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Diff for /OpenXM/src/asir-doc/parts/groebner.texi between version 1.6 and 1.10

version 1.6, 2003/04/20 09:55:18 version 1.10, 2003/04/28 03:09:23
Line 1 
Line 1 
 @comment $OpenXM: OpenXM/src/asir-doc/parts/groebner.texi,v 1.5 2003/04/20 08:01:25 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 449  If `on', various informations during a Groebner basis 
Line 449  If `on', various informations during a Groebner basis 
 displayed.  displayed.
 \E  \E
   
   @item PrintShort
   \JP on $B$G!"(BPrint $B$,(B off $B$N>l9g(B, $B%0%l%V%J4pDl7W;;$NESCf$N>pJs$rC;=L7A$GI=<($9$k(B.
   \BEG
   If `on' and Print is `off', short information during a Groebner basis computation is
   displayed.
   \E
   
 @item Stat  @item Stat
 \BJP  \BJP
 on $B$G(B @code{Print} $B$,(B off $B$J$i$P(B, @code{Print} $B$,(B on $B$N$H$-I=<($5(B  on $B$G(B @code{Print} $B$,(B off $B$J$i$P(B, @code{Print} $B$,(B on $B$N$H$-I=<($5(B
Line 471  is shown after every normal computation.  After comlet
Line 478  is shown after every normal computation.  After comlet
 computation the maximal value among the sums is shown.  computation the maximal value among the sums is shown.
 \E  \E
   
 @item Multiple  @item Content
   @itemx Multiple
 \BJP  \BJP
 0 $B$G$J$$@0?t$N;~(B, $BM-M}?t>e$N@55,7A7W;;$K$*$$$F(B, $B78?t$N%S%C%HD9$NOB$,(B  0 $B$G$J$$M-M}?t$N;~(B, $BM-M}?t>e$N@55,7A7W;;$K$*$$$F(B, $B78?t$N%S%C%HD9$NOB$,(B
 @code{Multiple} $BG\$K$J$k$4$H$K78?tA4BN$N(B GCD $B$,7W;;$5$l(B, $B$=$N(B GCD $B$G(B  @code{Content} $BG\$K$J$k$4$H$K78?tA4BN$N(B GCD $B$,7W;;$5$l(B, $B$=$N(B GCD $B$G(B
 $B3d$C$?B?9`<0$r4JLs$9$k(B. @code{Multiple} $B$,(B 1 $B$J$i$P(B, $B4JLs$9$k$4$H$K(B  $B3d$C$?B?9`<0$r4JLs$9$k(B. @code{Content} $B$,(B 1 $B$J$i$P(B, $B4JLs$9$k$4$H$K(B
 GCD $B7W;;$,9T$o$l0lHL$K$O8zN($,0-$/$J$k$,(B, @code{Multiple} $B$r(B 2 $BDxEY(B  GCD $B7W;;$,9T$o$l0lHL$K$O8zN($,0-$/$J$k$,(B, @code{Content} $B$r(B 2 $BDxEY(B
 $B$H$9$k$H(B, $B5pBg$J@0?t$,78?t$K8=$l$k>l9g(B, $B8zN($,NI$/$J$k>l9g$,$"$k(B.  $B$H$9$k$H(B, $B5pBg$J@0?t$,78?t$K8=$l$k>l9g(B, $B8zN($,NI$/$J$k>l9g$,$"$k(B.
   backward compatibility $B$N$?$a!"(B@code{Multiple} $B$G@0?tCM$r;XDj$G$-$k(B.
 \E  \E
 \BEG  \BEG
 If a non-zero integer, in a normal form computation  If a non-zero rational number, in a normal form computation
 over the rationals, the integer content of the polynomial being  over the rationals, the integer content of the polynomial being
 reduced is removed when its magnitude becomes @code{Multiple} times  reduced is removed when its magnitude becomes @code{Content} times
 larger than a registered value, which is set to the magnitude of the  larger than a registered value, which is set to the magnitude of the
 input polynomial. After each content removal the registered value is  input polynomial. After each content removal the registered value is
 set to the magnitude of the resulting polynomial. @code{Multiple} is  set to the magnitude of the resulting polynomial. @code{Content} is
 equal to 1, the simiplification is done after every normal form computation.  equal to 1, the simiplification is done after every normal form computation.
 It is empirically known that it is often efficient to set @code{Multiple} to 2  It is empirically known that it is often efficient to set @code{Content} to 2
 for the case where large integers appear during the computation.  for the case where large integers appear during the computation.
   An integer value can be set by the keyword @code{Multiple} for
   backward compatibility.
 \E  \E
   
