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version 1.3, 1999/12/24 04:38:04

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

Line 1

@comment $OpenXM: OpenXM/src/asir-doc/parts/groebner.texi,v 1.2 1999/12/21 02:47:31 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 17

* $B9`=g=x$N@_Dj(B::

* $BM-M}<0$r78?t$H$9$k%0%l%V%J4pDl7W;;(B::

* $B4pDlJQ49(B::

* Weyl $BBe?t(B::

* $B%0%l%V%J4pDl$K4X$9$kH!?t(B::

\BEG

Line 27

Line 28

* Setting term orderings::

* Groebner basis computation with rational function coefficients::

* Change of ordering::

* Weyl algebra::

* Functions for Groebner basis computation::

@end menu

Line 228 the head term and the head coefficient respectively.

Line 230 the head term and the head coefficient respectively.

@noindent

\BJP

$B%0%l%V%J4pDl$r7W;;$9$k$?$a$N4pK\E*$JH!?t$O(B @code{dp_gr_main()} $B$*$h$S(B

@code{dp_gr_mod_main()} $B$J$k(B 2 $B$D$NAH$_9~$_H!?t$G$"$k$,(B, $BDL>o$O(B, $B%Q%i%a%?(B

@code{dp_gr_mod_main()}, @code{dp_gr_f_main()}

$B$J$k(B 3 $B$D$NAH$_9~$_H!?t$G$"$k$,(B, $BDL>o$O(B, $B%Q%i%a%?(B

$B@_Dj$J$I$r9T$C$?$N$A$3$l$i$r8F$S=P$9%f!<%6H!?t$rMQ$$$k$N$,JXMx$G$"$k(B.

$B$3$l$i$N%f!<%6H!?t$O(B, $B%U%!%$%k(B @samp{gr} $B$r(B @code{load()} $B$K$h$jFI(B

$B$_9~$`$3$H$K$h$j;HMQ2DG=$H$J$k(B. @samp{gr} $B$O(B, @b{Asir} $B$NI8=`(B

$B%i%$%V%i%j%G%#%l%/%H%j$KCV$+$l$F$$$k(B. $B$h$C$F(B, $B4D6-JQ?t(B @code{ASIR_LIBDIR}

$B%i%$%V%i%j%G%#%l%/%H%j$KCV$+$l$F$$$k(B.

$B$rFC$K0[$J$k%Q%9$K@_Dj$7$J$$8B$j(B, $B%U%!%$%kL>$N$_$GFI$_9~$`$3$H$,$G$-$k(B.

\BEG

Facilities for computing Groebner bases are provided not by built-in

Facilities for computing Groebner bases are

functions but by a set of user functions written in @b{Asir}.

@code{dp_gr_main()}, @code{dp_gr_mod_main()}and @code{dp_gr_f_main()}.

The set of functions is provided as a file (sometimes called package),

To call these functions,

named @samp{gr}.

it is necessary to set several parameters correctly and it is convenient

to use a set of interface functions provided in the library file

@samp{gr}.

The facilities will be ready to use after you load the package by

@code{load()}. The package @samp{gr} is placed in the standard library

directory of @b{Asir}. Therefore, it is loaded simply by specifying

directory of @b{Asir}.

its file name, unless the environment variable @code{ASIR_LIBDIR}

is set to a non-standard one.

@example

Line 350 These parameters can be set and examined by a built-in

Line 352 These parameters can be set and examined by a built-in

@example

[100] dp_gr_flags();

[Demand,0,NoSugar,0,NoCriB,0,NoGC,0,NoMC,0,NoRA,0,NoGCD,0,Top,0,ShowMag,1,

[Demand,0,NoSugar,0,NoCriB,0,NoGC,0,NoMC,0,NoRA,0,NoGCD,0,Top,0,

Print,1,Stat,0,Reverse,0,InterReduce,0,Multiple,0]

ShowMag,1,Print,1,Stat,0,Reverse,0,InterReduce,0,Multiple,0]

[101]

@end example

Line 447 If `on', various informations during a Groebner basis

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 469 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 530 membercheck

Line 543 membercheck

(0,0)(0,0)(0,0)(0,0)

gbcheck total 8 pairs

........

