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Diff for /OpenXM/src/asir-doc/parts/groebner.texi between version 1.12 and 1.20

-version 1.12, 2003/12/27 11:52:07
+version 1.20, 2017/08/31 04:54:36
 Line 1
 Line 1
 Line 1
- @comment $OpenXM: OpenXM/src/asir-doc/parts/groebner.texi,v 1.11 2003/04/28 06:43:10 noro Exp $
+ @comment $OpenXM: OpenXM/src/asir-doc/parts/groebner.texi,v 1.19 2016/08/29 04:56:58 noro Exp $
  \BJP
  @node $B%0%l%V%J4pDl$N7W;;(B,,, Top
  @chapter $B%0%l%V%J4pDl$N7W;;(B
 Line 15
 Line 15
 Line 15
  * $B4pK\E*$JH!?t(B::
  * $B7W;;$*$h$SI=<($N@)8f(B::
  * $B9`=g=x$N@_Dj(B::
+ * Weight::
  * $BM-M}<0$r78?t$H$9$k%0%l%V%J4pDl7W;;(B::
  * $B4pDlJQ49(B::
  * Weyl $BBe?t(B::
-Line 26
+Line 27
 Line 26
 Line 27
  * Fundamental functions::
  * Controlling Groebner basis computations::
  * Setting term orderings::
+ * Weight::
  * Groebner basis computation with rational function coefficients::
  * Change of ordering::
  * Weyl algebra::
-Line 199  In an @b{Asir} session, it is displayed in the form li
+Line 201  In an @b{Asir} session, it is displayed in the form li
 Line 199  In an @b{Asir} session, it is displayed in the form li
 Line 201  In an @b{Asir} session, it is displayed in the form li
  \EG and also can be input in such a form.
  \BJP
- @itemx $BF,C19`<0(B (head monomial)
  @item $BF,9`(B (head term)
+ @itemx $BF,C19`<0(B (head monomial)
  @itemx $BF,78?t(B (head coefficient)
  $BJ,;6I=8=B?9`<0$K$*$1$k3FC19`<0$O(B, $B9`=g=x$K$h$j@0Ns$5$l$k(B. $B$3$N;~=g(B
  $B=x:GBg$NC19`<0$rF,C19`<0(B, $B$=$l$K8=$l$k9`(B, $B78?t$r$=$l$>$lF,9`(B, $BF,78?t(B
  $B$H8F$V(B.
  \E
  \BEG
- @itemx head monomial
  @item head term
+ @itemx head monomial
  @itemx head coefficient
  Monomials in a distributed polynomial is sorted by a total order.
-Line 218  the head term and the head coefficient respectively.
+Line 220  the head term and the head coefficient respectively.
 Line 218  the head term and the head coefficient respectively.
 Line 220  the head term and the head coefficient respectively.
  \E
  @end table
+ @noindent
+ ChangeLog
+ @itemize @bullet
  \BJP
+ @item $BJ,;6I=8=B?9`<0$OG$0U$N%*%V%8%'%/%H$r78?t$K$b$F$k$h$&$K$J$C$?(B.
+ $B$^$?2C72$N(Bk$B@.J,$NMWAG$r<!$N7A<0(B <<d0,d1,...:k>> $B$GI=8=$9$k$h$&$K$J$C$?(B (2017-08-31).
+ \E
+ \BEG
+ @item Distributed polynomials accept objects as coefficients.
+ The k-th element of a free module is expressed as <<d0,d1,...:k>> (2017-08-31).
+ \E
+ @item
+.15 algnum.c,
+.53 ctrl.c,
+.66 dp-supp.c,
+.105 dp.c,
+.73 gr.c,
+.4 reduct.c,
+.16 _distm.c,
+.17 dalg.c,
+.52 dist.c,
+.20 distm.c,
+.8  gmpq.c,
+.238 engine/nd.c,
+.102  ca.h,
+.411  version.h,
+.28 cpexpr.c,
+.42 pexpr.c,
+.20 pexpr_body.c,
+.40 spexpr.c,
+.27 arith.c,
+.77 eval.c,
+.56 parse.h,
+.37 parse.y,
+.8 stdio.c,
+.31 plotf.c
+ @end itemize
+ \BJP
  @node $B%U%!%$%k$NFI$_9~$_(B,,, $B%0%l%V%J4pDl$N7W;;(B
  @section $B%U%!%$%k$NFI$_9~$_(B
  \E
-Line 1055  beforehand, and some heuristic trial may be inevitable
+Line 1095  beforehand, and some heuristic trial may be inevitable
 Line 1055  beforehand, and some heuristic trial may be inevitable
 Line 1095  beforehand, and some heuristic trial may be inevitable
  \E
  \BJP
+ @node Weight ,,, $B%0%l%V%J4pDl$N7W;;(B
+ @section Weight
+ \E
+ \BEG
+ @node Weight,,, Groebner basis computation
+ @section Weight
+ \E
+ \BJP
+ $BA0@a$G>R2p$7$?9`=g=x$O(B, $B3FJQ?t$K(B weight ($B=E$_(B) $B$r@_Dj$9$k$3$H$G(B
+ $B$h$j0lHLE*$J$b$N$H$J$k(B.
