noindent
1.2 noro 918: \BJP
1.1 noro 919: $B$5$i$K(B, $B9TNs$K$h$j9`=g=x$r;XDj$9$k$3$H$,$G$-$k(B. $B0lHL$K(B, @code{n} $B9T(B
920: @code{m} $BNs$N<B?t9TNs(B @code{M} $B$,<!$N@-<A$r;}$D$H$9$k(B.
1.2 noro 921: \E
922: \BEG
923: Furthermore one can specify a term ordering by a matix.
924: Suppose that a real @code{n}, @code{m} matrix @code{M} has the
925: following properties.
926: \E
1.1 noro 927:
928: @enumerate
929: @item
1.2 noro 930: \JP $BD9$5(B @code{m} $B$N@0?t%Y%/%H%k(B @code{v} $B$KBP$7(B @code{Mv=0} $B$H(B @code{v=0} $B$OF1CM(B.
931: \BEG
932: For all integer verctors @code{v} of length @code{m} @code{Mv=0} is equivalent
933: to @code{v=0}.
934: \E
1.1 noro 935:
936: @item
1.2 noro 937: \BJP
1.1 noro 938: $BHsIi@.J,$r;}$DD9$5(B @code{m} $B$N(B 0 $B$G$J$$@0?t%Y%/%H%k(B @code{v} $B$KBP$7(B,
939: @code{Mv} $B$N(B 0 $B$G$J$$:G=i$N@.J,$OHsIi(B.
1.2 noro 940: \E
941: \BEG
942: For all non-negative integer vectors @code{v} the first non-zero component
943: of @code{Mv} is non-negative.
944: \E
1.1 noro 945: @end enumerate
946:
947: @noindent
1.2 noro 948: \BJP
1.1 noro 949: $B$3$N;~(B, 2 $B$D$N%Y%/%H%k(B @code{t}, @code{s} $B$KBP$7(B,
950: @code{t>s} $B$r(B, @code{M(t-s)} $B$N(B 0 $B$G$J$$:G=i$N@.J,$,HsIi(B,
951: $B$GDj5A$9$k$3$H$K$h$j9`=g=x$,Dj5A$G$-$k(B.
1.2 noro 952: \E
953: \BEG
954: Then we can define a term ordering such that, for two vectors
955: @code{t}, @code{s}, @code{t>s} means that the first non-zero component
956: of @code{M(t-s)} is non-negative.
957: \E
1.1 noro 958:
959: @noindent
1.2 noro 960: \BJP
1.1 noro 961: $B9`=g=x7?$O(B, @code{gr()} $B$J$I$N0z?t$H$7$F;XDj$5$l$kB>(B, $BAH$_9~$_H!?t(B
962: @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 963: \E
964: \BEG
965: Types of term orderings are used as arguments of functions such as
966: @code{gr()}. It is also set internally by @code{dp_ord()} and is used
967: during executions of various functions.
968: \E
1.1 noro 969:
970: @noindent
1.2 noro 971: \BJP
1.1 noro 972: $B$3$l$i$N=g=x$N6qBNE*$JDj5A$*$h$S%0%l%V%J4pDl$K4X$9$k99$K>\$7$$2r@b$O(B
973: @code{[Becker,Weispfenning]} $B$J$I$r;2>H$N$3$H(B.
1.2 noro 974: \E
975: \BEG
976: For concrete definitions of term ordering and more information
977: about Groebner basis, refer to, for example, the book
978: @code{[Becker,Weispfenning]}.
979: \E
1.1 noro 980:
981: @noindent
1.2 noro 982: \JP $B9`=g=x7?$N@_Dj$NB>$K(B, $BJQ?t$N=g=x<+BN$b7W;;;~4V$KBg$-$J1F6A$rM?$($k(B.
983: \BEG
984: Note that the variable ordering have strong effects on the computation
985: time as well as the choice of types of term orderings.
986: \E
1.1 noro 987:
988: @example
989: [90] B=[x^10-t,x^8-z,x^31-x^6-x-y]$
990: [91] gr(B,[x,y,z,t],2);
991: [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
992: -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
993: +(-12*t^16+72*t^13-28*t^11-180*t^10+112*t^8+240*t^7+28*t^6-127*t^5
994: -167*t^4-55*t^3+30*t^2+58*t-15)*z^4,
995: (y+t^2*z^2)*x+y^7+(20*t^2+6*t+1)*y^2+(-t^17+6*t^14-21*t^12-15*t^11+84*t^9
996: +20*t^8-35*t^7-126*t^6-15*t^5+70*t^4+84*t^3-t^2+5*t-9)*z^2*y+(6*t^16-36*t^13
997: +14*t^11+90*t^10-56*t^8-120*t^7-14*t^6+64*t^5+84*t^4+27*t^3-16*t^2-30*t+7)*z^4,
998: (t^3-1)*x-y^6+(-6*t^13+24*t^10-20*t^8-36*t^7+40*t^5+24*t^4-6*t^3-20*t^2-6*t-1)*y
999: +(t^17-6*t^14+9*t^12+15*t^11-36*t^9-20*t^8-5*t^7+54*t^6+15*t^5+10*t^4-36*t^3
1000: -11*t^2-5*t+9)*z^2,
1001: -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
1002: -56*t^6-336*t^5-112*t^4+112*t^3+224*t^2+24*t-56)*z^4*y+(t^24-8*t^21+20*t^19
1003: +28*t^18-120*t^16-56*t^15+14*t^14+300*t^13+70*t^12-56*t^11-400*t^10-84*t^9
1004: +84*t^8+268*t^7+84*t^6-56*t^5-63*t^4-36*t^3+46*t^2-12*t+1)*z,
1005: 2*t*y^5+z*y^2+(-2*t^11+8*t^8-20*t^6-12*t^5+40*t^3+8*t^2-10*t-20)*z^3*y+8*t^14
1006: -32*t^11+48*t^8-t^7-32*t^5-6*t^4+9*t^2-t,
1007: -z*y^3+(t^7-2*t^4+3*t^2+t)*y+(-2*t^6+4*t^3+2*t-2)*z^2,
1008: 2*t^2*y^3+z^2*y^2+(-2*t^5+4*t^2-6)*z^4*y+(4*t^8-t^7-8*t^5+2*t^4-4*t^3+5*t^2-t)*z,
1009: z^3*y^2+2*t^3*y+(-t^7+2*t^4+t^2-t)*z^2,
1010: -t*z*y^2-2*z^3*y+t^8-2*t^5-t^3+t^2,
1011: -t^3*y^2-2*t^2*z^2*y+(t^6-2*t^3-t+1)*z^4,
1012: z^5-t^4]
1013: [93] gr(B,[t,z,y,x],2);
1014: [x^10-t,x^8-z,x^31-x^6-x-y]
1015: @end example
1016:
1017: @noindent
1.2 noro 1018: \BJP
1.1 noro 1019: $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
1020: $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
1021: $B%0%l%V%J4pDl$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
1022: $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
1023: $B$3$H$G$"$j(B, $B$3$NNc$N>l9g$O(B, @code{t}, @code{z}, @code{y} $B$,4{$K(B
1024: @code{x} $B$GI=$5$l$F$$$k$3$H$+$i$3$N$h$&$J6KC<$J7k2L$H$J$C$?$o$1$G$"$k(B.
1025: $B<B:]$K8=$l$k7W;;$K$*$$$F$O(B, $B$3$N$h$&$KA*$V$Y$-JQ?t=g=x$,L@$i$+$G$"$k(B
1026: $B$3$H$O>/$J$/(B, $B;n9T:x8m$,I,MW$J>l9g$b$"$k(B.
1.2 noro 1027: \E
1028: \BEG
1029: As you see in the above example, the Groebner base under variable
1030: ordering @code{[x,y,z,t]} has a lot of bases and each base itself is
1031: large. Under variable ordering @code{[t,z,y,x]}, however, @code{B} itself
1032: is already the Groebner basis.
1033: Roughly speaking, to obtain a Groebner base under the lexicographic
1034: ordering is to express the variables on the left (having higher order)
1035: in terms of variables on the right (having lower order).
1036: In the example, variables @code{t}, @code{z}, and @code{y} are already
1037: expressed by variable @code{x}, and the above explanation justifies
1038: such a drastic experimental results.
1039: In practice, however, optimum ordering for variables may not known
1040: beforehand, and some heuristic trial may be inevitable.
1041: \E
1.1 noro 1042:
1.2 noro 1043: \BJP
1.1 noro 1044: @node $BM-M}<0$r78?t$H$9$k%0%l%V%J4pDl7W;;(B,,, $B%0%l%V%J4pDl$N7W;;(B
1045: @section $BM-M}<0$r78?t$H$9$k%0%l%V%J4pDl7W;;(B
1.2 noro 1046: \E
1047: \BEG
1048: @node Groebner basis computation with rational function coefficients,,, Groebner basis computation
1049: @section Groebner basis computation with rational function coefficients
1050: \E
1.1 noro 1051:
1052: @noindent
1.2 noro 1053: \BJP
1.1 noro 1054: @code{gr()} $B$J$I$N%H%C%W%l%Y%kH!?t$O(B, $B$$$:$l$b(B, $BF~NOB?9`<0%j%9%H$K(B
1055: $B8=$l$kJQ?t(B ($BITDj85(B) $B$H(B, $BJQ?t%j%9%H$K8=$l$kJQ?t$rHf3S$7$F(B, $BJQ?t%j%9%H$K(B
1056: $B$J$$JQ?t$,F~NOB?9`<0$K8=$l$F$$$k>l9g$K$O(B, $B<+F0E*$K(B, $B$=$NJQ?t$r(B, $B78?t(B
1057: $BBN$N85$H$7$F07$&(B.
1.2 noro 1058: \E
1059: \BEG
1060: Such variables that appear within the input polynomials but
1061: not appearing in the input variable list are automatically treated
1062: as elements in the coefficient field
1063: by top level functions, such as @code{gr()}.
1064: \E
1.1 noro 1065:
1066: @example
1067: [64] gr([a*x+b*y-c,d*x+e*y-f],[x,y],2);
1068: [(-e*a+d*b)*x-f*b+e*c,(-e*a+d*b)*y+f*a-d*c]
1069: @end example
1070:
1071: @noindent
1.2 noro 1072: \BJP
1.1 noro 1073: $B$3$NNc$G$O(B, @code{a}, @code{b}, @code{c}, @code{d} $B$,78?tBN$N85$H$7$F(B
1074: $B07$o$l$k(B. $B$9$J$o$A(B, $BM-M}H!?tBN(B
1075: @b{F} = @b{Q}(@code{a},@code{b},@code{c},@code{d}) $B>e$N(B 2 $BJQ?tB?9`<04D(B
1076: @b{F}[@code{x},@code{y}] $B$K$*$1$k%0%l%V%J4pDl$r5a$a$k$3$H$K$J$k(B.
1077: $BCm0U$9$Y$-$3$H$O(B,
1078: $B78?t$,BN$H$7$F07$o$l$F$$$k$3$H$G$"$k(B. $B$9$J$o$A(B, $B78?t$N4V$KB?9`<0(B
1079: $B$H$7$F$N6&DL0x;R$,$"$C$?>l9g$K$O(B, $B7k2L$+$i$=$N0x;R$O=|$+$l$F$$$k(B
1080: $B$?$a(B, $BM-M}?tBN>e$NB?9`<04D>e$NLdBj$H$7$F9M$($?>l9g$N7k2L$H$O0lHL(B
1081: $B$K$O0[$J$k(B. $B$^$?(B, $B<g$H$7$F7W;;8zN(>e$NLdBj$N$?$a(B, $BJ,;6I=8=B?9`<0(B
1082: $B$N78?t$H$7$F<B:]$K5v$5$l$k$N$OB?9`<0$^$G$G$"$k(B. $B$9$J$o$A(B, $BJ,Jl$r(B
1083: $B;}$DM-M}<0$OJ,;6I=8=B?9`<0$N78?t$H$7$F$O5v$5$l$J$$(B.
1.2 noro 1084: \E
1085: \BEG
1086: In this example, variables @code{a}, @code{b}, @code{c}, and @code{d}
1087: are treated as elements in the coefficient field.
1088: In this case, a Groebner basis is computed
1089: on a bi-variate polynomial ring
1090: @b{F}[@code{x},@code{y}]
1091: over rational function field
1092: @b{F} = @b{Q}(@code{a},@code{b},@code{c},@code{d}).
1093: Notice that coefficients are considered as a member in a field.
1094: As a consequence, polynomial factors common to the coefficients
1095: are removed so that the result, in general, is different from
1096: the result that would be obtained when the problem is considered
1097: as a computation of Groebner basis over a polynomial ring
1098: with rational function coefficients.
1099: And note that coefficients of a distributed polynomial are limited
1100: to numbers and polynomials because of efficiency.
1101: \E
1.1 noro 1102:
1.2 noro 1103: \BJP
1.1 noro 1104: @node $B4pDlJQ49(B,,, $B%0%l%V%J4pDl$N7W;;(B
1105: @section $B4pDlJQ49(B
1.2 noro 1106: \E
1107: \BEG
1108: @node Change of ordering,,, Groebner basis computation
1109: @section Change of orderng
1110: \E
1.1 noro 1111:
1112: @noindent
1.2 noro 1113: \BJP
1.1 noro 1114: $B<-=q<0=g=x$N%0%l%V%J4pDl$r5a$a$k>l9g(B, $BD>@\(B @code{gr()} $B$J$I$r5/F0$9$k(B
1115: $B$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,
1116: $B$=$l$rF~NO$H$7$F<-=q<0=g=x$N%0%l%V%J4pDl$r7W;;$9$kJ}$,8zN($,$h$$>l9g(B
1117: $B$,$"$k(B. $B$^$?(B, $BF~NO$,2?$i$+$N=g=x$G$N%0%l%V%J4pDl$K$J$C$F$$$k>l9g(B, $B4pDl(B
1118: $BJQ49$H8F$P$l$kJ}K!$K$h$j(B, Buchberger $B%"%k%4%j%:%`$K$h$i$:$K8zN(NI$/(B
1119: $B<-=q<0=g=x$N%0%l%V%J4pDl$,7W;;$G$-$k>l9g$,$"$k(B. $B$3$N$h$&$JL\E*$N$?$a$N(B
1120: $BH!?t$,(B, $B%f!<%6Dj5AH!?t$H$7$F(B @samp{gr} $B$K$$$/$D$+Dj5A$5$l$F$$$k(B.
1121: $B0J2<$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
1122: $B4{$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
1123: @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 1124: \E
1125: \BEG
1126: When we compute a lex order Groebner basis, it is often efficient to
1127: compute it via Groebner basis with respect to another order such as
1128: degree reverse lex order, rather than to compute it directory by
1129: @code{gr()} etc. If we know that an input is a Groebner basis with
1130: respect to an order, we can apply special methods called change of
1131: ordering for a Groebner basis computation with respect to another
1132: order, without using Buchberger algorithm. The following two functions
1133: are ones for change of ordering such that they convert a Groebner
1134: basis @var{gbase} with respect to the variable order @var{vlist1} and
1135: the order type @var{order} into a lex Groebner basis with respect
1136: to the variable order @var{vlist2}.
1137: \E
1.1 noro 1138:
1139: @table @code
1140: @item tolex(@var{gbase},@var{vlist1},@var{order},@var{vlist2})
1141:
1.2 noro 1142: \BJP
1.1 noro 1143: $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.
1144: $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
1145: $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 1146: \E
1147: \BEG
1148: This function can be used only when @var{gbase} is an ideal over the
1149: rationals. The input @var{gbase} must be a Groebner basis with respect
1150: to the variable order @var{vlist1} and the order type @var{order}. Moreover
1151: the ideal generated by @var{gbase} must be zero-dimensional.
1152: This computes the lex Groebner basis of @var{gbase}
1153: by using the modular change of ordering algorithm. The algorithm first
1154: computes the lex Groebner basis over a finite field. Then each element
1155: in the lex Groebner basis over the rationals is computed with undetermined
1156: coefficient method and linear equation solving by Hensel lifting.
1157: \E
1.1 noro 1158:
1159: @item tolex_tl(@var{gbase},@var{vlist1},@var{order},@var{vlist2},@var{homo})
1160:
1.2 noro 1161: \BJP
1.1 noro 1162: $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
1163: $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
1164: trace-lifting$B$K$*$1$k%0%l%V%J4pDl8uJd$NF,9`(B, $BF,78?t$N@-<A$rMxMQ$7$F(B,
1165: $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
1166: $B$k$?$a(B, $BC1$K(BBuchberger $B%"%k%4%j%:%`$r7+$jJV$9$h$j8zN($h$/7W;;$G$-$k(B.
1167: $B99$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
1168: $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,
1169: $BD>@\<-=q<0=g=x$N7W;;$r9T$&$h$j8zN($,$h$$(B. ($B$b$A$m$sNc30$"$j(B. )
1170: $B0z?t(B @var{homo} $B$,(B 0 $B$G$J$$;~(B, @code{hgr()} $B$HF1MM$K@F<!2=$r7PM3$7$F(B
1171: $B7W;;$r9T$&(B.
1.2 noro 1172: \E
1173: \BEG
1174: This function computes the lex Groebner basis of @var{gbase}. The
1175: input @var{gbase} must be a Groebner basis with respect to the
1176: variable order @var{vlist1} and the order type @var{order}.
1177: Buchberger algorithm with trace lifting is used to compute the lex
1178: Groebner basis, however the Groebner basis check and the ideal
1179: membership check can be omitted by using several properties derived
1180: from the fact that the input is a Groebner basis. So it is more
1181: efficient than simple repetition of Buchberger algorithm. If the input
1182: is zero-dimensional, this function inserts automatically a computation
1183: of Groebner basis with respect to an elimination order, which makes
1184: the whole computation more efficient for many cases. If @var{homo} is
1185: not equal to 0, homogenization is used in each step.
1186: \E
1.1 noro 1187: @end table
1188:
1189: @noindent
1.2 noro 1190: \BJP
1.1 noro 1191: $B$=$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
1192: $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
1193: @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 1194: \E
1195: \BEG
1196: For zero-dimensional systems, there are several fuctions to
1197: compute the minimal polynomial of a polynomial and or a more compact
1198: representation for zeros of the system. They are all defined in @samp{gr}.
1199: Refer to the sections for each functions.
