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Diff for /OpenXM/src/asir-doc/parts/groebner.texi between version 1.7 and 1.17

-version 1.7, 2003/04/21 03:07:32
+version 1.17, 2006/09/06 23:53:31
 Line 1
 Line 1
 Line 1
- @comment $OpenXM: OpenXM/src/asir-doc/parts/groebner.texi,v 1.6 2003/04/20 09:55:18 noro Exp $
+ @comment $OpenXM: OpenXM/src/asir-doc/parts/groebner.texi,v 1.16 2004/10/20 00:30:55 fujiwara Exp $
  \BJP
  @node $B%0%l%V%J4pDl$N7W;;(B,,, Top
  @chapter $B%0%l%V%J4pDl$N7W;;(B
 Line 15
 Line 15
 Line 15
  * $B4pK\E*$JH!?t(B::
  * $B7W;;$*$h$SI=<($N@)8f(B::
  * $B9`=g=x$N@_Dj(B::
+ * Weight::
  * $BM-M}<0$r78?t$H$9$k%0%l%V%J4pDl7W;;(B::
  * $B4pDlJQ49(B::
  * Weyl $BBe?t(B::
-Line 26
+Line 27
 Line 26
 Line 27
  * Fundamental functions::
  * Controlling Groebner basis computations::
  * Setting term orderings::
+ * Weight::
  * Groebner basis computation with rational function coefficients::
  * Change of ordering::
  * Weyl algebra::
-Line 1055  beforehand, and some heuristic trial may be inevitable
+Line 1057  beforehand, and some heuristic trial may be inevitable
 Line 1055  beforehand, and some heuristic trial may be inevitable
 Line 1057  beforehand, and some heuristic trial may be inevitable
  \E
  \BJP
+ @node Weight ,,, $B%0%l%V%J4pDl$N7W;;(B
+ @section Weight
+ \E
+ \BEG
+ @node Weight,,, Groebner basis computation
+ @section Weight
+ \E
+ \BJP
+ $BA0@a$G>R2p$7$?9`=g=x$O(B, $B3FJQ?t$K(B weight ($B=E$_(B) $B$r@_Dj$9$k$3$H$G(B
+ $B$h$j0lHLE*$J$b$N$H$J$k(B.
+ \E
+ \BEG
+ Term orderings introduced in the previous section can be generalized
+ by setting a weight for each variable.
+ \E
+ @example
+ [0] dp_td(<<1,1,1>>);
+ [1] dp_set_weight([1,2,3])$
+ [2] dp_td(<<1,1,1>>);
+ @end example
+ \BJP
+ $BC19`<0$NA4<!?t$r7W;;$9$k:](B, $B%G%U%)%k%H$G$O(B
+ $B3FJQ?t$N;X?t$NOB$rA4<!?t$H$9$k(B. $B$3$l$O3FJQ?t$N(B weight $B$r(B 1 $B$H(B
+ $B9M$($F$$$k$3$H$KAjEv$9$k(B. $B$3$NNc$G$O(B, $BBh0l(B, $BBhFs(B, $BBh;0JQ?t$N(B
+ weight $B$r$=$l$>$l(B 1,2,3 $B$H;XDj$7$F$$$k(B. $B$3$N$?$a(B, @code{<<1,1,1>>}
+ $B$NA4<!?t(B ($B0J2<$G$O$3$l$rC19`<0$N(B weight $B$H8F$V(B) $B$,(B @code{1*1+1*2+1*3=6} $B$H$J$k(B.
+ weight $B$r@_Dj$9$k$3$H$G(B, $BF1$89`=g=x7?$N$b$H$G0[$J$k9`=g=x$,Dj5A$G$-$k(B.
+ $BNc$($P(B, weight $B$r$&$^$/@_Dj$9$k$3$H$G(B, $BB?9`<0$r(B weighted homogeneous
+ $B$K$9$k$3$H$,$G$-$k>l9g$,$"$k(B.
+ \E
+ \BEG
+ By default, the total degree of a monomial is equal to
+ the sum of all exponents. This means that the weight for each variable
+ is set to 1.
+ In this example, the weights for the first, the second and the third
+ variable are set to 1, 2 and 3 respectively.
+ Therefore the total degree of @code{<<1,1,1>>} under this weight,
+ which is called the weight of the monomial, is @code{1*1+1*2+1*3=6}.