+ \E
+ \BEG
+ Term orderings introduced in the previous section can be generalized
+ by setting a weight for each variable.
+ \E
+ @example
+ [0] dp_td(<<1,1,1>>);
+ [1] dp_set_weight([1,2,3])$
+ [2] dp_td(<<1,1,1>>);
+ @end example
+ \BJP
+ $BC19`<0$NA4<!?t$r7W;;$9$k:](B, $B%G%U%)%k%H$G$O(B
+ $B3FJQ?t$N;X?t$NOB$rA4<!?t$H$9$k(B. $B$3$l$O3FJQ?t$N(B weight $B$r(B 1 $B$H(B
+ $B9M$($F$$$k$3$H$KAjEv$9$k(B. $B$3$NNc$G$O(B, $BBh0l(B, $BBhFs(B, $BBh;0JQ?t$N(B
+ weight $B$r$=$l$>$l(B 1,2,3 $B$H;XDj$7$F$$$k(B. $B$3$N$?$a(B, @code{<<1,1,1>>}
+ $B$NA4<!?t(B ($B0J2<$G$O$3$l$rC19`<0$N(B weight $B$H8F$V(B) $B$,(B @code{1*1+1*2+1*3=6} $B$H$J$k(B.
+ weight $B$r@_Dj$9$k$3$H$G(B, $BF1$89`=g=x7?$N$b$H$G0[$J$k9`=g=x$,Dj5A$G$-$k(B.
+ $BNc$($P(B, weight $B$r$&$^$/@_Dj$9$k$3$H$G(B, $BB?9`<0$r(B weighted homogeneous
+ $B$K$9$k$3$H$,$G$-$k>l9g$,$"$k(B.
+ \E
+ \BEG
+ By default, the total degree of a monomial is equal to
+ the sum of all exponents. This means that the weight for each variable
+ is set to 1.
+ In this example, the weights for the first, the second and the third
+ variable are set to 1, 2 and 3 respectively.
+ Therefore the total degree of @code{<<1,1,1>>} under this weight,
+ which is called the weight of the monomial, is @code{1*1+1*2+1*3=6}.
+ By setting weights, different term orderings can be set under a type of
+ term ordeing. In some case a polynomial can
+ be made weighted homogeneous by setting an appropriate weight.
+ \E
+ \BJP
+ $B3FJQ?t$KBP$9$k(B weight $B$r$^$H$a$?$b$N$r(B weight vector $B$H8F$V(B.
+ $B$9$Y$F$N@.J,$,@5$G$"$j(B, $B%0%l%V%J4pDl7W;;$K$*$$$F(B, $BA4<!?t$N(B
+ $BBe$o$j$KMQ$$$i$l$k$b$N$rFC$K(B sugar weight $B$H8F$V$3$H$K$9$k(B.
+ sugar strategy $B$K$*$$$F(B, $BA4<!?t$NBe$o$j$K;H$o$l$k$+$i$G$"$k(B.
+ $B0lJ}$G(B, $B3F@.J,$,I,$:$7$b@5$H$O8B$i$J$$(B weight vector $B$O(B,
+ sugar weight $B$H$7$F@_Dj$9$k$3$H$O$G$-$J$$$,(B, $B9`=g=x$N0lHL2=$K$O(B
+ $BM-MQ$G$"$k(B. $B$3$l$i$O(B, $B9TNs$K$h$k9`=g=x$N@_Dj$K$9$G$K8=$l$F(B
+ $B$$$k(B. $B$9$J$o$A(B, $B9`=g=x$rDj5A$9$k9TNs$N3F9T$,(B, $B0l$D$N(B weight vector
+ $B$H8+$J$5$l$k(B. $B$^$?(B, $B%V%m%C%/=g=x$O(B, $B3F%V%m%C%/$N(B
+ $BJQ?t$KBP1~$9$k@.J,$N$_(B 1 $B$GB>$O(B 0 $B$N(B weight vector $B$K$h$kHf3S$r(B
+ $B:G=i$K9T$C$F$+$i(B, $B3F%V%m%C%/Kh$N(B tie breaking $B$r9T$&$3$H$KAjEv$9$k(B.
+ \E
+ \BEG
+ A list of weights for all variables is called a weight vector.
+ A weight vector is called a sugar weight vector if
+ its elements are all positive and it is used for computing
+ a weighted total degree of a monomial, because such a weight
+ is used instead of total degree in sugar strategy.
+ On the other hand, a weight vector whose elements are not necessarily
+ positive cannot be set as a sugar weight, but it is useful for
+ generalizing term order. In fact, such a weight vector already
+ appeared in a matrix order. That is, each row of a matrix defining
+ a term order is regarded as a weight vector. A block order
+ is also considered as a refinement of comparison by weight vectors.
