[BACK]Return to groebner.texi CVS log [TXT][DIR] Up to [local] / OpenXM / src / asir-doc / parts

Diff for /OpenXM/src/asir-doc/parts/groebner.texi between version 1.5 and 1.10

version 1.5, 2003/04/20 08:01:25 version 1.10, 2003/04/28 03:09:23
Line 1 
Line 1 
 @comment $OpenXM: OpenXM/src/asir-doc/parts/groebner.texi,v 1.4 2003/04/19 15:44:56 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 1203  Refer to the sections for each functions.
Line 1214  Refer to the sections for each functions.
 \E  \E
   
 \BJP  \BJP
   @node Weyl $BBe?t(B,,, $B%0%l%V%J4pDl$N7W;;(B
   @section Weyl $BBe?t(B
   \E
   \BEG
   @node Weyl algebra,,, Groebner basis computation
   @section Weyl algebra
   \E
   
   @noindent
   
   \BJP
   $B$3$l$^$G$O(B, $BDL>o$N2D49$JB?9`<04D$K$*$1$k%0%l%V%J4pDl7W;;$K$D$$$F(B
   $B=R$Y$F$-$?$,(B, $B%0%l%V%J4pDl$NM}O@$O(B, $B$"$k>r7o$rK~$?$9Hs2D49$J(B
   $B4D$K$b3HD%$G$-$k(B. $B$3$N$h$&$J4D$NCf$G(B, $B1~MQ>e$b=EMW$J(B,
   Weyl $BBe?t(B, $B$9$J$o$AB?9`<04D>e$NHyJ,:nMQAG4D$N1i;;$*$h$S(B
   $B%0%l%V%J4pDl7W;;$,(B Risa/Asir $B$K<BAu$5$l$F$$$k(B.
   
   $BBN(B @code{K} $B>e$N(B @code{n} $B<!85(B Weyl $BBe?t(B
   @code{D=K<x1,@dots{},xn,D1,@dots{},Dn>} $B$O(B
   \E
   
   \BEG
   So far we have explained Groebner basis computation in
   commutative polynomial rings. However Groebner basis can be
   considered in more general non-commutative rings.
   Weyl algebra is one of such rings and
   Risa/Asir implements fundamental operations
   in Weyl algebra and Groebner basis computation in Weyl algebra.
   
   The @code{n} dimensional Weyl algebra over a field @code{K},
   @code{D=K<x1,@dots{},xn,D1,@dots{},Dn>} is a non-commutative
   algebra which has the following fundamental relations:
   \E
   
   @code{xi*xj-xj*xi=0}, @code{Di*Dj-Dj*Di=0}, @code{Di*xj-xj*Di=0} (@code{i!=j}),
   @code{Di*xi-xi*Di=1}
   
   \BJP
   $B$H$$$&4pK\4X78$r;}$D4D$G$"$k(B. @code{D} $B$O(B $BB?9`<04D(B @code{K[x1,@dots{},xn]} $B$r78?t(B
   $B$H$9$kHyJ,:nMQAG4D$G(B,  @code{Di} $B$O(B @code{xi} $B$K$h$kHyJ,$rI=$9(B. $B8r494X78$K$h$j(B,
   @code{D} $B$N85$O(B, @code{x1^i1*@dots{}*xn^in*D1^j1*@dots{}*Dn^jn} $B$J$kC19`(B
   $B<0$N(B @code{K} $B@~7A7k9g$H$7$F=q$-I=$9$3$H$,$G$-$k(B.
   Risa/Asir $B$K$*$$$F$O(B, $B$3$NC19`<0$r(B, $B2D49$JB?9`<0$HF1MM$K(B
   @code{<<i1,@dots{},in,j1,@dots{},jn>>} $B$GI=$9(B. $B$9$J$o$A(B, @code{D} $B$N85$b(B
   $BJ,;6I=8=B?9`<0$H$7$FI=$5$l$k(B. $B2C8:;;$O(B, $B2D49$N>l9g$HF1MM$K(B, @code{+}, @code{-}
   $B$K$h$j(B
   $B<B9T$G$-$k$,(B, $B>h;;$O(B, $BHs2D49@-$r9MN8$7$F(B @code{dp_weyl_mul()} $B$H$$$&4X?t(B
   $B$K$h$j<B9T$9$k(B.
   \E
   
