===================================================================
RCS file: /home/cvs/OpenXM/src/asir-doc/parts/groebner.texi,v
retrieving revision 1.12
retrieving revision 1.23
diff -u -p -r1.12 -r1.23
--- OpenXM/src/asir-doc/parts/groebner.texi	2003/12/27 11:52:07	1.12
+++ OpenXM/src/asir-doc/parts/groebner.texi	2019/09/13 09:31:00	1.23
@@ -1,4 +1,4 @@
-@comment $OpenXM: OpenXM/src/asir-doc/parts/groebner.texi,v 1.11 2003/04/28 06:43:10 noro Exp $
+@comment $OpenXM: OpenXM/src/asir-doc/parts/groebner.texi,v 1.22 2019/03/29 04:54:25 noro Exp $
 \BJP
 @node $B%0%l%V%J4pDl$N7W;;(B,,, Top
 @chapter $B%0%l%V%J4pDl$N7W;;(B
@@ -15,9 +15,11 @@
 * $B4pK\E*$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::
+* $BB?9`<04D>e$N2C72(B::
 * $B%0%l%V%J4pDl$K4X$9$kH!?t(B::
 \E
 \BEG
@@ -26,9 +28,11 @@
 * Fundamental functions::
 * Controlling Groebner basis computations::
 * Setting term orderings::
+* Weight::
 * Groebner basis computation with rational function coefficients::
 * Change of ordering::
 * Weyl algebra::
+* Module over a polynomial ring::
 * Functions for Groebner basis computation::
 \E
 @end menu
@@ -199,16 +203,16 @@ In an @b{Asir} session, it is displayed in the form li
 \EG and also can be input in such a form.
 
 \BJP
-@itemx $BF,C19`<0(B (head monomial)
 @item $BF,9`(B (head term)
+@itemx $BF,C19`<0(B (head monomial)
 @itemx $BF,78?t(B (head coefficient)
 $BJ,;6I=8=B?9`<0$K$*$1$k3FC19`<0$O(B, $B9`=g=x$K$h$j@0Ns$5$l$k(B. $B$3$N;~=g(B
 $B=x:GBg$NC19`<0$rF,C19`<0(B, $B$=$l$K8=$l$k9`(B, $B78?t$r$=$l$>$lF,9`(B, $BF,78?t(B
 $B$H8F$V(B. 
 \E
 \BEG
-@itemx head monomial
 @item head term
+@itemx head monomial
 @itemx head coefficient
       
 Monomials in a distributed polynomial is sorted by a total order.
@@ -218,7 +222,45 @@ the head term and the head coefficient respectively.
 \E
 @end table
 
+@noindent
+ChangeLog
+@itemize @bullet
 \BJP
+@item $BJ,;6I=8=B?9`<0$OG$0U$N%*%V%8%'%/%H$r78?t$K$b$F$k$h$&$K$J$C$?(B.
+$B$^$?2C72$N(Bk$B@.J,$NMWAG$r<!$N7A<0(B <<d0,d1,...:k>> $B$GI=8=$9$k$h$&$K$J$C$?(B (2017-08-31).
+\E
+\BEG
+@item Distributed polynomials accept objects as coefficients.
+The k-th element of a free module is expressed as <<d0,d1,...:k>> (2017-08-31).
+\E
+@item 
+1.15 algnum.c,
+1.53 ctrl.c,
+1.66 dp-supp.c,
+1.105 dp.c,
+1.73 gr.c,
+1.4 reduct.c,
+1.16 _distm.c,
+1.17 dalg.c,
+1.52 dist.c,
+1.20 distm.c,
+1.8  gmpq.c,
+1.238 engine/nd.c,
+1.102  ca.h,
+1.411  version.h,
+1.28 cpexpr.c,
+1.42 pexpr.c,
+1.20 pexpr_body.c,
+1.40 spexpr.c,
+1.27 arith.c,
+1.77 eval.c,
+1.56 parse.h,
+1.37 parse.y,
+1.8 stdio.c,
+1.31 plotf.c
+@end itemize
+
+\BJP
 @node $B%U%!%$%k$NFI$_9~$_(B,,, $B%0%l%V%J4pDl$N7W;;(B
 @section $B%U%!%$%k$NFI$_9~$_(B
 \E
@@ -1055,6 +1097,139 @@ 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
@@ -1313,6 +1488,57 @@ Computation of the global b function is implemented as
 \E
 
 \BJP
+@node $BB?9`<04D>e$N2C72(B,,, $B%0%l%V%J4pDl$N7W;;(B
+@section $BB?9`<04D>e$N2C72(B
+\E
+\BEG
+@node Module over a polynomial ring,,, Groebner basis computation
+@section Module over a polynomial ring
+\E
+
+@noindent
+
+\BJP
+$BB?9`<04D>e$N<+M32C72$N85$O(B, $B2C72C19`<0(B te_i $B$N@~7?OB$H$7$FFbItI=8=$5$l$k(B.
+$B$3$3$G(B t $B$OB?9`<04D$NC19`<0(B, e_i $B$O<+M32C72$NI8=`4pDl$G$"$k(B. $B2C72C19`<0$O(B, $BB?9`<04D$NC19`<0(B
+$B$K0LCV(B i $B$rDI2C$7$?(B @code{<<a,b,...,c:i>>} $B$GI=$9(B. $B2C72B?9`<0(B, $B$9$J$o$A2C72C19`<0$N@~7?OB$O(B,
+$B@_Dj$5$l$F$$$k2C729`=g=x$K$7$?$,$C$F9_=g$K@0Ns$5$l$k(B. $B2C729`=g=x$K$O0J2<$N(B3$B<oN`$,$"$k(B.
+
+@table @code
+@item TOP $B=g=x(B
+
+$B$3$l$O(B, te_i > se_j $B$H$J$k$N$O(B t>s $B$^$?$O(B (t=s $B$+$D(B i<j) $B$H$J$k$h$&$J9`=g=x$G$"$k(B. $B$3$3$G(B,
+t, s $B$NHf3S$OB?9`<04D$K@_Dj$5$l$F$$$k=g=x$G9T$&(B. 
+$B$3$N7?$N=g=x$O(B, @code{dp_ord([0,Ord])} $B$K(B
+$B$h$j@_Dj$9$k(B. $B$3$3$G(B, @code{Ord} $B$OB?9`<04D$N=g=x7?$G$"$k(B.
+
+@item POT $B=g=x(B
+
+$B$3$l$O(B, te_i > se_j $B$H$J$k$N$O(B i<j $B$^$?$O(B (i=j $B$+$D(B t>s) $B$H$J$k$h$&$J9`=g=x$G$"$k(B. $B$3$3$G(B,
+t, s $B$NHf3S$OB?9`<04D$K@_Dj$5$l$F$$$k=g=x$G9T$&(B.