+ By setting weights, different term orderings can be set under a type of
+ term ordeing. In some case a polynomial can
+ be made weighted homogeneous by setting an appropriate weight.
+ \E
+ \BJP
+ $B3FJQ?t$KBP$9$k(B weight $B$r$^$H$a$?$b$N$r(B weight vector $B$H8F$V(B.
+ $B$9$Y$F$N@.J,$,@5$G$"$j(B, $B%0%l%V%J4pDl7W;;$K$*$$$F(B, $BA4<!?t$N(B
+ $BBe$o$j$KMQ$$$i$l$k$b$N$rFC$K(B sugar weight $B$H8F$V$3$H$K$9$k(B.
+ sugar strategy $B$K$*$$$F(B, $BA4<!?t$NBe$o$j$K;H$o$l$k$+$i$G$"$k(B.
+ $B0lJ}$G(B, $B3F@.J,$,I,$:$7$b@5$H$O8B$i$J$$(B weight vector $B$O(B,
+ sugar weight $B$H$7$F@_Dj$9$k$3$H$O$G$-$J$$$,(B, $B9`=g=x$N0lHL2=$K$O(B
+ $BM-MQ$G$"$k(B. $B$3$l$i$O(B, $B9TNs$K$h$k9`=g=x$N@_Dj$K$9$G$K8=$l$F(B
+ $B$$$k(B. $B$9$J$o$A(B, $B9`=g=x$rDj5A$9$k9TNs$N3F9T$,(B, $B0l$D$N(B weight vector
+ $B$H8+$J$5$l$k(B. $B$^$?(B, $B%V%m%C%/=g=x$O(B, $B3F%V%m%C%/$N(B
+ $BJQ?t$KBP1~$9$k@.J,$N$_(B 1 $B$GB>$O(B 0 $B$N(B weight vector $B$K$h$kHf3S$r(B
+ $B:G=i$K9T$C$F$+$i(B, $B3F%V%m%C%/Kh$N(B tie breaking $B$r9T$&$3$H$KAjEv$9$k(B.
+ \E
+ \BEG
+ A list of weights for all variables is called a weight vector.
+ A weight vector is called a sugar weight vector if
+ its elements are all positive and it is used for computing
+ a weighted total degree of a monomial, because such a weight
+ is used instead of total degree in sugar strategy.
+ On the other hand, a weight vector whose elements are not necessarily
+ positive cannot be set as a sugar weight, but it is useful for
+ generalizing term order. In fact, such a weight vector already
+ appeared in a matrix order. That is, each row of a matrix defining
+ a term order is regarded as a weight vector. A block order
+ is also considered as a refinement of comparison by weight vectors.
+ It compares two terms by using a weight vector whose elements
+ corresponding to variables in a block is 1 and 0 otherwise,
+ then it applies a tie breaker.
+ \E
+ \BJP
+ weight vector $B$N@_Dj$O(B @code{dp_set_weight()} $B$G9T$&$3$H$,$G$-$k(B
+ $B$,(B, $B9`=g=x$r;XDj$9$k:]$NB>$N%Q%i%a%?(B ($B9`=g=x7?(B, $BJQ?t=g=x(B) $B$H(B
+ $B$^$H$a$F@_Dj$G$-$k$3$H$,K>$^$7$$(B. $B$3$N$?$a(B, $B<!$N$h$&$J7A$G$b(B
+ $B9`=g=x$,;XDj$G$-$k(B.
+ \E
+ \BEG
+ A weight vector can be set by using @code{dp_set_weight()}.
+ However it is more preferable if a weight vector can be set
+ together with other parapmeters such as a type of term ordering
+ and a variable order. This is realized as follows.
+ \E
+ @example
+ [64] B=[x+y+z-6,x*y+y*z+z*x-11,x*y*z-6]$
+ [65] dp_gr_main(B|v=[x,y,z],sugarweight=[3,2,1],order=0);
+ [z^3-6*z^2+11*z-6,x+y+z-6,-y^2+(-z+6)*y-z^2+6*z-11]
+ [66] dp_gr_main(B|v=[y,z,x],order=[[1,1,0],[0,1,0],[0,0,1]]);
+ [x^3-6*x^2+11*x-6,x+y+z-6,-x^2+(-y+6)*x-y^2+6*y-11]
+ [67] dp_gr_main(B|v=[y,z,x],order=[[x,1,y,2,z,3]]);
+ [x+y+z-6,x^3-6*x^2+11*x-6,-x^2+(-y+6)*x-y^2+6*y-11]
+ @end example
+ \BJP
+ $B$$$:$l$NNc$K$*$$$F$b(B, $B9`=g=x$O(B option $B$H$7$F;XDj$5$l$F$$$k(B.