1200: \E
1.1 noro 1201:
1.2 noro 1202: \BJP
1.1 noro 1203: @node $B%0%l%V%J4pDl$K4X$9$kH!?t(B,,, $B%0%l%V%J4pDl$N7W;;(B
1204: @section $B%0%l%V%J4pDl$K4X$9$kH!?t(B
1.2 noro 1205: \E
1206: \BEG
1207: @node Functions for Groebner basis computation,,, Groebner basis computation
1208: @section Functions for Groebner basis computation
1209: \E
1.1 noro 1210:
1211: @menu
1212: * gr hgr gr_mod::
1213: * lex_hensel lex_tl tolex tolex_d tolex_tl::
1214: * lex_hensel_gsl tolex_gsl tolex_gsl_d::
1215: * gr_minipoly minipoly::
1216: * tolexm minipolym::
1217: * dp_gr_main dp_gr_mod_main::
1218: * dp_f4_main dp_f4_mod_main::
1219: * dp_gr_flags dp_gr_print::
1220: * dp_ord::
1221: * dp_ptod::
1222: * dp_dtop::
1223: * dp_mod dp_rat::
1224: * dp_homo dp_dehomo::
1225: * dp_ptozp dp_prim::
1226: * dp_nf dp_nf_mod dp_true_nf dp_true_nf_mod::
1227: * dp_hm dp_ht dp_hc dp_rest::
1228: * dp_td dp_sugar::
1229: * dp_lcm::
1230: * dp_redble::
1231: * dp_subd::
1232: * dp_mbase::
1233: * dp_mag::
1234: * dp_red dp_red_mod::
1235: * dp_sp dp_sp_mod::
1236: * p_nf p_nf_mod p_true_nf p_true_nf_mod ::
1237: * p_terms::
1238: * gb_comp::
1239: * katsura hkatsura cyclic hcyclic::
1240: * dp_vtoe dp_etov::
1241: * lex_hensel_gsl tolex_gsl tolex_gsl_d::
1.3 noro 1242: * primadec primedec::
1.1 noro 1243: @end menu
1244:
1.2 noro 1245: \JP @node gr hgr gr_mod,,, $B%0%l%V%J4pDl$K4X$9$kH!?t(B
1246: \EG @node gr hgr gr_mod,,, Functions for Groebner basis computation
1.1 noro 1247: @subsection @code{gr}, @code{hgr}, @code{gr_mod}, @code{dgr}
1248: @findex gr
1249: @findex hgr
1250: @findex gr_mod
1251: @findex dgr
1252:
1253: @table @t
1254: @item gr(@var{plist},@var{vlist},@var{order})
1255: @itemx hgr(@var{plist},@var{vlist},@var{order})
1256: @itemx gr_mod(@var{plist},@var{vlist},@var{order},@var{p})
1257: @itemx dgr(@var{plist},@var{vlist},@var{order},@var{procs})
1.2 noro 1258: \JP :: $B%0%l%V%J4pDl$N7W;;(B
1259: \EG :: Groebner basis computation
1.1 noro 1260: @end table
1261:
1262: @table @var
1263: @item return
1.2 noro 1264: \JP $B%j%9%H(B
1265: \EG list
1.4 ! noro 1266: @item plist vlist procs
1.2 noro 1267: \JP $B%j%9%H(B
1268: \EG list
1.1 noro 1269: @item order
1.2 noro 1270: \JP $B?t(B, $B%j%9%H$^$?$O9TNs(B
1271: \EG number, list or matrix
1.1 noro 1272: @item p
1.2 noro 1273: \JP 2^27 $BL$K~$NAG?t(B
1274: \EG prime less than 2^27
1.1 noro 1275: @end table
1276:
1277: @itemize @bullet
1.2 noro 1278: \BJP
1.1 noro 1279: @item
1280: $BI8=`%i%$%V%i%j$N(B @samp{gr} $B$GDj5A$5$l$F$$$k(B.
1281: @item
1282: $B$$$:$l$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
1283: @var{order} $B$K4X$9$k%0%l%V%J4pDl$r5a$a$k(B. @code{gr()}, @code{hgr()}
1284: $B$O(B $BM-M}?t78?t(B, @code{gr_mod()} $B$O(B GF(@var{p}) $B78?t$H$7$F7W;;$9$k(B.
1285: @item
1286: @var{vlist} $B$OITDj85$N%j%9%H(B. @var{vlist} $B$K8=$l$J$$ITDj85$O(B,
1287: $B78?tBN$KB0$9$k$H8+$J$5$l$k(B.
1288: @item
1289: @code{gr()}, trace-lifting ($B%b%8%e%i1i;;$rMQ$$$?9bB.2=(B) $B$*$h$S(B sugar
1290: strategy $B$K$h$k7W;;(B, @code{hgr()} $B$O(B trace-lifting $B$*$h$S(B
1291: $B@F<!2=$K$h$k(B $B6:@5$5$l$?(B sugar strategy $B$K$h$k7W;;$r9T$&(B.
1292: @item
1293: @code{dgr()} $B$O(B, @code{gr()}, @code{dgr()} $B$r(B
1294: $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,
1295: $B@h$K7k2L$rJV$7$?J}$N7k2L$rJV$9(B. $B7k2L$OF10l$G$"$k$,(B, $B$I$A$i$NJ}K!$,(B
1296: $B9bB.$+0lHL$K$OITL@$N$?$a(B, $B<B:]$N7P2a;~4V$rC;=L$9$k$N$KM-8z$G$"$k(B.
1297: @item
1298: @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
1299: CPU $B;~4V$G$"$j(B, $B$3$NH!?t$N>l9g$O$[$H$s$IDL?.$N$?$a$N;~4V$G$"$k(B.
1.2 noro 1300: \E
1301: \BEG
1302: @item
1303: These functions are defined in @samp{gr} in the standard library
1304: directory.
1305: @item
1306: They compute a Groebner basis of a polynomial list @var{plist} with
1307: respect to the variable order @var{vlist} and the order type @var{order}.
1308: @code{gr()} and @code{hgr()} compute a Groebner basis over the rationals
1309: and @code{gr_mod} computes over GF(@var{p}).
1310: @item
1311: Variables not included in @var{vlist} are regarded as
1312: included in the ground field.
1313: @item
1314: @code{gr()} uses trace-lifting (an improvement by modular computation)
1315: and sugar strategy.
1316: @code{hgr()} uses trace-lifting and a cured sugar strategy
1317: by using homogenization.
1318: @item
1319: @code{dgr()} executes @code{gr()}, @code{dgr()} simultaneously on
1320: two process in a child process list @var{procs} and returns
1321: the result obtained first. The results returned from both the process
1322: should be equal, but it is not known in advance which method is faster.
1323: Therefore this function is useful to reduce the actual elapsed time.
1324: @item
1325: The CPU time shown after an exection of @code{dgr()} indicates
1326: that of the master process, and most of the time corresponds to the time
1327: for communication.
1328: \E
1.1 noro 1329: @end itemize
1330:
1331: @example
1332: [0] load("gr")$
1333: [64] load("cyclic")$
1334: [74] G=gr(cyclic(5),[c0,c1,c2,c3,c4],2);
1335: [c4^15+122*c4^10-122*c4^5-1,...]
1336: [75] GM=gr_mod(cyclic(5),[c0,c1,c2,c3,c4],2,31991)$
1337: 24628*c4^15+29453*c4^10+2538*c4^5+7363
1338: [76] (G[0]*24628-GM[0])%31991;
1339: 0
1340: @end example
1341:
1342: @table @t
1.2 noro 1343: \JP @item $B;2>H(B
1344: \EG @item References
1.1 noro 1345: @comment @fref{dp_gr_main dp_gr_mod_main},
1346: @fref{dp_gr_main dp_gr_mod_main},
1347: @fref{dp_ord}.
1348: @end table
1349:
1.2 noro 1350: \JP @node lex_hensel lex_tl tolex tolex_d tolex_tl,,, $B%0%l%V%J4pDl$K4X$9$kH!?t(B
1351: \EG @node lex_hensel lex_tl tolex tolex_d tolex_tl,,, Functions for Groebner basis computation
1.1 noro 1352: @subsection @code{lex_hensel}, @code{lex_tl}, @code{tolex}, @code{tolex_d}, @code{tolex_tl}
1353: @findex lex_hensel
1354: @findex lex_tl
1355: @findex tolex
1356: @findex tolex_d
1357: @findex tolex_tl
1358:
1359: @table @t
1360: @item lex_hensel(@var{plist},@var{vlist1},@var{order},@var{vlist2},@var{homo})
1361: @itemx lex_tl(@var{plist},@var{vlist1},@var{order},@var{vlist2},@var{homo})
1.2 noro 1362: \JP :: $B4pDlJQ49$K$h$k<-=q<0=g=x%0%l%V%J4pDl$N7W;;(B
1363: \EG:: Groebner basis computation with respect to a lex order by change of ordering
1.1 noro 1364: @item tolex(@var{plist},@var{vlist1},@var{order},@var{vlist2})
1365: @itemx tolex_d(@var{plist},@var{vlist1},@var{order},@var{vlist2},@var{procs})
1366: @itemx tolex_tl(@var{plist},@var{vlist1},@var{order},@var{vlist2},@var{homo})
1.2 noro 1367: \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
1368: \EG :: Groebner basis computation with respect to a lex order by change of ordering, starting from a Groebner basis
1.1 noro 1369: @end table
1370:
1371: @table @var
1372: @item return
1.2 noro 1373: \JP $B%j%9%H(B
1374: \EG list
1.4 ! noro 1375: @item plist vlist1 vlist2 procs
1.2 noro 1376: \JP $B%j%9%H(B
1377: \EG list
1.1 noro 1378: @item order
1.2 noro 1379: \JP $B?t(B, $B%j%9%H$^$?$O9TNs(B
1380: \EG number, list or matrix
1.1 noro 1381: @item homo
1.2 noro 1382: \JP $B%U%i%0(B
1383: \EG flag
1.1 noro 1384: @end table
1385:
1386: @itemize @bullet
1.2 noro 1387: \BJP
1.1 noro 1388: @item
1389: $BI8=`%i%$%V%i%j$N(B @samp{gr} $B$GDj5A$5$l$F$$$k(B.
1390: @item
1391: @code{lex_hensel()}, @code{lex_tl()} $B$O(B,
1392: $BB?9`<0%j%9%H(B @var{plist} $B$N(B, $BJQ?t=g=x(B @var{vlist1}, $B9`=g=x7?(B
1393: @var{order} $B$K4X$9$k%0%l%V%J4pDl$r5a$a(B, $B$=$l$r(B, $BJQ?t=g=x(B @var{vlist2}
1394: $B$N<-=q<0=g=x%0%l%V%J4pDl$KJQ49$9$k(B.
1395: @item
1396: @code{tolex()}, @code{tolex_tl()} $B$O(B,
1397: $BJQ?t=g=x(B @var{vlist1}, $B9`=g=x7?(B @var{order} $B$K4X$9$k%0%l%V%J4pDl$G$"$k(B
1398: $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
1399: $B4pDl$KJQ49$9$k(B.
1400: @code{tolex_d()} $B$O(B, @code{tolex()} $B$K$*$1$k(B, $B3F4pDl$N7W;;$r(B, $B;R%W%m%;%9(B
1401: $B%j%9%H(B @var{procs} $B$N3F%W%m%;%9$KJ,;67W;;$5$;$k(B.
1402: @item
1403: @code{lex_hensel()}, @code{lex_tl()} $B$K$*$$$F$O(B, $B<-=q<0=g=x%0%l%V%J4pDl$N(B
1404: $B7W;;$O<!$N$h$&$K9T$o$l$k(B. (@code{[Noro,Yokoyama]} $B;2>H(B.)
1405: @enumerate
1406: @item
1407: @var{vlist1}, @var{order} $B$K4X$9$k%0%l%V%J4pDl(B @var{G0} $B$r7W;;$9$k(B.
1408: (@code{lex_hensel()} $B$N$_(B. )
1409: @item
1410: @var{G0} $B$N3F85$N(B @var{vlist2} $B$K4X$9$k<-=q<0=g=x$K$*$1$kF,78?t$r3d$i$J$$(B
1411: $B$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
1412: @var{Gp} $B$r7W;;$9$k(B.
1413: @item
1414: @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.
1415: @item
1416: @var{Gp} $B$N3F85(B @var{f} $B$K$D$-(B, @var{f} $B$N78?t$rL$Dj78?t$G(B,
1417: @var{f} $B$N3F9`$rBP1~$9$k(B @var{NF} $B$N85$GCV$-49$((B, $B3F9`$N78?t$r(B 0 $B$HCV$$$?(B,
1418: $BL$Dj78?t$K4X$9$k@~7AJ}Dx<07O(B @var{Lf} $B$r:n$k(B.
1419: @item
1420: @var{Lf} $B$,(B, $BK!(B @var{p} $B$G0l0U2r$r;}$D$3$H$rMQ$$$F(B @var{Lf} $B$N2r$r(B
1421: $BK!(B @var{p}$B$N2r$+$i(B Hensel $B9=@.$K$h$j5a$a$k(B.
1422: @item
1423: $B$9$Y$F$N(B @var{Gp} $B$N85$K$D$-@~7AJ}Dx<0$,2r$1$?$i$=$N2rA4BN$,5a$a$k(B
1424: $B<-=q<0=g=x$G$N%0%l%V%J4pDl(B. $B$b$7$I$l$+$N@~7AJ}Dx<0$N5a2r$K<:GT$7$?$i(B,
1425: @var{p} $B$r$H$jD>$7$F$d$jD>$9(B.
1426: @end enumerate
1427:
1428: @item
1429: @code{lex_tl()}, @code{tolex_tl()} $B$K$*$$$F$O(B, $B<-=q<0=g=x%0%l%V%J4pDl$N(B
1430: $B7W;;$O<!$N$h$&$K9T$o$l$k(B.
1431:
1432: @enumerate
1433: @item
1434: @var{vlist1}, @var{order} $B$K4X$9$k%0%l%V%J4pDl(B @var{G0} $B$r7W;;$9$k(B.
1435: (@code{lex_hensel()} $B$N$_(B. )
1436: @item
1437: @var{G0} $B$,(B 0 $B<!85%7%9%F%`$G$J$$$H$-(B, @var{G0} $B$rF~NO$H$7$F(B,
1438: @var{G0} $B$N3F85$N(B @var{vlist2} $B$K4X$9$k<-=q<0=g=x$K$*$1$kF,78?t$r3d$i$J$$(B
1439: $B$h$&$JAG?t(B @var{p} $B$rA*$S(B, @var{p} $B$rMQ$$$?(B trace-lifting $B$K$h$j<-=q<0(B
1440: $B=g=x$N%0%l%V%J4pDl8uJd$r5a$a(B, $B$b$75a$^$C$?$J$i%A%'%C%/$J$7$K$=$l$,5a$a$k(B
1441: $B%0%l%V%J4pDl$H$J$k(B. $B$b$7<:GT$7$?$i(B, @var{p} $B$r$H$jD>$7$F$d$jD>$9(B.
1442: @item
1443: @var{G0} $B$,(B 0 $B<!85%7%9%F%`$N$H$-(B, @var{G0} $B$rF~NO$H$7$F(B,
1444: $B$^$:(B, @var{vlist2} $B$N:G8e$NJQ?t0J30$r>C5n$9$k>C5n=g=x$K$h$j(B
1445: $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
1446: $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
1447: $BF,78?t$r3d$i$J$$AG?t$rMQ$$$?(B trace-lifting $B$G%0%l%V%J4pDl8uJd$r5a$a(B,
1448: $B$b$75a$^$C$?$i%A%'%C%/$J$7$K$=$l$,$=$N=g=x$G$N%0%l%V%J4pDl$H$J$k(B.
1449: @end enumerate
1450:
1451: @item
1452: $BM-M}<078?t$N7W;;$O(B, @code{lex_tl()}, @code{tolex_tl()} $B$N$_<u$1IU$1$k(B.
1453: @item
1454: @code{homo} $B$,(B 0 $B$G$J$$>l9g(B, $BFbIt$G5/F0$5$l$k(B Buchberger $B%"%k%4%j%:%`$K(B
1455: $B$*$$$F(B, $B@F<!2=$,9T$o$l$k(B.
1456: @item
1457: @code{tolex_d()} $B$GI=<($5$l$k;~4V$O(B, $B$3$NH!?t$,<B9T$5$l$F$$$k%W%m%;%9$K(B
1458: $B$*$$$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 1459: \E
1460: \BEG
1461: @item
1462: These functions are defined in @samp{gr} in the standard library
1463: directory.
1464: @item
1465: @code{lex_hensel()} and @code{lex_tl()} first compute a Groebner basis
1466: with respect to the variable order @var{vlist1} and the order type @var{order}.
1467: Then the Groebner basis is converted into a lex order Groebner basis
1468: with respect to the varable order @var{vlist2}.
1469: @item
1470: @code{tolex()} and @code{tolex_tl()} convert a Groebner basis @var{plist}
1471: with respect to the variable order @var{vlist1} and the order type @var{order}
1472: into a lex order Groebner basis
1473: with respect to the varable order @var{vlist2}.
1474: @code{tolex_d()} does computations of basis elements in @code{tolex()}
1475: in parallel on the processes in a child process list @var{procs}.
1476: @item
1477: In @code{lex_hensel()} and @code{tolex_hensel()} a lex order Groebner basis
1478: is computed as follows.(Refer to @code{[Noro,Yokoyama]}.)
1479: @enumerate
1480: @item
1481: Compute a Groebner basis @var{G0} with respect to @var{vlist1} and @var{order}.
1482: (Only in @code{lex_hensel()}. )
1483: @item
1484: Choose a prime which does not divide head coefficients of elements in @var{G0}
1485: with respect to @var{vlist1} and @var{order}. Then compute a lex order
1486: Groebner basis @var{Gp} over GF(@var{p}) with respect to @var{vlist2}.
1487: @item
1488: Compute @var{NF}, the set of all the normal forms with respect to
1489: @var{G0} of terms appearing in @var{Gp}.
1490: @item
1491: For each element @var{f} in @var{Gp}, replace coefficients and terms in @var{f}
1492: with undetermined coefficients and the corresponding polynomials in @var{NF}
1493: respectively, and generate a system of liear equation @var{Lf} by equating
1494: the coefficients of terms in the replaced polynomial with 0.
1495: @item
1496: Solve @var{Lf} by Hensel lifting, starting from the unique mod @var{p}
1497: solution.
1498: @item
1499: If all the linear equations generated from the elements in @var{Gp}
1500: could be solved, then the set of solutions corresponds to a lex order
1501: Groebner basis. Otherwise redo the whole process with another @var{p}.
1502: @end enumerate
1503:
1504: @item
1505: In @code{lex_tl()} and @code{tolex_tl()} a lex order Groebner basis
1506: is computed as follows.(Refer to @code{[Noro,Yokoyama]}.)
1507:
1508: @enumerate
1509: @item
1510: Compute a Groebner basis @var{G0} with respect to @var{vlist1} and @var{order}.
1511: (Only in @code{lex_tl()}. )
1512: @item
1513: If @var{G0} is not zero-dimensional, choose a prime which does not divide
1514: head coefficients of elements in @var{G0} with respect to @var{vlist1} and
1515: @var{order}. Then compute a candidate of a lex order Groebner basis
1516: via trace lifting with @var{p}. If it succeeds the candidate is indeed
1517: a lex order Groebner basis without any check. Otherwise redo the whole
1518: process with another @var{p}.
1519: @item
1520:
1521: If @var{G0} is zero-dimensional, starting from @var{G0},
1522: compute a Groebner basis @var{G1} with respect to an elimination order
1523: to eliminate variables other than the last varibale in @var{vlist2}.
1524: Then compute a lex order Groebner basis stating from @var{G1}. These
1525: computations are done by trace lifting and the selection of a mudulus
1526: @var{p} is the same as in non zero-dimensional cases.
