===================================================================
RCS file: /home/cvs/OpenXM/src/asir-doc/parts/groebner.texi,v
retrieving revision 1.7
retrieving revision 1.12
diff -u -p -r1.7 -r1.12
--- OpenXM/src/asir-doc/parts/groebner.texi	2003/04/21 03:07:32	1.7
+++ OpenXM/src/asir-doc/parts/groebner.texi	2003/12/27 11:52:07	1.12
@@ -1,4 +1,4 @@
-@comment $OpenXM: OpenXM/src/asir-doc/parts/groebner.texi,v 1.6 2003/04/20 09:55:18 noro Exp $
+@comment $OpenXM: OpenXM/src/asir-doc/parts/groebner.texi,v 1.11 2003/04/28 06:43:10 noro Exp $
 \BJP
 @node $B%0%l%V%J4pDl$N7W;;(B,,, Top
 @chapter $B%0%l%V%J4pDl$N7W;;(B
@@ -1354,7 +1354,7 @@ Computation of the global b function is implemented as
 * lex_hensel_gsl tolex_gsl tolex_gsl_d::
 * primadec primedec::
 * primedec_mod::
-* bfunction generic_bfct::
+* bfunction bfct generic_bfct ann ann0::
 @end menu
 
 \JP @node gr hgr gr_mod,,, $B%0%l%V%J4pDl$K4X$9$kH!?t(B
@@ -1412,6 +1412,14 @@ CPU $B;~4V$G$"$j(B, $B$3$NH!?t$N>l9g$O$[$H$s$IDL?.$
 @item
 @code{dgr()} $B$GI=<($5$l$k;~4V$O(B, $B$3$NH!?t$,<B9T$5$l$F$$$k%W%m%;%9$G$N(B
 CPU $B;~4V$G$"$j(B, $B$3$NH!?t$N>l9g$O$[$H$s$IDL?.$N$?$a$N;~4V$G$"$k(B. 
+@item
+$BB?9`<0%j%9%H(B @var{plist} $B$NMWAG$,J,;6I=8=B?9`<0$N>l9g$O(B
+$B7k2L$bJ,;6I=8=B?9`<0$N%j%9%H$G$"$k(B.
+$B$3$N>l9g(B, $B0z?t$NJ,;6B?9`<0$OM?$($i$l$?=g=x$K=>$$(B @code{dp_sort} $B$G(B
+$B%=!<%H$5$l$F$+$i7W;;$5$l$k(B.
+$BB?9`<0%j%9%H$NMWAG$,J,;6I=8=B?9`<0$N>l9g$b(B
+$BJQ?t$N?tJ,$NITDj85$N%j%9%H$r(B @var{vlist} $B0z?t$H$7$FM?$($J$$$H$$$1$J$$(B
+($B%@%_!<(B).
 \E
 \BEG
 @item 
@@ -1440,6 +1448,13 @@ for communication.
 The CPU time shown after an exection of @code{dgr()} indicates
 that of the master process, and most of the time corresponds to the time
 for communication.
+@item
+When the elements of @var{plist} are distributed polynomials,
+the result is also a list of distributed polynomials.
+In this case, firstly  the elements of @var{plist} is sorted by @code{dp_sort}
+and the Grobner basis computation is started.
+Variables must be given in @var{vlist} even in this case
+(these variables are dummy).
 \E
 @end itemize
 
@@ -1690,8 +1705,8 @@ processes.
 @item lex_hensel_gsl(@var{plist},@var{vlist1},@var{order},@var{vlist2},@var{homo})
 \JP :: GSL $B7A<0$N%$%G%"%k4pDl$N7W;;(B
 \EG ::Computation of an GSL form ideal basis
-@item tolex_gsl(@var{plist},@var{vlist1},@var{order},@var{vlist2},@var{homo})
-@itemx tolex_gsl_d(@var{plist},@var{vlist1},@var{order},@var{vlist2},@var{homo},@var{procs})
+@item tolex_gsl(@var{plist},@var{vlist1},@var{order},@var{vlist2})
+@itemx tolex_gsl_d(@var{plist},@var{vlist1},@var{order},@var{vlist2},@var{procs})
 \JP :: $B%0%l%V%J4pDl$rF~NO$H$9$k(B, GSL $B7A<0$N%$%G%"%k4pDl$N7W;;(B
 \EG :: Computation of an GSL form ideal basis stating from a Groebner basis
 @end table
@@ -3918,21 +3933,32 @@ execute @code{dp_gr_print(2)} in advance.
 @fref{dp_gr_flags dp_gr_print}.
 @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}
+\JP @node bfunction bfct generic_bfct ann ann0,,, $B%0%l%V%J4pDl$K4X$9$kH!?t(B
+\EG @node bfunction bfct generic_bfct ann ann0,,, Functions for Groebner basis computation
+@subsection @code{bfunction}, @code{bfct}, @code{generic_bfct}, @code{ann}, @code{ann0}
 @findex bfunction
+@findex bfct
 @findex generic_bfct
+@findex ann
+@findex ann0
 