 @item Demand  @item Demand
Line 1343  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 1679  processes.
Line 1690  processes.
 @item lex_hensel_gsl(@var{plist},@var{vlist1},@var{order},@var{vlist2},@var{homo})  @item lex_hensel_gsl(@var{plist},@var{vlist1},@var{order},@var{vlist2},@var{homo})
 \JP :: GSL $B7A<0$N%$%G%"%k4pDl$N7W;;(B  \JP :: GSL $B7A<0$N%$%G%"%k4pDl$N7W;;(B
 \EG ::Computation of an GSL form ideal basis  \EG ::Computation of an GSL form ideal basis
 @item tolex_gsl(@var{plist},@var{vlist1},@var{order},@var{vlist2},@var{homo})  @item tolex_gsl(@var{plist},@var{vlist1},@var{order},@var{vlist2})
 @itemx tolex_gsl_d(@var{plist},@var{vlist1},@var{order},@var{vlist2},@var{homo},@var{procs})  @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  \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  \EG :: Computation of an GSL form ideal basis stating from a Groebner basis
 @end table  @end table
Line 2150  except for lack of the argument for controlling homoge
Line 2161  except for lack of the argument for controlling homoge
   
 @table @t  @table @t
 @item dp_gr_flags([@var{list}])  @item dp_gr_flags([@var{list}])
 @itemx dp_gr_print([@var{0|1}])  @itemx dp_gr_print([@var{i}])
 \JP :: $B7W;;$*$h$SI=<(MQ%Q%i%a%?$N@_Dj(B, $B;2>H(B  \JP :: $B7W;;$*$h$SI=<(MQ%Q%i%a%?$N@_Dj(B, $B;2>H(B
 \BEG :: Set and show various parameters for cotrolling computations  \BEG :: Set and show various parameters for cotrolling computations
 and showing informations.  and showing informations.
Line 2164  and showing informations.
Line 2175  and showing informations.
 @item list  @item list
 \JP $B%j%9%H(B  \JP $B%j%9%H(B
 \EG list  \EG list
   @item i
   \JP $B@0?t(B
   \EG integer
 @end table  @end table
   