+ It compares two terms by using a weight vector whose elements
+ corresponding to variables in a block is 1 and 0 otherwise,
+ then it applies a tie breaker.
+ \E
+ \BJP
+ weight vector $B$N@_Dj$O(B @code{dp_set_weight()} $B$G9T$&$3$H$,$G$-$k(B
+ $B$,(B, $B9`=g=x$r;XDj$9$k:]$NB>$N%Q%i%a%?(B ($B9`=g=x7?(B, $BJQ?t=g=x(B) $B$H(B
+ $B$^$H$a$F@_Dj$G$-$k$3$H$,K>$^$7$$(B. $B$3$N$?$a(B, $B<!$N$h$&$J7A$G$b(B
+ $B9`=g=x$,;XDj$G$-$k(B.
+ \E
+ \BEG
+ A weight vector can be set by using @code{dp_set_weight()}.
+ However it is more preferable if a weight vector can be set
+ together with other parapmeters such as a type of term ordering
+ and a variable order. This is realized as follows.
+ \E
+ @example
+ [64] B=[x+y+z-6,x*y+y*z+z*x-11,x*y*z-6]$
+ [65] dp_gr_main(B|v=[x,y,z],sugarweight=[3,2,1],order=0);
+ [z^3-6*z^2+11*z-6,x+y+z-6,-y^2+(-z+6)*y-z^2+6*z-11]
+ [66] dp_gr_main(B|v=[y,z,x],order=[[1,1,0],[0,1,0],[0,0,1]]);
+ [x^3-6*x^2+11*x-6,x+y+z-6,-x^2+(-y+6)*x-y^2+6*y-11]
+ [67] dp_gr_main(B|v=[y,z,x],order=[[x,1,y,2,z,3]]);
+ [x+y+z-6,x^3-6*x^2+11*x-6,-x^2+(-y+6)*x-y^2+6*y-11]
+ @end example
+ \BJP
+ $B$$$:$l$NNc$K$*$$$F$b(B, $B9`=g=x$O(B option $B$H$7$F;XDj$5$l$F$$$k(B.
+ $B:G=i$NNc$G$O(B @code{v} $B$K$h$jJQ?t=g=x$r(B, @code{sugarweight} $B$K$h$j(B
+ sugar weight vector $B$r(B, @code{order}$B$K$h$j9`=g=x7?$r;XDj$7$F$$$k(B.
+ $BFs$DL\$NNc$K$*$1$k(B @code{order} $B$N;XDj$O(B matrix order $B$HF1MM$G$"$k(B.
+ $B$9$J$o$A(B, $B;XDj$5$l$?(B weight vector $B$r:8$+$i=g$K;H$C$F(B weight $B$NHf3S(B
+ $B$r9T$&(B. $B;0$DL\$NNc$bF1MM$G$"$k$,(B, $B$3$3$G$O(B weight vector $B$NMWAG$r(B
+ $BJQ?tKh$K;XDj$7$F$$$k(B. $B;XDj$,$J$$$b$N$O(B 0 $B$H$J$k(B. $B;0$DL\$NNc$G$O(B,
+ @code{order} $B$K$h$k;XDj$G$O9`=g=x$,7hDj$7$J$$(B. $B$3$N>l9g$K$O(B,
+ tie breaker $B$H$7$FA4<!?t5U<-=q<0=g=x$,<+F0E*$K@_Dj$5$l$k(B.
+ $B$3$N;XDjJ}K!$O(B, @code{dp_gr_main}, @code{dp_gr_mod_main} $B$J$I(B
+ $B$NAH$_9~$_4X?t$G$N$_2DG=$G$"$j(B, @code{gr} $B$J$I$N%f!<%6Dj5A4X?t(B
+ $B$G$OL$BP1~$G$"$k(B.
+ \E
+ \BEG
+ In each example, a term ordering is specified as options.
+ In the first example, a variable order, a sugar weight vector
+ and a type of term ordering are specified by options @code{v},
+ @code{sugarweight} and @code{order} respectively.
+ In the second example, an option @code{order} is used
+ to set a matrix ordering. That is, the specified weight vectors
+ are used from left to right for comparing terms.
+ The third example shows a variant of specifying a weight vector,
+ where each component of a weight vector is specified variable by variable,
+ and unspecified components are set to zero. In this example,
+ a term order is not determined only by the specified weight vector.
+ In such a case a tie breaker by the graded reverse lexicographic ordering
+ is set automatically.
+ This type of a term ordering specification can be applied only to builtin
+ functions such as @code{dp_gr_main()}, @code{dp_gr_mod_main()}, not to
+ user defined functions such as @code{gr()}.
