===================================================================
RCS file: /home/cvs/OpenXM/src/asir-doc/parts/groebner.texi,v
retrieving revision 1.10
retrieving revision 1.13
diff -u -p -r1.10 -r1.13
--- OpenXM/src/asir-doc/parts/groebner.texi	2003/04/28 03:09:23	1.10
+++ OpenXM/src/asir-doc/parts/groebner.texi	2004/09/13 09:23:30	1.13
@@ -1,4 +1,4 @@
-@comment $OpenXM: OpenXM/src/asir-doc/parts/groebner.texi,v 1.9 2003/04/24 08:13:24 noro Exp $
+@comment $OpenXM: OpenXM/src/asir-doc/parts/groebner.texi,v 1.12 2003/12/27 11:52:07 takayama Exp $
 \BJP
 @node $B%0%l%V%J4pDl$N7W;;(B,,, Top
 @chapter $B%0%l%V%J4pDl$N7W;;(B
@@ -15,6 +15,7 @@
 * $B4pK\E*$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::
@@ -26,6 +27,7 @@
 * Fundamental functions::
 * Controlling Groebner basis computations::
 * Setting term orderings::
+* Weight::
 * Groebner basis computation with rational function coefficients::
 * Change of ordering::
 * Weyl algebra::
@@ -1055,6 +1057,84 @@ 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 orders introduced in the previous section can be generalized
+by setting a weight for each variable.
+\E
+@example
+[0] dp_td(<<1,1,1>>);
+3
+[1] dp_set_weight([1,2,3])$
+[2] dp_td(<<1,1,1>>);
+6
+@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$,$"$k(B. 
+\E
+\BEG
+By default, the total degree of a monomial is equal to
+the sum of all exponents. This means that the weight for each variable
+is set to 1.
+In this example, the weights for the first, the second and the third
+variable are set to 1, 2 and 3 respectively.
+Therefore the total degree of @code{<<1,1,1>>} under this weight,
+which is called the weight of the monomial, is @code{1*1+1*2+1*3=6}.
+By setting weights, different term orders can be set under a term
+order type. For example, a polynomial can be made weighted homogeneous
+by setting an appropriate weight.
+\E
+
+\BJP
+$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
 @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
@@ -1412,6 +1492,14 @@ CPU $B;~4V$G$"$j(B, $B$3$NH!?t$N>l9g$O$[$H$s$IDL?.$
 @item
 @code{dgr()} $B$GI=<($5$l$k;~4V$O(B, $B$3$NH!?t$,<B9T$5$l$F$$$k%W%m%;%9$G$N(B
 CPU $B;~4V$G$"$j(B, $B$3$NH!?t$N>l9g$O$[$H$s$IDL?.$N$?$a$N;~4V$G$"$k(B. 
+@item
+$BB?9`<0%j%9%H(B @var{plist} $B$NMWAG$,J,;6I=8=B?9`<0$N>l9g$O(B
+$B7k2L$bJ,;6I=8=B?9`<0$N%j%9%H$G$"$k(B.
+$B$3$N>l9g(B, $B0z?t$NJ,;6B?9`<0$OM?$($i$l$?=g=x$K=>$$(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
 \BEG
 @item 
@@ -1440,6 +1528,13 @@ for communication.
 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
 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
 @end itemize
 
@@ -3967,7 +4062,7 @@ execute @code{dp_gr_print(2)} in advance.
 @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$k(B.
+$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,