===================================================================
RCS file: /home/cvs/OpenXM/src/asir-doc/parts/groebner.texi,v
retrieving revision 1.20
retrieving revision 1.22
diff -u -p -r1.20 -r1.22
--- OpenXM/src/asir-doc/parts/groebner.texi	2017/08/31 04:54:36	1.20
+++ OpenXM/src/asir-doc/parts/groebner.texi	2019/03/29 04:54:25	1.22
@@ -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 1.21 2018/09/06 05:42:43 takayama Exp $
 \BJP
 @node $B%0%l%V%J4pDl$N7W;;(B,,, Top
 @chapter $B%0%l%V%J4pDl$N7W;;(B
@@ -1503,6 +1503,7 @@ 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::
@@ -1569,6 +1570,8 @@ 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 +1603,9 @@ 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}.
@@ -2488,6 +2494,67 @@ ndv_alloc=1477188
 \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