+ \E
+ \BJP
  @node $BM-M}<0$r78?t$H$9$k%0%l%V%J4pDl7W;;(B,,, $B%0%l%V%J4pDl$N7W;;(B
  @section $BM-M}<0$r78?t$H$9$k%0%l%V%J4pDl7W;;(B
  \E
-Line 1329  Computation of the global b function is implemented as
+Line 1502  Computation of the global b function is implemented as
 Line 1329  Computation of the global b function is implemented as
 Line 1502  Computation of the global b function is implemented as
  * tolexm minipolym::
  * 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_weyl_f4_main dp_weyl_f4_mod_main::
+ * nd_gr nd_gr_trace nd_f4 nd_f4_trace nd_weyl_gr nd_weyl_gr_trace::
  * dp_gr_flags dp_gr_print::
  * dp_ord::
+ * dp_set_weight dp_set_top_weight dp_weyl_set_weight::
  * dp_ptod::
  * dp_dtop::
  * dp_mod dp_rat::
  * dp_homo dp_dehomo::
  * dp_ptozp dp_prim::
- * dp_nf dp_nf_mod dp_true_nf dp_true_nf_mod::
+ * dp_nf dp_nf_mod dp_true_nf dp_true_nf_mod dp_weyl_nf dp_weyl_nf_mod::
  * dp_hm dp_ht dp_hc dp_rest::
  * dp_td dp_sugar::
  * dp_lcm::
-Line 1405  Computation of the global b function is implemented as
+Line 1580  Computation of the global b function is implemented as
 Line 1405  Computation of the global b function is implemented as
 Line 1580  Computation of the global b function is implemented as
  strategy $B$K$h$k7W;;(B, @code{hgr()} $B$O(B trace-lifting $B$*$h$S(B
  $B@F<!2=$K$h$k(B $B6:@5$5$l$?(B sugar strategy $B$K$h$k7W;;$r9T$&(B.
  @item
- @code{dgr()} $B$O(B, @code{gr()}, @code{dgr()} $B$r(B
+ @code{dgr()} $B$O(B, @code{gr()}, @code{hgr()} $B$r(B
  $B;R%W%m%;%9%j%9%H(B @var{procs} $B$N(B 2 $B$D$N%W%m%;%9$K$h$jF1;~$K7W;;$5$;(B,
  $B@h$K7k2L$rJV$7$?J}$N7k2L$rJV$9(B. $B7k2L$OF10l$G$"$k$,(B, $B$I$A$i$NJ}K!$,(B
  $B9bB.$+0lHL$K$OITL@$N$?$a(B, $B<B:]$N7P2a;~4V$rC;=L$9$k$N$KM-8z$G$"$k(B.
-Line 2168  except for lack of the argument for controlling homoge
+Line 2343  except for lack of the argument for controlling homoge
 Line 2168  except for lack of the argument for controlling homoge
 Line 2343  except for lack of the argument for controlling homoge
  \EG @fref{Controlling Groebner basis computations}
  @end table
+ \JP @node nd_gr nd_gr_trace nd_f4 nd_f4_trace nd_weyl_gr nd_weyl_gr_trace,,, $B%0%l%V%J4pDl$K4X$9$kH!?t(B
+ \EG @node nd_gr nd_gr_trace nd_f4 nd_f4_trace nd_weyl_gr nd_weyl_gr_trace,,, Functions for Groebner basis computation
+ @subsection @code{nd_gr}, @code{nd_gr_trace}, @code{nd_f4}, @code{nd_f4_trace}, @code{nd_weyl_gr}, @code{nd_weyl_gr_trace}
+ @findex nd_gr
+ @findex nd_gr_trace
+ @findex nd_f4
+ @findex nd_f4_trace
+ @findex nd_weyl_gr
+ @findex nd_weyl_gr_trace
+ @table @t
+ @item nd_gr(@var{plist},@var{vlist},@var{p},@var{order})
+ @itemx nd_gr_trace(@var{plist},@var{vlist},@var{homo},@var{p},@var{order})
+ @itemx nd_f4(@var{plist},@var{vlist},@var{modular},@var{order})
+ @itemx nd_f4_trace(@var{plist},@var{vlist},@var{homo},@var{p},@var{order})
+ @itemx nd_weyl_gr(@var{plist},@var{vlist},@var{p},@var{order})
+ @itemx nd_weyl_gr_trace(@var{plist},@var{vlist},@var{homo},@var{p},@var{order})
+ \JP :: $B%0%l%V%J4pDl$N7W;;(B ($BAH$_9~$_H!?t(B)
+ \EG :: Groebner basis computation (built-in functions)
+ @end table
+ @table @var
+ @item return
+ \JP $B%j%9%H(B
+ \EG list
+ @item plist  vlist
+ \JP $B%j%9%H(B
+ \EG list
+ @item order
+ \JP $B?t(B, $B%j%9%H$^$?$O9TNs(B
+ \EG number, list or matrix
+ @item homo
+ \JP $B%U%i%0(B
+ \EG flag
+ @item modular
+ \JP $B%U%i%0$^$?$OAG?t(B
+ \EG flag or prime
+ @end table
+ \BJP
+ @itemize @bullet
+ @item
+ $B$3$l$i$NH!?t$O(B, $B%0%l%V%J4pDl7W;;AH$_9~$_4X?t$N?7<BAu$G$"$k(B.