+ $B:G=i$NNc$G$O(B @code{v} $B$K$h$jJQ?t=g=x$r(B, @code{sugarweight} $B$K$h$j(B
+ sugar weight vector $B$r(B, @code{order}$B$K$h$j9`=g=x7?$r;XDj$7$F$$$k(B.
+ $BFs$DL\$NNc$K$*$1$k(B @code{order} $B$N;XDj$O(B matrix order $B$HF1MM$G$"$k(B.
+ $B$9$J$o$A(B, $B;XDj$5$l$?(B weight vector $B$r:8$+$i=g$K;H$C$F(B weight $B$NHf3S(B
+ $B$r9T$&(B. $B;0$DL\$NNc$bF1MM$G$"$k$,(B, $B$3$3$G$O(B weight vector $B$NMWAG$r(B
+ $BJQ?tKh$K;XDj$7$F$$$k(B. $B;XDj$,$J$$$b$N$O(B 0 $B$H$J$k(B. $B;0$DL\$NNc$G$O(B,
+ @code{order} $B$K$h$k;XDj$G$O9`=g=x$,7hDj$7$J$$(B. $B$3$N>l9g$K$O(B,
+ tie breaker $B$H$7$FA4<!?t5U<-=q<0=g=x$,<+F0E*$K@_Dj$5$l$k(B.
+ $B$3$N;XDjJ}K!$O(B, @code{dp_gr_main}, @code{dp_gr_mod_main} $B$J$I(B
+ $B$NAH$_9~$_4X?t$G$N$_2DG=$G$"$j(B, @code{gr} $B$J$I$N%f!<%6Dj5A4X?t(B
+ $B$G$OL$BP1~$G$"$k(B.
+ \E
+ \BEG
+ In each example, a term ordering is specified as options.
+ In the first example, a variable order, a sugar weight vector
+ and a type of term ordering are specified by options @code{v},
+ @code{sugarweight} and @code{order} respectively.
+ In the second example, an option @code{order} is used
+ to set a matrix ordering. That is, the specified weight vectors
+ are used from left to right for comparing terms.
+ The third example shows a variant of specifying a weight vector,
+ where each component of a weight vector is specified variable by variable,
+ and unspecified components are set to zero. In this example,
+ a term order is not determined only by the specified weight vector.
+ In such a case a tie breaker by the graded reverse lexicographic ordering
+ is set automatically.
+ This type of a term ordering specification can be applied only to builtin
+ functions such as @code{dp_gr_main()}, @code{dp_gr_mod_main()}, not to
+ user defined functions such as @code{gr()}.
+ \E
+ \BJP
  @node $BM-M}<0$r78?t$H$9$k%0%l%V%J4pDl7W;;(B,,, $B%0%l%V%J4pDl$N7W;;(B
  @section $BM-M}<0$r78?t$H$9$k%0%l%V%J4pDl7W;;(B
  \E
-Line 1329  Computation of the global b function is implemented as
+Line 1464  Computation of the global b function is implemented as
 Line 1329  Computation of the global b function is implemented as
 Line 1464  Computation of the global b function is implemented as
  * tolexm minipolym::
  * dp_gr_main dp_gr_mod_main dp_gr_f_main dp_weyl_gr_main dp_weyl_gr_mod_main dp_weyl_gr_f_main::
  * dp_f4_main dp_f4_mod_main dp_weyl_f4_main dp_weyl_f4_mod_main::
+ * nd_gr nd_gr_trace nd_f4 nd_f4_trace nd_weyl_gr nd_weyl_gr_trace::
  * dp_gr_flags dp_gr_print::
  * dp_ord::
  * dp_ptod::
-Line 1354  Computation of the global b function is implemented as
+Line 1490  Computation of the global b function is implemented as
 Line 1354  Computation of the global b function is implemented as
 Line 1490  Computation of the global b function is implemented as
  * lex_hensel_gsl tolex_gsl tolex_gsl_d::
  * primadec primedec::
  * primedec_mod::
- * bfunction generic_bfct::
+ * bfunction bfct generic_bfct ann ann0::
  @end menu
  \JP @node gr hgr gr_mod,,, $B%0%l%V%J4pDl$K4X$9$kH!?t(B
-Line 1405  Computation of the global b function is implemented as
+Line 1541  Computation of the global b function is implemented as
 Line 1405  Computation of the global b function is implemented as
 Line 1541  Computation of the global b function is implemented as
  strategy $B$K$h$k7W;;(B, @code{hgr()} $B$O(B trace-lifting $B$*$h$S(B
  $B@F<!2=$K$h$k(B $B6:@5$5$l$?(B sugar strategy $B$K$h$k7W;;$r9T$&(B.
