===================================================================
RCS file: /home/cvs/OpenXM/src/asir-doc/parts/groebner.texi,v
retrieving revision 1.5
retrieving revision 1.17
diff -u -p -r1.5 -r1.17
--- OpenXM/src/asir-doc/parts/groebner.texi	2003/04/20 08:01:25	1.5
+++ OpenXM/src/asir-doc/parts/groebner.texi	2006/09/06 23:53:31	1.17
@@ -1,4 +1,4 @@
-@comment $OpenXM: OpenXM/src/asir-doc/parts/groebner.texi,v 1.4 2003/04/19 15:44:56 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
@@ -15,6 +15,7 @@
 * $B4pK\E*$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::
@@ -26,6 +27,7 @@
 * Fundamental functions::
 * Controlling Groebner basis computations::
 * Setting term orderings::
+* Weight::
 * Groebner basis computation with rational function coefficients::
 * Change of ordering::
 * Weyl algebra::
@@ -449,6 +451,13 @@ If `on', various informations during a Groebner basis 
 displayed.
 \E
 
+@item PrintShort
+\JP on $B$G!"(BPrint $B$,(B off $B$N>l9g(B, $B%0%l%V%J4pDl7W;;$NESCf$N>pJs$rC;=L7A$GI=<($9$k(B. 
+\BEG
+If `on' and Print is `off', short information during a Groebner basis computation is
+displayed.
+\E
+
 @item Stat
 \BJP
 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
@@ -471,24 +480,28 @@ is shown after every normal computation.  After comlet
 computation the maximal value among the sums is shown.
 \E
 
-@item Multiple
+@item Content
+@itemx Multiple
 \BJP
-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
-@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
-$B3d$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
-GCD $B7W;;$,9T$o$l0lHL$K$O8zN($,0-$/$J$k$,(B, @code{Multiple} $B$r(B 2 $BDxEY(B
+0 $B$G$J$$M-M}?t$N;~(B, $BM-M}?t>e$N@55,7A7W;;$K$*$$$F(B, $B78?t$N%S%C%HD9$NOB$,(B
+@code{Content} $BG\$K$J$k$4$H$K78?tA4BN$N(B GCD $B$,7W;;$5$l(B, $B$=$N(B GCD $B$G(B
+$B3d$C$?B?9`<0$r4JLs$9$k(B. @code{Content} $B$,(B 1 $B$J$i$P(B, $B4JLs$9$k$4$H$K(B
+GCD $B7W;;$,9T$o$l0lHL$K$O8zN($,0-$/$J$k$,(B, @code{Content} $B$r(B 2 $BDxEY(B
 $B$H$9$k$H(B, $B5pBg$J@0?t$,78?t$K8=$l$k>l9g(B, $B8zN($,NI$/$J$k>l9g$,$"$k(B. 
+backward compatibility $B$N$?$a!"(B@code{Multiple} $B$G@0?tCM$r;XDj$G$-$k(B.
 \E
 \BEG
-If a non-zero integer, in a normal form computation
+If a non-zero rational number, in a normal form computation
 over the rationals, the integer content of the polynomial being
-reduced is removed when its magnitude becomes @code{Multiple} times
+reduced is removed when its magnitude becomes @code{Content} times
 larger than a registered value, which is set to the magnitude of the
 input polynomial. After each content removal the registered value is
-set to the magnitude of the resulting polynomial. @code{Multiple} is
+set to the magnitude of the resulting polynomial. @code{Content} is
 equal to 1, the simiplification is done after every normal form computation.
-It is empirically known that it is often efficient to set @code{Multiple} to 2
+It is empirically known that it is often efficient to set @code{Content} to 2
 for the case where large integers appear during the computation.
+An integer value can be set by the keyword @code{Multiple} for
+backward compatibility.
 \E
 
 @item Demand
@@ -1044,6 +1057,139 @@ 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>>);
+3
+[1] dp_set_weight([1,2,3])$
+[2] dp_td(<<1,1,1>>);
+6
+@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
@@ -1203,6 +1349,105 @@ Refer to the sections for each functions.
 \E
 