UP=(0,0)SP=(0,0)SPM=(0,0)NF=(0,0)NFM=(0.010002,0)ZNFM=(0.010002,0)PZ=(0,0)

UP=(0,0)SP=(0,0)SPM=(0,0)NF=(0,0)NFM=(0.010002,0)ZNFM=(0.010002,0)

NP=(0,0)MP=(0,0)RA=(0,0)MC=(0,0)GC=(0,0)T=40,B=0 M=8 F=6 D=12 ZR=5 NZR=6

PZ=(0,0)NP=(0,0)MP=(0,0)RA=(0,0)MC=(0,0)GC=(0,0)T=40,B=0 M=8 F=6

Max_mag=6

D=12 ZR=5 NZR=6 Max_mag=6

[94]

@end example

Line 992 time as well as the choice of types of term orderings.

Line 1005 time as well as the choice of types of term orderings.

-40*t^8+70*t^7+252*t^6+30*t^5-140*t^4-168*t^3+2*t^2-12*t+16)*z^2*y

+(-12*t^16+72*t^13-28*t^11-180*t^10+112*t^8+240*t^7+28*t^6-127*t^5

-167*t^4-55*t^3+30*t^2+58*t-15)*z^4,

(y+t^2*z^2)*x+y^7+(20*t^2+6*t+1)*y^2+(-t^17+6*t^14-21*t^12-15*t^11+84*t^9

(y+t^2*z^2)*x+y^7+(20*t^2+6*t+1)*y^2+(-t^17+6*t^14-21*t^12-15*t^11

+20*t^8-35*t^7-126*t^6-15*t^5+70*t^4+84*t^3-t^2+5*t-9)*z^2*y+(6*t^16-36*t^13

+84*t^9+20*t^8-35*t^7-126*t^6-15*t^5+70*t^4+84*t^3-t^2+5*t-9)*z^2*y

+14*t^11+90*t^10-56*t^8-120*t^7-14*t^6+64*t^5+84*t^4+27*t^3-16*t^2-30*t+7)*z^4,

+(6*t^16-36*t^13+14*t^11+90*t^10-56*t^8-120*t^7-14*t^6+64*t^5+84*t^4

(t^3-1)*x-y^6+(-6*t^13+24*t^10-20*t^8-36*t^7+40*t^5+24*t^4-6*t^3-20*t^2-6*t-1)*y

+27*t^3-16*t^2-30*t+7)*z^4,

+(t^17-6*t^14+9*t^12+15*t^11-36*t^9-20*t^8-5*t^7+54*t^6+15*t^5+10*t^4-36*t^3

(t^3-1)*x-y^6+(-6*t^13+24*t^10-20*t^8-36*t^7+40*t^5+24*t^4-6*t^3-20*t^2

-11*t^2-5*t+9)*z^2,

-6*t-1)*y+(t^17-6*t^14+9*t^12+15*t^11-36*t^9-20*t^8-5*t^7+54*t^6+15*t^5

+10*t^4-36*t^3-11*t^2-5*t+9)*z^2,

-y^8-8*t*y^3+16*z^2*y^2+(-8*t^16+48*t^13-56*t^11-120*t^10+224*t^8+160*t^7

-56*t^6-336*t^5-112*t^4+112*t^3+224*t^2+24*t-56)*z^4*y+(t^24-8*t^21+20*t^19

-56*t^6-336*t^5-112*t^4+112*t^3+224*t^2+24*t-56)*z^4*y+(t^24-8*t^21

+28*t^18-120*t^16-56*t^15+14*t^14+300*t^13+70*t^12-56*t^11-400*t^10-84*t^9

+20*t^19+28*t^18-120*t^16-56*t^15+14*t^14+300*t^13+70*t^12-56*t^11

+84*t^8+268*t^7+84*t^6-56*t^5-63*t^4-36*t^3+46*t^2-12*t+1)*z,

-400*t^10-84*t^9+84*t^8+268*t^7+84*t^6-56*t^5-63*t^4-36*t^3+46*t^2

2*t*y^5+z*y^2+(-2*t^11+8*t^8-20*t^6-12*t^5+40*t^3+8*t^2-10*t-20)*z^3*y+8*t^14

-12*t+1)*z,2*t*y^5+z*y^2+(-2*t^11+8*t^8-20*t^6-12*t^5+40*t^3+8*t^2

-32*t^11+48*t^8-t^7-32*t^5-6*t^4+9*t^2-t,

-10*t-20)*z^3*y+8*t^14-32*t^11+48*t^8-t^7-32*t^5-6*t^4+9*t^2-t,

-z*y^3+(t^7-2*t^4+3*t^2+t)*y+(-2*t^6+4*t^3+2*t-2)*z^2,

2*t^2*y^3+z^2*y^2+(-2*t^5+4*t^2-6)*z^4*y+(4*t^8-t^7-8*t^5+2*t^4-4*t^3+5*t^2-t)*z,

