===================================================================
RCS file: /home/cvs/OpenXM/src/asir-doc/parts/groebner.texi,v
retrieving revision 1.5
retrieving revision 1.6
diff -u -p -r1.5 -r1.6
--- OpenXM/src/asir-doc/parts/groebner.texi	2003/04/20 08:01:25	1.5
+++ OpenXM/src/asir-doc/parts/groebner.texi	2003/04/20 09:55:18	1.6
@@ -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 1.5 2003/04/20 08:01:25 noro Exp $
 \BJP
 @node $B%0%l%V%J4pDl$N7W;;(B,,, Top
 @chapter $B%0%l%V%J4pDl$N7W;;(B
@@ -1203,6 +1203,105 @@ 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 +1316,8 @@ 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::
 * dp_gr_flags dp_gr_print::
 * dp_ord::
 * dp_ptod::
@@ -1244,6 +1343,7 @@ Refer to the sections for each functions.
 * lex_hensel_gsl tolex_gsl tolex_gsl_d::
 * primadec primedec::
 * primedec_mod::
+* bfunction generic_bfct::
 @end menu
 
 \JP @node gr hgr gr_mod,,, $B%0%l%V%J4pDl$K4X$9$kH!?t(B
@@ -1346,8 +1446,7 @@ 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 +1663,7 @@ 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
@@ -1842,17 +1941,23 @@ 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 +1985,10 @@ 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 +2023,11 @@ 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 +2074,19 @@ 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 +2112,9 @@ 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 +2126,9 @@ 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
 
@@ -3665,105 +3781,6 @@ 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}
@@ -3857,11 +3874,83 @@ 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}.
 @end table
 
+\JP @node bfunction generic_bfct,,, $B%0%l%V%J4pDl$K4X$9$kH!?t(B
+\EG @node bfunction generic_bfct,,, Functions for Groebner basis computation
+@subsection @code{bfunction}, @code{generic_bfct}
+@findex bfunction
+@findex generic_bfct
 
+@table @t
+@item bfunction(@var{f})
+@item generic_bfct(@var{plist},@var{vlist},@var{dvlist},@var{weight})
+\JP :: b $B4X?t$N7W;;(B
+\EG :: Computes the global b function of a polynomial or an ideal
+@end table
+@table @var
+@item return
+@itemx 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$$$k(B. 
+@item @code{bfunction(@var{f})} $B$OB?9`<0(B @var{f} $B$N(B global b $B4X?t(B @code{b(s)} $B$r(B
+$B7W;;$9$k(B. @code{b(s)} $B$O(B, Weyl $BBe?t(B @code{D} $B>e$N0lJQ?tB?9`<04D(B @code{D[s]}
+$B$N85(B @code{P(x,s)} $B$,B8:_$7$F(B, @code{P(x,s)f^(s+1)=b(s)f^s} $B$rK~$?$9$h$&$J(B
+$BB?9`<0(B @code{b(s)} $B$NCf$G(B, $B<!?t$,:G$bDc$$$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. 
+@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 $B>\:Y$K$D$$$F$O(B, [SST] $B$r8+$h(B. 
+\E
+\BEG
+@item These functions are defined in @samp{bfct}.
+@item @code{bfunction(@var{f})} computes the global b-function @code{b(s)} of
+a polynomial @var{f}.
+@code{b(s)} is a polynomial of the minimal degree 
+such that there exists @code{P(x,s)} in D[s], which is a polynomial
+ring over Weyl algebra @code{D}, and @code{P(x,s)f^(s+1)=b(s)f^s} holds.
+@item @code{generic_bfct(@var{f},@var{vlist},@var{dvlist},@var{weight})}
+computes the global b-function of a left ideal @code{I} in @code{D}
+generated by @var{plist}, with respect to @var{weight}.
+@var{vlist} is the list of @code{x}-variables, 
+@var{vlist} is the list of corresponding @code{D}-variables.
+@item See [SST] for the details.
+\E
+@end itemize
+
+@example
+[0] load("bfct")$
+[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
+@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