   \BEG
   @code{D} is the ring of differential operators whose coefficients
   are polynomials in @code{K[x1,@dots{},xn]} and
   @code{Di} denotes the differentiation with respect to  @code{xi}.
   According to the commutation relation,
   elements of @code{D} can be represented as a @code{K}-linear combination
   of monomials @code{x1^i1*@dots{}*xn^in*D1^j1*@dots{}*Dn^jn}.
   In Risa/Asir, this type of monomial is represented
   by @code{<<i1,@dots{},in,j1,@dots{},jn>>} as in the case of commutative
   polynomial.
   That is, elements of @code{D} are represented by distributed polynomials.
   Addition and subtraction can be done by @code{+}, @code{-},
   but multiplication is done by calling @code{dp_weyl_mul()} because of
   the non-commutativity of @code{D}.
   \E
   
   @example
   [0] A=<<1,2,2,1>>;
   (1)*<<1,2,2,1>>
   [1] B=<<2,1,1,2>>;
   (1)*<<2,1,1,2>>
   [2] A*B;
   (1)*<<3,3,3,3>>
   [3] dp_weyl_mul(A,B);
   (1)*<<3,3,3,3>>+(1)*<<3,2,3,2>>+(4)*<<2,3,2,3>>+(4)*<<2,2,2,2>>
   +(2)*<<1,3,1,3>>+(2)*<<1,2,1,2>>
   @end example
   
   \BJP
   $B%0%l%V%J4pDl7W;;$K$D$$$F$b(B, Weyl $BBe?t@lMQ$N4X?t$H$7$F(B,
   $B<!$N4X?t$,MQ0U$7$F$"$k(B.
   \E
   \BEG
   The following functions are avilable for Groebner basis computation
   in Weyl algebra:
   \E
   @code{dp_weyl_gr_main()},
   @code{dp_weyl_gr_mod_main()},
   @code{dp_weyl_gr_f_main()},
   @code{dp_weyl_f4_main()},
   @code{dp_weyl_f4_mod_main()}.
   \BJP
   $B$^$?(B, $B1~MQ$H$7$F(B, global b $B4X?t$N7W;;$,<BAu$5$l$F$$$k(B.
   \E
   \BEG
   Computation of the global b function is implemented as an application.
   \E
   
   \BJP
 @node $B%0%l%V%J4pDl$K4X$9$kH!?t(B,,, $B%0%l%V%J4pDl$N7W;;(B  @node $B%0%l%V%J4pDl$K4X$9$kH!?t(B,,, $B%0%l%V%J4pDl$N7W;;(B
 @section $B%0%l%V%J4pDl$K4X$9$kH!?t(B  @section $B%0%l%V%J4pDl$K4X$9$kH!?t(B
 \E  \E
Line 1217  Refer to the sections for each functions.
Line 1327  Refer to the sections for each functions.
 * lex_hensel_gsl tolex_gsl tolex_gsl_d::  * lex_hensel_gsl tolex_gsl tolex_gsl_d::
 * gr_minipoly minipoly::  * gr_minipoly minipoly::
 * tolexm minipolym::  * tolexm minipolym::
 * dp_gr_main dp_gr_mod_main dp_gr_f_main::  * 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::
 * dp_f4_main dp_f4_mod_main::  * dp_f4_main dp_f4_mod_main dp_weyl_f4_main dp_weyl_f4_mod_main::
 * dp_gr_flags dp_gr_print::  * dp_gr_flags dp_gr_print::
 * dp_ord::  * dp_ord::
 * dp_ptod::  * dp_ptod::
Line 1244  Refer to the sections for each functions.
Line 1354  Refer to the sections for each functions.
 * 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 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 1346  for communication.
Line 1457  for communication.
 @table @t  @table @t
 \JP @item $B;2>H(B  \JP @item $B;2>H(B
 \EG @item References  \EG @item References
 @comment @fref{dp_gr_main dp_gr_mod_main dp_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},
 @fref{dp_gr_main dp_gr_mod_main dp_gr_f_main},  
 @fref{dp_ord}.  @fref{dp_ord}.
 @end table  @end table
   