  @item
- @code{dgr()} $B$O(B, @code{gr()}, @code{dgr()} $B$r(B
+ @code{dgr()} $B$O(B, @code{gr()}, @code{hgr()} $B$r(B
  $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,
  $B@h$K7k2L$rJV$7$?J}$N7k2L$rJV$9(B. $B7k2L$OF10l$G$"$k$,(B, $B$I$A$i$NJ}K!$,(B
  $B9bB.$+0lHL$K$OITL@$N$?$a(B, $B<B:]$N7P2a;~4V$rC;=L$9$k$N$KM-8z$G$"$k(B.
  @item
  @code{dgr()} $B$GI=<($5$l$k;~4V$O(B, $B$3$NH!?t$,<B9T$5$l$F$$$k%W%m%;%9$G$N(B
  CPU $B;~4V$G$"$j(B, $B$3$NH!?t$N>l9g$O$[$H$s$IDL?.$N$?$a$N;~4V$G$"$k(B.
+ @item
+ $BB?9`<0%j%9%H(B @var{plist} $B$NMWAG$,J,;6I=8=B?9`<0$N>l9g$O(B
+ $B7k2L$bJ,;6I=8=B?9`<0$N%j%9%H$G$"$k(B.
+ $B$3$N>l9g(B, $B0z?t$NJ,;6B?9`<0$OM?$($i$l$?=g=x$K=>$$(B @code{dp_sort} $B$G(B
+ $B%=!<%H$5$l$F$+$i7W;;$5$l$k(B.
+ $BB?9`<0%j%9%H$NMWAG$,J,;6I=8=B?9`<0$N>l9g$b(B
+ $BJQ?t$N?tJ,$NITDj85$N%j%9%H$r(B @var{vlist} $B0z?t$H$7$FM?$($J$$$H$$$1$J$$(B
+ ($B%@%_!<(B).
  \E
  \BEG
  @item
-Line 1440  Therefore this function is useful to reduce the actual
+Line 1584  Therefore this function is useful to reduce the actual
 Line 1440  Therefore this function is useful to reduce the actual
 Line 1584  Therefore this function is useful to reduce the actual
  The CPU time shown after an exection of @code{dgr()} indicates
  that of the master process, and most of the time corresponds to the time
  for communication.
+ @item
+ When the elements of @var{plist} are distributed polynomials,
+ the result is also a list of distributed polynomials.
+ In this case, firstly  the elements of @var{plist} is sorted by @code{dp_sort}
+ and the Grobner basis computation is started.
+ Variables must be given in @var{vlist} even in this case
+ (these variables are dummy).
  \E
  @end itemize
-Line 1690  processes.
+Line 1841  processes.
 Line 1690  processes.
 Line 1841  processes.