 \BJP
+@node Weyl $BBe?t(B,,, $B%0%l%V%J4pDl$N7W;;(B
+@section Weyl $BBe?t(B
+\E
+\BEG
+@node Weyl algebra,,, Groebner basis computation
+@section Weyl algebra
+\E
+
+@noindent
+
+\BJP
+$B$3$l$^$G$O(B, $BDL>o$N2D49$JB?9`<04D$K$*$1$k%0%l%V%J4pDl7W;;$K$D$$$F(B
+$B=R$Y$F$-$?$,(B, $B%0%l%V%J4pDl$NM}O@$O(B, $B$"$k>r7o$rK~$?$9Hs2D49$J(B
+$B4D$K$b3HD%$G$-$k(B. $B$3$N$h$&$J4D$NCf$G(B, $B1~MQ>e$b=EMW$J(B, 
+Weyl $BBe?t(B, $B$9$J$o$AB?9`<04D>e$NHyJ,:nMQAG4D$N1i;;$*$h$S(B
+$B%0%l%V%J4pDl7W;;$,(B Risa/Asir $B$K<BAu$5$l$F$$$k(B. 
+
+$BBN(B @code{K} $B>e$N(B @code{n} $B<!85(B Weyl $BBe?t(B 
+@code{D=K<x1,@dots{},xn,D1,@dots{},Dn>} $B$O(B
+\E
+
+\BEG
+So far we have explained Groebner basis computation in
+commutative polynomial rings. However Groebner basis can be
+considered in more general non-commutative rings.
+Weyl algebra is one of such rings and 
+Risa/Asir implements fundamental operations
+in Weyl algebra and Groebner basis computation in Weyl algebra.
+
+The @code{n} dimensional Weyl algebra over a field @code{K},
+@code{D=K<x1,@dots{},xn,D1,@dots{},Dn>} is a non-commutative
+algebra which has the following fundamental relations:
+\E
+
+@code{xi*xj-xj*xi=0}, @code{Di*Dj-Dj*Di=0}, @code{Di*xj-xj*Di=0} (@code{i!=j}),
+@code{Di*xi-xi*Di=1}
+
+\BJP
+$B$H$$$&4pK\4X78$r;}$D4D$G$"$k(B. @code{D} $B$O(B $BB?9`<04D(B @code{K[x1,@dots{},xn]} $B$r78?t(B
+$B$H$9$kHyJ,:nMQAG4D$G(B,  @code{Di} $B$O(B @code{xi} $B$K$h$kHyJ,$rI=$9(B. $B8r494X78$K$h$j(B, 
+@code{D} $B$N85$O(B, @code{x1^i1*@dots{}*xn^in*D1^j1*@dots{}*Dn^jn} $B$J$kC19`(B
+$B<0$N(B @code{K} $B@~7A7k9g$H$7$F=q$-I=$9$3$H$,$G$-$k(B. 
+Risa/Asir $B$K$*$$$F$O(B, $B$3$NC19`<0$r(B, $B2D49$JB?9`<0$HF1MM$K(B
+@code{<<i1,@dots{},in,j1,@dots{},jn>>} $B$GI=$9(B. $B$9$J$o$A(B, @code{D} $B$N85$b(B
+$BJ,;6I=8=B?9`<0$H$7$FI=$5$l$k(B. $B2C8:;;$O(B, $B2D49$N>l9g$HF1MM$K(B, @code{+}, @code{-}
+$B$K$h$j(B
+$B<B9T$G$-$k$,(B, $B>h;;$O(B, $BHs2D49@-$r9MN8$7$F(B @code{dp_weyl_mul()} $B$H$$$&4X?t(B
+$B$K$h$j<B9T$9$k(B.
+\E
+
+\BEG
+@code{D} is the ring of differential operators whose coefficients
+are polynomials in @code{K[x1,@dots{},xn]} and
+@code{Di} denotes the differentiation with respect to  @code{xi}.
+According to the commutation relation,
+elements of @code{D} can be represented as a @code{K}-linear combination 
+of monomials @code{x1^i1*@dots{}*xn^in*D1^j1*@dots{}*Dn^jn}.
+In Risa/Asir, this type of monomial is represented 
+by @code{<<i1,@dots{},in,j1,@dots{},jn>>} as in the case of commutative
+polynomial.
+That is, elements of @code{D} are represented by distributed polynomials.
+Addition and subtraction can be done by @code{+}, @code{-},
+but multiplication is done by calling @code{dp_weyl_mul()} because of
+the non-commutativity of @code{D}.
+\E
+
+@example
+[0] A=<<1,2,2,1>>;
+(1)*<<1,2,2,1>>
+[1] B=<<2,1,1,2>>;
+(1)*<<2,1,1,2>>
+[2] A*B;
+(1)*<<3,3,3,3>>
+[3] dp_weyl_mul(A,B);
+(1)*<<3,3,3,3>>+(1)*<<3,2,3,2>>+(4)*<<2,3,2,3>>+(4)*<<2,2,2,2>>
++(2)*<<1,3,1,3>>+(2)*<<1,2,1,2>>
+@end example
+
+\BJP
+$B%0%l%V%J4pDl7W;;$K$D$$$F$b(B, Weyl $BBe?t@lMQ$N4X?t$H$7$F(B, 
+$B<!$N4X?t$,MQ0U$7$F$"$k(B. 
+\E
+\BEG
+The following functions are avilable for Groebner basis computation
+in Weyl algebra:
+\E
+@code{dp_weyl_gr_main()},
+@code{dp_weyl_gr_mod_main()},
+@code{dp_weyl_gr_f_main()},
+@code{dp_weyl_f4_main()},
+@code{dp_weyl_f4_mod_main()}.
+\BJP
+$B$^$?(B, $B1~MQ$H$7$F(B, global b $B4X?t$N7W;;$,<BAu$5$l$F$$$k(B. 
+\E
+\BEG
+Computation of the global b function is implemented as an application.
+\E
+
+\BJP
 @node $B%0%l%V%J4pDl$K4X$9$kH!?t(B,,, $B%0%l%V%J4pDl$N7W;;(B
 @section $B%0%l%V%J4pDl$K4X$9$kH!?t(B
 \E
@@ -1217,8 +1462,9 @@ Refer to the sections for each functions.
 * lex_hensel_gsl tolex_gsl tolex_gsl_d::
 * gr_minipoly minipoly::
 * tolexm minipolym::
-* dp_gr_main dp_gr_mod_main dp_gr_f_main::
-* dp_f4_main dp_f4_mod_main::
+* 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::
@@ -1244,6 +1490,7 @@ Refer to the sections for each functions.
 * lex_hensel_gsl tolex_gsl tolex_gsl_d::
 * primadec primedec::
 * primedec_mod::
+* bfunction bfct generic_bfct ann ann0::
 @end menu
 