Line 1564  processes.
Line 1674  processes.
 @table @t  @table @t
 \JP @item $B;2>H(B  \JP @item $B;2>H(B
 \EG @item References  \EG @item References
 @fref{dp_gr_main dp_gr_mod_main dp_gr_f_main},  @fref{dp_gr_main dp_gr_mod_main dp_gr_f_main dp_weyl_gr_main dp_weyl_gr_mod_main dp_weyl_gr_f_main},
 \JP @fref{dp_ord}, @fref{$BJ,;67W;;(B}  \JP @fref{dp_ord}, @fref{$BJ,;67W;;(B}
 \EG @fref{dp_ord}, @fref{Distributed computation}  \EG @fref{dp_ord}, @fref{Distributed computation}
 @end table  @end table
Line 1580  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 1842  z^32+11405*z^31+20868*z^30+21602*z^29+...
Line 1952  z^32+11405*z^31+20868*z^30+21602*z^29+...
 @fref{gr_minipoly minipoly}.  @fref{gr_minipoly minipoly}.
 @end table  @end table
   
 \JP @node dp_gr_main dp_gr_mod_main dp_gr_f_main,,, $B%0%l%V%J4pDl$K4X$9$kH!?t(B  \JP @node 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,,, $B%0%l%V%J4pDl$K4X$9$kH!?t(B
 \EG @node dp_gr_main dp_gr_mod_main dp_gr_f_main,,, Functions for Groebner basis computation  \EG @node 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,,, Functions for Groebner basis computation
 @subsection @code{dp_gr_main}, @code{dp_gr_mod_main}, @code{dp_gr_f_main}  @subsection @code{dp_gr_main}, @code{dp_gr_mod_main}, @code{dp_gr_f_main}, @code{dp_weyl_gr_main}, @code{dp_weyl_gr_mod_main}, @code{dp_weyl_gr_f_main}
 @findex dp_gr_main  @findex dp_gr_main
 @findex dp_gr_mod_main  @findex dp_gr_mod_main
 @findex dp_gr_f_main  @findex dp_gr_f_main
   @findex dp_weyl_gr_main
   @findex dp_weyl_gr_mod_main
   @findex dp_weyl_gr_f_main
   
