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Diff for /OpenXM/src/asir-doc/parts/groebner.texi between version 1.6 and 1.12

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

version 1.12, 2003/12/27 11:52:07

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 1.11 2003/04/28 06:43:10 noro Exp $

\BJP

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

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

Line 449 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 478 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 1343 Computation of the global b function is implemented as

Line 1354 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 1412 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 1448 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 1705 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 2176 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 2190 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 2206 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 2232 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 3878 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 3905 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 3929 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 3972 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 4025 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

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