1527: @end enumerate
1528:
1529: @item
1530: Computations with rational function coefficients can be done only by
1531: @code{lex_tl()} and @code{tolex_tl()}.
1532: @item
1533: If @code{homo} is not equal to 0, homogenization is used in Buchberger
1534: algorithm.
1535: @item
1536: The CPU time shown after an execution of @code{tolex_d()} indicates
1537: that of the master process, and it does not include the time in child
1538: processes.
1539: \E
1.1 noro 1540: @end itemize
1541:
1542: @example
1543: [78] K=katsura(5)$
1544: 30msec + gc : 20msec
1545: [79] V=[u5,u4,u3,u2,u1,u0]$
1546: 0msec
1547: [80] G0=hgr(K,V,2)$
1548: 91.558sec + gc : 15.583sec
1549: [81] G1=lex_hensel(K,V,0,V,0)$
1550: 49.049sec + gc : 9.961sec
1551: [82] G2=lex_tl(K,V,0,V,1)$
1552: 31.186sec + gc : 3.500sec
1553: [83] gb_comp(G0,G1);
1554: 1
1555: 10msec
1556: [84] gb_comp(G0,G2);
1557: 1
1558: @end example
1559:
1560: @table @t
1.2 noro 1561: \JP @item $B;2>H(B
1562: \EG @item References
1.1 noro 1563: @fref{dp_gr_main dp_gr_mod_main},
1.2 noro 1564: \JP @fref{dp_ord}, @fref{$BJ,;67W;;(B}
1565: \EG @fref{dp_ord}, @fref{Distributed computation}
1.1 noro 1566: @end table
1567:
1.2 noro 1568: \JP @node lex_hensel_gsl tolex_gsl tolex_gsl_d,,, $B%0%l%V%J4pDl$K4X$9$kH!?t(B
1569: \EG @node lex_hensel_gsl tolex_gsl tolex_gsl_d,,, Functions for Groebner basis computation
1.1 noro 1570: @subsection @code{lex_hensel_gsl}, @code{tolex_gsl}, @code{tolex_gsl_d}
1571: @findex lex_hensel_gsl
1572: @findex tolex_gsl
1573: @findex tolex_gsl_d
1574:
1575: @table @t
1576: @item lex_hensel_gsl(@var{plist},@var{vlist1},@var{order},@var{vlist2},@var{homo})
1.2 noro 1577: \JP :: GSL $B7A<0$N%$%G%"%k4pDl$N7W;;(B
1578: \EG ::Computation of an GSL form ideal basis
1.1 noro 1579: @item tolex_gsl(@var{plist},@var{vlist1},@var{order},@var{vlist2},@var{homo})
1580: @itemx tolex_gsl_d(@var{plist},@var{vlist1},@var{order},@var{vlist2},@var{homo},@var{procs})
1.2 noro 1581: \JP :: $B%0%l%V%J4pDl$rF~NO$H$9$k(B, GSL $B7A<0$N%$%G%"%k4pDl$N7W;;(B
1582: \EG :: Computation of an GSL form ideal basis stating from a Groebner basis
1.1 noro 1583: @end table
1584:
1585: @table @var
1586: @item return
1.2 noro 1587: \JP $B%j%9%H(B
1588: \EG list
1.4 ! noro 1589: @item plist vlist1 vlist2 procs
1.2 noro 1590: \JP $B%j%9%H(B
1591: \EG list
1.1 noro 1592: @item order
1.2 noro 1593: \JP $B?t(B, $B%j%9%H$^$?$O9TNs(B
1594: \EG number, list or matrix
1.1 noro 1595: @item homo
1.2 noro 1596: \JP $B%U%i%0(B
1597: \EG flag
1.1 noro 1598: @end table
1599:
1600: @itemize @bullet
1.2 noro 1601: \BJP
1.1 noro 1602: @item
1603: @code{lex_hensel_gsl()} $B$O(B @code{lex_hensel()} $B$N(B, @code{tolex_gsl()} $B$O(B
1604: @code{tolex()} $B$NJQ<o$G(B, $B7k2L$N$_$,0[$J$k(B.
1605: @code{tolex_gsl_d()} $B$O(B, $B4pDl7W;;$r(B, @code{procs} $B$G;XDj$5$l$k;R%W%m%;%9$K(B
1606: $BJ,;67W;;$5$;$k(B.
1607: @item
1608: $BF~NO$,(B 0 $B<!85%7%9%F%`$G(B, $B$=$N<-=q<0=g=x%0%l%V%J4pDl$,(B
1609: @code{[f0,x1-f1,...,xn-fn]} (@code{f0},...,@code{fn} $B$O(B
1610: @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,
1611: @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)
1612: $B$rJV$9(B.
1.2 noro 1613: $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 1614: @code{x0} $B$N(B1 $BJQ?tB?9`<0$G(B,
1615: $B2r$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')]}
1616: $B$H$J$k(B. $B<-=q<0=g=x%0%l%V%J4pDl$,>e$N$h$&$J7A$G$J$$>l9g(B, @code{tolex()} $B$K(B
1617: $B$h$kDL>o$N%0%l%V%J4pDl$rJV$9(B.
1618: @item
1619: GSL $B7A<0$K$h$jI=$5$l$k4pDl$O%0%l%V%J4pDl$G$O$J$$$,(B, $B0lHL$K78?t$,(B SL $B7A<0(B
1620: $B$N%0%l%V%J4pDl$h$jHs>o$K>.$5$$$?$a7W;;$bB.$/(B, $B2r$b5a$a$d$9$$(B.
1621: @code{tolex_gsl_d()} $B$GI=<($5$l$k;~4V$O(B, $B$3$NH!?t$,<B9T$5$l$F$$$k%W%m%;%9$K(B
1622: $B$*$$$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 1623: \E
1624: \BEG
1625: @item
1626: @code{lex_hensel_gsl()} and @code{lex_hensel()} are variants of
1627: @code{tolex_gsl()} and @code{tolex()} respectively. The results are
1628: Groebner basis or a kind of ideal basis, called GSL form.
1629: @code{tolex_gsl_d()} does basis computations in parallel on child
1630: processes specified in @code{procs}.
1631:
1632: @item
1633: If the input is zero-dimensional and a lex order Groebner basis has
1634: the form @code{[f0,x1-f1,...,xn-fn]} (@code{f0},...,@code{fn} are
1635: univariate polynomials of @code{x0}; SL form), then this these
1636: functions return a list such as
1637: @code{[[x1,g1,d1],...,[xn,gn,dn],[x0,f0,f0']]} (GSL form). In this list
1638: @code{gi} is a univariate polynomial of @code{x0} such that
1639: @code{di*f0'*fi-gi} divides @code{f0} and the roots of the input ideal is
1640: @code{[x1=g1/(d1*f0'),...,xn=gn/(dn*f0')]} for @code{x0}
1641: such that @code{f0(x0)=0}.
1642: If the lex order Groebner basis does not have the above form,
1643: these functions return
1644: a lex order Groebner basis computed by @code{tolex()}.
1645: @item
1646: Though an ideal basis represented as GSL form is not a Groebner basis
1647: we can expect that the coefficients are much smaller than those in a Groebner
1648: basis and that the computation is efficient.
1649: The CPU time shown after an execution of @code{tolex_gsl_d()} indicates
1650: that of the master process, and it does not include the time in child
1651: processes.
1652: \E
1.1 noro 1653: @end itemize
1654:
1655: @example
1656: [103] K=katsura(5)$
1657: [104] V=[u5,u4,u3,u2,u1,u0]$
1658: [105] G0=gr(K,V,0)$
1659: [106] GSL=tolex_gsl(G0,V,0,V)$
1660: [107] GSL[0];
1661: [u1,8635837421130477667200000000*u0^31-...]
1662: [108] GSL[1];
1663: [u2,10352277157007342793600000000*u0^31-...]
1664: [109] GSL[5];
1665: [u0,11771021876193064124640000000*u0^32-...,376672700038178051988480000000*u0^31-...]
1666: @end example
1667:
1668: @table @t
1.2 noro 1669: \JP @item $B;2>H(B
1670: \EG @item References
1.1 noro 1671: @fref{lex_hensel lex_tl tolex tolex_d tolex_tl},
1.2 noro 1672: \JP @fref{$BJ,;67W;;(B}
1673: \EG @fref{Distributed computation}
1.1 noro 1674: @end table
1675:
1.2 noro 1676: \JP @node gr_minipoly minipoly,,, $B%0%l%V%J4pDl$K4X$9$kH!?t(B
1677: \EG @node gr_minipoly minipoly,,, Functions for Groebner basis computation
1.1 noro 1678: @subsection @code{gr_minipoly}, @code{minipoly}
1679: @findex gr_minipoly
1680: @findex minipoly
1681:
1682: @table @t
1683: @item gr_minipoly(@var{plist},@var{vlist},@var{order},@var{poly},@var{v},@var{homo})
1.2 noro 1684: \JP :: $BB?9`<0$N(B, $B%$%G%"%k$rK!$H$7$?:G>.B?9`<0$N7W;;(B
1685: \EG :: Computation of the minimal polynomial of a polynomial modulo an ideal
1.1 noro 1686: @item minipoly(@var{plist},@var{vlist},@var{order},@var{poly},@var{v})
1.2 noro 1687: \JP :: $B%0%l%V%J4pDl$rF~NO$H$9$k(B, $BB?9`<0$N:G>.B?9`<0$N7W;;(B
1688: \EG :: Computation of the minimal polynomial of a polynomial modulo an ideal
1.1 noro 1689: @end table
1690:
1691: @table @var
1692: @item return
1.2 noro 1693: \JP $BB?9`<0(B
1694: \EG polynomial
1.4 ! noro 1695: @item plist vlist
1.2 noro 1696: \JP $B%j%9%H(B
1697: \EG list
1.1 noro 1698: @item order
1.2 noro 1699: \JP $B?t(B, $B%j%9%H$^$?$O9TNs(B
1700: \EG number, list or matrix
1.1 noro 1701: @item poly
1.2 noro 1702: \JP $BB?9`<0(B
1703: \EG polynomial
1.1 noro 1704: @item v
1.2 noro 1705: \JP $BITDj85(B
1706: \EG indeterminate
1.1 noro 1707: @item homo
1.2 noro 1708: \JP $B%U%i%0(B
1709: \EG flag
1.1 noro 1710: @end table
1711:
1712: @itemize @bullet
1.2 noro 1713: \BJP
1.1 noro 1714: @item
1715: @code{gr_minipoly()} $B$O%0%l%V%J4pDl$N7W;;$+$i9T$$(B, @code{minipoly()} $B$O(B
1716: $BF~NO$r%0%l%V%J4pDl$H$_$J$9(B.
1717: @item
1718: $B%$%G%"%k(B I $B$,BN(B K $B>e$NB?9`<04D(B K[X] $B$N(B 0 $B<!85%$%G%"%k$N;~(B,
1719: K[@var{v}] $B$N85(B f(@var{v}) $B$K(B f(@var{p}) mod I $B$rBP1~$5$;$k(B
1720: $B4D=`F17?$N3K$O(B 0 $B$G$J$$B?9`<0$K$h$j@8@.$5$l$k(B. $B$3$N@8@.85$r(B @var{p}
1721: $B$N(B, $BK!(B @var{I} $B$G$N:G>.B?9`<0$H8F$V(B.
1722: @item
1723: @code{gr_minipoly()}, @code{minipoly()} $B$O(B, $BB?9`<0(B @var{p} $B$N:G>.B?9`<0(B
1724: $B$r5a$a(B, @var{v} $B$rJQ?t$H$9$kB?9`<0$H$7$FJV$9(B.
1725: @item
1726: $B:G>.B?9`<0$O(B, $B%0%l%V%J4pDl$N(B 1 $B$D$N85$H$7$F7W;;$9$k$3$H$b$G$-$k$,(B,
1727: $B:G>.B?9`<0$N$_$r5a$a$?$$>l9g(B, @code{minipoly()}, @code{gr_minipoly()} $B$O(B
1728: $B%0%l%V%J4pDl$rMQ$$$kJ}K!$KHf$Y$F8zN($,$h$$(B.
1729: @item
1730: @code{gr_minipoly()} $B$K;XDj$9$k9`=g=x$H$7$F$O(B, $BDL>oA4<!?t5U<-=q<0=g=x$r(B
1731: $BMQ$$$k(B.
1.2 noro 1732: \E
1733: \BEG
1734: @item
1735: @code{gr_minipoly()} begins by computing a Groebner basis.
1736: @code{minipoly()} regards an input as a Groebner basis with respect to
1737: the variable order @var{vlist} and the order type @var{order}.
1738: @item
1739: Let K be a field. If an ideal @var{I} in K[X] is zero-dimensional, then, for
1740: a polynomial @var{p} in K[X], the kernel of a homomorphism from
1741: K[@var{v}] to K[X]/@var{I} which maps f(@var{v}) to f(@var{p}) mod @var{I}
1742: is generated by a polynomial. The generator is called the minimal polynomial
1743: of @var{p} modulo @var{I}.
1744: @item
1745: @code{gr_minipoly()} and @code{minipoly()} computes the minimal polynomial
1746: of a polynomial @var{p} and returns it as a polynomial of @var{v}.
1747: @item
1748: The minimal polynomial can be computed as an element of a Groebner basis.
1749: But if we are only interested in the minimal polynomial,
1750: @code{minipoly()} and @code{gr_minipoly()} can compute it more efficiently
1751: than methods using Groebner basis computation.
1752: @item
1753: It is recommended to use a degree reverse lex order as a term order
1754: for @code{gr_minipoly()}.
1755: \E
1.1 noro 1756: @end itemize
1757:
1758: @example
1759: [117] G=tolex(G0,V,0,V)$
1760: 43.818sec + gc : 11.202sec
1761: [118] GSL=tolex_gsl(G0,V,0,V)$
1762: 17.123sec + gc : 2.590sec
1763: [119] MP=minipoly(G0,V,0,u0,z)$
1764: 4.370sec + gc : 780msec
1765: @end example
1766:
1767: @table @t
1.2 noro 1768: \JP @item $B;2>H(B
1769: \EG @item References
1.1 noro 1770: @fref{lex_hensel lex_tl tolex tolex_d tolex_tl}.
1771: @end table
1772:
1.2 noro 1773: \JP @node tolexm minipolym,,, $B%0%l%V%J4pDl$K4X$9$kH!?t(B
1774: \EG @node tolexm minipolym,,, Functions for Groebner basis computation
1.1 noro 1775: @subsection @code{tolexm}, @code{minipolym}
1776: @findex tolexm
1777: @findex minipolym
1778:
1779: @table @t
1780: @item tolexm(@var{plist},@var{vlist1},@var{order},@var{vlist2},@var{mod})
1.2 noro 1781: \JP :: $BK!(B @var{mod} $B$G$N4pDlJQ49$K$h$k%0%l%V%J4pDl7W;;(B
1782: \EG :: Groebner basis computation modulo @var{mod} by change of ordering.
1.1 noro 1783: @item minipolym(@var{plist},@var{vlist1},@var{order},@var{poly},@var{v},@var{mod})
1.2 noro 1784: \JP :: $BK!(B @var{mod} $B$G$N%0%l%V%J4pDl$K$h$kB?9`<0$N:G>.B?9`<0$N7W;;(B
1785: \EG :: Minimal polynomial computation modulo @var{mod} the same method as
1.1 noro 1786: @end table
1787:
1788: @table @var
1789: @item return
1.2 noro 1790: \JP @code{tolexm()} : $B%j%9%H(B, @code{minipolym()} : $BB?9`<0(B
1791: \EG @code{tolexm()} : list, @code{minipolym()} : polynomial
1.4 ! noro 1792: @item plist vlist1 vlist2
1.2 noro 1793: \JP $B%j%9%H(B
1794: \EG list
1.1 noro 1795: @item order
1.2 noro 1796: \JP $B?t(B, $B%j%9%H$^$?$O9TNs(B
1797: \EG number, list or matrix
1.1 noro 1798: @item mod
1.2 noro 1799: \JP $BAG?t(B
1800: \EG prime
1.1 noro 1801: @end table
1802:
1803: @itemize @bullet
1.2 noro 1804: \BJP
1.1 noro 1805: @item
1806: $BF~NO(B @var{plist} $B$O$$$:$l$b(B $BJQ?t=g=x(B @var{vlist1}, $B9`=g=x7?(B @var{order},
1807: $BK!(B @var{mod} $B$K$*$1$k%0%l%V%J4pDl$G$J$1$l$P$J$i$J$$(B.
1808: @item
1809: @code{minipolym()} $B$O(B @code{minipoly} $B$KBP1~$9$k7W;;$rK!(B @var{mod}$B$G9T$&(B.
1810: @item
1811: @code{tolexm()} $B$O(B FGLM $BK!$K$h$k4pDlJQ49$K$h$j(B @var{vlist2},
1812: $B<-=q<0=g=x$K$h$k%0%l%V%J4pDl$r7W;;$9$k(B.
1.2 noro 1813: \E
1814: \BEG
1815: @item
1816: An input @var{plist} must be a Groebner basis modulo @var{mod}
1817: with respect to the variable order @var{vlist1} and the order type @var{order}.
1818: @item
1819: @code{minipolym()} executes the same computation as in @code{minipoly}.
1820: @item
1821: @code{tolexm()} computes a lex order Groebner basis modulo @var{mod}
1822: with respect to the variable order @var{vlist2}, by using FGLM algorithm.
1823: \E
1.1 noro 1824: @end itemize
1825:
1826: @example
1827: [197] tolexm(G0,V,0,V,31991);
1828: [8271*u0^31+10435*u0^30+816*u0^29+26809*u0^28+...,...]
1829: [198] minipolym(G0,V,0,u0,z,31991);
1830: z^32+11405*z^31+20868*z^30+21602*z^29+...
1831: @end example
1832:
1833: @table @t
1.2 noro 1834: \JP @item $B;2>H(B
1835: \EG @item References
1.1 noro 1836: @fref{lex_hensel lex_tl tolex tolex_d tolex_tl},
1837: @fref{gr_minipoly minipoly}.