 \JP @node gr hgr gr_mod,,, $B%0%l%V%J4pDl$K4X$9$kH!?t(B
@@ -1294,13 +1541,21 @@ Refer to the sections for each functions.
 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 
@@ -1329,6 +1584,13 @@ 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
 
@@ -1346,8 +1608,7 @@ for communication.
 @table @t
 \JP @item $B;2>H(B
 \EG @item References
-@comment @fref{dp_gr_main dp_gr_mod_main dp_gr_f_main},
-@fref{dp_gr_main dp_gr_mod_main dp_gr_f_main},
+@fref{dp_gr_main dp_gr_mod_main dp_gr_f_main dp_weyl_gr_main dp_weyl_gr_mod_main dp_weyl_gr_f_main},
 @fref{dp_ord}.
 @end table
 
@@ -1564,7 +1825,7 @@ processes.
 @table @t
 \JP @item $B;2>H(B
 \EG @item References
-@fref{dp_gr_main dp_gr_mod_main dp_gr_f_main},
+@fref{dp_gr_main dp_gr_mod_main dp_gr_f_main dp_weyl_gr_main dp_weyl_gr_mod_main dp_weyl_gr_f_main},
 \JP @fref{dp_ord}, @fref{$BJ,;67W;;(B}
 \EG @fref{dp_ord}, @fref{Distributed computation}
 @end table
@@ -1580,8 +1841,8 @@ 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})
-@itemx tolex_gsl_d(@var{plist},@var{vlist1},@var{order},@var{vlist2},@var{homo},@var{procs})
+@item tolex_gsl(@var{plist},@var{vlist1},@var{order},@var{vlist2})
+@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
@@ -1842,17 +2103,23 @@ z^32+11405*z^31+20868*z^30+21602*z^29+...
 @fref{gr_minipoly minipoly}.
 @end table
 
-\JP @node dp_gr_main dp_gr_mod_main dp_gr_f_main,,, $B%0%l%V%J4pDl$K4X$9$kH!?t(B
-\EG @node dp_gr_main dp_gr_mod_main dp_gr_f_main,,, Functions for Groebner basis computation
-@subsection @code{dp_gr_main}, @code{dp_gr_mod_main}, @code{dp_gr_f_main}
+\JP @node dp_gr_main dp_gr_mod_main dp_gr_f_main dp_weyl_gr_main dp_weyl_gr_mod_main dp_weyl_gr_f_main,,, $B%0%l%V%J4pDl$K4X$9$kH!?t(B
+\EG @node dp_gr_main dp_gr_mod_main dp_gr_f_main dp_weyl_gr_main dp_weyl_gr_mod_main dp_weyl_gr_f_main,,, Functions for Groebner basis computation
+@subsection @code{dp_gr_main}, @code{dp_gr_mod_main}, @code{dp_gr_f_main}, @code{dp_weyl_gr_main}, @code{dp_weyl_gr_mod_main}, @code{dp_weyl_gr_f_main}
 @findex dp_gr_main
 @findex dp_gr_mod_main
 @findex dp_gr_f_main
+@findex dp_weyl_gr_main
+@findex dp_weyl_gr_mod_main
+@findex dp_weyl_gr_f_main
 
