===================================================================
RCS file: /home/cvs/OpenXM/src/asir-doc/parts/groebner.texi,v
retrieving revision 1.9
retrieving revision 1.10
diff -u -p -r1.9 -r1.10
--- OpenXM/src/asir-doc/parts/groebner.texi	2003/04/24 08:13:24	1.9
+++ OpenXM/src/asir-doc/parts/groebner.texi	2003/04/28 03:09:23	1.10
@@ -1,4 +1,4 @@
-@comment $OpenXM: OpenXM/src/asir-doc/parts/groebner.texi,v 1.8 2003/04/21 08:30:01 noro Exp $
+@comment $OpenXM: OpenXM/src/asir-doc/parts/groebner.texi,v 1.9 2003/04/24 08:13:24 noro Exp $
 \BJP
 @node グレブナ基底の計算,,, Top
 @chapter グレブナ基底の計算
@@ -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 bfct generic_bfct::
+* bfunction bfct generic_bfct ann ann0::
 @end menu
 
 \JP @node gr hgr gr_mod,,, グレブナ基底に関する函数
@@ -3918,23 +3918,32 @@ execute @code{dp_gr_print(2)} in advance.
 @fref{dp_gr_flags dp_gr_print}.
 @end table
 
-\JP @node bfunction bfct generic_bfct,,, グレブナ基底に関する函数
-\EG @node bfunction bfct generic_bfct,,, Functions for Groebner basis computation
-@subsection @code{bfunction}, @code{bfct}, @code{generic_bfct}
+\JP @node bfunction bfct generic_bfct ann ann0,,, グレブナ基底に関する函数
+\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 bfct(@var{f})
-@item generic_bfct(@var{plist},@var{vlist},@var{dvlist},@var{weight})
-\JP :: b 関数の計算
-\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} 関数の計算
+\EG :: Computes the global @var{b} function of a polynomial or an ideal
+@item ann(@var{f})
+@itemx ann0(@var{f})
+\JP :: 多項式のベキの annihilator の計算
+\EG :: Computes the annihilator of a power of polynomial
 @end table
+
 @table @var
 @item return
-@itemx f
+\JP 多項式またはリスト
+\EG polynomial or list
+@item f
 \JP 多項式
 \EG polynomial
 @item plist
@@ -3948,33 +3957,44 @@ execute @code{dp_gr_print(2)} in advance.
 @itemize @bullet
 \BJP
 @item @samp{bfct} で定義されている. 
-@item @code{bfunction(@var{f})}, @code{bfct(@var{f})} は多項式 @var{f} の global b 関数 @code{b(s)} を
+@item @code{bfunction(@var{f})}, @code{bfct(@var{f})} は多項式 @var{f} の global @var{b} 関数 @code{b(s)} を
 計算する. @code{b(s)} は, Weyl 代数 @code{D} 上の一変数多項式環 @code{D[s]}
 の元 @code{P(x,s)} が存在して, @code{P(x,s)f^(s+1)=b(s)f^s} を満たすような
 多項式 @code{b(s)} の中で, 次数が最も低いものである. 
 @item @code{generic_bfct(@var{f},@var{vlist},@var{dvlist},@var{weight})}
 は, @var{plist} で生成される @code{D} の左イデアル @code{I} の, 
-ウェイト @var{weight} に関する global b 関数を計算する. 
+ウェイト @var{weight} に関する global @var{b} 関数を計算する. 
 @var{vlist} は @code{x}-変数, @var{vlist} は対応する @code{D}-変数
 を順に並べる. 
 @item @code{bfunction} と @code{bfct} では用いているアルゴリズムが
 異なる. どちらが高速化は入力による.
+@item @code{ann(@var{f})} は, @code{@var{f}^s} の annihilator ideal
+の生成系を返す. @code{ann(@var{f})} は, @code{[@var{a},@var{list}]}
+なるリストを返す. ここで, @var{a} は @var{f} の @var{b} 関数の最小整数根,
+@var{list} は @code{ann(@var{f})} の結果の @code{s}$ に, @var{a} を
+代入したものである.
 @item 詳細については, [Saito,Sturmfels,Takayama] を見よ. 
 \E
 \BEG
 @item These functions are defined in @samp{bfct}.
-@item @code{bfunction(@var{f})} and @code{bfct(@var{f})} compute 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
@@ -3990,6 +4010,11 @@ 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
+[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