+$B$3$N7?$N=g=x$O(B, @code{dp_ord([1,Ord])} $B$K(B
+$B$h$j@_Dj$9$k(B. $B$3$3$G(B, @code{Ord} $B$OB?9`<04D$N=g=x7?$G$"$k(B.
+
+@item Schreyer $B7?=g=x(B
+
+$B3FI8=`4pDl(B e_i $B$KBP$7(B, $BJL$N<+M32C72$N2C72C19`<0(B T_i $B$,M?$($i$l$F$$$F(B, te_i > se_j $B$H$J$k$N$O(B
+tT_i > sT_j $B$^$?$O(B (tT_i=sT_j $B$+$D(B i<j) $B$H$J$k$h$&$J9`=g=x$G$"$k(B. $B$3$3$G(B tT_i, sT_j $B$N(B
+$BHf3S$O(B, $B$3$l$i$,=jB0$9$k<+M32C72$K@_Dj$5$l$F$$$k=g=x$G9T$&(B.
+$B$3$N7?$N=g=x$O(B, $BDL>o:F5"E*$K@_Dj$5$l$k(B. $B$9$J$o$A(B, T_i $B$,=jB0$9$k<+M32C72$N=g=x$b(B Schreyer $B7?(B
+$B$G$"$k$+(B, $B$^$?$O%\%H%`$H$J$k(B TOP, POT $B$J$I$N9`=g=x$H$J$k(B. 
+$B$3$N7?$N=g=x$O(B @code{dpm_set_schreyer([H_1,H_2,...])} $B$K$h$j;XDj$9$k(B. $B$3$3$G(B,
+@code{H_i=[T_1,T_2,...]} $B$O2C72C19`<0$N%j%9%H$G(B, @code{[H_2,...]} $B$GDj5A$5$l$k(B Schreyer $B7?9`=g=x$r(B
+@code{tT_i} $B$i$KE,MQ$9$k$H$$$&0UL#$G$"$k(B. 
+@end table
+
+$B2C72B?9`<0$rF~NO$9$kJ}K!$H$7$F$O(B, @code{<<a,b,...:i>>} $B$J$k7A<0$GD>@\F~NO$9$kB>$K(B,
+$BB?9`<0%j%9%H$r:n$j(B, @code{dpm_ltod()} $B$K$h$jJQ49$9$kJ}K!$b$"$k(B.
+\E
+\BEG
+not yet
+\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
@@ -1329,15 +1555,19 @@ 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::
+* nd_gr_postproc nd_weyl_gr_postproc::
 * dp_gr_flags dp_gr_print::
 * dp_ord::
+* dp_set_weight dp_set_top_weight dp_weyl_set_weight::
 * dp_ptod::
 * dp_dtop::
 * dp_mod dp_rat::
 * dp_homo dp_dehomo::
 * dp_ptozp dp_prim::
-* dp_nf dp_nf_mod dp_true_nf dp_true_nf_mod::
+* dp_nf dp_nf_mod dp_true_nf dp_true_nf_mod dp_weyl_nf dp_weyl_nf_mod::
 * dp_hm dp_ht dp_hc dp_rest::
+* dpm_hm dpm_ht dpm_hc dpm_hp dpm_rest::
 * dp_td dp_sugar::
 * dp_lcm::
 * dp_redble::
@@ -1394,6 +1624,8 @@ Computation of the global b function is implemented as
 @item
 $BI8=`%i%$%V%i%j$N(B @samp{gr} $B$GDj5A$5$l$F$$$k(B. 
 @item
+gr $B$rL>A0$K4^$`4X?t$O8=:_%a%s%F$5$l$F$$$J$$(B. @code{nd_gr}$B7O$N4X?t$rBe$o$j$KMxMQ$9$Y$-$G$"$k(B(@fref{nd_gr nd_gr_trace nd_f4 nd_f4_trace nd_weyl_gr nd_weyl_gr_trace}).
+@item
 $B$$$:$l$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
 @var{order} $B$K4X$9$k%0%l%V%J4pDl$r5a$a$k(B. @code{gr()}, @code{hgr()}
 $B$O(B $BM-M}?t78?t(B, @code{gr_mod()} $B$O(B GF(@var{p}) $B78?t$H$7$F7W;;$9$k(B. 
@@ -1405,7 +1637,7 @@ 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. 
@@ -1425,6 +1657,9 @@ CPU $B;~4V$G$"$j(B, $B$3$NH!?t$N>l9g$O$[$H$s$IDL?.$
 @item 
 These functions are defined in @samp{gr} in the standard library
 directory.
+@item
+Functions of which names contains gr are obsolted. 
+Functions of @code{nd_gr} families should be used (@fref{nd_gr nd_gr_trace nd_f4 nd_f4_trace nd_weyl_gr nd_weyl_gr_trace}).
 @item 
 They compute a Groebner basis of a polynomial list @var{plist} with
 respect to the variable order @var{vlist} and the order type @var{order}.
@@ -2168,6 +2403,234 @@ 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}[|@var{option=value,...}])
+@itemx nd_gr_trace(@var{plist},@var{vlist},@var{homo},@var{p},@var{order}[|@var{option=value,...}])
+@itemx nd_f4(@var{plist},@var{vlist},@var{modular},@var{order}[|@var{option=value,...}])
+@itemx nd_f4_trace(@var{plist},@var{vlist},@var{homo},@var{p},@var{order}[|@var{option=value,...}])
+@itemx nd_weyl_gr(@var{plist},@var{vlist},@var{p},@var{order}[|@var{option=value,...}])
+@itemx nd_weyl_gr_trace(@var{plist},@var{vlist},@var{homo},@var{p},@var{order}[|@var{option=value,...}])
+\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.
+@var{p} $B$,(B 0 $B$^$?$O(B 1 $B$N$H$-(B, $B<+F0E*$KA*$P$l$?AG?t$rMQ$$$F(B, $B@.8y$9$k(B
+$B$^$G(B trace $B%"%k%4%j%:%`$r<B9T$9$k(B.
+@var{p} $B$,(B 2 $B0J>e$N$H$-(B, trace $B$O(BGF(p) $B>e$G7W;;$5$l$k(B. trace $B%"%k%4%j%:%`(B
+$B$,<:GT$7$?>l9g(B 0 $B$,JV$5$l$k(B. @var{p} $B$,Ii$N>l9g(B, $B%0%l%V%J4pDl%A%'%C%/$O(B
+$B9T$o$J$$(B. $B$3$N>l9g(B, @var{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$$(B, $B$=$NA4<!?t$K$*$1$k4pDl$r@8@.$9$kJ}K!$G$"$k(B. $BF@$i$l$k(B
+$BB?9`<0=89g$O$d$O$j%0%l%V%J4pDl8uJd$G$"$j(B, @code{nd_gr_trace} $B$HF1MM$N(B
+$B%A%'%C%/$,9T$o$l$k(B.