 @table @t
 @item dp_gr_main(@var{plist},@var{vlist},@var{homo},@var{modular},@var{order})
 @itemx dp_gr_mod_main(@var{plist},@var{vlist},@var{homo},@var{modular},@var{order})
 @itemx dp_gr_f_main(@var{plist},@var{vlist},@var{homo},@var{order})
+@itemx dp_weyl_gr_main(@var{plist},@var{vlist},@var{homo},@var{modular},@var{order})
+@itemx dp_weyl_gr_mod_main(@var{plist},@var{vlist},@var{homo},@var{modular},@var{order})
+@itemx dp_weyl_gr_f_main(@var{plist},@var{vlist},@var{homo},@var{order})
 \JP :: $B%0%l%V%J4pDl$N7W;;(B ($BAH$_9~$_H!?t(B)
 \EG :: Groebner basis computation (built-in functions)
 @end table
@@ -1880,9 +2147,10 @@ z^32+11405*z^31+20868*z^30+21602*z^29+...
 @item
 $B$3$l$i$NH!?t$O(B, $B%0%l%V%J4pDl7W;;$N4pK\E*AH$_9~$_H!?t$G$"$j(B, @code{gr()},
 @code{hgr()}, @code{gr_mod()} $B$J$I$O$9$Y$F$3$l$i$NH!?t$r8F$S=P$7$F7W;;(B
-$B$r9T$C$F$$$k(B. 
+$B$r9T$C$F$$$k(B. $B4X?tL>$K(B weyl $B$,F~$C$F$$$k$b$N$O(B, Weyl $BBe?t>e$N7W;;(B
+$B$N$?$a$N4X?t$G$"$k(B. 
 @item
-@code{dp_gr_f_main()} $B$O(B, $B<o!9$NM-8BBN>e$N%0%l%V%J4pDl$r7W;;$9$k(B
+@code{dp_gr_f_main()}, @code{dp_weyl_f_main()} $B$O(B, $B<o!9$NM-8BBN>e$N%0%l%V%J4pDl$r7W;;$9$k(B
 $B>l9g$KMQ$$$k(B. $BF~NO$O(B, $B$"$i$+$8$a(B, @code{simp_ff()} $B$J$I$G(B, 
 $B9M$($kM-8BBN>e$K<M1F$5$l$F$$$kI,MW$,$"$k(B. 
 @item
@@ -1917,9 +2185,11 @@ z^32+11405*z^31+20868*z^30+21602*z^29+...
 @item
 These functions are fundamental built-in functions for Groebner basis
 computation and @code{gr()},@code{hgr()} and @code{gr_mod()}
-are all interfaces to these functions.
+are all interfaces to these functions. Functions whose names
+contain weyl are those for computation in Weyl algebra.
 @item
-@code{dp_gr_f_main()} is a function for Groebner basis computation
+@code{dp_gr_f_main()} and @code{dp_weyl_gr_f_main()}
+are functions for Groebner basis computation
 over various finite fields. Coefficients of input polynomials 
 must be converted to elements of a finite field 
 currently specified by @code{setmod_ff()}.
@@ -1966,15 +2236,19 @@ Actual computation is controlled by various parameters
 \EG @fref{Controlling Groebner basis computations}
 @end table
 
-\JP @node dp_f4_main dp_f4_mod_main,,, $B%0%l%V%J4pDl$K4X$9$kH!?t(B
-\EG @node dp_f4_main dp_f4_mod_main,,, Functions for Groebner basis computation
-@subsection @code{dp_f4_main}, @code{dp_f4_mod_main}
+\JP @node dp_f4_main dp_f4_mod_main dp_weyl_f4_main dp_weyl_f4_mod_main,,, $B%0%l%V%J4pDl$K4X$9$kH!?t(B
+\EG @node dp_f4_main dp_f4_mod_main dp_weyl_f4_main dp_weyl_f4_mod_main,,, Functions for Groebner basis computation
+@subsection @code{dp_f4_main}, @code{dp_f4_mod_main}, @code{dp_weyl_f4_main}, @code{dp_weyl_f4_mod_main}
 @findex dp_f4_main
 @findex dp_f4_mod_main
+@findex dp_weyl_f4_main
+@findex dp_weyl_f4_mod_main
 