+ @item @code{nd_gr} $B$O(B, @code{p} $B$,(B 0 $B$N$H$-M-M}?tBN>e$N(B Buchberger
+ $B%"%k%4%j%:%`$r<B9T$9$k(B. @code{p} $B$,(B 2 $B0J>e$N<+A3?t$N$H$-(B, GF(p) $B>e$N(B
+ Buchberger $B%"%k%4%j%:%`$r<B9T$9$k(B.
+ @item @code{nd_gr_trace} $B$*$h$S(B @code{nd_f4_trace}
+ $B$OM-M}?tBN>e$G(B trace $B%"%k%4%j%:%`$r<B9T$9$k(B.
+ @var{p} $B$,(B 0 $B$^$?$O(B 1 $B$N$H$-(B, $B<+F0E*$KA*$P$l$?AG?t$rMQ$$$F(B, $B@.8y$9$k(B
+ $B$^$G(B trace $B%"%k%4%j%:%`$r<B9T$9$k(B.
+ @var{p} $B$,(B 2 $B0J>e$N$H$-(B, trace $B$O(BGF(p) $B>e$G7W;;$5$l$k(B. trace $B%"%k%4%j%:%`(B
+ $B$,<:GT$7$?>l9g(B 0 $B$,JV$5$l$k(B. @var{p} $B$,Ii$N>l9g(B, $B%0%l%V%J4pDl%A%'%C%/$O(B
+ $B9T$o$J$$(B. $B$3$N>l9g(B, @var{p} $B$,(B -1 $B$J$i$P<+F0E*$KA*$P$l$?AG?t$,(B,
+ $B$=$l0J30$O;XDj$5$l$?AG?t$rMQ$$$F%0%l%V%J4pDl8uJd$N7W;;$,9T$o$l$k(B.
+ @code{nd_f4_trace} $B$O(B, $B3FA4<!?t$K$D$$$F(B, $B$"$kM-8BBN>e$G(B F4 $B%"%k%4%j%:%`(B
+ $B$G9T$C$?7k2L$r$b$H$K(B, $B$=$NM-8BBN>e$G(B 0 $B$G$J$$4pDl$rM?$($k(B S-$BB?9`<0$N$_$r(B
+ $BMQ$$$F9TNs@8@.$r9T$$(B, $B$=$NA4<!?t$K$*$1$k4pDl$r@8@.$9$kJ}K!$G$"$k(B. $BF@$i$l$k(B
+ $BB?9`<0=89g$O$d$O$j%0%l%V%J4pDl8uJd$G$"$j(B, @code{nd_gr_trace} $B$HF1MM$N(B
+ $B%A%'%C%/$,9T$o$l$k(B.
+ @item
+ @code{nd_f4} $B$O(B @code{modular} $B$,(B 0 $B$N$H$-M-M}?tBN>e$N(B, @code{modular} $B$,(B
+ $B%^%7%s%5%$%:AG?t$N$H$-M-8BBN>e$N(B F4 $B%"%k%4%j%:%`$r<B9T$9$k(B.
+ @item
+ @var{plist} $B$,B?9`<0%j%9%H$N>l9g(B, @var{plist}$B$G@8@.$5$l$k%$%G%"%k$N%0%l%V%J!<4pDl$,(B
+ $B7W;;$5$l$k(B. @var{plist} $B$,B?9`<0%j%9%H$N%j%9%H$N>l9g(B, $B3FMWAG$OB?9`<04D>e$N<+M32C72$N85$H8+$J$5$l(B,
+ $B$3$l$i$,@8@.$9$kItJ,2C72$N%0%l%V%J!<4pDl$,7W;;$5$l$k(B. $B8e<T$N>l9g(B, $B9`=g=x$O2C72$KBP$9$k9`=g=x$r(B
+ $B;XDj$9$kI,MW$,$"$k(B. $B$3$l$O(B @var{[s,ord]} $B$N7A$G;XDj$9$k(B. @var{s} $B$,(B 0 $B$J$i$P(B TOP (Term Over Position),
+$B$J$i$P(B POT (Position Over Term) $B$r0UL#$7(B, @var{ord} $B$OB?9`<04D$NC19`<0$KBP$9$k9`=g=x$G$"$k(B.
+ @item
+ @code{nd_weyl_gr}, @code{nd_weyl_gr_trace} $B$O(B Weyl $BBe?tMQ$G$"$k(B.
+ @item
+ @code{f4} $B7O4X?t0J30$O$9$Y$FM-M}4X?t78?t$N7W;;$,2DG=$G$"$k(B.
+ @item
+ $B0lHL$K(B @code{dp_gr_main}, @code{dp_gr_mod_main} $B$h$j9bB.$G$"$k$,(B,
+ $BFC$KM-8BBN>e$N>l9g82Cx$G$"$k(B.