+@item
+@code{nd_f4} $B$O(B @code{modular} $B$,(B 0 $B$N$H$-M-M}?tBN>e$N(B, @code{modular} $B$,(B
+$B%^%7%s%5%$%:AG?t$N$H$-M-8BBN>e$N(B F4 $B%"%k%4%j%:%`$r<B9T$9$k(B.
+@item
+@var{plist} $B$,B?9`<0%j%9%H$N>l9g(B, @var{plist}$B$G@8@.$5$l$k%$%G%"%k$N%0%l%V%J!<4pDl$,(B
+$B7W;;$5$l$k(B. @var{plist} $B$,B?9`<0%j%9%H$N%j%9%H$N>l9g(B, $B3FMWAG$OB?9`<04D>e$N<+M32C72$N85$H8+$J$5$l(B,
+$B$3$l$i$,@8@.$9$kItJ,2C72$N%0%l%V%J!<4pDl$,7W;;$5$l$k(B. $B8e<T$N>l9g(B, $B9`=g=x$O2C72$KBP$9$k9`=g=x$r(B
+$B;XDj$9$kI,MW$,$"$k(B. $B$3$l$O(B @var{[s,ord]} $B$N7A$G;XDj$9$k(B. @var{s} $B$,(B 0 $B$J$i$P(B TOP (Term Over Position),
+1 $B$J$i$P(B POT (Position Over Term) $B$r0UL#$7(B, @var{ord} $B$OB?9`<04D$NC19`<0$KBP$9$k9`=g=x$G$"$k(B.
+@item
+@code{nd_weyl_gr}, @code{nd_weyl_gr_trace} $B$O(B Weyl $BBe?tMQ$G$"$k(B.
+@item
+@code{f4} $B7O4X?t0J30$O$9$Y$FM-M}4X?t78?t$N7W;;$,2DG=$G$"$k(B.
+@item
+$B0lHL$K(B @code{dp_gr_main}, @code{dp_gr_mod_main} $B$h$j9bB.$G$"$k$,(B,
+$BFC$KM-8BBN>e$N>l9g82Cx$G$"$k(B.
+@item
+$B0J2<$N%*%W%7%g%s$,;XDj$G$-$k(B.
+@table @code
+@item homo
+1 $B$N$H$-(B, $B@F<!2=$r7PM3$7$F7W;;$9$k(B. (@code{nd_gr}, @code{nd_f4} $B$N$_(B)
+@item dp
+1 $B$N$H$-(B, $BJ,;6I=8=B?9`<0(B ($B2C72$N>l9g$K$O2C72B?9`<0(B) $B$r7k2L$H$7$FJV$9(B.
+@item nora
+1 $B$N$H$-(B, $B7k2L$NAj8_4JLs$r9T$o$J$$(B.
+@end table
+@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).
+If @var{plist} is a list of polynomials, then a Groebner basis of the ideal generated by @var{plist}
+is computed. If @var{plist} is a list of lists of polynomials, then each list of polynomials are regarded
+as an element of a free module over a polynomial ring and a Groebner basis of the sub-module generated by @var{plist}
+in the free module. In the latter case a term order in the free module should be specified.
+This is specified by @var{[s,ord]}. If @var{s} is 0 then it means TOP (Term Over Position).
+If @var{s} is 1 then it means POT 1 (Position Over Term). @var{ord} is a term order in the base polynomial ring.
+@item
+@code{nd_weyl_gr}, @code{nd_weyl_gr_trace} are for Weyl algebra computation.
+@item
+Functions except for F4 related ones can handle rational coeffient cases.
+@item
+In general these functions are more efficient than 
+@code{dp_gr_main}, @code{dp_gr_mod_main}, especially over finite fields.
+@item
+The fallowing options can be specified.
+@table @code
+@item homo
+If set to 1, the computation is done via homogenization. (only for @code{nd_gr} and @code{nd_f4})
+@item dp
+If set to 1, the functions return a list of distributed polynomials (a list of
+module polynomials when the input is a sub-module).
+@item nora
+If set to 1, the inter-reduction is not performed.
+@end table
+@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 nd_gr_postproc nd_weyl_gr_postproc,,, $B%0%l%V%J4pDl$K4X$9$kH!?t(B
+\EG @node nd_gr_postproc nd_weyl_gr_postproc,,, Functions for Groebner basis computation
+@subsection @code{nd_gr_postproc}, @code{nd_weyl_gr_postproc}
+@findex nd_gr_postproc
+@findex nd_weyl_gr_postproc
+
+@table @t
+@item nd_gr_postproc(@var{plist},@var{vlist},@var{p},@var{order},@var{check})
+@itemx nd_weyl_gr_postproc(@var{plist},@var{vlist},@var{p},@var{order},@var{check})
+\JP :: $B%0%l%V%J4pDl8uJd$N%A%'%C%/$*$h$SAj8_4JLs(B
+\EG :: Check of Groebner basis candidate and inter-reduction
+@end table
+
+@table @var
+@item return
+\JP $B%j%9%H(B $B$^$?$O(B 0
+\EG list or 0
+@item plist  vlist
+\JP $B%j%9%H(B
+\EG list
+@item p
+\JP $BAG?t$^$?$O(B 0
+\EG prime or 0
+@item order
+\JP $B?t(B, $B%j%9%H$^$?$O9TNs(B
+\EG number, list or matrix
+@item check
+\JP 0 $B$^$?$O(B 1
+\EG 0 or 1
+@end table
+
+@itemize @bullet
+\BJP
+@item
+$B%0%l%V%J4pDl(B($B8uJd(B)$B$NAj8_4JLs$r9T$&(B.
+@item
+@code{nd_weyl_gr_postproc} $B$O(B Weyl $BBe?tMQ$G$"$k(B.
+@item
+@var{check=1} $B$N>l9g(B, @var{plist} $B$,(B, @var{vlist}, @var{p}, @var{order} $B$G;XDj$5$l$kB?9`<04D(B, $B9`=g=x$G%0%l%V%J!<4pDl$K$J$C$F$$$k$+(B
+$B$N%A%'%C%/$b9T$&(B.
+@item
+$B@F<!2=$7$F7W;;$7$?%0%l%V%J!<4pDl$rHs@F<!2=$7$?$b$N$rAj8_4JLs$r9T$&(B, CRT $B$G7W;;$7$?%0%l%V%J!<4pDl8uJd$N%A%'%C%/$r9T$&$J$I$N>l9g$KMQ$$$k(B.
+\E
+\BEG
+@item
+Perform the inter-reduction for a Groebner basis (candidate).
+@item
+@code{nd_weyl_gr_postproc} is for Weyl algebra.
+@item
+If @var{check=1} then the check whether @var{plist} is a Groebner basis with respect to a term order in a polynomial ring 
+or Weyl algebra specified by @var{vlist}, @var{p} and @var{order}.