 @table @t
 @item dp_f4_main(@var{plist},@var{vlist},@var{order})
 @itemx dp_f4_mod_main(@var{plist},@var{vlist},@var{order})
+@itemx dp_weyl_f4_main(@var{plist},@var{vlist},@var{order})
+@itemx dp_weyl_f4_mod_main(@var{plist},@var{vlist},@var{order})
 \JP :: F4 $B%"%k%4%j%:%`$K$h$k%0%l%V%J4pDl$N7W;;(B ($BAH$_9~$_H!?t(B)
 \EG :: Groebner basis computation by F4 algorithm (built-in functions)
 @end table
@@ -2000,7 +2274,9 @@ F4 $B%"%k%4%j%:%`$O(B, J.C. Faugere $B$K$h$jDs>'$5$
 $B;;K!$G$"$j(B, $BK\<BAu$O(B, $BCf9q>jM>DjM}$K$h$k@~7AJ}Dx<05a2r$rMQ$$$?(B
 $B;n83E*$J<BAu$G$"$k(B. 
 @item
-$B0z?t$*$h$SF0:n$O$=$l$>$l(B @code{dp_gr_main()}, @code{dp_gr_mod_main()}
+$B@F<!2=$N0z?t$,$J$$$3$H$r=|$1$P(B, $B0z?t$*$h$SF0:n$O$=$l$>$l(B 
+@code{dp_gr_main()}, @code{dp_gr_mod_main()},
+@code{dp_weyl_gr_main()}, @code{dp_weyl_gr_mod_main()}
 $B$HF1MM$G$"$k(B. 
 \E
 \BEG
@@ -2012,7 +2288,9 @@ invented by J.C. Faugere. The current implementation o
 uses Chinese Remainder theorem and not highly optimized.
 @item
 Arguments and actions are the same as those of 
-@code{dp_gr_main()}, @code{dp_gr_mod_main()}.
+@code{dp_gr_main()}, @code{dp_gr_mod_main()},
+@code{dp_weyl_gr_main()}, @code{dp_weyl_gr_mod_main()},
+except for lack of the argument for controlling homogenization.
 \E
 @end itemize
 
@@ -2026,6 +2304,140 @@ Arguments and actions are the same as those of 
 \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)$
+26.06sec + gc : 0.313sec(26.4sec)
+[53] nd_gr(C,V,31991,0)$
+ndv_alloc=1477188
+5.737sec + gc : 0.1837sec(5.921sec)
+[54] dp_f4_mod_main(C,V,31991,0)$  
+3.51sec + gc : 0.7109sec(4.221sec)
+[55] nd_f4(C,V,31991,0)$
+1.906sec + 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}
@@ -2034,7 +2446,7 @@ Arguments and actions are the same as those of 
 
 @table @t
 @item dp_gr_flags([@var{list}])
-@itemx dp_gr_print([@var{0|1}])
+@itemx dp_gr_print([@var{i}])
 \JP :: $B7W;;$*$h$SI=<(MQ%Q%i%a%?$N@_Dj(B, $B;2>H(B
 \BEG :: Set and show various parameters for cotrolling computations 
 and showing informations.
@@ -2048,6 +2460,9 @@ and showing informations.
 @item list
 \JP $B%j%9%H(B
 \EG list
+@item i
+\JP $B@0?t(B
+\EG integer
 @end table
 
