OpenXM/src/asir-doc/parts/groebner.texi - diff

Return to groebner.texi CVS log

Up to [local] / OpenXM / src / asir-doc / parts

Diff for /OpenXM/src/asir-doc/parts/groebner.texi between version 1.6 and 1.14

version 1.6, 2003/04/20 09:55:18

version 1.14, 2004/09/14 01:32:34

Line 1

@comment $OpenXM: OpenXM/src/asir-doc/parts/groebner.texi,v 1.5 2003/04/20 08:01:25 noro Exp $

@comment $OpenXM: OpenXM/src/asir-doc/parts/groebner.texi,v 1.13 2004/09/13 09:23:30 noro Exp $

\BJP

@node $B%0%l%V%J4pDl$N7W;;(B,,, Top

@chapter $B%0%l%V%J4pDl$N7W;;(B

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

* Fundamental functions::

* Controlling Groebner basis computations::

* Setting term orderings::

* Weight::

* Groebner basis computation with rational function coefficients::

* Change of ordering::

* Weyl algebra::

Line 449 If `on', various informations during a Groebner basis

Line 451 If `on', various informations during a Groebner basis

displayed.

@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.

@item Stat

\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

Line 471 is shown after every normal computation. After comlet

Line 480 is shown after every normal computation. After comlet

computation the maximal value among the sums is shown.

@item Multiple

@item Content

@itemx Multiple

\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.

backward compatibility $B$N$?$a!"(B@code{Multiple} $B$G@0?tCM$r;XDj$G$-$k(B.

\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

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

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.

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.

An integer value can be set by the keyword @code{Multiple} for

backward compatibility.

@item Demand

Line 1044 beforehand, and some heuristic trial may be inevitable

Line 1057 beforehand, and some heuristic trial may be inevitable

\BJP

@node Weight ,,, $B%0%l%V%J4pDl$N7W;;(B

@section Weight

\BEG

@node Weight,,, Groebner basis computation

@section Weight

\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.

\BEG

Term orderings introduced in the previous section can be generalized

by setting a weight for each variable.

@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.

\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.

\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.

\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.

\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.

\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.

@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.

\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()}.

\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

Line 1343 Computation of the global b function is implemented as

Line 1489 Computation of the global b function is implemented as

* lex_hensel_gsl tolex_gsl tolex_gsl_d::

* primadec primedec::

* primedec_mod::

* bfunction generic_bfct::

* bfunction bfct generic_bfct ann ann0::

@end menu

\JP @node gr hgr gr_mod,,, $B%0%l%V%J4pDl$K4X$9$kH!?t(B

Line 1401 strategy $B$K$h$k7W;;(B, @code{hgr()} $B$O(B trace

Line 1547 strategy $B$K$h$k7W;;(B, @code{hgr()} $B$O(B trace

@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).

\BEG

@item

Line 1429 Therefore this function is useful to reduce the actual

Line 1583 Therefore this function is useful to reduce the actual

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).

@end itemize

Line 1679 processes.

Line 1840 processes.

@item lex_hensel_gsl(@var{plist},@var{vlist1},@var{order},@var{vlist2},@var{homo})

\JP :: GSL $B7A<0$N%$%G%"%k4pDl$N7W;;(B

\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

\EG :: Computation of an GSL form ideal basis stating from a Groebner basis

@end table

Line 2150 except for lack of the argument for controlling homoge

Line 2311 except for lack of the argument for controlling homoge

@table @t

@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

\BEG :: Set and show various parameters for cotrolling computations

and showing informations.

Line 2164 and showing informations.

Line 2325 and showing informations.

@item list

\JP $B%j%9%H(B

\EG list

@item i

\JP $B@0?t(B

\EG integer

@end table

@itemize @bullet

Line 2177 and showing informations.

Line 2341 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

$B@_Dj$5$l$k(B. $B%Q%i%a%?L>$OJ8;zNs$GM?$($kI,MW$,$"$k(B.

@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.

\BEG

Line 2194 Arguments must be specified as a list such as

Line 2367 Arguments must be specified as a list such as

strings.

@item

@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.

@end itemize

Line 3831 if an input ideal is not radical.

Line 4013 if an input ideal is not radical.

$BItJ,$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

$B>l9g$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.

\BEG

@item

Line 3855 is tried by computing the intersection of obtained com

Line 4040 is tried by computing the intersection of obtained com

incrementally. In general, this strategy is useful when the krull

dimension of the ideal is high, but it may add some overhead

if the dimension is small.

@item

If you want to see internal information during the computation,

execute @code{dp_gr_print(2)} in advance.

@end itemize

Line 3876 if the dimension is small.

Line 4064 if the dimension is small.

@fref{modfctr},

@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}.

\EG @fref{Setting term orderings}.

\EG @fref{Setting term orderings},

@fref{dp_gr_flags dp_gr_print}.

@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 bfct

@findex generic_bfct

@findex ann

@findex ann0

@table @t

@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

\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

@itemx f

\JP $BB?9`<0$^$?$O%j%9%H(B

\EG polynomial or list

@item f

\JP $BB?9`<0(B

\EG polynomial

@item plist

Line 3907 if the dimension is small.

Line 4107 if the dimension is small.

@itemize @bullet

\BJP

@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]}

$B$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 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

$B$r=g$KJB$Y$k(B.

@item $B>\:Y$K$D$$$F$O(B, [SST] $B$r8+$h(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$,9bB.$+$OF~NO$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.

\BEG

@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}.

@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 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}.

@var{vlist} is the list of @code{x}-variables,

@var{vlist} is the list of corresponding @code{D}-variables.

@item See [SST] for the details.

@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.

@end itemize

Line 3945 x*y*dt+5*z^4*dt+dz,-x^4-z*y*x-y^4-z^5+t]$

Line 4160 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

FreeBSD-CVSweb <freebsd-cvsweb@FreeBSD.org>