+ @end itemize
+ \E
+ \BEG
+ @itemize @bullet
+ @item
+ These functions are new implementations for computing Groebner bases.
+ @item @code{nd_gr} executes Buchberger algorithm over the rationals
+ if  @code{p} is 0, and that over GF(p) if @code{p} is a prime.
+ @item @code{nd_gr_trace} executes the trace algorithm over the rationals.
+ If @code{p} is 0 or 1, the trace algorithm is executed until it succeeds
+ by using automatically chosen primes.
+ If @code{p} a positive prime,
+ the trace is comuted over GF(p).
+ If the trace algorithm fails 0 is returned.
+ If @code{p} is negative,
+ the Groebner basis check and ideal-membership check are omitted.
+ In this case, an automatically chosen prime if @code{p} is 1,
+ otherwise the specified prime is used to compute a Groebner basis
+ candidate.
+ Execution of @code{nd_f4_trace} is done as follows:
+ For each total degree, an F4-reduction of S-polynomials over a finite field
+ is done, and S-polynomials which give non-zero basis elements are gathered.
+ Then F4-reduction over Q is done for the gathered S-polynomials.
+ The obtained polynomial set is a Groebner basis candidate and the same
+ check procedure as in the case of @code{nd_gr_trace} is done.
+ @item
+ @code{nd_f4} executes F4 algorithm over Q if @code{modular} is equal to 0,
+ or over a finite field GF(@code{modular})
+ if @code{modular} is a prime number of machine size (<2^29).
+ If @var{plist} is a list of polynomials, then a Groebner basis of the ideal generated by @var{plist}
+ is computed. If @var{plist} is a list of lists of polynomials, then each list of polynomials are regarded
+ as an element of a free module over a polynomial ring and a Groebner basis of the sub-module generated by @var{plist}
+ in the free module. In the latter case a term order in the free module should be specified.
+ This is specified by @var{[s,ord]}. If @var{s} is 0 then it means TOP (Term Over Position).
+ If @var{s} is 1 then it means POT 1 (Position Over Term). @var{ord} is a term order in the base polynomial ring.
+ @item
+ @code{nd_weyl_gr}, @code{nd_weyl_gr_trace} are for Weyl algebra computation.
+ @item
+ Functions except for F4 related ones can handle rational coeffient cases.
+ @item
+ In general these functions are more efficient than
+ @code{dp_gr_main}, @code{dp_gr_mod_main}, especially over finite fields.
+ @end itemize
+ \E
+ @example
+ [38] load("cyclic")$
+ [49] C=cyclic(7)$
+ [50] V=vars(C)$
+ [51] cputime(1)$
+ [52] dp_gr_mod_main(C,V,0,31991,0)$
+.06sec + gc : 0.313sec(26.4sec)
+ [53] nd_gr(C,V,31991,0)$
+ ndv_alloc=1477188
+.737sec + gc : 0.1837sec(5.921sec)
+ [54] dp_f4_mod_main(C,V,31991,0)$
+.51sec + gc : 0.7109sec(4.221sec)
+ [55] nd_f4(C,V,31991,0)$
+.906sec + gc : 0.126sec(2.032sec)
+ @end example
+ @table @t
+ \JP @item $B;2>H(B
+ \EG @item References
+ @fref{dp_ord},
+ @fref{dp_gr_flags dp_gr_print},
+ \JP @fref{$B7W;;$*$h$SI=<($N@)8f(B}.
+ \EG @fref{Controlling Groebner basis computations}
+ @end table
  \JP @node dp_gr_flags dp_gr_print,,, $B%0%l%V%J4pDl$K4X$9$kH!?t(B
  \EG @node dp_gr_flags dp_gr_print,,, Functions for Groebner basis computation
  @subsection @code{dp_gr_flags}, @code{dp_gr_print}
-Line 2344  when functions other than top level functions are call
+Line 2665  when functions other than top level functions are call
 Line 2344  when functions other than top level functions are call
 Line 2665  when functions other than top level functions are call
  \EG @fref{Setting term orderings}
  @end table
+ \JP @node dp_set_weight dp_set_top_weight dp_weyl_set_weight,,, $B%0%l%V%J4pDl$K4X$9$kH!?t(B
+ \EG @node dp_set_weight dp_set_top_weight dp_weyl_set_weight,,, Functions for Groebner basis computation
+ @subsection @code{dp_set_weight}, @code{dp_set_top_weight}, @code{dp_weyl_set_weight}
+ @findex dp_set_weight
+ @findex dp_set_top_weight
+ @findex dp_weyl_set_weight
+ @table @t
+ @item dp_set_weight([@var{weight}])
+ \JP :: sugar weight $B$N@_Dj(B, $B;2>H(B
+ \EG :: Set and show the sugar weight.
+ @item dp_set_top_weight([@var{weight}])
+ \JP :: top weight $B$N@_Dj(B, $B;2>H(B
+ \EG :: Set and show the top weight.