 @itemize @bullet
@@ -2061,9 +2476,18 @@ and showing informations.
 $B0z?t$O(B, @code{["Print",1,"NoSugar",1,...]} $B$J$k7A$N%j%9%H$G(B, $B:8$+$i=g$K(B
 $B@_Dj$5$l$k(B. $B%Q%i%a%?L>$OJ8;zNs$GM?$($kI,MW$,$"$k(B. 
 @item
-@code{dp_gr_print()} $B$O(B, $BFC$K%Q%i%a%?(B @code{Print} $B$NCM$rD>@\@_Dj(B, $B;2>H(B
-$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$$$k%f!<%6(B
-$BH!?t$K$*$$$F(B, @code{Print} $B$NCM$r8+$F(B, $B$=$N%5%V%k!<%A%s$,Cf4V>pJs$NI=<((B
+@code{dp_gr_print()} $B$O(B, $BFC$K%Q%i%a%?(B @code{Print}, @code{PrintShort} $B$NCM$rD>@\@_Dj(B, $B;2>H(B
+$B$G$-$k(B. $B@_Dj$5$l$kCM$O<!$NDL$j$G$"$k!#(B
+@table @var
+@item i=0
+@code{Print=0}, @code{PrintShort=0}
+@item i=1
+@code{Print=1}, @code{PrintShort=0}
+@item i=2
+@code{Print=0}, @code{PrintShort=1}
+@end table
+$B$3$l$O(B, @code{dp_gr_main()} $B$J$I$r%5%V%k!<%A%s$H$7$FMQ$$$k%f!<%6(B
+$BH!?t$K$*$$$F(B, $B$=$N%5%V%k!<%A%s$,Cf4V>pJs$NI=<((B
 $B$r9T$&:]$K(B, $B?WB.$K%U%i%0$r8+$k$3$H$,$G$-$k$h$&$KMQ0U$5$l$F$$$k(B. 
 \E
 \BEG
@@ -2078,8 +2502,17 @@ Arguments must be specified as a list such as
 strings.
 @item
 @code{dp_gr_print()} is used to set and show the value of a parameter
-@code{Print}. This functions is prepared to get quickly the value of
-@code{Print} when a user defined function calling @code{dp_gr_main()} etc.
+@code{Print} and @code{PrintShort}. 
+@table @var
+@item i=0
+@code{Print=0}, @code{PrintShort=0}
+@item i=1
+@code{Print=1}, @code{PrintShort=0}
+@item i=2
+@code{Print=0}, @code{PrintShort=1}
+@end table
+This functions is prepared to get quickly the value
+when a user defined function calling @code{dp_gr_main()} etc.
 uses the value as a flag for showing intermediate informations.
 \E
 @end itemize
@@ -3665,105 +4098,6 @@ if an input ideal is not radical.
 \EG @fref{Setting term orderings}.
 @end table
 