+@item
+This function is used for inter-reduction of a non-reduced Groebner basis that is obtained by dehomogenizing a Groebner basis
+computed via homogenization, or Groebner basis check of a Groebner basis candidate computed by CRT.
+\E
+@end itemize
+
+@example
+afo
+@end example
+
 \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}
@@ -2298,6 +2761,12 @@ uses the value as a flag for showing intermediate info
 @item
 $B%H%C%W%l%Y%kH!?t0J30$NH!?t$rD>@\8F$S=P$9>l9g$K$O(B, $B$3$NH!?t$K$h$j(B
 $BJQ?t=g=x7?$r@5$7$/@_Dj$7$J$1$l$P$J$i$J$$(B. 
+
+@item
+$B0z?t$,%j%9%H$N>l9g(B, $B<+M32C72$K$*$1$k9`=g=x7?$r@_Dj$9$k(B. $B0z?t$,(B@code{[0,Ord]} $B$N>l9g(B,
+$BB?9`<04D>e$G(B @code{Ord} $B$G;XDj$5$l$k9`=g=x$K4p$E$/(B TOP $B=g=x(B, $B0z?t$,(B @code{[1,Ord]} $B$N>l9g(B
+OPT $B=g=x$r@_Dj$9$k(B.
+
 \E
 \BEG
 @item
@@ -2325,6 +2794,12 @@ that such polynomials were generated under the same or
 @item
 Type of term ordering must be correctly set by this function
 when functions other than top level functions are called directly.
+
+@item
+If the argument is a list, then an ordering type in a free module is set.
+If the argument is @code{[0,Ord]} then a TOP ordering based on the ordering type specified
+by @code{Ord} is set. 
+If the argument is @code{[1,Ord]} then a POT ordering is set.
 \E
 @end itemize
 
@@ -2344,6 +2819,79 @@ when functions other than top level functions are call
 \EG @fref{Setting term orderings}
 @end table
 
+\JP @node dp_set_weight dp_set_top_weight dp_weyl_set_weight,,, $B%0%l%V%J4pDl$K4X$9$kH!?t(B
+\EG @node dp_set_weight dp_set_top_weight dp_weyl_set_weight,,, Functions for Groebner basis computation
+@subsection @code{dp_set_weight}, @code{dp_set_top_weight}, @code{dp_weyl_set_weight}
+@findex dp_set_weight
+@findex dp_set_top_weight
+@findex dp_weyl_set_weight
+
+@table @t
+@item dp_set_weight([@var{weight}])
+\JP :: sugar weight $B$N@_Dj(B, $B;2>H(B
+\EG :: Set and show the sugar weight.
+@item dp_set_top_weight([@var{weight}])
+\JP :: top weight $B$N@_Dj(B, $B;2>H(B
+\EG :: Set and show the top weight.
+@item dp_weyl_set_weight([@var{weight}])
+\JP :: weyl weight $B$N@_Dj(B, $B;2>H(B
+\EG :: Set and show the weyl weight.
+@end table
+
+@table @var
+@item return
+\JP $B%Y%/%H%k(B
+\EG a vector
+@item weight
+\JP $B@0?t$N%j%9%H$^$?$O%Y%/%H%k(B
+\EG a list or vector of integers
+@end table
+
+@itemize @bullet
+\BJP
+@item
+@code{dp_set_weight} $B$O(B sugar weight $B$r(B @var{weight} $B$K@_Dj$9$k(B. $B0z?t$,$J$$;~(B, 
+$B8=:_@_Dj$5$l$F$$$k(B sugar weight $B$rJV$9(B. sugar weight $B$O@5@0?t$r@.J,$H$9$k%Y%/%H%k$G(B,
+$B3FJQ?t$N=E$_$rI=$9(B. $B<!?t$D$-=g=x$K$*$$$F(B, $BC19`<0$N<!?t$r7W;;$9$k:]$KMQ$$$i$l$k(B.
+$B@F<!2=JQ?tMQ$K(B, $BKvHx$K(B 1 $B$rIU$12C$($F$*$/$H0BA4$G$"$k(B.
+@item
+@code{dp_set_top_weight} $B$O(B top weight $B$r(B @var{weight} $B$K@_Dj$9$k(B. $B0z?t$,$J$$;~(B, 
+$B8=:_@_Dj$5$l$F$$$k(B top weight $B$rJV$9(B. top weight $B$,@_Dj$5$l$F$$$k$H$-(B, 
+$B$^$:(B top weight $B$K$h$kC19`<0Hf3S$r@h$K9T$&(B. tie breaker $B$H$7$F8=:_@_Dj$5$l$F$$$k(B
+$B9`=g=x$,MQ$$$i$l$k$,(B, $B$3$NHf3S$K$O(B top weight $B$OMQ$$$i$l$J$$(B.
+
+@item
+@code{dp_weyl_set_weight} $B$O(B weyl weight $B$r(B @var{weight} $B$K@_Dj$9$k(B. $B0z?t$,$J$$;~(B, 
+$B8=:_@_Dj$5$l$F$$$k(B weyl weight $B$rJV$9(B. weyl weight w $B$r@_Dj$9$k$H(B,
+$B9`=g=x7?(B 11 $B$G$N7W;;$K$*$$$F(B, (-w,w) $B$r(B top weight, tie breaker $B$r(B graded reverse lex
+$B$H$7$?9`=g=x$,@_Dj$5$l$k(B.
+\E
+\BEG
+@item
+@code{dp_set_weight} sets the sugar weight=@var{weight}. It returns the current sugar weight.
+A sugar weight is a vector with positive integer components and it represents the weights of variables.
+It is used for computing the weight of a monomial in a graded ordering.
+It is recommended to append a component 1 at the end of the weight vector for a homogenizing variable.
+@item
+@code{dp_set_top_weight} sets the top weight=@var{weight}. It returns the current top weight.
+It a top weight is set, the weights of monomials under the top weight are firstly compared.
+If the the weights are equal then the current term ordering is applied as a tie breaker, but
+the top weight is not used in the tie breaker.
+
+@item
+@code{dp_weyl_set_weight} sets the weyl weigh=@var{weight}. It returns the current weyl weight.
+If a weyl weight w is set, in the comparsion by the term order type 11, a term order with 
+the top weight=(-w,w) and the tie breaker=graded reverse lex is applied.
+\E
+@end itemize
+
+@table @t
+\JP @item $B;2>H(B
+\EG @item References
+@fref{Weight}
+@end table
+
+
 \JP @node dp_ptod,,, $B%0%l%V%J4pDl$K4X$9$kH!?t(B
 \EG @node dp_ptod,,, Functions for Groebner basis computation
 @subsection @code{dp_ptod}
@@ -2403,6 +2951,171 @@ the coefficient field.
 @fref{dp_ord}.