 @table @t  @table @t
 @item dp_gr_main(@var{plist},@var{vlist},@var{homo},@var{modular},@var{order})  @item dp_gr_main(@var{plist},@var{vlist},@var{homo},@var{modular},@var{order})
 @itemx dp_gr_mod_main(@var{plist},@var{vlist},@var{homo},@var{modular},@var{order})  @itemx dp_gr_mod_main(@var{plist},@var{vlist},@var{homo},@var{modular},@var{order})
 @itemx dp_gr_f_main(@var{plist},@var{vlist},@var{homo},@var{order})  @itemx dp_gr_f_main(@var{plist},@var{vlist},@var{homo},@var{order})
   @itemx dp_weyl_gr_main(@var{plist},@var{vlist},@var{homo},@var{modular},@var{order})
   @itemx dp_weyl_gr_mod_main(@var{plist},@var{vlist},@var{homo},@var{modular},@var{order})
   @itemx dp_weyl_gr_f_main(@var{plist},@var{vlist},@var{homo},@var{order})
 \JP :: $B%0%l%V%J4pDl$N7W;;(B ($BAH$_9~$_H!?t(B)  \JP :: $B%0%l%V%J4pDl$N7W;;(B ($BAH$_9~$_H!?t(B)
 \EG :: Groebner basis computation (built-in functions)  \EG :: Groebner basis computation (built-in functions)
 @end table  @end table
Line 1880  z^32+11405*z^31+20868*z^30+21602*z^29+...
Line 1996  z^32+11405*z^31+20868*z^30+21602*z^29+...
 @item  @item
 $B$3$l$i$NH!?t$O(B, $B%0%l%V%J4pDl7W;;$N4pK\E*AH$_9~$_H!?t$G$"$j(B, @code{gr()},  $B$3$l$i$NH!?t$O(B, $B%0%l%V%J4pDl7W;;$N4pK\E*AH$_9~$_H!?t$G$"$j(B, @code{gr()},
 @code{hgr()}, @code{gr_mod()} $B$J$I$O$9$Y$F$3$l$i$NH!?t$r8F$S=P$7$F7W;;(B  @code{hgr()}, @code{gr_mod()} $B$J$I$O$9$Y$F$3$l$i$NH!?t$r8F$S=P$7$F7W;;(B
 $B$r9T$C$F$$$k(B.  $B$r9T$C$F$$$k(B. $B4X?tL>$K(B weyl $B$,F~$C$F$$$k$b$N$O(B, Weyl $BBe?t>e$N7W;;(B
   $B$N$?$a$N4X?t$G$"$k(B.
 @item  @item
 @code{dp_gr_f_main()} $B$O(B, $B<o!9$NM-8BBN>e$N%0%l%V%J4pDl$r7W;;$9$k(B  @code{dp_gr_f_main()}, @code{dp_weyl_f_main()} $B$O(B, $B<o!9$NM-8BBN>e$N%0%l%V%J4pDl$r7W;;$9$k(B
 $B>l9g$KMQ$$$k(B. $BF~NO$O(B, $B$"$i$+$8$a(B, @code{simp_ff()} $B$J$I$G(B,  $B>l9g$KMQ$$$k(B. $BF~NO$O(B, $B$"$i$+$8$a(B, @code{simp_ff()} $B$J$I$G(B,
 $B9M$($kM-8BBN>e$K<M1F$5$l$F$$$kI,MW$,$"$k(B.  $B9M$($kM-8BBN>e$K<M1F$5$l$F$$$kI,MW$,$"$k(B.
 @item  @item
Line 1917  z^32+11405*z^31+20868*z^30+21602*z^29+...
Line 2034  z^32+11405*z^31+20868*z^30+21602*z^29+...
 @item  @item
 These functions are fundamental built-in functions for Groebner basis  These functions are fundamental built-in functions for Groebner basis
 computation and @code{gr()},@code{hgr()} and @code{gr_mod()}  computation and @code{gr()},@code{hgr()} and @code{gr_mod()}
 are all interfaces to these functions.  are all interfaces to these functions. Functions whose names
   contain weyl are those for computation in Weyl algebra.
 @item  @item
 @code{dp_gr_f_main()} is a function for Groebner basis computation  @code{dp_gr_f_main()} and @code{dp_weyl_gr_f_main()}
   are functions for Groebner basis computation
 over various finite fields. Coefficients of input polynomials  over various finite fields. Coefficients of input polynomials
 must be converted to elements of a finite field  must be converted to elements of a finite field
 currently specified by @code{setmod_ff()}.  currently specified by @code{setmod_ff()}.
Line 1966  Actual computation is controlled by various parameters
Line 2085  Actual computation is controlled by various parameters
 \EG @fref{Controlling Groebner basis computations}  \EG @fref{Controlling Groebner basis computations}
 @end table  @end table
   
 \JP @node dp_f4_main dp_f4_mod_main,,, $B%0%l%V%J4pDl$K4X$9$kH!?t(B  \JP @node dp_f4_main dp_f4_mod_main dp_weyl_f4_main dp_weyl_f4_mod_main,,, $B%0%l%V%J4pDl$K4X$9$kH!?t(B
 \EG @node dp_f4_main dp_f4_mod_main,,, Functions for Groebner basis computation  \EG @node dp_f4_main dp_f4_mod_main dp_weyl_f4_main dp_weyl_f4_mod_main,,, Functions for Groebner basis computation
 @subsection @code{dp_f4_main}, @code{dp_f4_mod_main}  @subsection @code{dp_f4_main}, @code{dp_f4_mod_main}, @code{dp_weyl_f4_main}, @code{dp_weyl_f4_mod_main}
 @findex dp_f4_main  @findex dp_f4_main
 @findex dp_f4_mod_main  @findex dp_f4_mod_main
   @findex dp_weyl_f4_main
   @findex dp_weyl_f4_mod_main
   