-\BJP
-@node Weyl $BBe?t(B,,, $B%0%l%V%J4pDl$N7W;;(B
-@section Weyl $BBe?t(B
-\E
-\BEG
-@node Weyl algebra,,, Groebner basis computation
-@section Weyl algebra
-\E
-
-@noindent
-
-\BJP
-$B$3$l$^$G$O(B, $BDL>o$N2D49$JB?9`<04D$K$*$1$k%0%l%V%J4pDl7W;;$K$D$$$F(B
-$B=R$Y$F$-$?$,(B, $B%0%l%V%J4pDl$NM}O@$O(B, $B$"$k>r7o$rK~$?$9Hs2D49$J(B
-$B4D$K$b3HD%$G$-$k(B. $B$3$N$h$&$J4D$NCf$G(B, $B1~MQ>e$b=EMW$J(B, 
-Weyl $BBe?t(B, $B$9$J$o$AB?9`<04D>e$NHyJ,:nMQAG4D$N1i;;$*$h$S(B
-$B%0%l%V%J4pDl7W;;$,(B Risa/Asir $B$K<BAu$5$l$F$$$k(B. 
-
-$BBN(B @code{K} $B>e$N(B @code{n} $B<!85(B Weyl $BBe?t(B 
-@code{D=K<x1,@dots{},xn,D1,@dots{},Dn>} $B$O(B
-\E
-
-\BEG
-So far we have explained Groebner basis computation in
-commutative polynomial rings. However Groebner basis can be
-considered in more general non-commutative rings.
-Weyl algebra is one of such rings and 
-Risa/Asir implements fundamental operations
-in Weyl algebra and Groebner basis computation in Weyl algebra.
-
-The @code{n} dimensional Weyl algebra over a field @code{K},
-@code{D=K<x1,@dots{},xn,D1,@dots{},Dn>} is a non-commutative
-algebra which has the following fundamental relations:
-\E
-
-@code{xi*xj-xj*xi=0}, @code{Di*Dj-Dj*Di=0}, @code{Di*xj-xj*Di=0} (@code{i!=j}),
-@code{Di*xi-xi*Di=1}
-
-\BJP
-$B$H$$$&4pK\4X78$r;}$D4D$G$"$k(B. @code{D} $B$O(B $BB?9`<04D(B @code{K[x1,@dots{},xn]} $B$r78?t(B
-$B$H$9$kHyJ,:nMQAG4D$G(B,  @code{Di} $B$O(B @code{xi} $B$K$h$kHyJ,$rI=$9(B. $B8r494X78$K$h$j(B, 
-@code{D} $B$N85$O(B, @code{x1^i1*@dots{}*xn^in*D1^j1*@dots{}*Dn^jn} $B$J$kC19`(B
-$B<0$N(B @code{K} $B@~7A7k9g$H$7$F=q$-I=$9$3$H$,$G$-$k(B. 
-Risa/Asir $B$K$*$$$F$O(B, $B$3$NC19`<0$r(B, $B2D49$JB?9`<0$HF1MM$K(B
-@code{<<i1,@dots{},in,j1,@dots{},jn>>} $B$GI=$9(B. $B$9$J$o$A(B, @code{D} $B$N85$b(B
-$BJ,;6I=8=B?9`<0$H$7$FI=$5$l$k(B. $B2C8:;;$O(B, $B2D49$N>l9g$HF1MM$K(B, @code{+}, @code{-}
-$B$K$h$j(B
-$B<B9T$G$-$k$,(B, $B>h;;$O(B, $BHs2D49@-$r9MN8$7$F(B @code{dp_weyl_mul()} $B$H$$$&4X?t(B
-$B$K$h$j<B9T$9$k(B.
-\E
-
-\BEG
-@code{D} is the ring of differential operators whose coefficients
-are polynomials in @code{K[x1,@dots{},xn]} and
-@code{Di} denotes the differentiation with respect to  @code{xi}.
-According to the commutation relation,
-elements of @code{D} can be represented as a @code{K}-linear combination 
-of monomials @code{x1^i1*@dots{}*xn^in*D1^j1*@dots{}*Dn^jn}.
-In Risa/Asir, this type of monomial is represented 
-by @code{<<i1,@dots{},in,j1,@dots{},jn>>} as in the case of commutative
-polynomial.
-That is, elements of @code{D} are represented by distributed polynomials.
-Addition and subtraction can be done by @code{+}, @code{-},
-but multiplication is done by calling @code{dp_weyl_mul()} because of
-the non-commutativity of @code{D}.
-\E
-
-@example
-[0] A=<<1,2,2,1>>;
-(1)*<<1,2,2,1>>
-[1] B=<<2,1,1,2>>;
-(1)*<<2,1,1,2>>
-[2] A*B;
-(1)*<<3,3,3,3>>
-[3] dp_weyl_mul(A,B);
-(1)*<<3,3,3,3>>+(1)*<<3,2,3,2>>+(4)*<<2,3,2,3>>+(4)*<<2,2,2,2>>
-+(2)*<<1,3,1,3>>+(2)*<<1,2,1,2>>
-@end example
-
-\BJP
-$B%0%l%V%J4pDl7W;;$K$D$$$F$b(B, Weyl $BBe?t@lMQ$N4X?t$H$7$F(B, 
-$B<!$N4X?t$,MQ0U$7$F$"$k(B. 
-\E
-\BEG
-The following functions are avilable for Groebner basis computation
-in Weyl algebra:
-\E
-@code{dp_weyl_gr_main()},
-@code{dp_weyl_gr_mod_main()},
-@code{dp_weyl_gr_f_main()},
-@code{dp_weyl_f4_main()},
-@code{dp_weyl_f4_mod_main()}.
-\BJP
-$B$^$?(B, $B1~MQ$H$7$F(B, global b $B4X?t$N7W;;$,<BAu$5$l$F$$$k(B. 
-\E
-\BEG
-Computation of the global b function is implemented as an application.
-\E
-
 \JP @node primedec_mod,,, $B%0%l%V%J4pDl$K4X$9$kH!?t(B
 \EG @node primedec_mod,,, Functions for Groebner basis computation
 @subsection @code{primedec_mod}
@@ -3814,6 +4148,9 @@ Computation of the global b function is implemented as
 $BItJ,$r7W;;$9$k$3$H$K$h$k(B early termination $B$r9T$&(B. $B0lHL$K(B, 
 $B%$%G%"%k$N<!85$,9b$$>l9g$KM-8z$@$,(B, 0 $B<!85$N>l9g$J$I(B, $B<!85$,>.$5$$(B
 $B>l9g$K$O(B overhead $B$,Bg$-$$>l9g$,$"$k(B. 
+@item
+$B7W;;ESCf$GFbIt>pJs$r8+$?$$>l9g$K$O!"(B
+$BA0$b$C$F(B @code{dp_gr_print(2)} $B$r<B9T$7$F$*$1$P$h$$(B.
 \E
 \BEG
 @item
@@ -3838,6 +4175,9 @@ is tried by computing the intersection of obtained com
 incrementally. In general, this strategy is useful when the krull
 dimension of the ideal is high, but it may add some overhead
 if the dimension is small.
+@item
+If you want to see internal information during the computation,
+execute @code{dp_gr_print(2)} in advance.
 \E
 @end itemize
 