1838: @end table
1839:
1.2 noro 1840: \JP @node dp_gr_main dp_gr_mod_main,,, $B%0%l%V%J4pDl$K4X$9$kH!?t(B
1841: \EG @node dp_gr_main dp_gr_mod_main,,, Functions for Groebner basis computation
1.1 noro 1842: @subsection @code{dp_gr_main}, @code{dp_gr_mod_main}
1843: @findex dp_gr_main
1844: @findex dp_gr_mod_main
1845:
1846: @table @t
1847: @item dp_gr_main(@var{plist},@var{vlist},@var{homo},@var{modular},@var{order})
1848: @itemx dp_gr_mod_main(@var{plist},@var{vlist},@var{homo},@var{modular},@var{order})
1.2 noro 1849: \JP :: $B%0%l%V%J4pDl$N7W;;(B ($BAH$_9~$_H!?t(B)
1850: \EG :: Groebner basis computation (built-in functions)
1.1 noro 1851: @end table
1852:
1853: @table @var
1854: @item return
1.2 noro 1855: \JP $B%j%9%H(B
1856: \EG list
1.4 ! noro 1857: @item plist vlist
1.2 noro 1858: \JP $B%j%9%H(B
1859: \EG list
1.1 noro 1860: @item order
1.2 noro 1861: \JP $B?t(B, $B%j%9%H$^$?$O9TNs(B
1862: \EG number, list or matrix
1.1 noro 1863: @item homo
1.2 noro 1864: \JP $B%U%i%0(B
1865: \EG flag
1.1 noro 1866: @item modular
1.2 noro 1867: \JP $B%U%i%0$^$?$OAG?t(B
1868: \EG flag or prime
1.1 noro 1869: @end table
1870:
1871: @itemize @bullet
1.2 noro 1872: \BJP
1.1 noro 1873: @item
1874: $B$3$l$i$NH!?t$O(B, $B%0%l%V%J4pDl7W;;$N4pK\E*AH$_9~$_H!?t$G$"$j(B, @code{gr()},
1875: @code{hgr()}, @code{gr_mod()} $B$J$I$O$9$Y$F$3$l$i$NH!?t$r8F$S=P$7$F7W;;(B
1876: $B$r9T$C$F$$$k(B.
1877: @item
1878: $B%U%i%0(B @var{homo} $B$,(B 0 $B$G$J$$;~(B, $BF~NO$r@F<!2=$7$F$+$i(B Buchberger $B%"%k%4%j%:%`(B
1879: $B$r<B9T$9$k(B.
1880: @item
1881: @code{dp_gr_mod_main()} $B$KBP$7$F$O(B, @var{modular} $B$O(B, GF(@var{modular}) $B>e(B
1882: $B$G$N7W;;$r0UL#$9$k(B.
1883: @code{dp_gr_main()} $B$KBP$7$F$O(B, @var{modular} $B$O<!$N$h$&$J0UL#$r;}$D(B.
1884: @enumerate
1885: @item
1886: @var{modular} $B$,(B 1 $B$N;~(B, trace-lifting $B$K$h$k7W;;$r9T$&(B. $BAG?t$O(B
1887: @code{lprime(0)} $B$+$i=g$K@.8y$9$k$^$G(B @code{lprime()} $B$r8F$S=P$7$F@8@.$9$k(B.
1888: @item
1889: @var{modular} $B$,(B 2 $B0J>e$N<+A3?t$N;~(B, $B$=$NCM$rAG?t$H$_$J$7$F(B trace-lifting
1890: $B$r9T$&(B. $B$=$NAG?t$G<:GT$7$?>l9g(B, 0 $B$rJV$9(B.
1891: @item
1892: @var{modular} $B$,Ii$N>l9g(B,
1893: @var{-modular} $B$KBP$7$F>e=R$N5,B'$,E,MQ$5$l$k$,(B, trace-lifting $B$N:G=*(B
1894: $BCJ3,$N%0%l%V%J4pDl%A%'%C%/$H%$%G%"%k%a%s%P%7%C%W%A%'%C%/$,>JN,$5$l$k(B.
1895: @end enumerate
1896:
1897: @item
1898: @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
1899: @code{dp_gr_main(P,V,1,1,O)}, @code{gr_mod(P,V,O,M)} $B$O(B
1900: @code{dp_gr_mod_main(P,V,0,M,O)} $B$r$=$l$>$l<B9T$9$k(B.
1901: @item
1902: @var{homo}, @var{modular} $B$NB>$K(B, @code{dp_gr_flags()} $B$G@_Dj$5$l$k(B
1903: $B$5$^$6$^$J%U%i%0$K$h$j7W;;$,@)8f$5$l$k(B.
1.2 noro 1904: \E
1905: \BEG
1906: @item
1907: These functions are fundamental built-in functions for Groebner basis
1908: computation and @code{gr()},@code{hgr()} and @code{gr_mod()}
1909: are all interfaces to these functions.
1910: @item
1911: If @var{homo} is not equal to 0, homogenization is applied before entering
1912: Buchberger algorithm
1913: @item
1914: For @code{dp_gr_mod_main()}, @var{modular} means a computation over
1915: GF(@var{modular}).
1916: For @code{dp_gr_main()}, @var{modular} has the following mean.
1917: @enumerate
1918: @item
1919: If @var{modular} is 1 , trace lifting is used. Primes for trace lifting
1920: are generated by @code{lprime()}, starting from @code{lprime(0)}, until
1921: the computation succeeds.
1922: @item
1923: If @var{modular} is an integer greater than 1, the integer is regarded as a
1924: prime and trace lifting is executed by using the prime. If the computation
1925: fails then 0 is returned.
1926: @item
1927: If @var{modular} is negative, the above rule is applied for @var{-modular}
1928: but the Groebner basis check and ideal-membership check are omitted in
1929: the last stage of trace lifting.
1930: @end enumerate
1931:
1932: @item
1933: @code{gr(P,V,O)}, @code{hgr(P,V,O)} and @code{gr_mod(P,V,O,M)} execute
1934: @code{dp_gr_main(P,V,0,1,O)}, @code{dp_gr_main(P,V,1,1,O)}
1935: and @code{dp_gr_mod_main(P,V,0,M,O)} respectively.
1936: @item
1937: Actual computation is controlled by various parameters set by
1938: @code{dp_gr_flags()}, other then by @var{homo} and @var{modular}.
1939: \E
1.1 noro 1940: @end itemize
1941:
1942: @table @t
1.2 noro 1943: \JP @item $B;2>H(B
1944: \EG @item References
1.1 noro 1945: @fref{dp_ord},
1946: @fref{dp_gr_flags dp_gr_print},
1947: @fref{gr hgr gr_mod},
1.2 noro 1948: \JP @fref{$B7W;;$*$h$SI=<($N@)8f(B}.
1949: \EG @fref{Controlling Groebner basis computations}
1.1 noro 1950: @end table
1951:
1.2 noro 1952: \JP @node dp_f4_main dp_f4_mod_main,,, $B%0%l%V%J4pDl$K4X$9$kH!?t(B
1953: \EG @node dp_f4_main dp_f4_mod_main,,, Functions for Groebner basis computation
1.1 noro 1954: @subsection @code{dp_f4_main}, @code{dp_f4_mod_main}
1955: @findex dp_f4_main
1956: @findex dp_f4_mod_main
1957:
1958: @table @t
1959: @item dp_f4_main(@var{plist},@var{vlist},@var{order})
1960: @itemx dp_f4_mod_main(@var{plist},@var{vlist},@var{order})
1.2 noro 1961: \JP :: F4 $B%"%k%4%j%:%`$K$h$k%0%l%V%J4pDl$N7W;;(B ($BAH$_9~$_H!?t(B)
1962: \EG :: Groebner basis computation by F4 algorithm (built-in functions)
1.1 noro 1963: @end table
1964:
1965: @table @var
1966: @item return
1.2 noro 1967: \JP $B%j%9%H(B
1968: \EG list
1.4 ! noro 1969: @item plist vlist
1.2 noro 1970: \JP $B%j%9%H(B
1971: \EG list
1.1 noro 1972: @item order
1.2 noro 1973: \JP $B?t(B, $B%j%9%H$^$?$O9TNs(B
1974: \EG number, list or matrix
1.1 noro 1975: @end table
1976:
1977: @itemize @bullet
1.2 noro 1978: \BJP
1.1 noro 1979: @item
1980: F4 $B%"%k%4%j%:%`$K$h$j%0%l%V%J4pDl$N7W;;$r9T$&(B.
1981: @item
1982: F4 $B%"%k%4%j%:%`$O(B, J.C. Faugere $B$K$h$jDs>'$5$l$??7@$Be%0%l%V%J4pDl(B
1983: $B;;K!$G$"$j(B, $BK\<BAu$O(B, $BCf9q>jM>DjM}$K$h$k@~7AJ}Dx<05a2r$rMQ$$$?(B
1984: $B;n83E*$J<BAu$G$"$k(B.
1985: @item
1986: $B0z?t$*$h$SF0:n$O$=$l$>$l(B @code{dp_gr_main()}, @code{dp_gr_mod_main()}
1987: $B$HF1MM$G$"$k(B.
1.2 noro 1988: \E
1989: \BEG
1990: @item
1991: These functions compute Groebner bases by F4 algorithm.
1992: @item
1993: F4 is a new generation algorithm for Groebner basis computation
1994: invented by J.C. Faugere. The current implementation of @code{dp_f4_main()}
1995: uses Chinese Remainder theorem and not highly optimized.
1996: @item
1997: Arguments and actions are the same as those of
1998: @code{dp_gr_main()}, @code{dp_gr_mod_main()}.
1999: \E
1.1 noro 2000: @end itemize
2001:
2002: @table @t
1.2 noro 2003: \JP @item $B;2>H(B
2004: \EG @item References
1.1 noro 2005: @fref{dp_ord},
2006: @fref{dp_gr_flags dp_gr_print},
2007: @fref{gr hgr gr_mod},
1.2 noro 2008: \JP @fref{$B7W;;$*$h$SI=<($N@)8f(B}.
2009: \EG @fref{Controlling Groebner basis computations}
1.1 noro 2010: @end table
2011:
1.2 noro 2012: \JP @node dp_gr_flags dp_gr_print,,, $B%0%l%V%J4pDl$K4X$9$kH!?t(B
2013: \EG @node dp_gr_flags dp_gr_print,,, Functions for Groebner basis computation
1.1 noro 2014: @subsection @code{dp_gr_flags}, @code{dp_gr_print}
2015: @findex dp_gr_flags
2016: @findex dp_gr_print
2017:
2018: @table @t
2019: @item dp_gr_flags([@var{list}])
2020: @itemx dp_gr_print([@var{0|1}])
1.2 noro 2021: \JP :: $B7W;;$*$h$SI=<(MQ%Q%i%a%?$N@_Dj(B, $B;2>H(B
2022: \BEG :: Set and show various parameters for cotrolling computations
2023: and showing informations.
2024: \E
1.1 noro 2025: @end table
2026:
2027: @table @var
2028: @item return
1.2 noro 2029: \JP $B@_DjCM(B
2030: \EG value currently set
1.1 noro 2031: @item list
1.2 noro 2032: \JP $B%j%9%H(B
2033: \EG list
1.1 noro 2034: @end table
2035:
2036: @itemize @bullet
1.2 noro 2037: \BJP
1.1 noro 2038: @item
2039: @code{dp_gr_main()}, @code{dp_gr_mod_main()} $B<B9T;~$K$*$1$k$5$^$6$^(B
2040: $B$J%Q%i%a%?$r@_Dj(B, $B;2>H$9$k(B.
2041: @item
2042: $B0z?t$,$J$$>l9g(B, $B8=:_$N@_Dj$,JV$5$l$k(B.
2043: @item
2044: $B0z?t$O(B, @code{["Print",1,"NoSugar",1,...]} $B$J$k7A$N%j%9%H$G(B, $B:8$+$i=g$K(B
2045: $B@_Dj$5$l$k(B. $B%Q%i%a%?L>$OJ8;zNs$GM?$($kI,MW$,$"$k(B.
2046: @item
2047: @code{dp_gr_print()} $B$O(B, $BFC$K%Q%i%a%?(B @code{Print} $B$NCM$rD>@\@_Dj(B, $B;2>H(B
2048: $B$G$-$k(B. $B$3$l$O(B, @code{dp_gr_main()} $B$J$I$r%5%V%k!<%A%s$H$7$FMQ$$$k%f!<%6(B
2049: $BH!?t$K$*$$$F(B, @code{Print} $B$NCM$r8+$F(B, $B$=$N%5%V%k!<%A%s$,Cf4V>pJs$NI=<((B
2050: $B$r9T$&:]$K(B, $B?WB.$K%U%i%0$r8+$k$3$H$,$G$-$k$h$&$KMQ0U$5$l$F$$$k(B.
1.2 noro 2051: \E
2052: \BEG
2053: @item
2054: @code{dp_gr_flags()} sets and shows various parameters for Groebner basis
2055: computation.
2056: @item
2057: If no argument is specified the current settings are returned.
2058: @item
2059: Arguments must be specified as a list such as
2060: @code{["Print",1,"NoSugar",1,...]}. Names of parameters must be character
2061: strings.
2062: @item
2063: @code{dp_gr_print()} is used to set and show the value of a parameter
2064: @code{Print}. This functions is prepared to get quickly the value of
2065: @code{Print} when a user defined function calling @code{dp_gr_main()} etc.
2066: uses the value as a flag for showing intermediate informations.
2067: \E
1.1 noro 2068: @end itemize
2069:
2070: @table @t
1.2 noro 2071: \JP @item $B;2>H(B
2072: \EG @item References
2073: \JP @fref{$B7W;;$*$h$SI=<($N@)8f(B}
2074: \EG @fref{Controlling Groebner basis computations}
1.1 noro 2075: @end table
2076:
1.2 noro 2077: \JP @node dp_ord,,, $B%0%l%V%J4pDl$K4X$9$kH!?t(B
2078: \EG @node dp_ord,,, Functions for Groebner basis computation
1.1 noro 2079: @subsection @code{dp_ord}
2080: @findex dp_ord
2081:
2082: @table @t
2083: @item dp_ord([@var{order}])
1.2 noro 2084: \JP :: $BJQ?t=g=x7?$N@_Dj(B, $B;2>H(B
2085: \EG :: Set and show the ordering type.
1.1 noro 2086: @end table
2087:
2088: @table @var
2089: @item return
1.2 noro 2090: \JP $BJQ?t=g=x7?(B ($B?t(B, $B%j%9%H$^$?$O9TNs(B)
2091: \EG ordering type (number, list or matrix)
1.1 noro 2092: @item order
1.2 noro 2093: \JP $B?t(B, $B%j%9%H$^$?$O9TNs(B
2094: \EG number, list or matrix
1.1 noro 2095: @end table
2096:
2097: @itemize @bullet
1.2 noro 2098: \BJP
1.1 noro 2099: @item
2100: $B0z?t$,$"$k;~(B, $BJQ?t=g=x7?$r(B @var{order} $B$K@_Dj$9$k(B. $B0z?t$,$J$$;~(B,
2101: $B8=:_@_Dj$5$l$F$$$kJQ?t=g=x7?$rJV$9(B.
2102:
2103: @item
2104: $BJ,;6I=8=B?9`<0$K4X$9$kH!?t(B, $B1i;;$O0z?t$H$7$FJQ?t=g=x7?$r$H$k$b$N$H$H$i$J$$$b$N(B
2105: $B$,$"$j(B, $B$H$i$J$$$b$N$K4X$7$F$O(B, $B$=$N;~E@$G@_Dj$5$l$F$$$kCM$rMQ$$$F7W;;$,(B
2106: $B9T$o$l$k(B.
2107:
2108: @item
2109: @code{gr()} $B$J$I(B, $B0z?t$H$7$FJQ?t=g=x7?$r$H$k$b$N$O(B, $BFbIt$G(B @code{dp_ord()}
2110: $B$r8F$S=P$7(B, $BJQ?t=g=x7?$r@_Dj$9$k(B. $B$3$N@_Dj$O(B, $B7W;;=*N;8e$b@8$-;D$k(B.
2111:
2112: @item
2113: $BJ,;6I=8=B?9`<0$N;MB'1i;;$b(B, $B@_Dj$5$l$F$$$kCM$rMQ$$$F7W;;$5$l$k(B. $B=>$C$F(B,
2114: $B$=$NB?9`<0$,@8@.$5$l$?;~E@$K$*$1$kJQ?t=g=x7?$,(B, $B;MB'1i;;;~$K@5$7$/@_Dj(B
2115: $B$5$l$F$$$J$1$l$P$J$i$J$$(B. $B$^$?(B, $B1i;;BP>]$H$J$kB?9`<0$O(B, $BF10l$NJQ?t=g=x(B
2116: $B7?$K4p$E$$$F@8@.$5$l$?$b$N$G$J$1$l$P$J$i$J$$(B.
2117:
2118: @item
2119: $B%H%C%W%l%Y%kH!?t0J30$NH!?t$rD>@\8F$S=P$9>l9g$K$O(B, $B$3$NH!?t$K$h$j(B
2120: $BJQ?t=g=x7?$r@5$7$/@_Dj$7$J$1$l$P$J$i$J$$(B.
1.2 noro 2121: \E
2122: \BEG
2123: @item
2124: If an argument is specified, the function
2125: sets the current ordering type to @var{order}.
2126: If no argument is specified, the function returns the ordering
2127: type currently set.
2128:
2129: @item
2130: There are two types of functions concerning distributed polynomial,
2131: functions which take a ordering type and those which don't take it.
2132: The latter ones use the current setting.
2133:
2134: @item
2135: Functions such as @code{gr()}, which need a ordering type as an argument,
2136: call @code{dp_ord()} internally during the execution.
2137: The setting remains after the execution.
2138:
2139: Fundamental arithmetics for distributed polynomial also use the current
2140: setting. Therefore, when such arithmetics for distributed polynomials
2141: are done, the current setting must coincide with the ordering type
2142: which was used upon the creation of the polynomials. It is assumed
2143: that such polynomials were generated under the same ordering type.
2144:
2145: @item
2146: Type of term ordering must be correctly set by this function
2147: when functions other than top level functions are called directly.
2148: \E
1.1 noro 2149: @end itemize
2150:
2151: @example
2152: [19] dp_ord(0)$
2153: [20] <<1,2,3>>+<<3,1,1>>;
2154: (1)*<<1,2,3>>+(1)*<<3,1,1>>
2155: [21] dp_ord(2)$
2156: [22] <<1,2,3>>+<<3,1,1>>;
2157: (1)*<<3,1,1>>+(1)*<<1,2,3>>
2158: @end example
2159:
2160: @table @t
1.2 noro 2161: \JP @item $B;2>H(B
2162: \EG @item References
2163: \JP @fref{$B9`=g=x$N@_Dj(B}
2164: \EG @fref{Setting term orderings}
1.1 noro 2165: @end table
2166:
1.2 noro 2167: \JP @node dp_ptod,,, $B%0%l%V%J4pDl$K4X$9$kH!?t(B
2168: \EG @node dp_ptod,,, Functions for Groebner basis computation
1.1 noro 2169: @subsection @code{dp_ptod}
2170: @findex dp_ptod
2171:
2172: @table @t
2173: @item dp_ptod(@var{poly},@var{vlist})
1.2 noro 2174: \JP :: $BB?9`<0$rJ,;6I=8=B?9`<0$KJQ49$9$k(B.
2175: \EG :: Converts an ordinary polynomial into a distributed polynomial.
1.1 noro 2176: @end table
2177:
2178: @table @var
2179: @item return
1.2 noro 2180: \JP $BJ,;6I=8=B?9`<0(B
2181: \EG distributed polynomial
1.1 noro 2182: @item poly
1.2 noro 2183: \JP $BB?9`<0(B
2184: \EG polynomial
1.1 noro 2185: @item vlist
1.2 noro 2186: \JP $B%j%9%H(B
2187: \EG list
1.1 noro 2188: @end table
2189:
2190: @itemize @bullet
1.2 noro 2191: \BJP
1.1 noro 2192: @item
2193: $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.