  @item lex_hensel_gsl(@var{plist},@var{vlist1},@var{order},@var{vlist2},@var{homo})
  \JP :: GSL $B7A<0$N%$%G%"%k4pDl$N7W;;(B
  \EG ::Computation of an GSL form ideal basis
- @item tolex_gsl(@var{plist},@var{vlist1},@var{order},@var{vlist2},@var{homo})
+ @item tolex_gsl(@var{plist},@var{vlist1},@var{order},@var{vlist2})
- @itemx tolex_gsl_d(@var{plist},@var{vlist1},@var{order},@var{vlist2},@var{homo},@var{procs})
+ @itemx tolex_gsl_d(@var{plist},@var{vlist1},@var{order},@var{vlist2},@var{procs})
  \JP :: $B%0%l%V%J4pDl$rF~NO$H$9$k(B, GSL $B7A<0$N%$%G%"%k4pDl$N7W;;(B
  \EG :: Computation of an GSL form ideal basis stating from a Groebner basis
  @end table
-Line 2153  except for lack of the argument for controlling homoge
+Line 2304  except for lack of the argument for controlling homoge
 Line 2153  except for lack of the argument for controlling homoge
 Line 2304  except for lack of the argument for controlling homoge
  \EG @fref{Controlling Groebner basis computations}
  @end table
+ \JP @node nd_gr nd_gr_trace nd_f4 nd_f4_trace nd_weyl_gr nd_weyl_gr_trace,,, $B%0%l%V%J4pDl$K4X$9$kH!?t(B
+ \EG @node nd_gr nd_gr_trace nd_f4 nd_f4_trace nd_weyl_gr nd_weyl_gr_trace,,, Functions for Groebner basis computation
+ @subsection @code{nd_gr}, @code{nd_gr_trace}, @code{nd_f4}, @code{nd_f4_trace}, @code{nd_weyl_gr}, @code{nd_weyl_gr_trace}
+ @findex nd_gr
+ @findex nd_gr_trace
+ @findex nd_f4
+ @findex nd_f4_trace
+ @findex nd_weyl_gr
+ @findex nd_weyl_gr_trace
+ @table @t
+ @item nd_gr(@var{plist},@var{vlist},@var{p},@var{order})
+ @itemx nd_gr_trace(@var{plist},@var{vlist},@var{homo},@var{p},@var{order})
+ @itemx nd_f4(@var{plist},@var{vlist},@var{modular},@var{order})
+ @itemx nd_f4_trace(@var{plist},@var{vlist},@var{homo},@var{p},@var{order})
+ @item nd_weyl_gr(@var{plist},@var{vlist},@var{p},@var{order})
+ @itemx nd_weyl_gr_trace(@var{plist},@var{vlist},@var{homo},@var{p},@var{order})
+ \JP :: $B%0%l%V%J4pDl$N7W;;(B ($BAH$_9~$_H!?t(B)
+ \EG :: Groebner basis computation (built-in functions)
+ @end table
+ @table @var
+ @item return
+ \JP $B%j%9%H(B
+ \EG list
+ @item plist  vlist
+ \JP $B%j%9%H(B
+ \EG list
+ @item order
+ \JP $B?t(B, $B%j%9%H$^$?$O9TNs(B
+ \EG number, list or matrix
+ @item homo
+ \JP $B%U%i%0(B
+ \EG flag
+ @item modular
+ \JP $B%U%i%0$^$?$OAG?t(B
+ \EG flag or prime
+ @end table
+ \BJP
+ @itemize @bullet
+ @item
+ $B$3$l$i$NH!?t$O(B, $B%0%l%V%J4pDl7W;;AH$_9~$_4X?t$N?7<BAu$G$"$k(B.
+ @item @code{nd_gr} $B$O(B, @code{p} $B$,(B 0 $B$N$H$-M-M}?tBN>e$N(B Buchberger
+ $B%"%k%4%j%:%`$r<B9T$9$k(B. @code{p} $B$,(B 2 $B0J>e$N<+A3?t$N$H$-(B, GF(p) $B>e$N(B
+ Buchberger $B%"%k%4%j%:%`$r<B9T$9$k(B.
+ @item @code{nd_gr_trace} $B$*$h$S(B @code{nd_f4_trace}
+ $B$OM-M}?tBN>e$G(B trace $B%"%k%4%j%:%`$r<B9T$9$k(B.
+ @code{p} $B$,(B 0 $B$^$?$O(B 1 $B$N$H$-(B, $B<+F0E*$KA*$P$l$?AG?t$rMQ$$$F(B, $B@.8y$9$k(B
+ $B$^$G(B trace $B%"%k%4%j%:%`$r<B9T$9$k(B.
+ @code{p} $B$,(B 2 $B0J>e$N$H$-(B, trace $B$O(BGF(p) $B>e$G7W;;$5$l$k(B. trace $B%"%k%4%j%:%`(B
+ $B$,<:GT$7$?>l9g(B 0 $B$,JV$5$l$k(B. @code{p} $B$,Ii$N>l9g(B, $B%0%l%V%J4pDl%A%'%C%/$O(B
+ $B9T$o$J$$(B. $B$3$N>l9g(B, @code{p} $B$,(B -1 $B$J$i$P<+F0E*$KA*$P$l$?AG?t$,(B,
+ $B$=$l0J30$O;XDj$5$l$?AG?t$rMQ$$$F%0%l%V%J4pDl8uJd$N7W;;$,9T$o$l$k(B.