+ @item dp_weyl_set_weight([@var{weight}])
+ \JP :: weyl weight $B$N@_Dj(B, $B;2>H(B
+ \EG :: Set and show the weyl weight.
+ @end table
+ @table @var
+ @item return
+ \JP $B%Y%/%H%k(B
+ \EG a vector
+ @item weight
+ \JP $B@0?t$N%j%9%H$^$?$O%Y%/%H%k(B
+ \EG a list or vector of integers
+ @end table
+ @itemize @bullet
+ \BJP
+ @item
+ @code{dp_set_weight} $B$O(B sugar weight $B$r(B @var{weight} $B$K@_Dj$9$k(B. $B0z?t$,$J$$;~(B,
+ $B8=:_@_Dj$5$l$F$$$k(B sugar weight $B$rJV$9(B. sugar weight $B$O@5@0?t$r@.J,$H$9$k%Y%/%H%k$G(B,
+ $B3FJQ?t$N=E$_$rI=$9(B. $B<!?t$D$-=g=x$K$*$$$F(B, $BC19`<0$N<!?t$r7W;;$9$k:]$KMQ$$$i$l$k(B.
+ $B@F<!2=JQ?tMQ$K(B, $BKvHx$K(B 1 $B$rIU$12C$($F$*$/$H0BA4$G$"$k(B.
+ @item
+ @code{dp_set_top_weight} $B$O(B top weight $B$r(B @var{weight} $B$K@_Dj$9$k(B. $B0z?t$,$J$$;~(B,
+ $B8=:_@_Dj$5$l$F$$$k(B top weight $B$rJV$9(B. top weight $B$,@_Dj$5$l$F$$$k$H$-(B,
+ $B$^$:(B top weight $B$K$h$kC19`<0Hf3S$r@h$K9T$&(B. tie breaker $B$H$7$F8=:_@_Dj$5$l$F$$$k(B
+ $B9`=g=x$,MQ$$$i$l$k$,(B, $B$3$NHf3S$K$O(B top weight $B$OMQ$$$i$l$J$$(B.
+ @item
+ @code{dp_weyl_set_weight} $B$O(B weyl weight $B$r(B @var{weight} $B$K@_Dj$9$k(B. $B0z?t$,$J$$;~(B,
+ $B8=:_@_Dj$5$l$F$$$k(B weyl weight $B$rJV$9(B. weyl weight w $B$r@_Dj$9$k$H(B,
+ $B9`=g=x7?(B 11 $B$G$N7W;;$K$*$$$F(B, (-w,w) $B$r(B top weight, tie breaker $B$r(B graded reverse lex
+ $B$H$7$?9`=g=x$,@_Dj$5$l$k(B.
+ \E
+ \BEG
+ @item
+ @code{dp_set_weight} sets the sugar weight=@var{weight}. It returns the current sugar weight.
+ A sugar weight is a vector with positive integer components and it represents the weights of variables.
+ It is used for computing the weight of a monomial in a graded ordering.
+ It is recommended to append a component 1 at the end of the weight vector for a homogenizing variable.
+ @item
+ @code{dp_set_top_weight} sets the top weight=@var{weight}. It returns the current top weight.
+ It a top weight is set, the weights of monomials under the top weight are firstly compared.
+ If the the weights are equal then the current term ordering is applied as a tie breaker, but
+ the top weight is not used in the tie breaker.
+ @item
+ @code{dp_weyl_set_weight} sets the weyl weigh=@var{weight}. It returns the current weyl weight.
+ If a weyl weight w is set, in the comparsion by the term order type 11, a term order with
+ the top weight=(-w,w) and the tie breaker=graded reverse lex is applied.
+ \E
+ @end itemize
+ @table @t
+ \JP @item $B;2>H(B
+ \EG @item References
+ @fref{Weight}
+ @end table
  \JP @node dp_ptod,,, $B%0%l%V%J4pDl$K4X$9$kH!?t(B
  \EG @node dp_ptod,,, Functions for Groebner basis computation
  @subsection @code{dp_ptod}
-Line 2526  converting the coefficients into elements of a finite
+Line 2920  converting the coefficients into elements of a finite
 Line 2526  converting the coefficients into elements of a finite
 Line 2920  converting the coefficients into elements of a finite
  @table @t
  \JP @item $B;2>H(B
  \EG @item References
- @fref{dp_nf dp_nf_mod dp_true_nf dp_true_nf_mod},
+ @fref{dp_nf dp_nf_mod dp_true_nf dp_true_nf_mod dp_weyl_nf dp_weyl_nf_mod},
  @fref{subst psubst},
  @fref{setmod}.
  @end table
-Line 2617  These are used internally in @code{hgr()} etc.
+Line 3011  These are used internally in @code{hgr()} etc.
 Line 2617  These are used internally in @code{hgr()} etc.
 Line 3011  These are used internally in @code{hgr()} etc.
  into an integral distributed polynomial such that GCD of all its coefficients
  is 1.