 @table @t  @table @t
 @item dp_f4_main(@var{plist},@var{vlist},@var{order})  @item dp_f4_main(@var{plist},@var{vlist},@var{order})
 @itemx dp_f4_mod_main(@var{plist},@var{vlist},@var{order})  @itemx dp_f4_mod_main(@var{plist},@var{vlist},@var{order})
   @itemx dp_weyl_f4_main(@var{plist},@var{vlist},@var{order})
   @itemx dp_weyl_f4_mod_main(@var{plist},@var{vlist},@var{order})
 \JP :: F4 $B%"%k%4%j%:%`$K$h$k%0%l%V%J4pDl$N7W;;(B ($BAH$_9~$_H!?t(B)  \JP :: F4 $B%"%k%4%j%:%`$K$h$k%0%l%V%J4pDl$N7W;;(B ($BAH$_9~$_H!?t(B)
 \EG :: Groebner basis computation by F4 algorithm (built-in functions)  \EG :: Groebner basis computation by F4 algorithm (built-in functions)
 @end table  @end table
Line 2000  F4 $B%"%k%4%j%:%`$O(B, J.C. Faugere $B$K$h$jDs>'$5$
Line 2123  F4 $B%"%k%4%j%:%`$O(B, J.C. Faugere $B$K$h$jDs>'$5$
 $B;;K!$G$"$j(B, $BK\<BAu$O(B, $BCf9q>jM>DjM}$K$h$k@~7AJ}Dx<05a2r$rMQ$$$?(B  $B;;K!$G$"$j(B, $BK\<BAu$O(B, $BCf9q>jM>DjM}$K$h$k@~7AJ}Dx<05a2r$rMQ$$$?(B
 $B;n83E*$J<BAu$G$"$k(B.  $B;n83E*$J<BAu$G$"$k(B.
 @item  @item
 $B0z?t$*$h$SF0:n$O$=$l$>$l(B @code{dp_gr_main()}, @code{dp_gr_mod_main()}  $B@F<!2=$N0z?t$,$J$$$3$H$r=|$1$P(B, $B0z?t$*$h$SF0:n$O$=$l$>$l(B
   @code{dp_gr_main()}, @code{dp_gr_mod_main()},
   @code{dp_weyl_gr_main()}, @code{dp_weyl_gr_mod_main()}
 $B$HF1MM$G$"$k(B.  $B$HF1MM$G$"$k(B.
 \E  \E
 \BEG  \BEG
Line 2012  invented by J.C. Faugere. The current implementation o
Line 2137  invented by J.C. Faugere. The current implementation o
 uses Chinese Remainder theorem and not highly optimized.  uses Chinese Remainder theorem and not highly optimized.
 @item  @item
 Arguments and actions are the same as those of  Arguments and actions are the same as those of
 @code{dp_gr_main()}, @code{dp_gr_mod_main()}.  @code{dp_gr_main()}, @code{dp_gr_mod_main()},
   @code{dp_weyl_gr_main()}, @code{dp_weyl_gr_mod_main()},
   except for lack of the argument for controlling homogenization.
 \E  \E
 @end itemize  @end itemize
   
Line 2034  Arguments and actions are the same as those of 
Line 2161  Arguments and actions are the same as those of 
   
 @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 2048  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 2061  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 2078  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 3665  if an input ideal is not radical.
Line 3813  if an input ideal is not radical.
 \EG @fref{Setting term orderings}.  \EG @fref{Setting term orderings}.
 @end table  @end table
   
 \BJP  
 @node Weyl $BBe?t(B,,, $B%0%l%V%J4pDl$N7W;;(B  
 @section Weyl $BBe?t(B  
 \E  
 \BEG  
 @node Weyl algebra,,, Groebner basis computation  
 @section Weyl algebra  
 \E  
   
 @noindent  
   
 \BJP  
 $B$3$l$^$G$O(B, $BDL>o$N2D49$JB?9`<04D$K$*$1$k%0%l%V%J4pDl7W;;$K$D$$$F(B  
 $B=R$Y$F$-$?$,(B, $B%0%l%V%J4pDl$NM}O@$O(B, $B$"$k>r7o$rK~$?$9Hs2D49$J(B  
 $B4D$K$b3HD%$G$-$k(B. $B$3$N$h$&$J4D$NCf$G(B, $B1~MQ>e$b=EMW$J(B,  
 Weyl $BBe?t(B, $B$9$J$o$AB?9`<04D>e$NHyJ,:nMQAG4D$N1i;;$*$h$S(B  
 $B%0%l%V%J4pDl7W;;$,(B Risa/Asir $B$K<BAu$5$l$F$$$k(B.  
   