 @itemize @bullet  @itemize @bullet
Line 2177  and showing informations.
Line 2191  and showing informations.
 $B0z?t$O(B, @code{["Print",1,"NoSugar",1,...]} $B$J$k7A$N%j%9%H$G(B, $B:8$+$i=g$K(B  $B0z?t$O(B, @code{["Print",1,"NoSugar",1,...]} $B$J$k7A$N%j%9%H$G(B, $B:8$+$i=g$K(B
 $B@_Dj$5$l$k(B. $B%Q%i%a%?L>$OJ8;zNs$GM?$($kI,MW$,$"$k(B.  $B@_Dj$5$l$k(B. $B%Q%i%a%?L>$OJ8;zNs$GM?$($kI,MW$,$"$k(B.
 @item  @item
 @code{dp_gr_print()} $B$O(B, $BFC$K%Q%i%a%?(B @code{Print} $B$NCM$rD>@\@_Dj(B, $B;2>H(B  @code{dp_gr_print()} $B$O(B, $BFC$K%Q%i%a%?(B @code{Print}, @code{PrintShort} $B$NCM$rD>@\@_Dj(B, $B;2>H(B
 $B$G$-$k(B. $B$3$l$O(B, @code{dp_gr_main()} $B$J$I$r%5%V%k!<%A%s$H$7$FMQ$$$k%f!<%6(B  $B$G$-$k(B. $B@_Dj$5$l$kCM$O<!$NDL$j$G$"$k!#(B
 $BH!?t$K$*$$$F(B, @code{Print} $B$NCM$r8+$F(B, $B$=$N%5%V%k!<%A%s$,Cf4V>pJs$NI=<((B  @table @var
   @item i=0
   @code{Print=0}, @code{PrintShort=0}
   @item i=1
   @code{Print=1}, @code{PrintShort=0}
   @item i=2
   @code{Print=0}, @code{PrintShort=1}
   @end table
   $B$3$l$O(B, @code{dp_gr_main()} $B$J$I$r%5%V%k!<%A%s$H$7$FMQ$$$k%f!<%6(B
   $BH!?t$K$*$$$F(B, $B$=$N%5%V%k!<%A%s$,Cf4V>pJs$NI=<((B
 $B$r9T$&:]$K(B, $B?WB.$K%U%i%0$r8+$k$3$H$,$G$-$k$h$&$KMQ0U$5$l$F$$$k(B.  $B$r9T$&:]$K(B, $B?WB.$K%U%i%0$r8+$k$3$H$,$G$-$k$h$&$KMQ0U$5$l$F$$$k(B.
 \E  \E
 \BEG  \BEG
Line 2194  Arguments must be specified as a list such as
Line 2217  Arguments must be specified as a list such as
 strings.  strings.
 @item  @item
 @code{dp_gr_print()} is used to set and show the value of a parameter  @code{dp_gr_print()} is used to set and show the value of a parameter
 @code{Print}. This functions is prepared to get quickly the value of  @code{Print} and @code{PrintShort}.
 @code{Print} when a user defined function calling @code{dp_gr_main()} etc.  @table @var
   @item i=0
   @code{Print=0}, @code{PrintShort=0}
   @item i=1
   @code{Print=1}, @code{PrintShort=0}
   @item i=2
   @code{Print=0}, @code{PrintShort=1}
   @end table
   This functions is prepared to get quickly the value
   when a user defined function calling @code{dp_gr_main()} etc.
 uses the value as a flag for showing intermediate informations.  uses the value as a flag for showing intermediate informations.
 \E  \E
 @end itemize  @end itemize
Line 3831  if an input ideal is not radical.
Line 3863  if an input ideal is not radical.
 $BItJ,$r7W;;$9$k$3$H$K$h$k(B early termination $B$r9T$&(B. $B0lHL$K(B,  $BItJ,$r7W;;$9$k$3$H$K$h$k(B early termination $B$r9T$&(B. $B0lHL$K(B,
 $B%$%G%"%k$N<!85$,9b$$>l9g$KM-8z$@$,(B, 0 $B<!85$N>l9g$J$I(B, $B<!85$,>.$5$$(B  $B%$%G%"%k$N<!85$,9b$$>l9g$KM-8z$@$,(B, 0 $B<!85$N>l9g$J$I(B, $B<!85$,>.$5$$(B
 $B>l9g$K$O(B overhead $B$,Bg$-$$>l9g$,$"$k(B.  $B>l9g$K$O(B overhead $B$,Bg$-$$>l9g$,$"$k(B.
   @item
   $B7W;;ESCf$GFbIt>pJs$r8+$?$$>l9g$K$O!"(B
   $BA0$b$C$F(B @code{dp_gr_print(2)} $B$r<B9T$7$F$*$1$P$h$$(B.
 \E  \E
 \BEG  \BEG
 @item  @item
Line 3855  is tried by computing the intersection of obtained com
Line 3890  is tried by computing the intersection of obtained com
 incrementally. In general, this strategy is useful when the krull  incrementally. In general, this strategy is useful when the krull
 dimension of the ideal is high, but it may add some overhead  dimension of the ideal is high, but it may add some overhead
 if the dimension is small.  if the dimension is small.
   @item
   If you want to see internal information during the computation,
   execute @code{dp_gr_print(2)} in advance.
 \E  \E
 @end itemize  @end itemize
   
Line 3876  if the dimension is small.
Line 3914  if the dimension is small.
 @fref{modfctr},  @fref{modfctr},
 @fref{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},  @fref{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},
 \JP @fref{$B9`=g=x$N@_Dj(B}.  \JP @fref{$B9`=g=x$N@_Dj(B}.
 \EG @fref{Setting term orderings}.  \EG @fref{Setting term orderings},
   @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 3907  if the dimension is small.
Line 3957  if the dimension is small.
 @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 $B>\:Y$K$D$$$F$O(B, [SST] $B$r8+$h(B.  @item @code{bfunction} $B$H(B @code{bfct} $B$G$OMQ$$$F$$$k%"%k%4%j%:%`$,(B
   $B0[$J$k(B. $B$I$A$i$,9bB.2=$OF~NO$K$h$k(B.
   @item @code{ann(@var{f})} $B$O(B, @code{@var{f}^s} $B$N(B annihilator ideal
   $B$N@8@.7O$rJV$9(B. @code{ann(@var{f})} $B$O(B, @code{[@var{a},@var{list}]}
   $B$J$k%j%9%H$rJV$9(B. $B$3$3$G(B, @var{a} $B$O(B @var{f} $B$N(B @var{b} $B4X?t$N:G>.@0?t:,(B,
   @var{list} $B$O(B @code{ann(@var{f})} $B$N7k2L$N(B @code{s}$ $B$K(B, @var{a} $B$r(B
   $BBeF~$7$?$b$N$G$"$k(B.
   @item $B>\:Y$K$D$$$F$O(B, [Saito,Sturmfels,Takayama] $B$r8+$h(B.
 \E  \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 See [SST] for the details.  @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  \E
 @end itemize  @end itemize
   
Line 3945  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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