 @end table
 
+\JP @node dpm_dptodpm,,, $B%0%l%V%J4pDl$K4X$9$kH!?t(B
+\EG @node dpm_dptodpm,,, Functions for Groebner basis computation
+@subsection @code{dpm_dptodpm}
+@findex dpm_dptodpm
+
+@table @t
+@item dpm_dptodpm(@var{dpoly},@var{pos})
+\JP :: $BJ,;6I=8=B?9`<0$r2C72B?9`<0$KJQ49$9$k(B. 
+\EG :: Converts a distributed polynomial into a module polynomial.
+@end table
+
+@table @var
+@item return
+\JP $B2C72B?9`<0(B
+\EG module polynomial
+@item dpoly
+\JP $BJ,;6I=8=B?9`<0(B
+\EG distributed polynomial
+@item pos
+\JP $B@5@0?t(B
+\EG positive integer
+@end table
+
+@itemize @bullet
+\BJP
+@item
+$BJ,;6I=8=B?9`<0$r2C72B?9`<0$KJQ49$9$k(B.
+@item
+$B=PNO$O2C72B?9`<0(B @code{dpoly e_pos} $B$G$"$k(B.
+\E
+\BEG
+@item
+This function converts a distributed polynomial into a module polynomial.
+@item
+The output is @code{dpoly e_pos}.
+\E
+@end itemize
+
+@example
+[50] dp_ord([0,0])$
+[51] D=dp_ptod((x+y+z)^2,[x,y,z]);
+(1)*<<2,0,0>>+(2)*<<1,1,0>>+(1)*<<0,2,0>>+(2)*<<1,0,1>>+(2)*<<0,1,1>>
++(1)*<<0,0,2>>
+[52] dp_dptodpm(D,2);
+(1)*<<2,0,0:2>>+(2)*<<1,1,0:2>>+(1)*<<0,2,0:2>>+(2)*<<1,0,1:2>>
++(2)*<<0,1,1:2>>+(1)*<<0,0,2:2>>
+@end example
+
+@table @t
+\JP @item $B;2>H(B
+\EG @item References
+@fref{dp_ptod},
+@fref{dp_ord}.
+@end table
+
+\JP @node dpm_ltod,,, $B%0%l%V%J4pDl$K4X$9$kH!?t(B
+\EG @node dpm_ltod,,, Functions for Groebner basis computation
+@subsection @code{dpm_ltod}
+@findex dpm_ltod
+
+@table @t
+@item dpm_dptodpm(@var{plist},@var{vlist})
+\JP :: $BB?9`<0%j%9%H$r2C72B?9`<0$KJQ49$9$k(B. 
+\EG :: Converts a list of polynomials into a module polynomial.
+@end table
+
+@table @var
+@item return
+\JP $B2C72B?9`<0(B
+\EG module polynomial
+@item plist
+\JP $BB?9`<0%j%9%H(B
+\EG list of polynomials
+@item vlist
+\JP $BJQ?t%j%9%H(B
+\EG list of variables
+@end table
+
+@itemize @bullet
+\BJP
+@item
+$BB?9`<0%j%9%H$r2C72B?9`<0$KJQ49$9$k(B.
+@item
+@code{[p1,...,pm]} $B$O(B @code{p1 e1+...+pm em} $B$KJQ49$5$l$k(B.
+\E
+\BEG
+@item
+This function converts a list of polynomials into a module polynomial.
+@item
+@code{[p1,...,pm]} is converted into @code{p1 e1+...+pm em}.
+\E
+@end itemize
+
+@example
+[2126] dp_ord([0,0])$
+[2127] dpm_ltod([x^2+y^2,x,y-z],[x,y,z]);
+(1)*<<2,0,0:1>>+(1)*<<0,2,0:1>>+(1)*<<1,0,0:2>>+(1)*<<0,1,0:3>>
++(-1)*<<0,0,1:3>>
+@end example
+
+@table @t
+\JP @item $B;2>H(B
+\EG @item References
+@fref{dpm_dtol},
+@fref{dp_ord}.
+@end table
+
+\JP @node dpm_dtol,,, $B%0%l%V%J4pDl$K4X$9$kH!?t(B
+\EG @node dpm_dtol,,, Functions for Groebner basis computation
+@subsection @code{dpm_dtol}
+@findex dpm_dtol
+
+@table @t
+@item dpm_dptodpm(@var{poly},@var{vlist})
+\JP :: $B2C72B?9`<0$rB?9`<0%j%9%H$KJQ49$9$k(B. 
+\EG :: Converts a module polynomial into a list of polynomials.
+@end table
+
+@table @var
+@item return
+\JP $BB?9`<0%j%9%H(B
+\EG list of polynomials
+@item poly
+\JP $B2C72B?9`<0(B
+\EG module polynomial
+@item vlist
+\JP $BJQ?t%j%9%H(B
+\EG list of variables
+@end table
+
+@itemize @bullet
+\BJP
+@item
+$B2C72B?9`<0$rB?9`<0%j%9%H$KJQ49$9$k(B.
+@item
+@code{p1 e1+...+pm em} $B$O(B @code{[p1,...,pm]} $B$KJQ49$5$l$k(B.
+@item
+$B=PNO%j%9%H$ND9$5$O(B, @code{poly} $B$K4^$^$l$kI8=`4pDl$N:GBg%$%s%G%C%/%9$H$J$k(B.
+\E
+\BEG
+@item
+This function converts a module polynomial into a list of polynomials.
+@item
+@code{p1 e1+...+pm em} is converted into @code{[p1,...,pm]}.
+@item
+The length of the output list is equal to the largest index among those of the standard bases
+containd in @code{poly}.
+\E
+@end itemize
+
+@example
+[2126] dp_ord([0,0])$
+[2127] D=(1)*<<2,0,0:1>>+(1)*<<0,2,0:1>>+(1)*<<1,0,0:2>>+(1)*<<0,1,0:3>>
++(-1)*<<0,0,1:3>>$
+[2128] dpm_dtol(D,[x,y,z]);
+[x^2+y^2,x,y-z]
+@end example
+
+@table @t
+\JP @item $B;2>H(B
+\EG @item References
+@fref{dpm_ltod},
+@fref{dp_ord}.
+@end table
+
 \JP @node dp_dtop,,, $B%0%l%V%J4pDl$K4X$9$kH!?t(B
 \EG @node dp_dtop,,, Functions for Groebner basis computation
 @subsection @code{dp_dtop}
@@ -2526,7 +3239,7 @@ converting the coefficients into elements of a finite 
 @table @t
 \JP @item $B;2>H(B
 \EG @item References
-@fref{dp_nf dp_nf_mod dp_true_nf dp_true_nf_mod},
+@fref{dp_nf dp_nf_mod dp_true_nf dp_true_nf_mod dp_weyl_nf dp_weyl_nf_mod},
 @fref{subst psubst},
 @fref{setmod}.
 @end table
@@ -2617,7 +3330,7 @@ These are used internally in @code{hgr()} etc.
 into an integral distributed polynomial such that GCD of all its coefficients
 is 1.