2194: @item
2195: @var{vlist} $B$K4^$^$l$J$$ITDj85$O(B, $B78?tBN$KB0$9$k$H$7$FJQ49$5$l$k(B.
1.2 noro 2196: \E
2197: \BEG
2198: @item
2199: According to the variable ordering @var{vlist} and current
2200: type of term ordering, this function converts an ordinary
2201: polynomial into a distributed polynomial.
2202: @item
2203: Indeterminates not included in @var{vlist} are regarded to belong to
2204: the coefficient field.
2205: \E
1.1 noro 2206: @end itemize
2207:
2208: @example
2209: [50] dp_ord(0);
2210: 1
2211: [51] dp_ptod((x+y+z)^2,[x,y,z]);
2212: (1)*<<2,0,0>>+(2)*<<1,1,0>>+(1)*<<0,2,0>>+(2)*<<1,0,1>>+(2)*<<0,1,1>>
2213: +(1)*<<0,0,2>>
2214: [52] dp_ptod((x+y+z)^2,[x,y]);
2215: (1)*<<2,0>>+(2)*<<1,1>>+(1)*<<0,2>>+(2*z)*<<1,0>>+(2*z)*<<0,1>>+(z^2)*<<0,0>>
2216: @end example
2217:
2218: @table @t
1.2 noro 2219: \JP @item $B;2>H(B
2220: \EG @item References
1.1 noro 2221: @fref{dp_dtop},
2222: @fref{dp_ord}.
2223: @end table
2224:
1.2 noro 2225: \JP @node dp_dtop,,, $B%0%l%V%J4pDl$K4X$9$kH!?t(B
2226: \EG @node dp_dtop,,, Functions for Groebner basis computation
1.1 noro 2227: @subsection @code{dp_dtop}
2228: @findex dp_dtop
2229:
2230: @table @t
2231: @item dp_dtop(@var{dpoly},@var{vlist})
1.2 noro 2232: \JP :: $BJ,;6I=8=B?9`<0$rB?9`<0$KJQ49$9$k(B.
2233: \EG :: Converts a distributed polynomial into an ordinary polynomial.
1.1 noro 2234: @end table
2235:
2236: @table @var
2237: @item return
1.2 noro 2238: \JP $BB?9`<0(B
2239: \EG polynomial
1.1 noro 2240: @item dpoly
1.2 noro 2241: \JP $BJ,;6I=8=B?9`<0(B
2242: \EG distributed polynomial
1.1 noro 2243: @item vlist
1.2 noro 2244: \JP $B%j%9%H(B
2245: \EG list
1.1 noro 2246: @end table
2247:
2248: @itemize @bullet
1.2 noro 2249: \BJP
1.1 noro 2250: @item
2251: $BJ,;6I=8=B?9`<0$r(B, $BM?$($i$l$?ITDj85%j%9%H$rMQ$$$FB?9`<0$KJQ49$9$k(B.
2252: @item
2253: $BITDj85%j%9%H$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 2254: \E
2255: \BEG
2256: @item
2257: This function converts a distributed polynomial into an ordinary polynomial
2258: according to a list of indeterminates @var{vlist}.
2259: @item
2260: @var{vlist} is such a list that its length coincides with the number of
2261: variables of @var{dpoly}.
2262: \E
1.1 noro 2263: @end itemize
2264:
2265: @example
2266: [53] T=dp_ptod((x+y+z)^2,[x,y]);
2267: (1)*<<2,0>>+(2)*<<1,1>>+(1)*<<0,2>>+(2*z)*<<1,0>>+(2*z)*<<0,1>>+(z^2)*<<0,0>>
2268: [54] P=dp_dtop(T,[a,b]);
2269: z^2+(2*a+2*b)*z+a^2+2*b*a+b^2
2270: @end example
2271:
1.2 noro 2272: \JP @node dp_mod dp_rat,,, $B%0%l%V%J4pDl$K4X$9$kH!?t(B
2273: \EG @node dp_mod dp_rat,,, Functions for Groebner basis computation
1.1 noro 2274: @subsection @code{dp_mod}, @code{dp_rat}
2275: @findex dp_mod
2276: @findex dp_rat
2277:
2278: @table @t
2279: @item dp_mod(@var{p},@var{mod},@var{subst})
1.2 noro 2280: \JP :: $BM-M}?t78?tJ,;6I=8=B?9`<0$NM-8BBN78?t$X$NJQ49(B
2281: \EG :: Converts a disributed polynomial into one with coefficients in a finite field.
1.1 noro 2282: @item dp_rat(@var{p})
1.2 noro 2283: \JP :: $BM-8BBN78?tJ,;6I=8=B?9`<0$NM-M}?t78?t$X$NJQ49(B
2284: \BEG
2285: :: Converts a distributed polynomial with coefficients in a finite field into
2286: one with coefficients in the rationals.
2287: \E
1.1 noro 2288: @end table
2289:
2290: @table @var
2291: @item return
1.2 noro 2292: \JP $BJ,;6I=8=B?9`<0(B
2293: \EG distributed polynomial
1.1 noro 2294: @item p
1.2 noro 2295: \JP $BJ,;6I=8=B?9`<0(B
2296: \EG distributed polynomial
1.1 noro 2297: @item mod
1.2 noro 2298: \JP $BAG?t(B
2299: \EG prime
1.1 noro 2300: @item subst
1.2 noro 2301: \JP $B%j%9%H(B
2302: \EG list
1.1 noro 2303: @end table
2304:
2305: @itemize @bullet
1.2 noro 2306: \BJP
1.1 noro 2307: @item
2308: @code{dp_nf_mod()}, @code{dp_true_nf_mod()} $B$O(B, $BF~NO$H$7$FM-8BBN78?t$N(B
2309: $BJ,;6I=8=B?9`<0$rI,MW$H$9$k(B. $B$3$N$h$&$J>l9g(B, @code{dp_mod()} $B$K$h$j(B
2310: $BM-M}?t78?tJ,;6I=8=B?9`<0$rJQ49$7$FMQ$$$k$3$H$,$G$-$k(B. $B$^$?(B, $BF@$i$l$?(B
2311: $B7k2L$O(B, $BM-8BBN78?tB?9`<0$H$O1i;;$G$-$k$,(B, $BM-M}?t78?tB?9`<0$H$O1i;;$G$-$J$$(B
2312: $B$?$a(B, @code{dp_rat()} $B$K$h$jJQ49$9$kI,MW$,$"$k(B.
2313: @item
2314: $BM-8BBN78?t$N1i;;$K$*$$$F$O(B, $B$"$i$+$8$a(B @code{setmod()} $B$K$h$jM-8BBN$N85$N(B
2315: $B8D?t$r;XDj$7$F$*$/I,MW$,$"$k(B.
2316: @item
2317: @var{subst} $B$O(B, $B78?t$,M-M}<0$N>l9g(B, $B$=$NM-M}<0$NJQ?t$K$"$i$+$8$a?t$rBeF~(B
2318: $B$7$?8eM-8BBN78?t$KJQ49$9$k$H$$$&A`:n$r9T$&:]$N(B, $BBeF~CM$r;XDj$9$k$b$N$G(B,
2319: @code{[[@var{var},@var{value}],...]} $B$N7A$N%j%9%H$G$"$k(B.
1.2 noro 2320: \E
2321: \BEG
2322: @item
2323: @code{dp_nf_mod()} and @code{dp_true_nf_mod()} require
2324: distributed polynomials with coefficients in a finite field as arguments.
2325: @code{dp_mod()} is used to convert distributed polynomials with rational
2326: number coefficients into appropriate ones.
2327: Polynomials with coefficients in a finite field
2328: cannot be used as inputs of operations with polynomials
2329: with rational number coefficients. @code{dp_rat()} is used for such cases.
2330: @item
2331: The ground finite field must be set in advance by using @code{setmod()}.
2332: @item
2333: @var{subst} is such a list as @code{[[@var{var},@var{value}],...]}.
2334: This is valid when the ground field of the input polynomial is a
2335: rational function field. @var{var}'s are variables in the ground field and
2336: the list means that @var{value} is substituted for @var{var} before
2337: converting the coefficients into elements of a finite field.
2338: \E
1.1 noro 2339: @end itemize
2340:
2341: @example
2342: @end example
2343:
2344: @table @t
1.2 noro 2345: \JP @item $B;2>H(B
2346: \EG @item References
1.1 noro 2347: @fref{dp_nf dp_nf_mod dp_true_nf dp_true_nf_mod},
2348: @fref{subst psubst},
2349: @fref{setmod}.
2350: @end table
2351:
1.2 noro 2352: \JP @node dp_homo dp_dehomo,,, $B%0%l%V%J4pDl$K4X$9$kH!?t(B
2353: \EG @node dp_homo dp_dehomo,,, Functions for Groebner basis computation
1.1 noro 2354: @subsection @code{dp_homo}, @code{dp_dehomo}
2355: @findex dp_homo
2356: @findex dp_dehomo
2357:
2358: @table @t
2359: @item dp_homo(@var{dpoly})
1.2 noro 2360: \JP :: $BJ,;6I=8=B?9`<0$N@F<!2=(B
2361: \EG :: Homogenize a distributed polynomial
1.1 noro 2362: @item dp_dehomo(@var{dpoly})
1.2 noro 2363: \JP :: $B@F<!J,;6I=8=B?9`<0$NHs@F<!2=(B
2364: \EG :: Dehomogenize a homogenious distributed polynomial
1.1 noro 2365: @end table
2366:
2367: @table @var
2368: @item return
1.2 noro 2369: \JP $BJ,;6I=8=B?9`<0(B
2370: \EG distributed polynomial
1.1 noro 2371: @item dpoly
1.2 noro 2372: \JP $BJ,;6I=8=B?9`<0(B
2373: \EG distributed polynomial
1.1 noro 2374: @end table
2375:
2376: @itemize @bullet
1.2 noro 2377: \BJP
1.1 noro 2378: @item
2379: @code{dp_homo()} $B$O(B, @var{dpoly} $B$N(B $B3F9`(B @var{t} $B$K$D$$$F(B, $B;X?t%Y%/%H%k$ND9$5$r(B
2380: 1 $B?-$P$7(B, $B:G8e$N@.J,$NCM$r(B @var{d}-@code{deg(@var{t})}
2381: (@var{d} $B$O(B @var{dpoly} $B$NA4<!?t(B) $B$H$7$?J,;6I=8=B?9`<0$rJV$9(B.
2382: @item
2383: @code{dp_dehomo()} $B$O(B, @var{dpoly} $B$N3F9`$K$D$$$F(B, $B;X?t%Y%/%H%k$N:G8e$N@.J,(B
2384: $B$r<h$j=|$$$?J,;6B?9`<0$rJV$9(B.
2385: @item
2386: $B$$$:$l$b(B, $B@8@.$5$l$?B?9`<0$rMQ$$$?1i;;$r9T$&>l9g(B, $B$=$l$i$KE,9g$9$k9`=g=x$r(B
2387: $B@5$7$/@_Dj$9$kI,MW$,$"$k(B.
2388: @item
2389: @code{hgr()} $B$J$I$K$*$$$F(B, $BFbItE*$KMQ$$$i$l$F$$$k(B.
1.2 noro 2390: \E
2391: \BEG
2392: @item
2393: @code{dp_homo()} makes a copy of @var{dpoly}, extends
2394: the length of the exponent vector of each term @var{t} in the copy by 1,
2395: and sets the value of the newly appended
2396: component to @var{d}-@code{deg(@var{t})}, where @var{d} is the total
2397: degree of @var{dpoly}.
2398: @item
2399: @code{dp_dehomo()} make a copy of @var{dpoly} and removes the last component
2400: of each terms in the copy.
2401: @item
2402: Appropriate term orderings must be set when the results are used as inputs
2403: of some operations.
2404: @item
2405: These are used internally in @code{hgr()} etc.
2406: \E
1.1 noro 2407: @end itemize
2408:
2409: @example
2410: [202] X=<<1,2,3>>+3*<<1,2,1>>;
2411: (1)*<<1,2,3>>+(3)*<<1,2,1>>
2412: [203] dp_homo(X);
2413: (1)*<<1,2,3,0>>+(3)*<<1,2,1,2>>
2414: [204] dp_dehomo(@@);
2415: (1)*<<1,2,3>>+(3)*<<1,2,1>>
2416: @end example
2417:
2418: @table @t
1.2 noro 2419: \JP @item $B;2>H(B
2420: \EG @item References
1.1 noro 2421: @fref{gr hgr gr_mod}.
2422: @end table
2423:
1.2 noro 2424: \JP @node dp_ptozp dp_prim,,, $B%0%l%V%J4pDl$K4X$9$kH!?t(B
2425: \EG @node dp_ptozp dp_prim,,, Functions for Groebner basis computation
1.1 noro 2426: @subsection @code{dp_ptozp}, @code{dp_prim}
2427: @findex dp_ptozp
2428: @findex dp_prim
2429:
2430: @table @t
2431: @item dp_ptozp(@var{dpoly})
1.2 noro 2432: \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.
2433: \BEG
2434: :: Converts a distributed polynomial @var{poly} with rational coefficients
2435: into an integral distributed polynomial such that GCD of all its coefficients
2436: is 1.
2437: \E
1.1 noro 2438: @itemx dp_prim(@var{dpoly})
1.2 noro 2439: \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.
2440: \BEG
2441: :: Converts a distributed polynomial @var{poly} with rational function
2442: coefficients into an integral distributed polynomial such that polynomial
2443: GCD of all its coefficients is 1.
2444: \E
1.1 noro 2445: @end table
2446:
2447: @table @var
2448: @item return
1.2 noro 2449: \JP $BJ,;6I=8=B?9`<0(B
2450: \EG distributed polynomial
1.1 noro 2451: @item dpoly
1.2 noro 2452: \JP $BJ,;6I=8=B?9`<0(B
2453: \EG distributed polynomial
1.1 noro 2454: @end table
2455:
2456: @itemize @bullet
1.2 noro 2457: \BJP
1.1 noro 2458: @item
2459: @code{dp_ptozp()} $B$O(B, @code{ptozp()} $B$KAjEv$9$kA`:n$rJ,;6I=8=B?9`<0$K(B
2460: $BBP$7$F9T$&(B. $B78?t$,B?9`<0$r4^$`>l9g(B, $B78?t$K4^$^$l$kB?9`<06&DL0x;R$O(B
2461: $B<h$j=|$+$J$$(B.
2462: @item
2463: @code{dp_prim()} $B$O(B, $B78?t$,B?9`<0$r4^$`>l9g(B, $B78?t$K4^$^$l$kB?9`<06&DL0x;R(B
2464: $B$r<h$j=|$/(B.
1.2 noro 2465: \E
2466: \BEG
2467: @item
2468: @code{dp_ptozp()} executes the same operation as @code{ptozp()} for
2469: a distributed polynomial. If the coefficients include polynomials,
2470: polynomial contents included in the coefficients are not removed.
2471: @item
2472: @code{dp_prim()} removes polynomial contents.
2473: \E
1.1 noro 2474: @end itemize
2475:
2476: @example
2477: [208] X=dp_ptod(3*(x-y)*(y-z)*(z-x),[x]);
2478: (-3*y+3*z)*<<2>>+(3*y^2-3*z^2)*<<1>>+(-3*z*y^2+3*z^2*y)*<<0>>
2479: [209] dp_ptozp(X);
2480: (-y+z)*<<2>>+(y^2-z^2)*<<1>>+(-z*y^2+z^2*y)*<<0>>
2481: [210] dp_prim(X);
2482: (1)*<<2>>+(-y-z)*<<1>>+(z*y)*<<0>>
2483: @end example
2484:
2485: @table @t
1.2 noro 2486: \JP @item $B;2>H(B
2487: \EG @item References
1.1 noro 2488: @fref{ptozp}.
2489: @end table
2490:
1.2 noro 2491: \JP @node dp_nf dp_nf_mod dp_true_nf dp_true_nf_mod,,, $B%0%l%V%J4pDl$K4X$9$kH!?t(B
2492: \EG @node dp_nf dp_nf_mod dp_true_nf dp_true_nf_mod,,, Functions for Groebner basis computation
1.1 noro 2493: @subsection @code{dp_nf}, @code{dp_nf_mod}, @code{dp_true_nf}, @code{dp_true_nf_mod}
2494: @findex dp_nf
2495: @findex dp_true_nf
2496: @findex dp_nf_mod
2497: @findex dp_true_nf_mod
2498:
2499: @table @t
2500: @item dp_nf(@var{indexlist},@var{dpoly},@var{dpolyarray},@var{fullreduce})
2501: @item dp_nf_mod(@var{indexlist},@var{dpoly},@var{dpolyarray},@var{fullreduce},@var{mod})
1.2 noro 2502: \JP :: $BJ,;6I=8=B?9`<0$N@55,7A$r5a$a$k(B. ($B7k2L$ODj?tG\$5$l$F$$$k2DG=@-$"$j(B)
1.1 noro 2503:
1.2 noro 2504: \BEG
2505: :: Computes the normal form of a distributed polynomial.
2506: (The result may be multiplied by a constant in the ground field.)
2507: \E
1.1 noro 2508: @item dp_true_nf(@var{indexlist},@var{dpoly},@var{dpolyarray},@var{fullreduce})
2509: @item dp_true_nf_mod(@var{indexlist},@var{dpoly},@var{dpolyarray},@var{fullreduce},@var{mod})
1.2 noro 2510: \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)
2511: \BEG
2512: :: Computes the normal form of a distributed polynomial. (The true result
2513: is returned in such a list as @code{[numerator, denominator]})
2514: \E
1.1 noro 2515: @end table
2516:
2517: @table @var
2518: @item return
1.2 noro 2519: \JP @code{dp_nf()} : $BJ,;6I=8=B?9`<0(B, @code{dp_true_nf()} : $B%j%9%H(B
2520: \EG @code{dp_nf()} : distributed polynomial, @code{dp_true_nf()} : list
1.1 noro 2521: @item indexlist
1.2 noro 2522: \JP $B%j%9%H(B
2523: \EG list
1.1 noro 2524: @item dpoly
1.2 noro 2525: \JP $BJ,;6I=8=B?9`<0(B
2526: \EG distributed polynomial
1.1 noro 2527: @item dpolyarray
1.2 noro 2528: \JP $BG[Ns(B
2529: \EG array of distributed polynomial
1.1 noro 2530: @item fullreduce
1.2 noro 2531: \JP $B%U%i%0(B
2532: \EG flag
1.1 noro 2533: @item mod
1.2 noro 2534: \JP $BAG?t(B
2535: \EG prime
1.1 noro 2536: @end table
2537:
2538: @itemize @bullet
1.2 noro 2539: \BJP
1.1 noro 2540: @item
2541: $BJ,;6I=8=B?9`<0(B @var{dpoly} $B$N@55,7A$r5a$a$k(B.
2542: @item
2543: @code{dp_nf_mod()}, @code{dp_true_nf_mod()} $B$NF~NO$O(B, @code{dp_mod()} $B$J$I(B
2544: $B$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.