+ @code{nd_f4_trace} $B$O(B, $B3FA4<!?t$K$D$$$F(B, $B$"$kM-8BBN>e$G(B F4 $B%"%k%4%j%:%`(B
+ $B$G9T$C$?7k2L$r$b$H$K(B, $B$=$NM-8BBN>e$G(B 0 $B$G$J$$4pDl$rM?$($k(B S-$BB?9`<0$N$_$r(B
+ $BMQ$$$F9TNs@8@.$r9T$$(B, $B$=$NA4<!?t$K$*$1$k4pDl$r@8@.$9$kJ}K!$G$"$k(B. $BF@$i$l$k(B
+ $BB?9`<0=89g$O$d$O$j%0%l%V%J4pDl8uJd$G$"$j(B, @code{nd_gr_trace} $B$HF1MM$N(B
+ $B%A%'%C%/$,9T$o$l$k(B.
+ @item
+ @code{nd_f4} $B$O(B @code{modular} $B$,(B 0 $B$N$H$-M-M}?tBN>e$N(B, @code{modular} $B$,(B
+ $B%^%7%s%5%$%:AG?t$N$H$-M-8BBN>e$N(B F4 $B%"%k%4%j%:%`$r<B9T$9$k(B.
+ @item
+ @code{nd_weyl_gr}, @code{nd_weyl_gr_trace} $B$O(B Weyl $BBe?tMQ$G$"$k(B.
+ @item
+ $B$$$:$l$N4X?t$b(B, $BM-M}4X?tBN>e$N7W;;$OL$BP1~$G$"$k(B.
+ @item
+ $B0lHL$K(B @code{dp_gr_main}, @code{dp_gr_mod_main} $B$h$j9bB.$G$"$k$,(B,
+ $BFC$KM-8BBN>e$N>l9g82Cx$G$"$k(B.
+ @end itemize
+ \E
+ \BEG
+ @itemize @bullet
+ @item
+ These functions are new implementations for computing Groebner bases.
+ @item @code{nd_gr} executes Buchberger algorithm over the rationals
+ if  @code{p} is 0, and that over GF(p) if @code{p} is a prime.
+ @item @code{nd_gr_trace} executes the trace algorithm over the rationals.
+ If @code{p} is 0 or 1, the trace algorithm is executed until it succeeds
+ by using automatically chosen primes.
+ If @code{p} a positive prime,
+ the trace is comuted over GF(p).
+ If the trace algorithm fails 0 is returned.
+ If @code{p} is negative,
+ the Groebner basis check and ideal-membership check are omitted.
+ In this case, an automatically chosen prime if @code{p} is 1,
+ otherwise the specified prime is used to compute a Groebner basis
+ candidate.
+ Execution of @code{nd_f4_trace} is done as follows:
+ For each total degree, an F4-reduction of S-polynomials over a finite field
+ is done, and S-polynomials which give non-zero basis elements are gathered.
+ Then F4-reduction over Q is done for the gathered S-polynomials.
+ The obtained polynomial set is a Groebner basis candidate and the same
+ check procedure as in the case of @code{nd_gr_trace} is done.
+ @item
+ @code{nd_f4} executes F4 algorithm over Q if @code{modular} is equal to 0,
+ or over a finite field GF(@code{modular})
+ if @code{modular} is a prime number of machine size (<2^29).
+ @item
+ @code{nd_weyl_gr}, @code{nd_weyl_gr_trace} are for Weyl algebra computation.
+ @item
+ Each function cannot handle rational function coefficient cases.
+ @item
+ In general these functions are more efficient than
+ @code{dp_gr_main}, @code{dp_gr_mod_main}, especially over finite fields.
+ @end itemize
+ \E
+ @example
+ [38] load("cyclic")$
+ [49] C=cyclic(7)$
+ [50] V=vars(C)$
+ [51] cputime(1)$
+ [52] dp_gr_mod_main(C,V,0,31991,0)$
+.06sec + gc : 0.313sec(26.4sec)
+ [53] nd_gr(C,V,31991,0)$
+ ndv_alloc=1477188
+.737sec + gc : 0.1837sec(5.921sec)
+ [54] dp_f4_mod_main(C,V,31991,0)$
+.51sec + gc : 0.7109sec(4.221sec)
+ [55] nd_f4(C,V,31991,0)$
+.906sec + gc : 0.126sec(2.032sec)
+ @end example
+ @table @t
+ \JP @item $B;2>H(B
+ \EG @item References
+ @fref{dp_ord},
+ @fref{dp_gr_flags dp_gr_print},
+ \JP @fref{$B7W;;$*$h$SI=<($N@)8f(B}.