 \E
-@itemx dp_prim(@var{dpoly})
+@item dp_prim(@var{dpoly})
 \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. 
 \BEG
 :: Converts a distributed polynomial @var{poly} with rational function
@@ -2670,17 +3383,21 @@ polynomial contents included in the coefficients are n
 @fref{ptozp}.
 @end table
 
-\JP @node dp_nf dp_nf_mod dp_true_nf dp_true_nf_mod,,, $B%0%l%V%J4pDl$K4X$9$kH!?t(B
-\EG @node dp_nf dp_nf_mod dp_true_nf dp_true_nf_mod,,, Functions for Groebner basis computation
+\JP @node dp_nf dp_nf_mod dp_true_nf dp_true_nf_mod dp_weyl_nf dp_weyl_nf_mod,,, $B%0%l%V%J4pDl$K4X$9$kH!?t(B
+\EG @node dp_nf dp_nf_mod dp_true_nf dp_true_nf_mod dp_weyl_nf dp_weyl_nf_mod,,, Functions for Groebner basis computation
 @subsection @code{dp_nf}, @code{dp_nf_mod}, @code{dp_true_nf}, @code{dp_true_nf_mod}
 @findex dp_nf
 @findex  dp_true_nf
 @findex dp_nf_mod
 @findex  dp_true_nf_mod
+@findex dp_weyl_nf
+@findex dp_weyl_nf_mod
 
 @table @t
 @item dp_nf(@var{indexlist},@var{dpoly},@var{dpolyarray},@var{fullreduce})
+@item dp_weyl_nf(@var{indexlist},@var{dpoly},@var{dpolyarray},@var{fullreduce})
 @item dp_nf_mod(@var{indexlist},@var{dpoly},@var{dpolyarray},@var{fullreduce},@var{mod})
+@item dp_weyl_nf_mod(@var{indexlist},@var{dpoly},@var{dpolyarray},@var{fullreduce},@var{mod})
 \JP :: $BJ,;6I=8=B?9`<0$N@55,7A$r5a$a$k(B. ($B7k2L$ODj?tG\$5$l$F$$$k2DG=@-$"$j(B)
 
 \BEG
@@ -2722,6 +3439,8 @@ is returned in such a list as @code{[numerator, denomi
 @item
 $BJ,;6I=8=B?9`<0(B @var{dpoly} $B$N@55,7A$r5a$a$k(B. 
 @item
+$BL>A0$K(B weyl $B$r4^$`4X?t$O%o%$%kBe?t$K$*$1$k@55,7A7W;;$r9T$&(B. $B0J2<$N@bL@$O(B weyl $B$r4^$`$b$N$KBP$7$F$bF1MM$K@.N)$9$k(B.
+@item
 @code{dp_nf_mod()}, @code{dp_true_nf_mod()} $B$NF~NO$O(B, @code{dp_mod()} $B$J$I(B
 $B$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. 
 @item
@@ -2754,6 +3473,9 @@ is returned in such a list as @code{[numerator, denomi
 @item
 Computes the normal form of a distributed polynomial.
 @item
+Functions whose name contain @code{weyl} compute normal forms in Weyl algebra. The description below also applies to
+the functions for Weyl algebra.
+@item
 @code{dp_nf_mod()} and @code{dp_true_nf_mod()} require
 distributed polynomials with coefficients in a finite field as arguments.
 @item
@@ -2824,6 +3546,126 @@ u4^2+(6*u3+2*u2+6*u1-2)*u4+9*u3^2+(6*u2+18*u1-6)*u3+u2
 @fref{p_nf p_nf_mod p_true_nf p_true_nf_mod}.
 @end table
 
+\JP @node dpm_nf dpm_nf_and_quotient,,, $B%0%l%V%J4pDl$K4X$9$kH!?t(B
+\EG @node dpm_nf dpm_nf_and_quotient,,, Functions for Groebner basis computation
+@subsection @code{dpm_nf}, @code{dpm_nf_and_quotient}
+@findex dpm_nf
+@findex dpm_nf_and_quotient
+
+@table @t
+@item dpm_nf([@var{indexlist},]@var{dpoly},@var{dpolyarray},@var{fullreduce})
+\JP :: $B2C72B?9`<0$N@55,7A$r5a$a$k(B. ($B7k2L$ODj?tG\$5$l$F$$$k2DG=@-$"$j(B)
+
+\BEG
+:: Computes the normal form of a module polynomial.
+(The result may be multiplied by a constant in the ground field.)
+\E
+@item dpm_nf_and_quotient([@var{indexlist},]@var{dpoly},@var{dpolyarray})
+\JP :: $B2C72B?9`<0$N@55,7A$H>&$r5a$a$k(B.
+\BEG
+:: Computes the normal form of a module polynomial and the quotient.
+\E
+@end table
+
+@table @var
+@item return
+\JP @code{dpm_nf()} : $B2C72B?9`<0(B, @code{dpm_nf_and_quotient()} : $B%j%9%H(B
+\EG @code{dpm_nf()} : module polynomial, @code{dpm_nf_and_quotient()} : list
+@item indexlist
+\JP $B%j%9%H(B
+\EG list
+@item dpoly
+\JP $B2C72B?9`<0(B
+\EG module polynomial
+@item dpolyarray
+\JP $BG[Ns(B
+\EG array of module polynomial
+@end table
+
+@itemize @bullet
+\BJP
+@item
+$B2C72B?9`<0(B @var{dpoly} $B$N@55,7A$r5a$a$k(B. 
+@item
+$B7k2L$KM-M}?t(B, $BM-M}<0$,4^$^$l$k$N$rHr$1$k$?$a(B, @code{dpm_nf()} $B$O(B
+$B??$NCM$NDj?tG\$NCM$rJV$9(B. 
+@item
+@var{dpolyarray} $B$O2C72B?9`<0$rMWAG$H$9$k%Y%/%H%k(B, 
+@var{indexlist} $B$O@55,2=7W;;$KMQ$$$k(B @var{dpolyarray} $B$NMWAG$N%$%s%G%C%/%9(B
+@item
+@var{indexlist} $B$,M?$($i$l$F$$$k>l9g(B, @var{dpolyarray} $B$NCf$G(B, @var{indexlist} $B$G;XDj$5$l$?$b$N$N$_$,(B, $BA0$NJ}$+$iM%@hE*$K;H$o$l$k(B. 
+@var{indexlist} $B$,M?$($i$l$F$$$J$$>l9g$K$O(B, @var{dpolyarray} $B$NCf$NA4$F$NB?9`<0$,A0$NJ}$+$iM%@hE*$K;H$o$l$k(B.
+@item
+@code{dpm_nf_and_quotient()} $B$O(B, 
+@code{[@var{nm},@var{dn},@var{quo}]} $B$J$k7A$N%j%9%H$rJV$9(B. 