2545: @item
2546: $B7k2L$KM-M}?t(B, $BM-M}<0$,4^$^$l$k$N$rHr$1$k$?$a(B, @code{dp_nf()} $B$O(B
2547: $B??$NCM$NDj?tG\$NCM$rJV$9(B. $BM-M}<078?t$N>l9g$N(B @code{dp_nf_mod()} $B$bF1MM(B
2548: $B$G$"$k$,(B, $B78?tBN$,M-8BBN$N>l9g(B @code{dp_nf_mod()} $B$O??$NCM$rJV$9(B.
2549: @item
2550: @code{dp_true_nf()}, @code{dp_true_nf_mod()} $B$O(B,
2551: @code{[@var{nm},@var{dn}]} $B$J$k7A$N%j%9%H$rJV$9(B.
2552: $B$?$@$7(B, @var{nm} $B$O78?t$KJ,?t(B, $BM-M}<0$r4^$^$J$$J,;6I=8=B?9`<0(B, @var{dn} $B$O(B
2553: $B?t$^$?$OB?9`<0$G(B @var{nm}/@var{dn} $B$,??$NCM$H$J$k(B.
2554: @item
2555: @var{dpolyarray} $B$OJ,;6I=8=B?9`<0$rMWAG$H$9$k%Y%/%H%k(B,
2556: @var{indexlist} $B$O@55,2=7W;;$KMQ$$$k(B @var{dpolyarray} $B$NMWAG$N%$%s%G%C%/%9(B
2557: $B$N%j%9%H(B.
2558: @item
2559: @var{fullreduce} $B$,(B 0 $B$G$J$$$H$-A4$F$N9`$KBP$7$F4JLs$r9T$&(B. @var{fullreduce}
2560: $B$,(B 0 $B$N$H$-F,9`$N$_$KBP$7$F4JLs$r9T$&(B.
2561: @item
2562: @var{indexlist} $B$G;XDj$5$l$?B?9`<0$O(B, $BA0$NJ}$N$b$N$,M%@hE*$K;H$o$l$k(B.
2563: @item
2564: $B0lHL$K$O(B @var{indexlist} $B$NM?$(J}$K$h$jH!?t$NCM$O0[$J$k2DG=@-$,$"$k$,(B,
2565: $B%0%l%V%J4pDl$KBP$7$F$O0l0UE*$KDj$^$k(B.
2566: @item
2567: $BJ,;6I=8=$G$J$$8GDj$5$l$?B?9`<0=89g$K$h$k@55,7A$rB??t5a$a$kI,MW$,$"$k>l9g(B
2568: $B$KJXMx$G$"$k(B. $BC10l$N1i;;$K4X$7$F$O(B, @code{p_nf}, @code{p_true_nf} $B$r(B
2569: $BMQ$$$k$H$h$$(B.
1.2 noro 2570: \E
2571: \BEG
2572: @item
2573: Computes the normal form of a distributed polynomial.
2574: @item
2575: @code{dp_nf_mod()} and @code{dp_true_nf_mod()} require
2576: distributed polynomials with coefficients in a finite field as arguments.
2577: @item
2578: The result of @code{dp_nf()} may be multiplied by a constant in the
2579: ground field in order to make the result integral. The same is true
2580: for @code{dp_nf_mod()}, but it returns the true normal form if
2581: the ground field is a finite field.
2582: @item
2583: @code{dp_true_nf()} and @code{dp_true_nf_mod()} return
2584: such a list as @code{[@var{nm},@var{dn}]}.
2585: Here @var{nm} is a distributed polynomial whose coefficients are integral
2586: in the ground field, @var{dn} is an integral element in the ground
2587: field and @var{nm}/@var{dn} is the true normal form.
2588: @item
2589: @var{dpolyarray} is a vector whose components are distributed polynomials
2590: and @var{indexlist} is a list of indices which is used for the normal form
2591: computation.
2592: @item
2593: When argument @var{fullreduce} has non-zero value,
2594: all terms are reduced. When it has value 0,
2595: only the head term is reduced.
2596: @item
2597: As for the polynomials specified by @var{indexlist}, one specified by
2598: an index placed at the preceding position has priority to be selected.
2599: @item
2600: In general, the result of the function may be different depending on
2601: @var{indexlist}. However, the result is unique for Groebner bases.
2602: @item
2603: These functions are useful when a fixed non-distributed polynomial set
2604: is used as a set of reducers to compute normal forms of many polynomials.
2605: For single computation @code{p_nf} and @code{p_true_nf} are sufficient.
2606: \E
1.1 noro 2607: @end itemize
2608:
2609: @example
2610: [0] load("gr")$
2611: [64] load("katsura")$
2612: [69] K=katsura(4)$
2613: [70] dp_ord(2)$
2614: [71] V=[u0,u1,u2,u3,u4]$
2615: [72] DP1=newvect(length(K),map(dp_ptod,K,V))$
2616: [73] G=gr(K,V,2)$
2617: [74] DP2=newvect(length(G),map(dp_ptod,G,V))$
2618: [75] T=dp_ptod((u0-u1+u2-u3+u4)^2,V)$
2619: [76] dp_dtop(dp_nf([0,1,2,3,4],T,DP1,1),V);
2620: u4^2+(6*u3+2*u2+6*u1-2)*u4+9*u3^2+(6*u2+18*u1-6)*u3+u2^2+(6*u1-2)*u2+9*u1^2-6*u1+1
2621: [77] dp_dtop(dp_nf([4,3,2,1,0],T,DP1,1),V);
2622: -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
2623: [78] dp_dtop(dp_nf([0,1,2,3,4],T,DP2,1),V);
2624: -1138087976845165778088612297273078520347097001020471455633353049221045677593
2625: 0005716505560062087150928400876150217079820311439477560587583488*u4^15+...
2626: [79] dp_dtop(dp_nf([4,3,2,1,0],T,DP2,1),V);
2627: -1138087976845165778088612297273078520347097001020471455633353049221045677593
2628: 0005716505560062087150928400876150217079820311439477560587583488*u4^15+...
2629: [80] @@78==@@79;
2630: 1
2631: @end example
2632:
2633: @table @t
1.2 noro 2634: \JP @item $B;2>H(B
2635: \EG @item References
1.1 noro 2636: @fref{dp_dtop},
2637: @fref{dp_ord},
2638: @fref{dp_mod dp_rat},
2639: @fref{p_nf p_nf_mod p_true_nf p_true_nf_mod}.
2640: @end table
2641:
1.2 noro 2642: \JP @node dp_hm dp_ht dp_hc dp_rest,,, $B%0%l%V%J4pDl$K4X$9$kH!?t(B
2643: \EG @node dp_hm dp_ht dp_hc dp_rest,,, Functions for Groebner basis computation
1.1 noro 2644: @subsection @code{dp_hm}, @code{dp_ht}, @code{dp_hc}, @code{dp_rest}
2645: @findex dp_hm
2646: @findex dp_ht
2647: @findex dp_hc
2648: @findex dp_rest
2649:
2650: @table @t
2651: @item dp_hm(@var{dpoly})
1.2 noro 2652: \JP :: $BF,C19`<0$r<h$j=P$9(B.
2653: \EG :: Gets the head monomial.
1.1 noro 2654: @item dp_ht(@var{dpoly})
1.2 noro 2655: \JP :: $BF,9`$r<h$j=P$9(B.
2656: \EG :: Gets the head term.
1.1 noro 2657: @item dp_hc(@var{dpoly})
1.2 noro 2658: \JP :: $BF,78?t$r<h$j=P$9(B.
2659: \EG :: Gets the head coefficient.
1.1 noro 2660: @item dp_rest(@var{dpoly})
1.2 noro 2661: \JP :: $BF,C19`<0$r<h$j=|$$$?;D$j$rJV$9(B.
2662: \EG :: Gets the remainder of the polynomial where the head monomial is removed.
1.1 noro 2663: @end table
2664:
2665: @table @var
1.2 noro 2666: \BJP
1.1 noro 2667: @item return
2668: @code{dp_hm()}, @code{dp_ht()}, @code{dp_rest()} : $BJ,;6I=8=B?9`<0(B,
2669: @code{dp_hc()} : $B?t$^$?$OB?9`<0(B
2670: @item dpoly
2671: $BJ,;6I=8=B?9`<0(B
1.2 noro 2672: \E
2673: \BEG
2674: @item return
2675: @code{dp_hm()}, @code{dp_ht()}, @code{dp_rest()} : distributed polynomial
2676: @code{dp_hc()} : number or polynomial
2677: @item dpoly
2678: distributed polynomial
2679: \E
1.1 noro 2680: @end table
2681:
2682: @itemize @bullet
1.2 noro 2683: \BJP
1.1 noro 2684: @item
2685: $B$3$l$i$O(B, $BJ,;6I=8=B?9`<0$N3FItJ,$r<h$j=P$9$?$a$NH!?t$G$"$k(B.
2686: @item
2687: $BJ,;6I=8=B?9`<0(B @var{p} $B$KBP$7<!$,@.$jN)$D(B.
1.2 noro 2688: \E
2689: \BEG
2690: @item
2691: These are used to get various parts of a distributed polynomial.
2692: @item
2693: The next equations hold for a distributed polynomial @var{p}.
2694: \E
1.1 noro 2695: @table @code
2696: @item @var{p} = dp_hm(@var{p}) + dp_rest(@var{p})
2697: @item dp_hm(@var{p}) = dp_hc(@var{p}) dp_ht(@var{p})
2698: @end table
2699: @end itemize
2700:
2701: @example
2702: [87] dp_ord(0)$
2703: [88] X=ptozp((a46^2+7/10*a46+7/48)*u3^4-50/27*a46^2-35/27*a46-49/216)$
2704: [89] T=dp_ptod(X,[u3,u4,a46])$
2705: [90] dp_hm(T);
2706: (2160)*<<4,0,2>>
2707: [91] dp_ht(T);
2708: (1)*<<4,0,2>>
2709: [92] dp_hc(T);
2710: 2160
2711: [93] dp_rest(T);
2712: (1512)*<<4,0,1>>+(315)*<<4,0,0>>+(-4000)*<<0,0,2>>+(-2800)*<<0,0,1>>
2713: +(-490)*<<0,0,0>>
2714: @end example
2715:
1.2 noro 2716: \JP @node dp_td dp_sugar,,, $B%0%l%V%J4pDl$K4X$9$kH!?t(B
2717: \EG @node dp_td dp_sugar,,, Functions for Groebner basis computation
1.1 noro 2718: @subsection @code{dp_td}, @code{dp_sugar}
2719: @findex dp_td
2720: @findex dp_sugar
2721:
2722: @table @t
2723: @item dp_td(@var{dpoly})
1.2 noro 2724: \JP :: $BF,9`$NA4<!?t$rJV$9(B.
2725: \EG :: Gets the total degree of the head term.
1.1 noro 2726: @item dp_sugar(@var{dpoly})
1.2 noro 2727: \JP :: $BB?9`<0$N(B @code{sugar} $B$rJV$9(B.
2728: \EG :: Gets the @code{sugar} of a polynomial.
1.1 noro 2729: @end table
2730:
2731: @table @var
2732: @item return
1.2 noro 2733: \JP $B<+A3?t(B
2734: \EG non-negative integer
1.1 noro 2735: @item dpoly
1.2 noro 2736: \JP $BJ,;6I=8=B?9`<0(B
2737: \EG distributed polynomial
1.1 noro 2738: @item onoff
1.2 noro 2739: \JP $B%U%i%0(B
2740: \EG flag
1.1 noro 2741: @end table
2742:
2743: @itemize @bullet
1.2 noro 2744: \BJP
1.1 noro 2745: @item
2746: @code{dp_td()} $B$O(B, $BF,9`$NA4<!?t(B, $B$9$J$o$A3FJQ?t$N;X?t$NOB$rJV$9(B.
2747: @item
2748: $BJ,;6I=8=B?9`<0$,@8@.$5$l$k$H(B, @code{sugar} $B$H8F$P$l$k$"$k@0?t$,IUM?(B
2749: $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.
2750: @item
2751: @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
2752: $B7hDj$9$k$?$a$N=EMW$J;X?K$H$J$k(B.
1.2 noro 2753: \E
2754: \BEG
2755: @item
2756: Function @code{dp_td()} returns the total degree of the head term,
2757: i.e., the sum of all exponent of variables in that term.
2758: @item
2759: Upon creation of a distributed polynomial, an integer called @code{sugar}
2760: is associated. This value is
2761: the total degree of the virtually homogenized one of the original
2762: polynomial.
2763: @item
2764: The quantity @code{sugar} is an important guide to determine the
2765: selection strategy of critical pairs in Groebner basis computation.
2766: \E
1.1 noro 2767: @end itemize
2768:
2769: @example
2770: [74] dp_ord(0)$
2771: [75] X=<<1,2>>+<<0,1>>$
2772: [76] Y=<<1,2>>+<<1,0>>$
2773: [77] Z=X-Y;
2774: (-1)*<<1,0>>+(1)*<<0,1>>
2775: [78] dp_sugar(T);
2776: 3
2777: @end example
2778:
1.2 noro 2779: \JP @node dp_lcm,,, $B%0%l%V%J4pDl$K4X$9$kH!?t(B
2780: \EG @node dp_lcm,,, Functions for Groebner basis computation
1.1 noro 2781: @subsection @code{dp_lcm}
2782: @findex dp_lcm
2783:
2784: @table @t
2785: @item dp_lcm(@var{dpoly1},@var{dpoly2})
1.2 noro 2786: \JP :: $B:G>.8xG\9`$rJV$9(B.
2787: \EG :: Returns the least common multiple of the head terms of the given two polynomials.
1.1 noro 2788: @end table
2789:
2790: @table @var
2791: @item return
1.2 noro 2792: \JP $BJ,;6I=8=B?9`<0(B
2793: \EG distributed polynomial
1.4 ! noro 2794: @item dpoly1 dpoly2
1.2 noro 2795: \JP $BJ,;6I=8=B?9`<0(B
2796: \EG distributed polynomial
1.1 noro 2797: @end table
2798:
2799: @itemize @bullet
1.2 noro 2800: \BJP
1.1 noro 2801: @item
2802: $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 2803: \E
2804: \BEG
2805: @item
2806: Returns the least common multiple of the head terms of the given
2807: two polynomials, where coefficient is always set to 1.
2808: \E
1.1 noro 2809: @end itemize
2810:
2811: @example
2812: [100] dp_lcm(<<1,2,3,4,5>>,<<5,4,3,2,1>>);
2813: (1)*<<5,4,3,4,5>>
2814: @end example
2815:
2816: @table @t
1.2 noro 2817: \JP @item $B;2>H(B
2818: \EG @item References
1.1 noro 2819: @fref{p_nf p_nf_mod p_true_nf p_true_nf_mod}.
2820: @end table
2821:
1.2 noro 2822: \JP @node dp_redble,,, $B%0%l%V%J4pDl$K4X$9$kH!?t(B
2823: \EG @node dp_redble,,, Functions for Groebner basis computation
1.1 noro 2824: @subsection @code{dp_redble}
2825: @findex dp_redble
2826:
2827: @table @t
2828: @item dp_redble(@var{dpoly1},@var{dpoly2})
1.2 noro 2829: \JP :: $BF,9`$I$&$7$,@0=|2DG=$+$I$&$+D4$Y$k(B.
2830: \EG :: Checks whether one head term is divisible by the other head term.
1.1 noro 2831: @end table
2832:
2833: @table @var
2834: @item return
1.2 noro 2835: \JP $B@0?t(B
2836: \EG integer
1.4 ! noro 2837: @item dpoly1 dpoly2
1.2 noro 2838: \JP $BJ,;6I=8=B?9`<0(B
2839: \EG distributed polynomial
1.1 noro 2840: @end table
2841:
2842: @itemize @bullet
1.2 noro 2843: \BJP
1.1 noro 2844: @item
2845: @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
2846: 0 $B$rJV$9(B.
2847: @item
2848: $BB?9`<0$N4JLs$r9T$&:](B, $B$I$N9`$r4JLs$G$-$k$+$rC5$9$N$KMQ$$$k(B.
1.2 noro 2849: \E
2850: \BEG
2851: @item
2852: Returns 1 if the head term of @var{dpoly2} divides the head term of
2853: @var{dpoly1}; otherwise 0.
2854: @item
2855: Used for finding candidate terms at reduction of polynomials.
2856: \E
1.1 noro 2857: @end itemize
2858:
2859: @example
2860: [148] C;
2861: (1)*<<1,1,1,0,0>>+(1)*<<0,1,1,1,0>>+(1)*<<1,1,0,0,1>>+(1)*<<1,0,0,1,1>>
2862: [149] T;
2863: (3)*<<2,1,0,0,0>>+(3)*<<1,2,0,0,0>>+(1)*<<0,3,0,0,0>>+(6)*<<1,1,1,0,0>>
2864: [150] for ( ; T; T = dp_rest(T)) print(dp_redble(T,C));
2865: 0
2866: 0
2867: 0
2868: 1
2869: @end example
2870:
2871: @table @t
1.2 noro 2872: \JP @item $B;2>H(B
2873: \EG @item References
1.1 noro 2874: @fref{dp_red dp_red_mod}.
2875: @end table
2876:
1.2 noro 2877: \JP @node dp_subd,,, $B%0%l%V%J4pDl$K4X$9$kH!?t(B
2878: \EG @node dp_subd,,, Functions for Groebner basis computation
1.1 noro 2879: @subsection @code{dp_subd}
2880: @findex dp_subd
2881:
2882: @table @t
2883: @item dp_subd(@var{dpoly1},@var{dpoly2})
1.2 noro 2884: \JP :: $BF,9`$N>&C19`<0$rJV$9(B.
2885: \EG :: Returns the quotient monomial of the head terms.
1.1 noro 2886: @end table
2887:
2888: @table @var
2889: @item return
1.2 noro 2890: \JP $BJ,;6I=8=B?9`<0(B
2891: \EG distributed polynomial
1.4 ! noro 2892: @item dpoly1 dpoly2
1.2 noro 2893: \JP $BJ,;6I=8=B?9`<0(B
2894: \EG distributed polynomial
1.1 noro 2895: @end table
2896:
2897: @itemize @bullet
1.2 noro 2898: \BJP
1.1 noro 2899: @item
2900: @code{dp_ht(@var{dpoly1})/dp_ht(@var{dpoly2})} $B$r5a$a$k(B. $B7k2L$N78?t$O(B 1
2901: $B$G$"$k(B.
2902: @item
2903: $B3d$j@Z$l$k$3$H$,$"$i$+$8$a$o$+$C$F$$$kI,MW$,$"$k(B.
1.2 noro 2904: \E
2905: \BEG
2906: @item
2907: Gets @code{dp_ht(@var{dpoly1})/dp_ht(@var{dpoly2})}.
2908: The coefficient of the result is always set to 1.
2909: @item
2910: Divisibility assumed.
2911: \E
1.1 noro 2912: @end itemize
2913:
2914: @example
2915: [162] dp_subd(<<1,2,3,4,5>>,<<1,1,2,3,4>>);
2916: (1)*<<0,1,1,1,1>>
2917: @end example
2918:
2919: @table @t
1.2 noro 2920: \JP @item $B;2>H(B
2921: \EG @item References
1.1 noro 2922: @fref{dp_red dp_red_mod}.