2*t^2*y^3+z^2*y^2+(-2*t^5+4*t^2-6)*z^4*y

+(4*t^8-t^7-8*t^5+2*t^4-4*t^3+5*t^2-t)*z,

z^3*y^2+2*t^3*y+(-t^7+2*t^4+t^2-t)*z^2,

-t*z*y^2-2*z^3*y+t^8-2*t^5-t^3+t^2,

-t^3*y^2-2*t^2*z^2*y+(t^6-2*t^3-t+1)*z^4,

-t^3*y^2-2*t^2*z^2*y+(t^6-2*t^3-t+1)*z^4,z^5-t^4]

z^5-t^4]

[93] gr(B,[t,z,y,x],2);

[x^10-t,x^8-z,x^31-x^6-x-y]

@end example

Line 1200 Refer to the sections for each functions.

Line 1214 Refer to the sections for each functions.

\BJP

@node Weyl $BBe?t(B,,, $B%0%l%V%J4pDl$N7W;;(B

@section Weyl $BBe?t(B

\BEG

@node Weyl algebra,,, Groebner basis computation

@section Weyl algebra

@noindent

\BJP

$B$3$l$^$G$O(B, $BDL>o$N2D49$JB?9`<04D$K$*$1$k%0%l%V%J4pDl7W;;$K$D$$$F(B

$B=R$Y$F$-$?$,(B, $B%0%l%V%J4pDl$NM}O@$O(B, $B$"$k>r7o$rK~$?$9Hs2D49$J(B

$B4D$K$b3HD%$G$-$k(B. $B$3$N$h$&$J4D$NCf$G(B, $B1~MQ>e$b=EMW$J(B,

Weyl $BBe?t(B, $B$9$J$o$AB?9`<04D>e$NHyJ,:nMQAG4D$N1i;;$*$h$S(B

$B%0%l%V%J4pDl7W;;$,(B Risa/Asir $B$K<BAu$5$l$F$$$k(B.

$BBN(B @code{K} $B>e$N(B @code{n} $B<!85(B Weyl $BBe?t(B

@code{D=K<x1,@dots{},xn,D1,@dots{},Dn>} $B$O(B

\BEG

So far we have explained Groebner basis computation in

commutative polynomial rings. However Groebner basis can be

considered in more general non-commutative rings.

Weyl algebra is one of such rings and

Risa/Asir implements fundamental operations

in Weyl algebra and Groebner basis computation in Weyl algebra.

The @code{n} dimensional Weyl algebra over a field @code{K},

@code{D=K<x1,@dots{},xn,D1,@dots{},Dn>} is a non-commutative

algebra which has the following fundamental relations:

@code{xi*xj-xj*xi=0}, @code{Di*Dj-Dj*Di=0}, @code{Di*xj-xj*Di=0} (@code{i!=j}),

@code{Di*xi-xi*Di=1}

\BJP

$B$H$$$&4pK\4X78$r;}$D4D$G$"$k(B. @code{D} $B$O(B $BB?9`<04D(B @code{K[x1,@dots{},xn]} $B$r78?t(B

$B$H$9$kHyJ,:nMQAG4D$G(B, @code{Di} $B$O(B @code{xi} $B$K$h$kHyJ,$rI=$9(B. $B8r494X78$K$h$j(B,

@code{D} $B$N85$O(B, @code{x1^i1*@dots{}*xn^in*D1^j1*@dots{}*Dn^jn} $B$J$kC19`(B

$B<0$N(B @code{K} $B@~7A7k9g$H$7$F=q$-I=$9$3$H$,$G$-$k(B.

Risa/Asir $B$K$*$$$F$O(B, $B$3$NC19`<0$r(B, $B2D49$JB?9`<0$HF1MM$K(B

@code{<<i1,@dots{},in,j1,@dots{},jn>>} $B$GI=$9(B. $B$9$J$o$A(B, @code{D} $B$N85$b(B

$BJ,;6I=8=B?9`<0$H$7$FI=$5$l$k(B. $B2C8:;;$O(B, $B2D49$N>l9g$HF1MM$K(B, @code{+}, @code{-}

$B$K$h$j(B

$B<B9T$G$-$k$,(B, $B>h;;$O(B, $BHs2D49@-$r9MN8$7$F(B @code{dp_weyl_mul()} $B$H$$$&4X?t(B

$B$K$h$j<B9T$9$k(B.