+$B$?$@$7(B, @var{nm} $B$O78?t$KJ,?t$r4^$^$J$$2C72B?9`<0(B, @var{dn} $B$O(B
+$B?t$^$?$OB?9`<0$G(B @var{nm}/@var{dn} $B$,??$NCM$H$J$k(B. 
+@var{quo} $B$O=|;;$N>&$rI=$9G[Ns$G(B, @var{dn}@var{dpoly}=@var{nm}+@var{quo[0]dpolyarray[0]+...} $B$,@.$jN)$D(B.
+$B$N%j%9%H(B. 
+@item
+@var{fullreduce} $B$,(B 0 $B$G$J$$$H$-A4$F$N9`$KBP$7$F4JLs$r9T$&(B. @var{fullreduce}
+$B$,(B 0 $B$N$H$-F,9`$N$_$KBP$7$F4JLs$r9T$&(B. 
+\E
+\BEG
+@item
+Computes the normal form of a module polynomial.
+@item
+The result of @code{dpm_nf()} may be multiplied by a constant in the
+ground field in order to make the result integral.
+@item
+@var{dpolyarray} is a vector whose components are module polynomials
+and @var{indexlist} is a list of indices which is used for the normal form
+computation.
+@item
+If @var{indexlist} is given, only the polynomials in @var{dpolyarray} specified in @var{indexlist}
+is used in the division. An index placed at the preceding position has priority to be selected.
+If @var{indexlist} is not given, all the polynomials in @var{dpolyarray} are used.
+@item
+@code{dpm_nf_and_quotient()} returns
+such a list as @code{[@var{nm},@var{dn},@var{quo}]}.
+Here @var{nm} is a module polynomial whose coefficients are integral
+in the ground field, @var{dn} is an integral element in the ground
+field and @var{nm}/@var{dn} is the true normal form.
+@var{quo} is an array containing the quotients of the division satisfying
+@var{dn}@var{dpoly}=@var{nm}+@var{quo[0]dpolyarray[0]+...}.
+@item
+When argument @var{fullreduce} has non-zero value,
+all terms are reduced. When it has value 0,
+only the head term is reduced.
+\E
+@end itemize
+
+@example
+[2126] dp_ord([1,0])$
+[2127] S=ltov([(1)*<<0,0,2,0:1>>+(1)*<<0,0,1,1:1>>+(1)*<<0,0,0,2:1>>
++(-1)*<<3,0,0,0:2>>+(-1)*<<0,0,2,1:2>>+(-1)*<<0,0,1,2:2>>
++(1)*<<3,0,1,0:3>>+(1)*<<3,0,0,1:3>>+(1)*<<0,0,2,2:3>>,
+(-1)*<<0,1,0,0:1>>+(-1)*<<0,0,1,0:1>>+(-1)*<<0,0,0,1:1>>
++(-1)*<<3,0,0,0:3>>+(1)*<<0,1,1,1:3>>,(1)*<<0,1,0,0:2>>
++(1)*<<0,0,1,0:2>>+(1)*<<0,0,0,1:2>>+(-1)*<<0,1,1,0:3>>
++(-1)*<<0,1,0,1:3>>+(-1)*<<0,0,1,1:3>>])$
+[2128] U=dpm_sp(S[0],S[1]);
+(1)*<<0,0,3,0:1>>+(-1)*<<0,1,1,1:1>>+(1)*<<0,0,2,1:1>>
++(-1)*<<0,1,0,2:1>>+(1)*<<3,1,0,0:2>>+(1)*<<0,1,2,1:2>>
++(1)*<<0,1,1,2:2>>+(-1)*<<3,1,1,0:3>>+(1)*<<3,0,2,0:3>>
++(-1)*<<3,1,0,1:3>>+(-1)*<<0,1,3,1:3>>+(-1)*<<0,1,2,2:3>>
+[2129] dpm_nf(U,S,1);
+0
+[2130] L=dpm_nf_and_quotient(U,S)$
+[2131] Q=L[2]$
+[2132] D=L[1]$
+[2133] D*U-(Q[1]*S[1]+Q[2]*S[2]);
+0
+@end example
+
+@table @t
+\JP @item $B;2>H(B
+\EG @item References
+@fref{dpm_sp},
+@fref{dp_ord}.
+@end table
+
+
 \JP @node dp_hm dp_ht dp_hc dp_rest,,, $B%0%l%V%J4pDl$K4X$9$kH!?t(B
 \EG @node dp_hm dp_ht dp_hc dp_rest,,, Functions for Groebner basis computation
 @subsection @code{dp_hm}, @code{dp_ht}, @code{dp_hc}, @code{dp_rest}
@@ -2898,6 +3740,88 @@ The next equations hold for a distributed polynomial @
 +(-490)*<<0,0,0>>
 @end example
 
+\JP @node dpm_hm dpm_ht dpm_hc dpm_hp dpm_rest,,, $B%0%l%V%J4pDl$K4X$9$kH!?t(B
+\EG @node dpm_hm dpm_ht dpm_hc dpm_hp dpm_rest,,, Functions for Groebner basis computation
+@subsection @code{dpm_hm}, @code{dpm_ht}, @code{dpm_hc}, @code{dpm_hp}, @code{dpm_rest}
+@findex dpm_hm
+@findex dpm_ht
+@findex dpm_hc
+@findex dpm_hp
+@findex dpm_rest
+
+@table @t
+@item dpm_hm(@var{dpoly})
+\JP :: $B2C72B?9`<0$NF,C19`<0$r<h$j=P$9(B. 
+\EG :: Gets the head monomial of a module polynomial.
+@item dpm_ht(@var{dpoly})
+\JP :: $B2C72B?9`<0$NF,9`$r<h$j=P$9(B. 
+\EG :: Gets the head term of a module polynomial.
+@item dpm_hc(@var{dpoly})
+\JP :: $B2C72B?9`<0$NF,78?t$r<h$j=P$9(B. 
+\EG :: Gets the head coefficient of a module polynomial.
+@item dpm_hp(@var{dpoly})
+\JP :: $B2C72B?9`<0$NF,0LCV$r<h$j=P$9(B. 
+\EG :: Gets the head position of a module polynomial.
+@item dpm_rest(@var{dpoly})
+\JP :: $B2C72B?9`<0$NF,C19`<0$r<h$j=|$$$?;D$j$rJV$9(B. 
+\EG :: Gets the remainder of a module polynomial where the head monomial is removed.
+@end table
+
+@table @var
+\BJP
+@item return
+@code{dp_hm()}, @code{dp_ht()}, @code{dp_rest()} : $B2C72B?9`<0(B,
+@code{dp_hc()} : $B?t$^$?$OB?9`<0(B
+@item dpoly
+$B2C72B?9`<0(B
+\E
+\BEG
+@item return
+@code{dpm_hm()}, @code{dpm_ht()}, @code{dpm_rest()} : module polynomial
+@code{dpm_hc()} : monomial
+@item dpoly
+distributed polynomial
+\E
+@end table
+
+@itemize @bullet
+\BJP
+@item
+$B$3$l$i$O(B, $B2C72B?9`<0$N3FItJ,$r<h$j=P$9$?$a$NH!?t$G$"$k(B. 