+ \EG @fref{Controlling Groebner basis computations}
+ @end table
  \JP @node dp_gr_flags dp_gr_print,,, $B%0%l%V%J4pDl$K4X$9$kH!?t(B
  \EG @node dp_gr_flags dp_gr_print,,, Functions for Groebner basis computation
  @subsection @code{dp_gr_flags}, @code{dp_gr_print}
-Line 3918  execute @code{dp_gr_print(2)} in advance.
+Line 4203  execute @code{dp_gr_print(2)} in advance.
 Line 3918  execute @code{dp_gr_print(2)} in advance.
 Line 4203  execute @code{dp_gr_print(2)} in advance.
  @fref{dp_gr_flags dp_gr_print}.
  @end table
- \JP @node bfunction generic_bfct,,, $B%0%l%V%J4pDl$K4X$9$kH!?t(B
+ \JP @node bfunction bfct generic_bfct ann ann0,,, $B%0%l%V%J4pDl$K4X$9$kH!?t(B
- \EG @node bfunction generic_bfct,,, Functions for Groebner basis computation
+ \EG @node bfunction bfct generic_bfct ann ann0,,, Functions for Groebner basis computation
- @subsection @code{bfunction}, @code{generic_bfct}
+ @subsection @code{bfunction}, @code{bfct}, @code{generic_bfct}, @code{ann}, @code{ann0}
  @findex bfunction
+ @findex bfct
  @findex generic_bfct
+ @findex ann
+ @findex ann0
  @table @t
  @item bfunction(@var{f})
- @item generic_bfct(@var{plist},@var{vlist},@var{dvlist},@var{weight})
+ @itemx bfct(@var{f})
- \JP :: b $B4X?t$N7W;;(B
+ @itemx generic_bfct(@var{plist},@var{vlist},@var{dvlist},@var{weight})
- \EG :: Computes the global b function of a polynomial or an ideal
+ \JP :: @var{b} $B4X?t$N7W;;(B
+ \EG :: Computes the global @var{b} function of a polynomial or an ideal
+ @item ann(@var{f})
+ @itemx ann0(@var{f})
+ \JP :: $BB?9`<0$N%Y%-$N(B annihilator $B$N7W;;(B
+ \EG :: Computes the annihilator of a power of polynomial
  @end table
  @table @var
  @item return
- @itemx f
+ \JP $BB?9`<0$^$?$O%j%9%H(B
+ \EG polynomial or list
+ @item f
  \JP $BB?9`<0(B
  \EG polynomial
  @item plist
-Line 3946  execute @code{dp_gr_print(2)} in advance.
+Line 4242  execute @code{dp_gr_print(2)} in advance.
 Line 3946  execute @code{dp_gr_print(2)} in advance.
 Line 4242  execute @code{dp_gr_print(2)} in advance.
  @itemize @bullet
  \BJP
  @item @samp{bfct} $B$GDj5A$5$l$F$$$k(B.
- @item @code{bfunction(@var{f})} $B$OB?9`<0(B @var{f} $B$N(B global b $B4X?t(B @code{b(s)} $B$r(B
+ @item @code{bfunction(@var{f})}, @code{bfct(@var{f})} $B$OB?9`<0(B @var{f} $B$N(B global @var{b} $B4X?t(B @code{b(s)} $B$r(B
  $B7W;;$9$k(B. @code{b(s)} $B$O(B, Weyl $BBe?t(B @code{D} $B>e$N0lJQ?tB?9`<04D(B @code{D[s]}
  $B$N85(B @code{P(x,s)} $B$,B8:_$7$F(B, @code{P(x,s)f^(s+1)=b(s)f^s} $B$rK~$?$9$h$&$J(B
  $BB?9`<0(B @code{b(s)} $B$NCf$G(B, $B<!?t$,:G$bDc$$$b$N$G$"$k(B.
  @item @code{generic_bfct(@var{f},@var{vlist},@var{dvlist},@var{weight})}
  $B$O(B, @var{plist} $B$G@8@.$5$l$k(B @code{D} $B$N:8%$%G%"%k(B @code{I} $B$N(B,
- $B%&%'%$%H(B @var{weight} $B$K4X$9$k(B global b $B4X?t$r7W;;$9$k(B.
