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Diff for /OpenXM/src/asir-doc/parts/groebner.texi between version 1.8 and 1.12

version 1.8, 2003/04/21 08:30:01 version 1.12, 2003/12/27 11:52:07
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 @comment $OpenXM: OpenXM/src/asir-doc/parts/groebner.texi,v 1.7 2003/04/21 03:07:32 noro Exp $  @comment $OpenXM: OpenXM/src/asir-doc/parts/groebner.texi,v 1.11 2003/04/28 06:43:10 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 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 1412  strategy $B$K$h$k7W;;(B, @code{hgr()} $B$O(B trace
Line 1412  strategy $B$K$h$k7W;;(B, @code{hgr()} $B$O(B trace
 @item  @item
 @code{dgr()} $B$GI=<($5$l$k;~4V$O(B, $B$3$NH!?t$,<B9T$5$l$F$$$k%W%m%;%9$G$N(B  @code{dgr()} $B$GI=<($5$l$k;~4V$O(B, $B$3$NH!?t$,<B9T$5$l$F$$$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.  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  \E
 \BEG  \BEG
 @item  @item
Line 1440  Therefore this function is useful to reduce the actual
Line 1448  Therefore this function is useful to reduce the actual
 The CPU time shown after an exection of @code{dgr()} indicates  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  that of the master process, and most of the time corresponds to the time
 for communication.  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  \E
 @end itemize  @end itemize
   
Line 3918  execute @code{dp_gr_print(2)} in advance.
Line 3933  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 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 generic_bfct,,, Functions for Groebner basis computation  \EG @node bfunction bfct generic_bfct ann ann0,,, Functions for Groebner basis computation
 @subsection @code{bfunction}, @code{generic_bfct}  @subsection @code{bfunction}, @code{bfct}, @code{generic_bfct}, @code{ann}, @code{ann0}
 @findex bfunction  @findex bfunction
   @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 generic_bfct(@var{plist},@var{vlist},@var{dvlist},@var{weight})  @itemx bfct(@var{f})
 \JP :: b $B4X?t$N7W;;(B  @itemx generic_bfct(@var{plist},@var{vlist},@var{dvlist},@var{weight})
 \EG :: Computes the global b function of a polynomial or an ideal  \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  @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 3946  execute @code{dp_gr_print(2)} in advance.
Line 3972  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})} $B$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
   $B0[$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$$$F$O(B, [Saito,Sturmfels,Takayama] $B$r8+$h(B.  @item $B>\:Y$K$D$$$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})} 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}.  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
   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 3984  x*y*dt+5*z^4*dt+dz,-x^4-z*y*x-y^4-z^5+t]$
Line 4025  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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