+@item
+@code{dpm_hc()} $B$O(B, @code{dpm_hm()} $B$N(B, $BI8=`4pDl$K4X$9$k78?t$G$"$kC19`<0$rJV$9(B.
+$B%9%+%i!<78?t$r<h$j=P$9$K$O(B, $B$5$i$K(B @code{dp_hc()} $B$r<B9T$9$k(B.
+@item
+@code{dpm_hp()} $B$O(B, $BF,2C72C19`<0$K4^$^$l$kI8=`4pDl$N%$%s%G%C%/%9$rJV$9(B.
+\E
+\BEG
+@item
+These are used to get various parts of a module polynomial.
+@item
+@code{dpm_hc()} returns the monomial that is the coefficient of @code{dpm_hm()} with respect to the
+standard base.
+For getting its scalar coefficient apply @code{dp_hc()}.
+@item
+@code{dpm_hp()} returns the index of the standard base conteind in the head module monomial.
+\E
+@end itemize
+
+@example
+[2126] dp_ord([1,0]); 
+[1,0]
+[2127] F=2*<<1,2,0:2>>-3*<<1,0,2:3>>+<<2,1,0:2>>;
+(1)*<<2,1,0:2>>+(2)*<<1,2,0:2>>+(-3)*<<1,0,2:3>>
+[2128] M=dpm_hm(F);
+(1)*<<2,1,0:2>>
+[2129] C=dpm_hc(F);
+(1)*<<2,1,0>>
+[2130] R=dpm_rest(F);
+(2)*<<1,2,0:2>>+(-3)*<<1,0,2:3>>
+[2131] dpm_hp(F);
+2
+@end example
+
+
 \JP @node dp_td dp_sugar,,, $B%0%l%V%J4pDl$K4X$9$kH!?t(B
 \EG @node dp_td dp_sugar,,, Functions for Groebner basis computation
 @subsection @code{dp_td}, @code{dp_sugar}
@@ -3059,6 +3983,43 @@ Used for finding candidate terms at reduction of polyn
 @fref{dp_red dp_red_mod}.
 @end table
 
+\JP @node dpm_redble,,, $B%0%l%V%J4pDl$K4X$9$kH!?t(B
+\EG @node dpm_redble,,, Functions for Groebner basis computation
+@subsection @code{dpm_redble}
+@findex dpm_redble
+
+@table @t
+@item dpm_redble(@var{dpoly1},@var{dpoly2})
+\JP :: $BF,9`$I$&$7$,@0=|2DG=$+$I$&$+D4$Y$k(B. 
+\EG :: Checks whether one head term is divisible by the other head term.
+@end table
+
+@table @var
+@item return
+\JP $B@0?t(B
+\EG integer
+@item dpoly1  dpoly2
+\JP $B2C72B?9`<0(B
+\EG module polynomial
+@end table
+
+@itemize @bullet
+\BJP
+@item
+@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
+0 $B$rJV$9(B. 
+@item
+$BB?9`<0$N4JLs$r9T$&:](B, $B$I$N9`$r4JLs$G$-$k$+$rC5$9$N$KMQ$$$k(B. 
+\E
+\BEG
+@item
+Returns 1 if the head term of @var{dpoly2} divides the head term of
+@var{dpoly1}; otherwise 0.
+@item
+Used for finding candidate terms at reduction of polynomials.
+\E
+@end itemize
+
 \JP @node dp_subd,,, $B%0%l%V%J4pDl$K4X$9$kH!?t(B
 \EG @node dp_subd,,, Functions for Groebner basis computation
 @subsection @code{dp_subd}
@@ -3426,6 +4387,46 @@ make the result integral.
 \EG @item References
 @fref{dp_mod dp_rat}.
 @end table
+
+\JP @node dpm_sp,,, $B%0%l%V%J4pDl$K4X$9$kH!?t(B
+\EG @node dmp_sp,,, Functions for Groebner basis computation
+@subsection @code{dpm_sp}
+@findex dpm_sp
+
+@table @t
+@item dpm_sp(@var{dpoly1},@var{dpoly2}[|coef=1])
+\JP :: S-$BB?9`<0$N7W;;(B
+\EG :: Computation of an S-polynomial
+@end table
+
+@table @var
+@item return
+\JP $B2C72B?9`<0$^$?$O%j%9%H(B
+\EG module polynomial or list
+@item dpoly1  dpoly2
+\JP $B2C72B?9`<0(B
+\EG module polynomial
+\JP $BJ,;6I=8=B?9`<0(B
+@end table
+
+@itemize @bullet
+\BJP
+@item
+@var{dpoly1}, @var{dpoly2} $B$N(B S-$BB?9`<0$r7W;;$9$k(B. 
+@item
+$B%*%W%7%g%s(B @var{coef=1} $B$,;XDj$5$l$F$$$k>l9g(B, @code{[S,t1,t2]} $B$J$k%j%9%H$rJV$9(B.
+$B$3$3$G(B, @code{t1}, @code{t2} $B$O(BS-$BB?9`<0$r:n$k:]$N78?tC19`<0$G(B @code{S=t1 dpoly1-t2 dpoly2}
+$B$rK~$?$9(B.
+\E
+\BEG
+@item
+This function computes the S-polynomial of @var{dpoly1} and @var{dpoly2}.
+@item
+If an option @var{coef=1} is specified, it returns a list @code{[S,t1,t2]}, 
+where @code{S} is the S-polynmial and @code{t1}, @code{t2} are monomials satisfying @code{S=t1 dpoly1-t2 dpoly2}.
+\E
+@end itemize
+
 \JP @node p_nf p_nf_mod p_true_nf p_true_nf_mod,,, $B%0%l%V%J4pDl$K4X$9$kH!?t(B
 \EG @node p_nf p_nf_mod p_true_nf p_true_nf_mod,,, Functions for Groebner basis computation
 @subsection @code{p_nf}, @code{p_nf_mod}, @code{p_true_nf}, @code{p_true_nf_mod}
@@ -3535,7 +4536,7 @@ refer to @code{dp_true_nf()} and @code{dp_true_nf_mod(
 @fref{dp_ptod},
 @fref{dp_dtop},
 @fref{dp_ord},
-@fref{dp_nf dp_nf_mod dp_true_nf dp_true_nf_mod}.
+@fref{dp_nf dp_nf_mod dp_true_nf dp_true_nf_mod dp_weyl_nf dp_weyl_nf_mod}.
 @end table
 
 \JP @node p_terms,,, $B%0%l%V%J4pDl$K4X$9$kH!?t(B