 $BBN(B @code{K} $B>e$N(B @code{n} $B<!85(B Weyl $BBe?t(B  
 @code{D=K<x1,@dots{},xn,D1,@dots{},Dn>} $B$O(B  
 \E  
   
 \BEG  
 So far we have explained Groebner basis computation in  
 commutative polynomial rings. However Groebner basis can be  
 considered in more general non-commutative rings.  
 Weyl algebra is one of such rings and  
 Risa/Asir implements fundamental operations  
 in Weyl algebra and Groebner basis computation in Weyl algebra.  
   
 The @code{n} dimensional Weyl algebra over a field @code{K},  
 @code{D=K<x1,@dots{},xn,D1,@dots{},Dn>} is a non-commutative  
 algebra which has the following fundamental relations:  
 \E  
   
 @code{xi*xj-xj*xi=0}, @code{Di*Dj-Dj*Di=0}, @code{Di*xj-xj*Di=0} (@code{i!=j}),  
 @code{Di*xi-xi*Di=1}  
   
 \BJP  
 $B$H$$$&4pK\4X78$r;}$D4D$G$"$k(B. @code{D} $B$O(B $BB?9`<04D(B @code{K[x1,@dots{},xn]} $B$r78?t(B  
 $B$H$9$kHyJ,:nMQAG4D$G(B,  @code{Di} $B$O(B @code{xi} $B$K$h$kHyJ,$rI=$9(B. $B8r494X78$K$h$j(B,  
 @code{D} $B$N85$O(B, @code{x1^i1*@dots{}*xn^in*D1^j1*@dots{}*Dn^jn} $B$J$kC19`(B  
 $B<0$N(B @code{K} $B@~7A7k9g$H$7$F=q$-I=$9$3$H$,$G$-$k(B.  
 Risa/Asir $B$K$*$$$F$O(B, $B$3$NC19`<0$r(B, $B2D49$JB?9`<0$HF1MM$K(B  
 @code{<<i1,@dots{},in,j1,@dots{},jn>>} $B$GI=$9(B. $B$9$J$o$A(B, @code{D} $B$N85$b(B  
 $BJ,;6I=8=B?9`<0$H$7$FI=$5$l$k(B. $B2C8:;;$O(B, $B2D49$N>l9g$HF1MM$K(B, @code{+}, @code{-}  
 $B$K$h$j(B  
 $B<B9T$G$-$k$,(B, $B>h;;$O(B, $BHs2D49@-$r9MN8$7$F(B @code{dp_weyl_mul()} $B$H$$$&4X?t(B  
 $B$K$h$j<B9T$9$k(B.  
 \E  
   
 \BEG  
 @code{D} is the ring of differential operators whose coefficients  
 are polynomials in @code{K[x1,@dots{},xn]} and  
 @code{Di} denotes the differentiation with respect to  @code{xi}.  
 According to the commutation relation,  
 elements of @code{D} can be represented as a @code{K}-linear combination  
 of monomials @code{x1^i1*@dots{}*xn^in*D1^j1*@dots{}*Dn^jn}.  
 In Risa/Asir, this type of monomial is represented  
 by @code{<<i1,@dots{},in,j1,@dots{},jn>>} as in the case of commutative  
 polynomial.  
 That is, elements of @code{D} are represented by distributed polynomials.  
 Addition and subtraction can be done by @code{+}, @code{-},  
 but multiplication is done by calling @code{dp_weyl_mul()} because of  
 the non-commutativity of @code{D}.  
 \E  
   
 @example  
 [0] A=<<1,2,2,1>>;  
 (1)*<<1,2,2,1>>  
 [1] B=<<2,1,1,2>>;  
 (1)*<<2,1,1,2>>  
 [2] A*B;  
 (1)*<<3,3,3,3>>  
 [3] dp_weyl_mul(A,B);  
 (1)*<<3,3,3,3>>+(1)*<<3,2,3,2>>+(4)*<<2,3,2,3>>+(4)*<<2,2,2,2>>  
 +(2)*<<1,3,1,3>>+(2)*<<1,2,1,2>>  
 @end example  
   