2923: @end table
2924:
1.2 noro 2925: \JP @node dp_vtoe dp_etov,,, $B%0%l%V%J4pDl$K4X$9$kH!?t(B
2926: \EG @node dp_vtoe dp_etov,,, Functions for Groebner basis computation
1.1 noro 2927: @subsection @code{dp_vtoe}, @code{dp_etov}
2928: @findex dp_vtoe
2929: @findex dp_etov
2930:
2931: @table @t
2932: @item dp_vtoe(@var{vect})
1.2 noro 2933: \JP :: $B;X?t%Y%/%H%k$r9`$KJQ49(B
2934: \EG :: Converts an exponent vector into a term.
1.1 noro 2935: @item dp_etov(@var{dpoly})
1.2 noro 2936: \JP :: $BF,9`$r;X?t%Y%/%H%k$KJQ49(B
2937: \EG :: Convert the head term of a distributed polynomial into an exponent vector.
1.1 noro 2938: @end table
2939:
2940: @table @var
2941: @item return
1.2 noro 2942: \JP @code{dp_vtoe} : $BJ,;6I=8=B?9`<0(B, @code{dp_etov} : $B%Y%/%H%k(B
2943: \EG @code{dp_vtoe} : distributed polynomial, @code{dp_etov} : vector
1.1 noro 2944: @item vect
1.2 noro 2945: \JP $B%Y%/%H%k(B
2946: \EG vector
1.1 noro 2947: @item dpoly
1.2 noro 2948: \JP $BJ,;6I=8=B?9`<0(B
2949: \EG distributed polynomial
1.1 noro 2950: @end table
2951:
2952: @itemize @bullet
1.2 noro 2953: \BJP
1.1 noro 2954: @item
2955: @code{dp_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.
2956: @item
2957: @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
2958: $B%Y%/%H%k$KJQ49$9$k(B.
1.2 noro 2959: \E
2960: \BEG
2961: @item
2962: @code{dp_vtoe()} generates a term whose exponent vector is @var{vect}.
2963: @item
2964: @code{dp_etov()} generates a vector which is the exponent vector of the
2965: head term of @code{dpoly}.
2966: \E
1.1 noro 2967: @end itemize
2968:
2969: @example
2970: [211] X=<<1,2,3>>;
2971: (1)*<<1,2,3>>
2972: [212] V=dp_etov(X);
2973: [ 1 2 3 ]
2974: [213] V[2]++$
2975: [214] Y=dp_vtoe(V);
2976: (1)*<<1,2,4>>
2977: @end example
2978:
1.2 noro 2979: \JP @node dp_mbase,,, $B%0%l%V%J4pDl$K4X$9$kH!?t(B
2980: \EG @node dp_mbase,,, Functions for Groebner basis computation
1.1 noro 2981: @subsection @code{dp_mbase}
2982: @findex dp_mbase
2983:
2984: @table @t
2985: @item dp_mbase(@var{dplist})
1.2 noro 2986: \JP :: monomial $B4pDl$N7W;;(B
2987: \EG :: Computes the monomial basis
1.1 noro 2988: @end table
2989:
2990: @table @var
2991: @item return
1.2 noro 2992: \JP $BJ,;6I=8=B?9`<0$N%j%9%H(B
2993: \EG list of distributed polynomial
1.1 noro 2994: @item dplist
1.2 noro 2995: \JP $BJ,;6I=8=B?9`<0$N%j%9%H(B
2996: \EG list of distributed polynomial
1.1 noro 2997: @end table
2998:
2999: @itemize @bullet
1.2 noro 3000: \BJP
1.1 noro 3001: @item
3002: $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
3003: $B$G$"$k(B @var{dplist} $B$K$D$$$F(B,
3004: @var{dplist} $B$,(B K[X] $BCf$G@8@.$9$k%$%G%"%k(B I $B$,(B 0 $B<!85$N;~(B,
3005: K $B>eM-8B<!85@~7A6u4V$G$"$k(B K[X]/I $B$N(B monomial $B$K$h$k4pDl$r5a$a$k(B.
3006: @item
3007: $BF@$i$l$?4pDl$N8D?t$,(B, K[X]/I $B$N(B K-$B@~7A6u4V$H$7$F$N<!85$KEy$7$$(B.
1.2 noro 3008: \E
3009: \BEG
3010: @item
3011: Assuming that @var{dplist} is a list of distributed polynomials which
3012: is a Groebner basis with respect to the current ordering type and
3013: that the ideal @var{I} generated by @var{dplist} in K[X] is zero-dimensional,
3014: this function computes the monomial basis of a finite dimenstional K-vector
3015: space K[X]/I.
3016: @item
3017: The number of elements in the monomial basis is equal to the
3018: K-dimenstion of K[X]/I.
3019: \E
1.1 noro 3020: @end itemize
3021:
3022: @example
3023: [215] K=katsura(5)$
3024: [216] V=[u5,u4,u3,u2,u1,u0]$
3025: [217] G0=gr(K,V,0)$
3026: [218] H=map(dp_ptod,G0,V)$
3027: [219] map(dp_ptod,dp_mbase(H),V)$
3028: [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,
3029: 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,
3030: u1*u2,u1^2,u4*u0,u3*u0,u2*u0,u1*u0,u0^2,u4,u3,u2,u1,u0,1]
3031: @end example
3032:
3033: @table @t
1.2 noro 3034: \JP @item $B;2>H(B
3035: \EG @item References
1.1 noro 3036: @fref{gr hgr gr_mod}.
3037: @end table
3038:
1.2 noro 3039: \JP @node dp_mag,,, $B%0%l%V%J4pDl$K4X$9$kH!?t(B
3040: \EG @node dp_mag,,, Functions for Groebner basis computation
1.1 noro 3041: @subsection @code{dp_mag}
3042: @findex dp_mag
3043:
3044: @table @t
3045: @item dp_mag(@var{p})
1.2 noro 3046: \JP :: $B78?t$N%S%C%HD9$NOB$rJV$9(B
3047: \EG :: Computes the sum of bit lengths of coefficients of a distributed polynomial.
1.1 noro 3048: @end table
3049:
3050: @table @var
3051: @item return
1.2 noro 3052: \JP $B?t(B
3053: \EG integer
1.1 noro 3054: @item p
1.2 noro 3055: \JP $BJ,;6I=8=B?9`<0(B
3056: \EG distributed polynomial
1.1 noro 3057: @end table
3058:
3059: @itemize @bullet
1.2 noro 3060: \BJP
1.1 noro 3061: @item
3062: $BJ,;6I=8=B?9`<0$N78?t$K8=$l$kM-M}?t$K$D$-(B, $B$=$NJ,JlJ,;R(B ($B@0?t$N>l9g$OJ,;R(B)
3063: $B$N%S%C%HD9$NAmOB$rJV$9(B.
3064: @item
3065: $BBP>]$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$*$$$F$O(B
3066: $B78?tKDD%$,LdBj$H$J$j(B, $BESCf@8@.$5$l$kB?9`<0$,78?tKDD%$r5/$3$7$F$$$k$+$I$&$+(B
3067: $B$NH=Dj$KLrN)$D(B.
3068: @item
3069: @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
3070: $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 3071: \E
3072: \BEG
3073: @item
3074: This function computes the sum of bit lengths of coefficients of a
3075: distributed polynomial @var{p}. If a coefficient is non integral,
3076: the sum of bit lengths of the numerator and the denominator is taken.
3077: @item
3078: This is a measure of the size of a polynomial. Especially for
3079: zero-dimensional system coefficient swells are often serious and
3080: the returned value is useful to detect such swells.
3081: @item
3082: If @code{ShowMag} and @code{Print} for @code{dp_gr_flags()} are on,
3083: values of @code{dp_mag()} for intermediate basis elements are shown.
3084: \E
1.1 noro 3085: @end itemize
3086:
3087: @example
3088: [221] X=dp_ptod((x+2*y)^10,[x,y])$
3089: [222] dp_mag(X);
3090: 115
3091: @end example
3092:
3093: @table @t
1.2 noro 3094: \JP @item $B;2>H(B
3095: \EG @item References
1.1 noro 3096: @fref{dp_gr_flags dp_gr_print}.
3097: @end table
3098:
1.2 noro 3099: \JP @node dp_red dp_red_mod,,, $B%0%l%V%J4pDl$K4X$9$kH!?t(B
3100: \EG @node dp_red dp_red_mod,,, Functions for Groebner basis computation
1.1 noro 3101: @subsection @code{dp_red}, @code{dp_red_mod}
3102: @findex dp_red
3103: @findex dp_red_mod
3104:
3105: @table @t
3106: @item dp_red(@var{dpoly1},@var{dpoly2},@var{dpoly3})
3107: @item dp_red_mod(@var{dpoly1},@var{dpoly2},@var{dpoly3},@var{mod})
1.2 noro 3108: \JP :: $B0l2s$N4JLsA`:n(B
3109: \EG :: Single reduction operation
1.1 noro 3110: @end table
3111:
3112: @table @var
3113: @item return
1.2 noro 3114: \JP $B%j%9%H(B
3115: \EG list
1.4 ! noro 3116: @item dpoly1 dpoly2 dpoly3
1.2 noro 3117: \JP $BJ,;6I=8=B?9`<0(B
3118: \EG distributed polynomial
1.1 noro 3119: @item vlist
1.2 noro 3120: \JP $B%j%9%H(B
3121: \EG list
1.1 noro 3122: @item mod
1.2 noro 3123: \JP $BAG?t(B
3124: \EG prime
1.1 noro 3125: @end table
3126:
3127: @itemize @bullet
1.2 noro 3128: \BJP
1.1 noro 3129: @item
3130: @var{dpoly1} + @var{dpoly2} $B$J$kJ,;6I=8=B?9`<0$r(B @var{dpoly3} $B$G(B
3131: 1 $B2s4JLs$9$k(B.
3132: @item
3133: @code{dp_red_mod()} $B$NF~NO$O(B, $BA4$FM-8BBN78?t$KJQ49$5$l$F$$$kI,MW$,$"$k(B.
3134: @item
3135: $B4JLs$5$l$k9`$O(B @var{dpoly2} $B$NF,9`$G$"$k(B. $B=>$C$F(B, @var{dpoly2} $B$N(B
3136: $BF,9`$,(B @var{dpoly3} $B$NF,9`$G3d$j@Z$l$k$3$H$,$"$i$+$8$a$o$+$C$F$$$J$1$l$P(B
3137: $B$J$i$J$$(B.
3138: @item
3139: $B0z?t$,@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 3140: $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 3141: @item
3142: $B7k2L$O(B, @code{[@var{a dpoly1},@var{a dpoly2 - bt dpoly3}]} $B$J$k%j%9%H$G$"$k(B.
1.2 noro 3143: \E
3144: \BEG
3145: @item
3146: Reduces a distributed polynomial, @var{dpoly1} + @var{dpoly2},
3147: by @var{dpoly3} for single time.
3148: @item
3149: An input for @code{dp_red_mod()} must be converted into a distributed
3150: polynomial with coefficients in a finite field.
3151: @item
3152: This implies that
3153: the divisibility of the head term of @var{dpoly2} by the head term of
3154: @var{dpoly3} is assumed.
3155: @item
3156: When integral coefficients, computation is so carefully performed that
3157: no rational operations appear in the reduction procedure.
3158: It is computed for integers @var{a} and @var{b}, and a term @var{t} as:
1.4 ! noro 3159: @var{a}(@var{dpoly1} + @var{dpoly2})-@var{bt} @var{dpoly3}.
1.2 noro 3160: @item
3161: The result is a list @code{[@var{a dpoly1},@var{a dpoly2 - bt dpoly3}]}.
3162: \E
1.1 noro 3163: @end itemize
3164:
3165: @example
3166: [157] D=(3)*<<2,1,0,0,0>>+(3)*<<1,2,0,0,0>>+(1)*<<0,3,0,0,0>>;
3167: (3)*<<2,1,0,0,0>>+(3)*<<1,2,0,0,0>>+(1)*<<0,3,0,0,0>>
3168: [158] R=(6)*<<1,1,1,0,0>>;
3169: (6)*<<1,1,1,0,0>>
3170: [159] C=12*<<1,1,1,0,0>>+(1)*<<0,1,1,1,0>>+(1)*<<1,1,0,0,1>>;
3171: (12)*<<1,1,1,0,0>>+(1)*<<0,1,1,1,0>>+(1)*<<1,1,0,0,1>>
3172: [160] dp_red(D,R,C);
3173: [(6)*<<2,1,0,0,0>>+(6)*<<1,2,0,0,0>>+(2)*<<0,3,0,0,0>>,(-1)*<<0,1,1,1,0>>
3174: +(-1)*<<1,1,0,0,1>>]
3175: @end example
3176:
3177: @table @t
1.2 noro 3178: \JP @item $B;2>H(B
3179: \EG @item References
1.1 noro 3180: @fref{dp_mod dp_rat}.
3181: @end table
3182:
1.2 noro 3183: \JP @node dp_sp dp_sp_mod,,, $B%0%l%V%J4pDl$K4X$9$kH!?t(B
3184: \EG @node dp_sp dp_sp_mod,,, Functions for Groebner basis computation
1.1 noro 3185: @subsection @code{dp_sp}, @code{dp_sp_mod}
3186: @findex dp_sp
3187: @findex dp_sp_mod
3188:
3189: @table @t
3190: @item dp_sp(@var{dpoly1},@var{dpoly2})
3191: @item dp_sp_mod(@var{dpoly1},@var{dpoly2},@var{mod})
1.2 noro 3192: \JP :: S-$BB?9`<0$N7W;;(B
3193: \EG :: Computation of an S-polynomial
1.1 noro 3194: @end table
3195:
3196: @table @var
3197: @item return
1.2 noro 3198: \JP $BJ,;6I=8=B?9`<0(B
3199: \EG distributed polynomial
1.4 ! noro 3200: @item dpoly1 dpoly2
1.2 noro 3201: \JP $BJ,;6I=8=B?9`<0(B
3202: \EG distributed polynomial
1.1 noro 3203: @item mod
1.2 noro 3204: \JP $BAG?t(B
3205: \EG prime
1.1 noro 3206: @end table
3207:
3208: @itemize @bullet
1.2 noro 3209: \BJP
1.1 noro 3210: @item
3211: @var{dpoly1}, @var{dpoly2} $B$N(B S-$BB?9`<0$r7W;;$9$k(B.
3212: @item
3213: @code{dp_sp_mod()} $B$NF~NO$O(B, $BA4$FM-8BBN78?t$KJQ49$5$l$F$$$kI,MW$,$"$k(B.
3214: @item
3215: $B7k2L$KM-M}?t(B, $BM-M}<0$,F~$k$N$rHr$1$k$?$a(B, $B7k2L$,Dj?tG\(B, $B$"$k$$$OB?9`<0(B
3216: $BG\$5$l$F$$$k2DG=@-$,$"$k(B.
1.2 noro 3217: \E
3218: \BEG
3219: @item
3220: This function computes the S-polynomial of @var{dpoly1} and @var{dpoly2}.
3221: @item
3222: Inputs of @code{dp_sp_mod()} must be polynomials with coefficients in a
3223: finite field.
3224: @item
3225: The result may be multiplied by a constant in the ground field in order to
3226: make the result integral.
3227: \E
1.1 noro 3228: @end itemize
3229:
3230: @example
3231: [227] X=dp_ptod(x^2*y+x*y,[x,y]);
3232: (1)*<<2,1>>+(1)*<<1,1>>
3233: [228] Y=dp_ptod(x*y^2+x*y,[x,y]);
3234: (1)*<<1,2>>+(1)*<<1,1>>
3235: [229] dp_sp(X,Y);
3236: (-1)*<<2,1>>+(1)*<<1,2>>
3237: @end example
3238:
3239: @table @t
1.2 noro 3240: \JP @item $B;2>H(B
3241: \EG @item References
1.1 noro 3242: @fref{dp_mod dp_rat}.
3243: @end table
1.2 noro 3244: \JP @node p_nf p_nf_mod p_true_nf p_true_nf_mod,,, $B%0%l%V%J4pDl$K4X$9$kH!?t(B
3245: \EG @node p_nf p_nf_mod p_true_nf p_true_nf_mod,,, Functions for Groebner basis computation
1.1 noro 3246: @subsection @code{p_nf}, @code{p_nf_mod}, @code{p_true_nf}, @code{p_true_nf_mod}
3247: @findex p_nf
3248: @findex p_nf_mod
3249: @findex p_true_nf
3250: @findex p_true_nf_mod
3251:
3252: @table @t
3253: @item p_nf(@var{poly},@var{plist},@var{vlist},@var{order})
3254: @itemx p_nf_mod(@var{poly},@var{plist},@var{vlist},@var{order},@var{mod})
1.2 noro 3255: \JP :: $BI=8=B?9`<0$N@55,7A$r5a$a$k(B. ($B7k2L$ODj?tG\$5$l$F$$$k2DG=@-$"$j(B)
3256: \BEG
3257: :: Computes the normal form of the given polynomial.
3258: (The result may be multiplied by a constant.)
3259: \E
1.1 noro 3260: @item p_true_nf(@var{poly},@var{plist},@var{vlist},@var{order})
3261: @itemx p_true_nf_mod(@var{poly},@var{plist},@var{vlist},@var{order},@var{mod})
1.2 noro 3262: \JP :: $BI=8=B?9`<0$N@55,7A$r5a$a$k(B. ($B??$N7k2L$r(B @code{[$BJ,;R(B, $BJ,Jl(B]} $B$N7A$GJV$9(B)
3263: \BEG
3264: :: Computes the normal form of the given polynomial. (The result is returned
3265: as a form of @code{[numerator, denominator]})
3266: \E
1.1 noro 3267: @end table
3268:
3269: @table @var
3270: @item return
1.2 noro 3271: \JP @code{p_nf} : $BB?9`<0(B, @code{p_true_nf} : $B%j%9%H(B
3272: \EG @code{p_nf} : polynomial, @code{p_true_nf} : list
1.1 noro 3273: @item poly
1.2 noro 3274: \JP $BB?9`<0(B
3275: \EG polynomial
1.4 ! noro 3276: @item plist vlist
1.2 noro 3277: \JP $B%j%9%H(B
3278: \EG list
1.1 noro 3279: @item order
1.2 noro 3280: \JP $B?t(B, $B%j%9%H$^$?$O9TNs(B
3281: \EG number, list or matrix
1.1 noro 3282: @item mod
1.2 noro 3283: \JP $BAG?t(B
3284: \EG prime
1.1 noro 3285: @end table
3286:
3287: @itemize @bullet
1.2 noro 3288: \BJP
1.1 noro 3289: @item
3290: @samp{gr} $B$GDj5A$5$l$F$$$k(B.
3291: @item
3292: $BB?9`<0$N(B, $BB?9`<0%j%9%H$K$h$k@55,7A$r5a$a$k(B.
3293: @item
3294: @code{dp_nf()}, @code{dp_true_nf()}, @code{dp_nf_mod()}, @code{dp_true_nf_mod}
3295: $B$KBP$9$k%$%s%?%U%'!<%9$G$"$k(B.
