| version 1.7, 2003/04/21 03:07:32 |
version 1.13, 2004/09/13 09:23:30 |
|
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| @comment $OpenXM: OpenXM/src/asir-doc/parts/groebner.texi,v 1.6 2003/04/20 09:55:18 noro Exp $ |
@comment $OpenXM: OpenXM/src/asir-doc/parts/groebner.texi,v 1.12 2003/12/27 11:52:07 takayama 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 |
|
|
| * $B4pK\E*$JH!?t(B:: |
* $B4pK\E*$JH!?t(B:: |
| * $B7W;;$*$h$SI=<($N@)8f(B:: |
* $B7W;;$*$h$SI=<($N@)8f(B:: |
| * $B9`=g=x$N@_Dj(B:: |
* $B9`=g=x$N@_Dj(B:: |
| |
* Weight:: |
| * $BM-M}<0$r78?t$H$9$k%0%l%V%J4pDl7W;;(B:: |
* $BM-M}<0$r78?t$H$9$k%0%l%V%J4pDl7W;;(B:: |
| * $B4pDlJQ49(B:: |
* $B4pDlJQ49(B:: |
| * Weyl $BBe?t(B:: |
* Weyl $BBe?t(B:: |
|
|
| * Fundamental functions:: |
* Fundamental functions:: |
| * Controlling Groebner basis computations:: |
* Controlling Groebner basis computations:: |
| * Setting term orderings:: |
* Setting term orderings:: |
| |
* Weight:: |
| * Groebner basis computation with rational function coefficients:: |
* Groebner basis computation with rational function coefficients:: |
| * Change of ordering:: |
* Change of ordering:: |
| * Weyl algebra:: |
* Weyl algebra:: |
| Line 1055 beforehand, and some heuristic trial may be inevitable |
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| Line 1057 beforehand, and some heuristic trial may be inevitable |
|
| \E |
\E |
| |
|
| \BJP |
\BJP |
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@node Weight ,,, $B%0%l%V%J4pDl$N7W;;(B |
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@section Weight |
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\E |
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\BEG |
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@node Weight,,, Groebner basis computation |
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@section Weight |
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\E |
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\BJP |
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$BA0@a$G>R2p$7$?9`=g=x$O(B, $B3FJQ?t$K(B weight ($B=E$_(B) $B$r@_Dj$9$k$3$H$G(B |
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$B$h$j0lHLE*$J$b$N$H$J$k(B. |
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\E |
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\BEG |
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Term orders introduced in the previous section can be generalized |
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by setting a weight for each variable. |
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\E |
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@example |
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[0] dp_td(<<1,1,1>>); |
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3 |
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[1] dp_set_weight([1,2,3])$ |
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[2] dp_td(<<1,1,1>>); |
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6 |
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@end example |
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\BJP |
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$BC19`<0$NA4<!?t$r7W;;$9$k:](B, $B%G%U%)%k%H$G$O(B |
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$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 |
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$B9M$($F$$$k$3$H$KAjEv$9$k(B. $B$3$NNc$G$O(B, $BBh0l(B, $BBhFs(B, $BBh;0JQ?t$N(B |
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weight $B$r$=$l$>$l(B 1,2,3 $B$H;XDj$7$F$$$k(B. $B$3$N$?$a(B, @code{<<1,1,1>>} |
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$B$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. |
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weight $B$r@_Dj$9$k$3$H$G(B, $BF1$89`=g=x7?$N$b$H$G0[$J$k9`=g=x$,Dj5A$G$-$k(B. |
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$BNc$($P(B, weight $B$r$&$^$/@_Dj$9$k$3$H$G(B, $BB?9`<0$r(B weighted homogeneous |
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$B$K$9$k$3$H$,$G$-$k>l9g$,$"$k(B. |
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\E |
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\BEG |
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By default, the total degree of a monomial is equal to |
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the sum of all exponents. This means that the weight for each variable |
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is set to 1. |
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In this example, the weights for the first, the second and the third |
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variable are set to 1, 2 and 3 respectively. |
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Therefore the total degree of @code{<<1,1,1>>} under this weight, |
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which is called the weight of the monomial, is @code{1*1+1*2+1*3=6}. |
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By setting weights, different term orders can be set under a term |
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order type. For example, a polynomial can be made weighted homogeneous |