@@ -3857,11 +4197,115 @@ if the dimension is small.
 \JP @item $B;2>H(B
 \EG @item References
 @fref{modfctr},
-@fref{dp_gr_main dp_gr_mod_main dp_gr_f_main},
+@fref{dp_gr_main dp_gr_mod_main dp_gr_f_main dp_weyl_gr_main dp_weyl_gr_mod_main dp_weyl_gr_f_main},
 \JP @fref{$B9`=g=x$N@_Dj(B}.
-\EG @fref{Setting term orderings}.
+\EG @fref{Setting term orderings},
+@fref{dp_gr_flags dp_gr_print}.
 @end table
 
+\JP @node bfunction bfct generic_bfct ann ann0,,, $B%0%l%V%J4pDl$K4X$9$kH!?t(B
+\EG @node bfunction bfct generic_bfct ann ann0,,, Functions for Groebner basis computation
+@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})
+@itemx bfct(@var{f})
+@itemx generic_bfct(@var{plist},@var{vlist},@var{dvlist},@var{weight})
+\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
+\JP $BB?9`<0$^$?$O%j%9%H(B
+\EG polynomial or list
+@item f
+\JP $BB?9`<0(B
+\EG polynomial
+@item plist
+\JP $BB?9`<0%j%9%H(B
+\EG list of polynomials
+@item vlist dvlist
+\JP $BJQ?t%j%9%H(B
+\EG list of variables
+@end table
+
+@itemize @bullet
+\BJP
+@item @samp{bfct} $B$GDj5A$5$l$F$$$k(B. 
+@item @code{bfunction(@var{f})}, @code{bfct(@var{f})} $B$OB?9`<0(B @var{f} $B$N(B global @var{b} $B4X?t(B @code{b(s)} $B$r(B
+$B7W;;$9$k(B. @code{b(s)} $B$O(B, Weyl $BBe?t(B @code{D} $B>e$N0lJQ?tB?9`<04D(B @code{D[s]}
+$B$N85(B @code{P(x,s)} $B$,B8:_$7$F(B, @code{P(x,s)f^(s+1)=b(s)f^s} $B$rK~$?$9$h$&$J(B
+$BB?9`<0(B @code{b(s)} $B$NCf$G(B, $B<!?t$,:G$bDc$$$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 @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+$h(B. 
+\E
+\BEG
+@item These functions are defined in @samp{bfct}.
+@item @code{bfunction(@var{f})} and @code{bfct(@var{f})} compute the global @var{b}-function @code{b(s)} of
+a polynomial @var{f}.
+@code{b(s)} is a polynomial of the minimal degree 
+such that there exists @code{P(x,s)} in D[s], which is a polynomial
+ring over Weyl algebra @code{D}, and @code{P(x,s)f^(s+1)=b(s)f^s} holds.
+@item @code{generic_bfct(@var{f},@var{vlist},@var{dvlist},@var{weight})}
+computes the global @var{b}-function of a left ideal @code{I} in @code{D}
+generated by @var{plist}, with respect to @var{weight}.
+@var{vlist} is the list of @code{x}-variables, 
+@var{vlist} is the list of corresponding @code{D}-variables.
+@item @code{bfunction(@var{f})} and @code{bfct(@var{f})} implement
+different algorithms and the efficiency depends on inputs.
+@item @code{ann(@var{f})} returns the generator set of the annihilator
+ideal of @code{@var{f}^s}.
+@code{ann(@var{f})} returns a list @code{[@var{a},@var{list}]},
+where @var{a} is the minimal integral root of the global @var{b}-function
+of @var{f}, and @var{list} is a list of polynomials obtained by
+substituting @code{s} in @code{ann(@var{f})} with @var{a}.
+@item See [Saito,Sturmfels,Takayama] for the details.
+\E
+@end itemize
+
+@example
+[0] load("bfct")$
+[216] bfunction(x^3+y^3+z^3+x^2*y^2*z^2+x*y*z);
+-9*s^5-63*s^4-173*s^3-233*s^2-154*s-40
+[217] fctr(@@);
+[[-1,1],[s+2,1],[3*s+4,1],[3*s+5,1],[s+1,2]]
+[218] F = [4*x^3*dt+y*z*dt+dx,x*z*dt+4*y^3*dt+dy,
+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]);
+20000*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
+\JP @item $B;2>H(B
+\EG @item References
+\JP @fref{Weyl $BBe?t(B}.
+\EG @fref{Weyl algebra}.
+@end table