\BEG

@code{D} is the ring of differential operators whose coefficients

are polynomials in @code{K[x1,@dots{},xn]} and

@code{Di} denotes the differentiation with respect to @code{xi}.

According to the commutation relation,

elements of @code{D} can be represented as a @code{K}-linear combination

of monomials @code{x1^i1*@dots{}*xn^in*D1^j1*@dots{}*Dn^jn}.

In Risa/Asir, this type of monomial is represented

by @code{<<i1,@dots{},in,j1,@dots{},jn>>} as in the case of commutative

polynomial.

That is, elements of @code{D} are represented by distributed polynomials.

Addition and subtraction can be done by @code{+}, @code{-},

but multiplication is done by calling @code{dp_weyl_mul()} because of

the non-commutativity of @code{D}.

@example

[0] A=<<1,2,2,1>>;

(1)*<<1,2,2,1>>

[1] B=<<2,1,1,2>>;

(1)*<<2,1,1,2>>

[2] A*B;

(1)*<<3,3,3,3>>

[3] dp_weyl_mul(A,B);

(1)*<<3,3,3,3>>+(1)*<<3,2,3,2>>+(4)*<<2,3,2,3>>+(4)*<<2,2,2,2>>

+(2)*<<1,3,1,3>>+(2)*<<1,2,1,2>>

@end example

\BJP

$B%0%l%V%J4pDl7W;;$K$D$$$F$b(B, Weyl $BBe?t@lMQ$N4X?t$H$7$F(B,

$B<!$N4X?t$,MQ0U$7$F$"$k(B.

\BEG

The following functions are avilable for Groebner basis computation

in Weyl algebra:

@code{dp_weyl_gr_main()},

@code{dp_weyl_gr_mod_main()},

@code{dp_weyl_gr_f_main()},

@code{dp_weyl_f4_main()},

@code{dp_weyl_f4_mod_main()}.

\BJP

$B$^$?(B, $B1~MQ$H$7$F(B, global b $B4X?t$N7W;;$,<BAu$5$l$F$$$k(B.

\BEG

Computation of the global b function is implemented as an application.

\BJP

@node $B%0%l%V%J4pDl$K4X$9$kH!?t(B,,, $B%0%l%V%J4pDl$N7W;;(B

@section $B%0%l%V%J4pDl$K4X$9$kH!?t(B

Line 1214 Refer to the sections for each functions.

Line 1327 Refer to the sections for each functions.

* lex_hensel_gsl tolex_gsl tolex_gsl_d::

* gr_minipoly minipoly::

* tolexm minipolym::

* dp_gr_main dp_gr_mod_main::

* 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::

* dp_f4_main dp_f4_mod_main::

* dp_f4_main dp_f4_mod_main dp_weyl_f4_main dp_weyl_f4_mod_main::

* dp_gr_flags dp_gr_print::

* dp_ord::

* dp_ptod::

Line 1240 Refer to the sections for each functions.

Line 1353 Refer to the sections for each functions.

* dp_vtoe dp_etov::

* lex_hensel_gsl tolex_gsl tolex_gsl_d::

* primadec primedec::

* primedec_mod::

* 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 1263 Refer to the sections for each functions.

Line 1378 Refer to the sections for each functions.

@item return

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

\EG list

@item plist, vlist, procs

@item plist vlist procs

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

\EG list

@item order

Line 1297 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 1325 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 1342 for communication.

Line 1472 for communication.

@table @t

\JP @item $B;2>H(B

\EG @item References

@comment @fref{dp_gr_main dp_gr_mod_main},

@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},

@fref{dp_gr_main dp_gr_mod_main},

@fref{dp_ord}.

@end table

Line 1372 for communication.

Line 1501 for communication.

@item return

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

\EG list

@item plist, vlist1, vlist2, procs

@item plist vlist1 vlist2 procs

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

\EG list

@item order

Line 1560 processes.

Line 1689 processes.

@table @t

\JP @item $B;2>H(B

\EG @item References

@fref{dp_gr_main dp_gr_mod_main},

@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{dp_ord}, @fref{$BJ,;67W;;(B}

\EG @fref{dp_ord}, @fref{Distributed computation}

@end table

Line 1576 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 1586 processes.