+ $B%&%'%$%H(B @var{weight} $B$K4X$9$k(B global @var{b} $B4X?t$r7W;;$9$k(B.
  @var{vlist} $B$O(B @code{x}-$BJQ?t(B, @var{vlist} $B$OBP1~$9$k(B @code{D}-$BJQ?t(B
  $B$r=g$KJB$Y$k(B.
+ @item @code{bfunction} $B$H(B @code{bfct} $B$G$OMQ$$$F$$$k%"%k%4%j%:%`$,(B
+ $B0[$J$k(B. $B$I$A$i$,9bB.$+$OF~NO$K$h$k(B.
+ @item @code{ann(@var{f})} $B$O(B, @code{@var{f}^s} $B$N(B annihilator ideal
+ $B$N@8@.7O$rJV$9(B. @code{ann(@var{f})} $B$O(B, @code{[@var{a},@var{list}]}
+ $B$J$k%j%9%H$rJV$9(B. $B$3$3$G(B, @var{a} $B$O(B @var{f} $B$N(B @var{b} $B4X?t$N:G>.@0?t:,(B,
+ @var{list} $B$O(B @code{ann(@var{f})} $B$N7k2L$N(B @code{s}$ $B$K(B, @var{a} $B$r(B
+ $BBeF~$7$?$b$N$G$"$k(B.
  @item $B>\:Y$K$D$$$F$O(B, [Saito,Sturmfels,Takayama] $B$r8+$h(B.
  \E
  \BEG
  @item These functions are defined in @samp{bfct}.
- @item @code{bfunction(@var{f})} computes the global b-function @code{b(s)} of
+ @item @code{bfunction(@var{f})} and @code{bfct(@var{f})} compute the global @var{b}-function @code{b(s)} of
  a polynomial @var{f}.
  @code{b(s)} is a polynomial of the minimal degree
  such that there exists @code{P(x,s)} in D[s], which is a polynomial
  ring over Weyl algebra @code{D}, and @code{P(x,s)f^(s+1)=b(s)f^s} holds.
  @item @code{generic_bfct(@var{f},@var{vlist},@var{dvlist},@var{weight})}
- computes the global b-function of a left ideal @code{I} in @code{D}
+ computes the global @var{b}-function of a left ideal @code{I} in @code{D}
  generated by @var{plist}, with respect to @var{weight}.
  @var{vlist} is the list of @code{x}-variables,
  @var{vlist} is the list of corresponding @code{D}-variables.
+ @item @code{bfunction(@var{f})} and @code{bfct(@var{f})} implement
+ different algorithms and the efficiency depends on inputs.
+ @item @code{ann(@var{f})} returns the generator set of the annihilator
+ ideal of @code{@var{f}^s}.
+ @code{ann(@var{f})} returns a list @code{[@var{a},@var{list}]},
+ where @var{a} is the minimal integral root of the global @var{b}-function
+ of @var{f}, and @var{list} is a list of polynomials obtained by
+ substituting @code{s} in @code{ann(@var{f})} with @var{a}.
  @item See [Saito,Sturmfels,Takayama] for the details.
  \E
  @end itemize
-Line 3984  x*y*dt+5*z^4*dt+dz,-x^4-z*y*x-y^4-z^5+t]$
+Line 4295  x*y*dt+5*z^4*dt+dz,-x^4-z*y*x-y^4-z^5+t]$
 Line 3984  x*y*dt+5*z^4*dt+dz,-x^4-z*y*x-y^4-z^5+t]$
 Line 4295  x*y*dt+5*z^4*dt+dz,-x^4-z*y*x-y^4-z^5+t]$
  [219] generic_bfct(F,[t,z,y,x],[dt,dz,dy,dx],[1,0,0,0]);
 *s^10-70000*s^9+101750*s^8-79375*s^7+35768*s^6-9277*s^5
  +1278*s^4-72*s^3
+ [220] P=x^3-y^2$
+ [221] ann(P);
+ [2*dy*x+3*dx*y^2,-3*dx*x-2*dy*y+6*s]
+ [222] ann0(P);
+ [-1,[2*dy*x+3*dx*y^2,-3*dx*x-2*dy*y-6]]
  @end example
  @table @t

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