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by setting an appropriate weight. |
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\E |
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|
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\BJP |
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$B3FJQ?t$KBP$9$k(B weight $B$r$^$H$a$?$b$N$r(B weight vector $B$H8F$V(B. |
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$B$9$Y$F$N@.J,$,@5$G$"$j(B, $B%0%l%V%J4pDl7W;;$K$*$$$F(B, $BA4<!?t$N(B |
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$BBe$o$j$KMQ$$$i$l$k$b$N$rFC$K(B sugar weight $B$H8F$V$3$H$K$9$k(B. |
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sugar strategy $B$K$*$$$F(B, $BA4<!?t$NBe$o$j$K;H$o$l$k$+$i$G$"$k(B. |
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$B0lJ}$G(B, $B3F@.J,$,I,$:$7$b@5$H$O8B$i$J$$(B weight vector $B$O(B, |
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sugar weight $B$H$7$F@_Dj$9$k$3$H$O$G$-$J$$$,(B, $B9`=g=x$N0lHL2=$K$O(B |
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$BM-MQ$G$"$k(B. $B$3$l$i$O(B, $B9TNs$K$h$k9`=g=x$N@_Dj$K$9$G$K8=$l$F(B |
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$B$$$k(B. $B$9$J$o$A(B, $B9`=g=x$rDj5A$9$k9TNs$N3F9T$,(B, $B0l$D$N(B weight vector |
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$B$H8+$J$5$l$k(B. $B$^$?(B, $B%V%m%C%/=g=x$O(B, $B3F%V%m%C%/$N(B |
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$BJQ?t$KBP1~$9$k@.J,$N$_(B 1 $B$GB>$O(B 0 $B$N(B weight vector $B$K$h$kHf3S$r(B |
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$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. |
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\E |
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|
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\BEG |
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A list of weights for all variables is called a weight vector. |
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A weight vector is called a sugar weight vector if |
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its elements are all positive and it is used for computing |
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a weighted total degree of a monomial, because such a weight |
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is used instead of total degree in sugar strategy. |
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On the other hand, a weight vector whose elements are not necessarily |
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positive cannot be set as a sugar weight, but it is useful for |
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generalizing term order. In fact, such a weight vector already |
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appeared in a matrix order. That is, each row of a matrix defining |
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a term order is regarded as a weight vector. A block order |
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is also considered as a refinement of comparison by weight vectors. |
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It compares two terms by using a weight vector whose elements |
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corresponding to variables in a block is 1 and 0 otherwise, |
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then it applies a tie breaker. |
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|
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\E |
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|
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\BJP |
| @node $BM-M}<0$r78?t$H$9$k%0%l%V%J4pDl7W;;(B,,, $B%0%l%V%J4pDl$N7W;;(B |
@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 |
@section $BM-M}<0$r78?t$H$9$k%0%l%V%J4pDl7W;;(B |
| \E |
\E |
| Line 1354 Computation of the global b function is implemented as |
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| Line 1434 Computation of the global b function is implemented as |
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| * 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 1412 strategy $B$K$h$k7W;;(B, @code{hgr()} $B$O(B trace |
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| Line 1492 strategy $B$K$h$k7W;;(B, @code{hgr()} $B$O(B trace |
|
| @item |
@item |
| @code{dgr()} $B$GI=<($5$l$k;~4V$O(B, $B$3$NH!?t$,<B9T$5$l$F$$$k%W%m%;%9$G$N(B |
@code{dgr()} $B$GI=<($5$l$k;~4V$O(B, $B$3$NH!?t$,<B9T$5$l$F$$$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. |
CPU $B;~4V$G$"$j(B, $B$3$NH!?t$N>l9g$O$[$H$s$IDL?.$N$?$a$N;~4V$G$"$k(B. |
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@item |
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$BB?9`<0%j%9%H(B @var{plist} $B$NMWAG$,J,;6I=8=B?9`<0$N>l9g$O(B |
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$B7k2L$bJ,;6I=8=B?9`<0$N%j%9%H$G$"$k(B. |
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$B$3$N>l9g(B, $B0z?t$NJ,;6B?9`<0$OM?$($i$l$?=g=x$K=>$$(B @code{dp_sort} $B$G(B |
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$B%=!<%H$5$l$F$+$i7W;;$5$l$k(B. |
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$BB?9`<0%j%9%H$NMWAG$,J,;6I=8=B?9`<0$N>l9g$b(B |