 @table @t
 @item bfunction(@var{f})
-@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
+@itemx bfct(@var{f})
+@itemx generic_bfct(@var{plist},@var{vlist},@var{dvlist},@var{weight})
+\JP :: @var{b} $B4X?t$N7W;;(B
+\EG :: Computes the global @var{b} function of a polynomial or an ideal
+@item ann(@var{f})
+@itemx ann0(@var{f})
+\JP :: $BB?9`<0$N%Y%-$N(B annihilator $B$N7W;;(B
+\EG :: Computes the annihilator of a power of polynomial
 @end table
+
 @table @var
 @item return
-@itemx f
+\JP $BB?9`<0$^$?$O%j%9%H(B
+\EG polynomial or list
+@item f
 \JP $BB?9`<0(B
 \EG polynomial
 @item plist
@@ -3946,29 +3972,44 @@ execute @code{dp_gr_print(2)} in advance.
 @itemize @bullet
 \BJP
 @item @samp{bfct} $B$GDj5A$5$l$F$$$k(B. 
-@item @code{bfunction(@var{f})} $B$OB?9`<0(B @var{f} $B$N(B global b $B4X?t(B @code{b(s)} $B$r(B
+@item @code{bfunction(@var{f})}, @code{bfct(@var{f})} $B$OB?9`<0(B @var{f} $B$N(B global @var{b} $B4X?t(B @code{b(s)} $B$r(B
 $B7W;;$9$k(B. @code{b(s)} $B$O(B, Weyl $BBe?t(B @code{D} $B>e$N0lJQ?tB?9`<04D(B @code{D[s]}
 $B$N85(B @code{P(x,s)} $B$,B8:_$7$F(B, @code{P(x,s)f^(s+1)=b(s)f^s} $B$rK~$?$9$h$&$J(B
 $BB?9`<0(B @code{b(s)} $B$NCf$G(B, $B<!?t$,:G$bDc$$$b$N$G$"$k(B. 
 @item @code{generic_bfct(@var{f},@var{vlist},@var{dvlist},@var{weight})}
 $B$O(B, @var{plist} $B$G@8@.$5$l$k(B @code{D} $B$N:8%$%G%"%k(B @code{I} $B$N(B, 
-$B%&%'%$%H(B @var{weight} $B$K4X$9$k(B global b $B4X?t$r7W;;$9$k(B. 
+$B%&%'%$%H(B @var{weight} $B$K4X$9$k(B global @var{b} $B4X?t$r7W;;$9$k(B. 
 @var{vlist} $B$O(B @code{x}-$BJQ?t(B, @var{vlist} $B$OBP1~$9$k(B @code{D}-$BJQ?t(B
 $B$r=g$KJB$Y$k(B. 
+@item @code{bfunction} $B$H(B @code{bfct} $B$G$OMQ$$$F$$$k%"%k%4%j%:%`$,(B
+$B0[$J$k(B. $B$I$A$i$,9bB.$+$OF~NO$K$h$k(B.
+@item @code{ann(@var{f})} $B$O(B, @code{@var{f}^s} $B$N(B annihilator ideal
+$B$N@8@.7O$rJV$9(B. @code{ann(@var{f})} $B$O(B, @code{[@var{a},@var{list}]}
+$B$J$k%j%9%H$rJV$9(B. $B$3$3$G(B, @var{a} $B$O(B @var{f} $B$N(B @var{b} $B4X?t$N:G>.@0?t:,(B,
+@var{list} $B$O(B @code{ann(@var{f})} $B$N7k2L$N(B @code{s}$ $B$K(B, @var{a} $B$r(B
+$BBeF~$7$?$b$N$G$"$k(B.
 @item $B>\:Y$K$D$$$F$O(B, [Saito,Sturmfels,Takayama] $B$r8+$h(B. 
 \E
 \BEG
 @item These functions are defined in @samp{bfct}.
-@item @code{bfunction(@var{f})} computes the global b-function @code{b(s)} of
+@item @code{bfunction(@var{f})} and @code{bfct(@var{f})} compute the global @var{b}-function @code{b(s)} of
 a polynomial @var{f}.
 @code{b(s)} is a polynomial of the minimal degree 
 such that there exists @code{P(x,s)} in D[s], which is a polynomial
 ring over Weyl algebra @code{D}, and @code{P(x,s)f^(s+1)=b(s)f^s} holds.
 @item @code{generic_bfct(@var{f},@var{vlist},@var{dvlist},@var{weight})}
-computes the global b-function of a left ideal @code{I} in @code{D}
+computes the global @var{b}-function of a left ideal @code{I} in @code{D}
 generated by @var{plist}, with respect to @var{weight}.
 @var{vlist} is the list of @code{x}-variables, 
 @var{vlist} is the list of corresponding @code{D}-variables.
+@item @code{bfunction(@var{f})} and @code{bfct(@var{f})} implement
+different algorithms and the efficiency depends on inputs.
+@item @code{ann(@var{f})} returns the generator set of the annihilator
+ideal of @code{@var{f}^s}.
+@code{ann(@var{f})} returns a list @code{[@var{a},@var{list}]},
+where @var{a} is the minimal integral root of the global @var{b}-function
+of @var{f}, and @var{list} is a list of polynomials obtained by
+substituting @code{s} in @code{ann(@var{f})} with @var{a}.
 @item See [Saito,Sturmfels,Takayama] for the details.
 \E
 @end itemize
@@ -3984,6 +4025,11 @@ +1278*s^4-72*s^3
 [219] generic_bfct(F,[t,z,y,x],[dt,dz,dy,dx],[1,0,0,0]);
 20000*s^10-70000*s^9+101750*s^8-79375*s^7+35768*s^6-9277*s^5
 +1278*s^4-72*s^3
+[220] P=x^3-y^2$
+[221] ann(P);
+[2*dy*x+3*dx*y^2,-3*dx*x-2*dy*y+6*s]
+[222] ann0(P);
+[-1,[2*dy*x+3*dx*y^2,-3*dx*x-2*dy*y-6]]
 @end example
 
 @table @t