  \E
- @itemx dp_prim(@var{dpoly})
+ @item dp_prim(@var{dpoly})
  \JP :: $BM-M}<0G\$7$F78?t$r@0?t78?tB?9`<078?t$+$D78?t$NB?9`<0(B GCD $B$r(B 1 $B$K$9$k(B.
  \BEG
  :: Converts a distributed polynomial @var{poly} with rational function
-Line 2670  polynomial contents included in the coefficients are n
+Line 3064  polynomial contents included in the coefficients are n
 Line 2670  polynomial contents included in the coefficients are n
 Line 3064  polynomial contents included in the coefficients are n
  @fref{ptozp}.
  @end table
- \JP @node dp_nf dp_nf_mod dp_true_nf dp_true_nf_mod,,, $B%0%l%V%J4pDl$K4X$9$kH!?t(B
+ \JP @node dp_nf dp_nf_mod dp_true_nf dp_true_nf_mod dp_weyl_nf dp_weyl_nf_mod,,, $B%0%l%V%J4pDl$K4X$9$kH!?t(B
- \EG @node dp_nf dp_nf_mod dp_true_nf dp_true_nf_mod,,, Functions for Groebner basis computation
+ \EG @node dp_nf dp_nf_mod dp_true_nf dp_true_nf_mod dp_weyl_nf dp_weyl_nf_mod,,, Functions for Groebner basis computation
  @subsection @code{dp_nf}, @code{dp_nf_mod}, @code{dp_true_nf}, @code{dp_true_nf_mod}
  @findex dp_nf
  @findex  dp_true_nf
  @findex dp_nf_mod
  @findex  dp_true_nf_mod
+ @findex dp_weyl_nf
+ @findex dp_weyl_nf_mod
  @table @t
  @item dp_nf(@var{indexlist},@var{dpoly},@var{dpolyarray},@var{fullreduce})
+ @item dp_weyl_nf(@var{indexlist},@var{dpoly},@var{dpolyarray},@var{fullreduce})
  @item dp_nf_mod(@var{indexlist},@var{dpoly},@var{dpolyarray},@var{fullreduce},@var{mod})
+ @item dp_weyl_nf_mod(@var{indexlist},@var{dpoly},@var{dpolyarray},@var{fullreduce},@var{mod})
  \JP :: $BJ,;6I=8=B?9`<0$N@55,7A$r5a$a$k(B. ($B7k2L$ODj?tG\$5$l$F$$$k2DG=@-$"$j(B)
  \BEG
-Line 2722  is returned in such a list as @code{[numerator, denomi
+Line 3120  is returned in such a list as @code{[numerator, denomi
 Line 2722  is returned in such a list as @code{[numerator, denomi
 Line 3120  is returned in such a list as @code{[numerator, denomi
  @item
  $BJ,;6I=8=B?9`<0(B @var{dpoly} $B$N@55,7A$r5a$a$k(B.
  @item
+ $BL>A0$K(B weyl $B$r4^$`4X?t$O%o%$%kBe?t$K$*$1$k@55,7A7W;;$r9T$&(B. $B0J2<$N@bL@$O(B weyl $B$r4^$`$b$N$KBP$7$F$bF1MM$K@.N)$9$k(B.
+ @item
  @code{dp_nf_mod()}, @code{dp_true_nf_mod()} $B$NF~NO$O(B, @code{dp_mod()} $B$J$I(B
  $B$K$h$j(B, $BM-8BBN>e$NJ,;6I=8=B?9`<0$K$J$C$F$$$J$1$l$P$J$i$J$$(B.
  @item
-Line 2754  is returned in such a list as @code{[numerator, denomi
+Line 3154  is returned in such a list as @code{[numerator, denomi
 Line 2754  is returned in such a list as @code{[numerator, denomi
 Line 3154  is returned in such a list as @code{[numerator, denomi
  @item
  Computes the normal form of a distributed polynomial.
  @item
+ Functions whose name contain @code{weyl} compute normal forms in Weyl algebra. The description below also applies to
+ the functions for Weyl algebra.
+ @item
  @code{dp_nf_mod()} and @code{dp_true_nf_mod()} require
  distributed polynomials with coefficients in a finite field as arguments.
  @item
-Line 3535  refer to @code{dp_true_nf()} and @code{dp_true_nf_mod(
+Line 3938  refer to @code{dp_true_nf()} and @code{dp_true_nf_mod(
 Line 3535  refer to @code{dp_true_nf()} and @code{dp_true_nf_mod(
 Line 3938  refer to @code{dp_true_nf()} and @code{dp_true_nf_mod(
  @fref{dp_ptod},
  @fref{dp_dtop},
  @fref{dp_ord},
- @fref{dp_nf dp_nf_mod dp_true_nf dp_true_nf_mod}.
+ @fref{dp_nf dp_nf_mod dp_true_nf dp_true_nf_mod dp_weyl_nf dp_weyl_nf_mod}.
  @end table
  \JP @node p_terms,,, $B%0%l%V%J4pDl$K4X$9$kH!?t(B

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