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$BJQ?t$N?tJ,$NITDj85$N%j%9%H$r(B @var{vlist} $B0z?t$H$7$FM?$($J$$$H$$$1$J$$(B |
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($B%@%_!<(B). |
| \E |
\E |
| \BEG |
\BEG |
| @item |
@item |
| Line 1440 Therefore this function is useful to reduce the actual |
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| Line 1528 Therefore this function is useful to reduce the actual |
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| The CPU time shown after an exection of @code{dgr()} indicates |
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 |
that of the master process, and most of the time corresponds to the time |
| for communication. |
for communication. |
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@item |
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When the elements of @var{plist} are distributed polynomials, |
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the result is also a list of distributed polynomials. |
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In this case, firstly the elements of @var{plist} is sorted by @code{dp_sort} |
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and the Grobner basis computation is started. |
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Variables must be given in @var{vlist} even in this case |
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(these variables are dummy). |
| \E |
\E |
| @end itemize |
@end itemize |
| |
|
|
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| @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 3918 execute @code{dp_gr_print(2)} in advance. |
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| Line 4013 execute @code{dp_gr_print(2)} in advance. |
|
| @fref{dp_gr_flags dp_gr_print}. |
@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 |
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@findex ann |
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@findex ann0 |
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|
| @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 |
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\EG :: Computes the global @var{b} function of a polynomial or an ideal |
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@item ann(@var{f}) |
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@itemx ann0(@var{f}) |
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\JP :: $BB?9`<0$N%Y%-$N(B annihilator $B$N7W;;(B |
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\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 |
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\EG polynomial or list |
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@item f |
| \JP $BB?9`<0(B |
\JP $BB?9`<0(B |
| \EG polynomial |
\EG polynomial |
| @item plist |
@item plist |
| Line 3946 execute @code{dp_gr_print(2)} in advance. |
|
| Line 4052 execute @code{dp_gr_print(2)} in advance. |
|
| @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. |
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@item @code{bfunction} $B$H(B @code{bfct} $B$G$OMQ$$$F$$$k%"%k%4%j%:%`$,(B |
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$B0[$J$k(B. $B$I$A$i$,9bB.$+$OF~NO$K$h$k(B. |
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@item @code{ann(@var{f})} $B$O(B, @code{@var{f}^s} $B$N(B annihilator ideal |
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$B$N@8@.7O$rJV$9(B. @code{ann(@var{f})} $B$O(B, @code{[@var{a},@var{list}]} |
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$B$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, |
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@var{list} $B$O(B @code{ann(@var{f})} $B$N7k2L$N(B @code{s}$ $B$K(B, @var{a} $B$r(B |
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$BBeF~$7$?$b$N$G$"$k(B. |
| @item $B>\:Y$K$D$$$F$O(B, [Saito,Sturmfels,Takayama] $B$r8+$h(B. |
@item $B>\:Y$K$D$$$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 @code{bfunction(@var{f})} and @code{bfct(@var{f})} implement |
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different algorithms and the efficiency depends on inputs. |
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@item @code{ann(@var{f})} returns the generator set of the annihilator |
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ideal of @code{@var{f}^s}. |
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@code{ann(@var{f})} returns a list @code{[@var{a},@var{list}]}, |
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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. |
@item See [Saito,Sturmfels,Takayama] for the details. |
| \E |
\E |
| @end itemize |
@end itemize |
| Line 3984 x*y*dt+5*z^4*dt+dz,-x^4-z*y*x-y^4-z^5+t]$ |
|
| Line 4105 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 |
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[220] P=x^3-y^2$ |
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[221] ann(P); |
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[2*dy*x+3*dx*y^2,-3*dx*x-2*dy*y+6*s] |
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[222] ann0(P); |
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[-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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