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Diff for /OpenXM/src/asir-doc/parts/groebner.texi between version 1.9 and 1.10

version 1.9, 2003/04/24 08:13:24 version 1.10, 2003/04/28 03:09:23
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 @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  \BJP
 @node $B%0%l%V%J4pDl$N7W;;(B,,, Top  @node $B%0%l%V%J4pDl$N7W;;(B,,, Top
 @chapter $B%0%l%V%J4pDl$N7W;;(B  @chapter $B%0%l%V%J4pDl$N7W;;(B
Line 1354  Computation of the global b function is implemented as
Line 1354  Computation of the global b function is implemented as
 * lex_hensel_gsl tolex_gsl tolex_gsl_d::  * lex_hensel_gsl tolex_gsl tolex_gsl_d::
 * primadec primedec::  * primadec primedec::
 * primedec_mod::  * primedec_mod::
 * bfunction bfct generic_bfct::  * bfunction bfct generic_bfct ann ann0::
 @end menu  @end menu
   
 \JP @node gr hgr gr_mod,,, $B%0%l%V%J4pDl$K4X$9$kH!?t(B  \JP @node gr hgr gr_mod,,, $B%0%l%V%J4pDl$K4X$9$kH!?t(B
Line 3918  execute @code{dp_gr_print(2)} in advance.
Line 3918  execute @code{dp_gr_print(2)} in advance.
 @fref{dp_gr_flags dp_gr_print}.  @fref{dp_gr_flags dp_gr_print}.
 @end table  @end table
   
 \JP @node bfunction bfct generic_bfct,,, $B%0%l%V%J4pDl$K4X$9$kH!?t(B  \JP @node bfunction bfct generic_bfct ann ann0,,, $B%0%l%V%J4pDl$K4X$9$kH!?t(B
 \EG @node bfunction bfct generic_bfct,,, Functions for Groebner basis computation  \EG @node bfunction bfct generic_bfct ann ann0,,, Functions for Groebner basis computation
 @subsection @code{bfunction}, @code{bfct}, @code{generic_bfct}  @subsection @code{bfunction}, @code{bfct}, @code{generic_bfct}, @code{ann}, @code{ann0}
 @findex bfunction  @findex bfunction
 @findex bfct  @findex bfct
 @findex generic_bfct  @findex generic_bfct
   @findex ann
   @findex ann0
   
 @table @t  @table @t
 @item bfunction(@var{f})  @item bfunction(@var{f})
 @item bfct(@var{f})  @itemx bfct(@var{f})
 @item generic_bfct(@var{plist},@var{vlist},@var{dvlist},@var{weight})  @itemx generic_bfct(@var{plist},@var{vlist},@var{dvlist},@var{weight})
 \JP :: b $B4X?t$N7W;;(B  \JP :: @var{b} $B4X?t$N7W;;(B
 \EG :: Computes the global b function of a polynomial or an ideal  \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  @end table
   
 @table @var  @table @var
 @item return  @item return
 @itemx f  \JP $BB?9`<0$^$?$O%j%9%H(B
   \EG polynomial or list
   @item f
 \JP $BB?9`<0(B  \JP $BB?9`<0(B
 \EG polynomial  \EG polynomial
 @item plist  @item plist
Line 3948  execute @code{dp_gr_print(2)} in advance.
Line 3957  execute @code{dp_gr_print(2)} in advance.
 @itemize @bullet  @itemize @bullet
 \BJP  \BJP
 @item @samp{bfct} $B$GDj5A$5$l$F$$$k(B.  @item @samp{bfct} $B$GDj5A$5$l$F$$$k(B.
 @item @code{bfunction(@var{f})}, @code{bfct(@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]}  $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  $B$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$$$b$N$G$"$k(B.  $BB?9`<0(B @code{b(s)} $B$NCf$G(B, $B<!?t$,:G$bDc$$$b$N$G$"$k(B.
 @item @code{generic_bfct(@var{f},@var{vlist},@var{dvlist},@var{weight})}  @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,  $B$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  @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.  $B$r=g$KJB$Y$k(B.
 @item @code{bfunction} $B$H(B @code{bfct} $B$G$OMQ$$$F$$$k%"%k%4%j%:%`$,(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$,9bB.2=$OF~NO$K$h$k(B.  $B0[$J$k(B. $B$I$A$i$,9bB.2=$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$$$F$O(B, [Saito,Sturmfels,Takayama] $B$r8+$h(B.  @item $B>\:Y$K$D$$$F$O(B, [Saito,Sturmfels,Takayama] $B$r8+$h(B.
 \E  \E
 \BEG  \BEG
 @item These functions are defined in @samp{bfct}.  @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}.  a polynomial @var{f}.
 @code{b(s)} is a polynomial of the minimal degree  @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  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.  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})}  @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}.  generated by @var{plist}, with respect to @var{weight}.
 @var{vlist} is the list of @code{x}-variables,  @var{vlist} is the list of @code{x}-variables,
 @var{vlist} is the list of corresponding @code{D}-variables.  @var{vlist} is the list of corresponding @code{D}-variables.
 @item @code{bfunction(@var{f})} and @code{bfct(@var{f})} implement  @item @code{bfunction(@var{f})} and @code{bfct(@var{f})} implement
 different algorithms and the efficiency depends on inputs.  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.  @item See [Saito,Sturmfels,Takayama] for the details.
 \E  \E
 @end itemize  @end itemize
Line 3990  x*y*dt+5*z^4*dt+dz,-x^4-z*y*x-y^4-z^5+t]$
Line 4010  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]);  [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  20000*s^10-70000*s^9+101750*s^8-79375*s^7+35768*s^6-9277*s^5
 +1278*s^4-72*s^3  +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  @end example
   
 @table @t  @table @t
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