3296: @item
3297: @var{poly} $B$*$h$S(B @var{plist} $B$O(B, $BJQ?t=g=x(B @var{vlist} $B$*$h$S(B
3298: $BJQ?t=g=x7?(B @var{otype} $B$K=>$C$FJ,;6I=8=B?9`<0$KJQ49$5$l(B,
3299: @code{dp_nf()}, @code{dp_true_nf()}, @code{dp_nf_mod()},
3300: @code{dp_true_nf_mod()} $B$KEO$5$l$k(B.
3301: @item
3302: @code{dp_nf()}, @code{dp_true_nf()}, @code{dp_nf_mod()},
3303: @code{dp_true_nf_mod()} $B$O(B @var{fullreduce} $B$,(B 1 $B$G8F$S=P$5$l$k(B.
3304: @item
3305: $B7k2L$OB?9`<0$KJQ49$5$l$F=PNO$5$l$k(B.
3306: @item
3307: @code{p_true_nf()}, @code{p_true_nf_mod()} $B$N=PNO$K4X$7$F$O(B,
3308: @code{dp_true_nf()}, @code{dp_true_nf_mod()} $B$N9`$r;2>H(B.
1.2 noro 3309: \E
3310: \BEG
3311: @item
3312: Defined in the package @samp{gr}.
3313: @item
3314: Obtains the normal form of a polynomial by a polynomial list.
3315: @item
3316: These are interfaces to @code{dp_nf()}, @code{dp_true_nf()}, @code{dp_nf_mod()},
3317: @code{dp_true_nf_mod}
3318: @item
3319: The polynomial @var{poly} and the polynomials in @var{plist} is
3320: converted, according to the variable ordering @var{vlist} and
3321: type of term ordering @var{otype}, into their distributed polynomial
3322: counterparts and passed to @code{dp_nf()}.
3323: @item
3324: @code{dp_nf()}, @code{dp_true_nf()}, @code{dp_nf_mod()} and
3325: @code{dp_true_nf_mod()}
3326: is called with value 1 for @var{fullreduce}.
3327: @item
3328: The result is converted back into an ordinary polynomial.
3329: @item
3330: As for @code{p_true_nf()}, @code{p_true_nf_mod()}
3331: refer to @code{dp_true_nf()} and @code{dp_true_nf_mod()}.
3332: \E
1.1 noro 3333: @end itemize
3334:
3335: @example
3336: [79] K = katsura(5)$
3337: [80] V = [u5,u4,u3,u2,u1,u0]$
3338: [81] G = hgr(K,V,2)$
3339: [82] p_nf(K[1],G,V,2);
3340: 0
3341: [83] L = p_true_nf(K[1]+1,G,V,2);
3342: [-1503...,-1503...]
3343: [84] L[0]/L[1];
3344: 1
3345: @end example
3346:
3347: @table @t
1.2 noro 3348: \JP @item $B;2>H(B
3349: \EG @item References
1.1 noro 3350: @fref{dp_ptod},
3351: @fref{dp_dtop},
3352: @fref{dp_ord},
3353: @fref{dp_nf dp_nf_mod dp_true_nf dp_true_nf_mod}.
3354: @end table
3355:
1.2 noro 3356: \JP @node p_terms,,, $B%0%l%V%J4pDl$K4X$9$kH!?t(B
3357: \EG @node p_terms,,, Functions for Groebner basis computation
1.1 noro 3358: @subsection @code{p_terms}
3359: @findex p_terms
3360:
3361: @table @t
3362: @item p_terms(@var{poly},@var{vlist},@var{order})
1.2 noro 3363: \JP :: $BB?9`<0$K$"$i$o$l$kC19`$r%j%9%H$K$9$k(B.
3364: \EG :: Monomials appearing in the given polynomial is collected into a list.
1.1 noro 3365: @end table
3366:
3367: @table @var
3368: @item return
1.2 noro 3369: \JP $B%j%9%H(B
3370: \EG list
1.1 noro 3371: @item poly
1.2 noro 3372: \JP $BB?9`<0(B
3373: \EG polynomial
1.1 noro 3374: @item vlist
1.2 noro 3375: \JP $B%j%9%H(B
3376: \EG list
1.1 noro 3377: @item order
1.2 noro 3378: \JP $B?t(B, $B%j%9%H$^$?$O9TNs(B
3379: \EG number, list or matrix
1.1 noro 3380: @end table
3381:
3382: @itemize @bullet
1.2 noro 3383: \BJP
1.1 noro 3384: @item
3385: @samp{gr} $B$GDj5A$5$l$F$$$k(B.
3386: @item
3387: $BB?9`<0$rC19`$KE83+$7$?;~$K8=$l$k9`$r%j%9%H$K$7$FJV$9(B.
3388: @var{vlist} $B$*$h$S(B @var{order} $B$K$h$jDj$^$k9`=g=x$K$h$j(B, $B=g=x$N9b$$$b$N(B
3389: $B$,%j%9%H$N@hF,$KMh$k$h$&$K%=!<%H$5$l$k(B.
3390: @item
3391: $B%0%l%V%J4pDl$O$7$P$7$P78?t$,5pBg$K$J$k$?$a(B, $B<B:]$K$I$N9`$,8=$l$F(B
3392: $B$$$k$N$+$r8+$k$?$a$J$I$KMQ$$$k(B.
1.2 noro 3393: \E
3394: \BEG
3395: @item
3396: Defined in the package @samp{gr}.
3397: @item
3398: This returns a list which contains all non-zero monomials in the given
3399: polynomial. The monomials are ordered according to the current
3400: type of term ordering and @var{vlist}.
3401: @item
3402: Since polynomials in a Groebner base often have very large coefficients,
3403: examining a polynomial as it is may sometimes be difficult to perform.
3404: For such a case, this function enables to examine which term is really
3405: exists.
3406: \E
1.1 noro 3407: @end itemize
3408:
3409: @example
3410: [233] G=gr(katsura(5),[u5,u4,u3,u2,u1,u0],2)$
3411: [234] p_terms(G[0],[u5,u4,u3,u2,u1,u0],2);
3412: [u5,u0^31,u0^30,u0^29,u0^28,u0^27,u0^26,u0^25,u0^24,u0^23,u0^22,u0^21,u0^20,
3413: u0^19,u0^18,u0^17,u0^16,u0^15,u0^14,u0^13,u0^12,u0^11,u0^10,u0^9,u0^8,u0^7,
3414: u0^6,u0^5,u0^4,u0^3,u0^2,u0,1]
3415: @end example
3416:
1.2 noro 3417: \JP @node gb_comp,,, $B%0%l%V%J4pDl$K4X$9$kH!?t(B
3418: \EG @node gb_comp,,, Functions for Groebner basis computation
1.1 noro 3419: @subsection @code{gb_comp}
3420: @findex gb_comp
3421:
3422: @table @t
3423: @item gb_comp(@var{plist1}, @var{plist2})
1.2 noro 3424: \JP :: $BB?9`<0%j%9%H$,(B, $BId9f$r=|$$$F=89g$H$7$FEy$7$$$+$I$&$+D4$Y$k(B.
3425: \EG :: Checks whether two polynomial lists are equal or not as a set
1.1 noro 3426: @end table
3427:
3428: @table @var
1.2 noro 3429: \JP @item return 0 $B$^$?$O(B 1
3430: \EG @item return 0 or 1
1.4 ! noro 3431: @item plist1 plist2
1.1 noro 3432: @end table
3433:
3434: @itemize @bullet
1.2 noro 3435: \BJP
1.1 noro 3436: @item
3437: @var{plist1}, @var{plist2} $B$K$D$$$F(B, $BId9f$r=|$$$F=89g$H$7$FEy$7$$$+$I$&$+(B
3438: $BD4$Y$k(B.
3439: @item
3440: $B0[$J$kJ}K!$G5a$a$?%0%l%V%J4pDl$O(B, $B4pDl$N=g=x(B, $BId9f$,0[$J$k>l9g$,$"$j(B,
3441: $B$=$l$i$,Ey$7$$$+$I$&$+$rD4$Y$k$?$a$KMQ$$$k(B.
1.2 noro 3442: \E
3443: \BEG
3444: @item
3445: This function checks whether @var{plist1} and @var{plist2} are equal or
3446: not as a set .
3447: @item
3448: For the same input and the same term ordering different
3449: functions for Groebner basis computations may produce different outputs
3450: as lists. This function compares such lists whether they are equal
3451: as a generating set of an ideal.
3452: \E
1.1 noro 3453: @end itemize
3454:
3455: @example
3456: [243] C=cyclic(6)$
3457: [244] V=[c0,c1,c2,c3,c4,c5]$
3458: [245] G0=gr(C,V,0)$
3459: [246] G=tolex(G0,V,0,V)$
3460: [247] GG=lex_tl(C,V,0,V,0)$
3461: [248] gb_comp(G,GG);
3462: 1
3463: @end example
3464:
1.2 noro 3465: \JP @node katsura hkatsura cyclic hcyclic,,, $B%0%l%V%J4pDl$K4X$9$kH!?t(B
3466: \EG @node katsura hkatsura cyclic hcyclic,,, Functions for Groebner basis computation
1.1 noro 3467: @subsection @code{katsura}, @code{hkatsura}, @code{cyclic}, @code{hcyclic}
3468: @findex katsura
3469: @findex hkatsura
3470: @findex cyclic
3471: @findex hcyclic
3472:
3473: @table @t
3474: @item katsura(@var{n})
3475: @item hkatsura(@var{n})
3476: @item cyclic(@var{n})
3477: @item hcyclic(@var{n})
1.2 noro 3478: \JP :: $BB?9`<0%j%9%H$N@8@.(B
3479: \EG :: Generates a polynomial list of standard benchmark.
1.1 noro 3480: @end table
3481:
3482: @table @var
3483: @item return
1.2 noro 3484: \JP $B%j%9%H(B
3485: \EG list
1.1 noro 3486: @item n
1.2 noro 3487: \JP $B@0?t(B
3488: \EG integer
1.1 noro 3489: @end table
3490:
3491: @itemize @bullet
1.2 noro 3492: \BJP
1.1 noro 3493: @item
3494: @code{katsura()} $B$O(B @samp{katsura}, @code{cyclic()} $B$O(B @samp{cyclic}
3495: $B$GDj5A$5$l$F$$$k(B.
3496: @item
3497: $B%0%l%V%J4pDl7W;;$G$7$P$7$P%F%9%H(B, $B%Y%s%A%^!<%/$KMQ$$$i$l$k(B @code{katsura},
3498: @code{cyclic} $B$*$h$S$=$N@F<!2=$r@8@.$9$k(B.
3499: @item
3500: @code{cyclic} $B$O(B @code{Arnborg}, @code{Lazard}, @code{Davenport} $B$J$I$N(B
3501: $BL>$G8F$P$l$k$3$H$b$"$k(B.
1.2 noro 3502: \E
3503: \BEG
3504: @item
3505: Function @code{katsura()} is defined in @samp{katsura}, and
3506: function @code{cyclic()} in @samp{cyclic}.
3507: @item
3508: These functions generate a series of polynomial sets, respectively,
3509: which are often used for testing and bench marking:
3510: @code{katsura}, @code{cyclic} and their homogenized versions.
3511: @item
3512: Polynomial set @code{cyclic} is sometimes called by other name:
3513: @code{Arnborg}, @code{Lazard}, and @code{Davenport}.
3514: \E
1.1 noro 3515: @end itemize
3516:
3517: @example
3518: [74] load("katsura")$
3519: [79] load("cyclic")$
3520: [89] katsura(5);
3521: [u0+2*u4+2*u3+2*u2+2*u1+2*u5-1,2*u4*u0-u4+2*u1*u3+u2^2+2*u5*u1,
3522: 2*u3*u0+2*u1*u4-u3+(2*u1+2*u5)*u2,2*u2*u0+2*u2*u4+(2*u1+2*u5)*u3-u2+u1^2,
3523: 2*u1*u0+(2*u3+2*u5)*u4+2*u2*u3+2*u1*u2-u1,
3524: u0^2-u0+2*u4^2+2*u3^2+2*u2^2+2*u1^2+2*u5^2]
3525: [90] hkatsura(5);
3526: [-t+u0+2*u4+2*u3+2*u2+2*u1+2*u5,
3527: -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,
3528: -u2*t+2*u2*u0+2*u2*u4+(2*u1+2*u5)*u3+u1^2,
3529: -u1*t+2*u1*u0+(2*u3+2*u5)*u4+2*u2*u3+2*u1*u2,
3530: -u0*t+u0^2+2*u4^2+2*u3^2+2*u2^2+2*u1^2+2*u5^2]
3531: [91] cyclic(6);
3532: [c5*c4*c3*c2*c1*c0-1,
3533: ((((c4+c5)*c3+c5*c4)*c2+c5*c4*c3)*c1+c5*c4*c3*c2)*c0+c5*c4*c3*c2*c1,
3534: (((c3+c5)*c2+c5*c4)*c1+c5*c4*c3)*c0+c4*c3*c2*c1+c5*c4*c3*c2,
3535: ((c2+c5)*c1+c5*c4)*c0+c3*c2*c1+c4*c3*c2+c5*c4*c3,
3536: (c1+c5)*c0+c2*c1+c3*c2+c4*c3+c5*c4,c0+c1+c2+c3+c4+c5]
3537: [92] hcyclic(6);
3538: [-c^6+c5*c4*c3*c2*c1*c0,
3539: ((((c4+c5)*c3+c5*c4)*c2+c5*c4*c3)*c1+c5*c4*c3*c2)*c0+c5*c4*c3*c2*c1,
3540: (((c3+c5)*c2+c5*c4)*c1+c5*c4*c3)*c0+c4*c3*c2*c1+c5*c4*c3*c2,
3541: ((c2+c5)*c1+c5*c4)*c0+c3*c2*c1+c4*c3*c2+c5*c4*c3,
3542: (c1+c5)*c0+c2*c1+c3*c2+c4*c3+c5*c4,c0+c1+c2+c3+c4+c5]
3543: @end example
3544:
3545: @table @t
1.2 noro 3546: \JP @item $B;2>H(B
3547: \EG @item References
1.1 noro 3548: @fref{dp_dtop}.
3549: @end table
3550:
1.3 noro 3551: \JP @node primadec primedec,,, $B%0%l%V%J4pDl$K4X$9$kH!?t(B
3552: \EG @node primadec primedec,,, Functions for Groebner basis computation
3553: @subsection @code{primadec}, @code{primedec}
3554: @findex primadec
3555: @findex primedec
3556:
3557: @table @t
3558: @item primadec(@var{plist},@var{vlist})
3559: @item primedec(@var{plist},@var{vlist})
3560: \JP :: $B%$%G%"%k$NJ,2r(B
3561: \EG :: Computes decompositions of ideals.
3562: @end table
3563:
3564: @table @var
3565: @item return
3566: @itemx plist
3567: \JP $BB?9`<0%j%9%H(B
3568: \EG list of polynomials
3569: @item vlist
3570: \JP $BJQ?t%j%9%H(B
3571: \EG list of variables
3572: @end table
3573:
3574: @itemize @bullet
3575: \BJP
3576: @item
3577: @code{primadec()}, @code{primedec} $B$O(B @samp{primdec} $B$GDj5A$5$l$F$$$k(B.
3578: @item
3579: @code{primadec()}, @code{primedec()} $B$O$=$l$>$lM-M}?tBN>e$G$N%$%G%"%k$N(B
3580: $B=`AGJ,2r(B, $B:,4p$NAG%$%G%"%kJ,2r$r9T$&(B.
3581: @item
3582: $B0z?t$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.
3583: @item
3584: @code{primadec} $B$O(B @code{[$B=`AG@.J,(B, $BIUB0AG%$%G%"%k(B]} $B$N%j%9%H$rJV$9(B.
3585: @item
3586: @code{primadec} $B$O(B $BAG0x;R$N%j%9%H$rJV$9(B.
3587: @item
3588: $B7k2L$K$*$$$F(B, $BB?9`<0%j%9%H$H$7$FI=<($5$l$F$$$k3F%$%G%"%k$OA4$F(B
3589: $B%0%l%V%J4pDl$G$"$k(B. $BBP1~$9$k9`=g=x$O(B, $B$=$l$>$l(B
3590: $BJQ?t(B @code{PRIMAORD}, @code{PRIMEORD} $B$K3JG<$5$l$F$$$k(B.
3591: @item
3592: @code{primadec} $B$O(B @code{[Shimoyama,Yokoyama]} $B$N=`AGJ,2r%"%k%4%j%:%`(B
3593: $B$r<BAu$7$F$$$k(B.
3594: @item
3595: $B$b$7AG0x;R$N$_$r5a$a$?$$$J$i(B, @code{primedec} $B$r;H$&J}$,$h$$(B.
3596: $B$3$l$O(B, $BF~NO%$%G%"%k$,:,4p%$%G%"%k$G$J$$>l9g$K(B, @code{primadec}
3597: $B$N7W;;$KM>J,$J%3%9%H$,I,MW$H$J$k>l9g$,$"$k$+$i$G$"$k(B.
3598: \E
3599: \BEG
3600: @item
3601: Function @code{primadec()} and @code{primedec} are defined in @samp{primdec}.
3602: @item
3603: @code{primadec()}, @code{primedec()} are the function for primary
3604: ideal decomposition and prime decomposition of the radical over the
3605: rationals respectively.
3606: @item
3607: The arguments are a list of polynomials and a list of variables.
3608: These functions accept ideals with rational function coefficients only.
3609: @item
3610: @code{primadec} returns the list of pair lists consisting a primary component
3611: and its associated prime.
3612: @item
3613: @code{primedec} returns the list of prime components.
3614: @item
3615: Each component is a Groebner basis and the corresponding term order
3616: is indicated by the global variables @code{PRIMAORD}, @code{PRIMEORD}
3617: respectively.
3618: @item
3619: @code{primadec} implements the primary decompostion algorithm
3620: in @code{[Shimoyama,Yokoyama]}.
3621: @item
3622: If one only wants to know the prime components of an ideal, then
3623: use @code{primedec} because @code{primadec} may need additional costs
3624: if an input ideal is not radical.
3625: \E
3626: @end itemize
3627:
3628: @example
3629: [84] load("primdec")$
3630: [102] primedec([p*q*x-q^2*y^2+q^2*y,-p^2*x^2+p^2*x+p*q*y,
3631: (q^3*y^4-2*q^3*y^3+q^3*y^2)*x-q^3*y^4+q^3*y^3,
3632: -q^3*y^4+2*q^3*y^3+(-q^3+p*q^2)*y^2],[p,q,x,y]);
3633: [[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]]
3634: [103] primadec([x,z*y,w*y^2,w^2*y-z^3,y^3],[x,y,z,w]);
3635: [[[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]]]
3636: @end example
3637:
3638: @table @t
3639: \JP @item $B;2>H(B
3640: \EG @item References
3641: @fref{fctr sqfr},
3642: \JP @fref{$B9`=g=x$N@_Dj(B}.
3643: \EG @fref{Setting term orderings}.
3644: @end table
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