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version 1.13, 2004/09/13 09:23:30 version 1.14, 2004/09/14 01:32:34
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 @comment $OpenXM: OpenXM/src/asir-doc/parts/groebner.texi,v 1.12 2003/12/27 11:52:07 takayama Exp $  @comment $OpenXM: OpenXM/src/asir-doc/parts/groebner.texi,v 1.13 2004/09/13 09:23:30 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 1069  beforehand, and some heuristic trial may be inevitable
Line 1069  beforehand, and some heuristic trial may be inevitable
 $B$h$j0lHLE*$J$b$N$H$J$k(B.  $B$h$j0lHLE*$J$b$N$H$J$k(B.
 \E  \E
 \BEG  \BEG
 Term orders introduced in the previous section can be generalized  Term orderings introduced in the previous section can be generalized
 by setting a weight for each variable.  by setting a weight for each variable.
 \E  \E
 @example  @example
Line 1097  In this example, the weights for the first, the second
Line 1097  In this example, the weights for the first, the second
 variable are set to 1, 2 and 3 respectively.  variable are set to 1, 2 and 3 respectively.
 Therefore the total degree of @code{<<1,1,1>>} under this weight,  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}.  which is called the weight of the monomial, is @code{1*1+1*2+1*3=6}.
 By setting weights, different term orders can be set under a term  By setting weights, different term orderings can be set under a type of
 order type. For example, a polynomial can be made weighted homogeneous  term ordeing. In some case a polynomial can
 by setting an appropriate weight.  be made weighted homogeneous by setting an appropriate weight.
 \E  \E
   
 \BJP  \BJP
Line 1131  is also considered as a refinement of comparison by we
Line 1131  is also considered as a refinement of comparison by we
 It compares two terms by using a weight vector whose elements  It compares two terms by using a weight vector whose elements
 corresponding to variables in a block is 1 and 0 otherwise,  corresponding to variables in a block is 1 and 0 otherwise,
 then it applies a tie breaker.  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  \E
   
 \BJP  \BJP
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