===================================================================
RCS file: /home/cvs/OpenXM/src/asir-doc/parts/groebner.texi,v
retrieving revision 1.20
retrieving revision 1.24
diff -u -p -r1.20 -r1.24
--- OpenXM/src/asir-doc/parts/groebner.texi	2017/08/31 04:54:36	1.20
+++ OpenXM/src/asir-doc/parts/groebner.texi	2020/09/01 09:25:32	1.24
@@ -1,4 +1,4 @@
-@comment $OpenXM: OpenXM/src/asir-doc/parts/groebner.texi,v 1.19 2016/08/29 04:56:58 noro Exp $
+@comment $OpenXM: OpenXM/src/asir-doc/parts/groebner.texi,v 1.23 2019/09/13 09:31:00 noro Exp $
 \BJP
 @node $B%0%l%V%J4pDl$N7W;;(B,,, Top
 @chapter $B%0%l%V%J4pDl$N7W;;(B
@@ -19,6 +19,7 @@
 * $BM-M}<0$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
@@ -31,6 +32,7 @@
 * 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
@@ -1486,6 +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
@@ -1503,6 +1556,7 @@ Computation of the global b function is implemented as
 * 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::
@@ -1513,6 +1567,13 @@ Computation of the global b function is implemented as
 * dp_ptozp dp_prim::
 * 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::
+* dpm_sp::
+* dpm_redble::
+* dpm_nf dpm_nf_and_quotient::
+* dpm_dtol::
+* dpm_ltod::
+* dpm_dptodpm::
 * dp_td dp_sugar::
 * dp_lcm::
 * dp_redble::
@@ -1569,6 +1630,8 @@ 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. 
@@ -1600,6 +1663,9 @@ 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}.
@@ -2354,12 +2420,12 @@ except for lack of the argument for controlling homoge
 @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})
-@itemx 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})
+@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
@@ -2418,6 +2484,16 @@ Buchberger $B%"%k%4%j%:%`$r<B9T$9$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
 
@@ -2461,6 +2537,17 @@ Functions except for F4 related ones can handle ration
 @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
 
@@ -2489,6 +2576,67 @@ ndv_alloc=1477188
 \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}
@@ -2619,6 +2767,12 @@ 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
@@ -2646,6 +2800,12 @@ 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
 
@@ -2797,6 +2957,171 @@ 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}
@@ -3227,6 +3552,126 @@ 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}
@@ -3301,6 +3746,88 @@ 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}
@@ -3462,6 +3989,43 @@ 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$$$k(B. 
+\E
+\BEG
+@item
+Returns 1 if the head term of @var{dpoly2} divides the head term of
+@var{dpoly1}; otherwise 0.
+@item
+Used for finding candidate terms at reduction of polynomials.
+\E
+@end itemize
+
 \JP @node dp_subd,,, $B%0%l%V%J4pDl$K4X$9$kH!?t(B
 \EG @node dp_subd,,, Functions for Groebner basis computation
 @subsection @code{dp_subd}
@@ -3829,6 +4393,46 @@ make the result integral.
 \EG @item References
 @fref{dp_mod dp_rat}.
 @end table
+
+\JP @node dpm_sp,,, $B%0%l%V%J4pDl$K4X$9$kH!?t(B
+\EG @node dmp_sp,,, Functions for Groebner basis computation
+@subsection @code{dpm_sp}
+@findex dpm_sp
+
+@table @t
+@item dpm_sp(@var{dpoly1},@var{dpoly2}[|coef=1])
+\JP :: S-$BB?9`<0$N7W;;(B
+\EG :: Computation of an S-polynomial
+@end table
+
+@table @var
+@item return
+\JP $B2C72B?9`<0$^$?$O%j%9%H(B
+\EG module polynomial or list
+@item dpoly1  dpoly2
+\JP $B2C72B?9`<0(B
+\EG module polynomial
+\JP $BJ,;6I=8=B?9`<0(B
+@end table
+
+@itemize @bullet
+\BJP
+@item
+@var{dpoly1}, @var{dpoly2} $B$N(B S-$BB?9`<0$r7W;;$9$k(B. 
+@item
+$B%*%W%7%g%s(B @var{coef=1} $B$,;XDj$5$l$F$$$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}