 \BJP  
 $B%0%l%V%J4pDl7W;;$K$D$$$F$b(B, Weyl $BBe?t@lMQ$N4X?t$H$7$F(B,  
 $B<!$N4X?t$,MQ0U$7$F$"$k(B.  
 \E  
 \BEG  
 The following functions are avilable for Groebner basis computation  
 in Weyl algebra:  
 \E  
 @code{dp_weyl_gr_main()},  
 @code{dp_weyl_gr_mod_main()},  
 @code{dp_weyl_gr_f_main()},  
 @code{dp_weyl_f4_main()},  
 @code{dp_weyl_f4_mod_main()}.  
 \BJP  
 $B$^$?(B, $B1~MQ$H$7$F(B, global b $B4X?t$N7W;;$,<BAu$5$l$F$$$k(B.  
 \E  
 \BEG  
 Computation of the global b function is implemented as an application.  
 \E  
   
 \JP @node primedec_mod,,, $B%0%l%V%J4pDl$K4X$9$kH!?t(B  \JP @node primedec_mod,,, $B%0%l%V%J4pDl$K4X$9$kH!?t(B
 \EG @node primedec_mod,,, Functions for Groebner basis computation  \EG @node primedec_mod,,, Functions for Groebner basis computation
 @subsection @code{primedec_mod}  @subsection @code{primedec_mod}
Line 3814  Computation of the global b function is implemented as
Line 3863  Computation of the global b function is implemented as
 $BItJ,$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 3838  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 3857  if the dimension is small.
Line 3912  if the dimension is small.
 \JP @item $B;2>H(B  \JP @item $B;2>H(B
 \EG @item References  \EG @item References
 @fref{modfctr},  @fref{modfctr},
 @fref{dp_gr_main dp_gr_mod_main dp_gr_f_main},  @fref{dp_gr_main dp_gr_mod_main dp_gr_f_main dp_weyl_gr_main dp_weyl_gr_mod_main dp_weyl_gr_f_main},
 \JP @fref{$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 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})
   @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
   \JP $BB?9`<0$^$?$O%j%9%H(B
   \EG polynomial or list
   @item f
   \JP $BB?9`<0(B
   \EG polynomial
   @item plist
   \JP $BB?9`<0%j%9%H(B
   \EG list of polynomials
   @item vlist dvlist
   \JP $BJQ?t%j%9%H(B
   \EG list of variables
   @end table
   
   @itemize @bullet
   \BJP
   @item @samp{bfct} $B$GDj5A$5$l$F$$$k(B.
   @item @code{bfunction(@var{f})}, @code{bfct(@var{f})} $B$OB?9`<0(B @var{f} $B$N(B global @var{b} $B4X?t(B @code{b(s)} $B$r(B
   $B7W;;$9$k(B. @code{b(s)} $B$O(B, Weyl $BBe?t(B @code{D} $B>e$N0lJQ?tB?9`<04D(B @code{D[s]}
   $B$N85(B @code{P(x,s)} $B$,B8:_$7$F(B, @code{P(x,s)f^(s+1)=b(s)f^s} $B$rK~$?$9$h$&$J(B
   $BB?9`<0(B @code{b(s)} $B$NCf$G(B, $B<!?t$,:G$bDc$$$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 @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$,9bB.2=$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
   \BEG
   @item These functions are defined in @samp{bfct}.
   @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 @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
   
   @example
   [0] load("bfct")$
   [216] bfunction(x^3+y^3+z^3+x^2*y^2*z^2+x*y*z);
   -9*s^5-63*s^4-173*s^3-233*s^2-154*s-40
   [217] fctr(@@);
   [[-1,1],[s+2,1],[3*s+4,1],[3*s+5,1],[s+1,2]]
   [218] F = [4*x^3*dt+y*z*dt+dx,x*z*dt+4*y^3*dt+dy,
   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
   \JP @item $B;2>H(B
   \EG @item References
   \JP @fref{Weyl $BBe?t(B}.
   \EG @fref{Weyl algebra}.
   @end table
   
Legend:
Removed from v.1.5  
changed lines
  Added in v.1.10
FreeBSD-CVSweb <freebsd-cvsweb@FreeBSD.org>