Line 1715 processes.

@item return

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

\EG list

@item plist, vlist1, vlist2, procs

@item plist vlist1 vlist2 procs

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

\EG list

@item order

Line 1662 processes.

Line 1791 processes.

[108] GSL[1];

[u2,10352277157007342793600000000*u0^31-...]

[109] GSL[5];

[u0,11771021876193064124640000000*u0^32-...,376672700038178051988480000000*u0^31-...]

[u0,11771021876193064124640000000*u0^32-...,

376672700038178051988480000000*u0^31-...]

@end example

@table @t

Line 1692 processes.

Line 1822 processes.

@item return

\JP $BB?9`<0(B

\EG polynomial

@item plist, vlist

@item plist vlist

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

\EG list

@item order

Line 1789 for @code{gr_minipoly()}.

Line 1919 for @code{gr_minipoly()}.

@item return

\JP @code{tolexm()} : $B%j%9%H(B, @code{minipolym()} : $BB?9`<0(B

\EG @code{tolexm()} : list, @code{minipolym()} : polynomial

@item plist, vlist1, vlist2

@item plist vlist1 vlist2

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

\EG list

@item order

Line 1837 z^32+11405*z^31+20868*z^30+21602*z^29+...

Line 1967 z^32+11405*z^31+20868*z^30+21602*z^29+...

@fref{gr_minipoly minipoly}.

@end table

\JP @node dp_gr_main dp_gr_mod_main,,, $B%0%l%V%J4pDl$K4X$9$kH!?t(B

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

\EG @node dp_gr_main dp_gr_mod_main,,, Functions for Groebner basis computation

\EG @node 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,,, Functions for Groebner basis computation

@subsection @code{dp_gr_main}, @code{dp_gr_mod_main}

@subsection @code{dp_gr_main}, @code{dp_gr_mod_main}, @code{dp_gr_f_main}, @code{dp_weyl_gr_main}, @code{dp_weyl_gr_mod_main}, @code{dp_weyl_gr_f_main}

@findex dp_gr_main

@findex dp_gr_mod_main

@findex dp_gr_f_main

@findex dp_weyl_gr_main

@findex dp_weyl_gr_mod_main

@findex dp_weyl_gr_f_main

@table @t

@item dp_gr_main(@var{plist},@var{vlist},@var{homo},@var{modular},@var{order})

@itemx dp_gr_mod_main(@var{plist},@var{vlist},@var{homo},@var{modular},@var{order})

@itemx dp_gr_f_main(@var{plist},@var{vlist},@var{homo},@var{order})

@itemx dp_weyl_gr_main(@var{plist},@var{vlist},@var{homo},@var{modular},@var{order})

@itemx dp_weyl_gr_mod_main(@var{plist},@var{vlist},@var{homo},@var{modular},@var{order})

@itemx dp_weyl_gr_f_main(@var{plist},@var{vlist},@var{homo},@var{order})

\JP :: $B%0%l%V%J4pDl$N7W;;(B ($BAH$_9~$_H!?t(B)

\EG :: Groebner basis computation (built-in functions)

@end table

Line 1854 z^32+11405*z^31+20868*z^30+21602*z^29+...

Line 1992 z^32+11405*z^31+20868*z^30+21602*z^29+...

@item return

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

\EG list

@item plist, vlist

@item plist vlist

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

\EG list

@item order

Line 1873 z^32+11405*z^31+20868*z^30+21602*z^29+...

Line 2011 z^32+11405*z^31+20868*z^30+21602*z^29+...

@item

$B$3$l$i$NH!?t$O(B, $B%0%l%V%J4pDl7W;;$N4pK\E*AH$_9~$_H!?t$G$"$j(B, @code{gr()},

@code{hgr()}, @code{gr_mod()} $B$J$I$O$9$Y$F$3$l$i$NH!?t$r8F$S=P$7$F7W;;(B

$B$r9T$C$F$$$k(B.

$B$r9T$C$F$$$k(B. $B4X?tL>$K(B weyl $B$,F~$C$F$$$k$b$N$O(B, Weyl $BBe?t>e$N7W;;(B

$B$N$?$a$N4X?t$G$"$k(B.

@item

@code{dp_gr_f_main()}, @code{dp_weyl_f_main()} $B$O(B, $B<o!9$NM-8BBN>e$N%0%l%V%J4pDl$r7W;;$9$k(B

$B>l9g$KMQ$$$k(B. $BF~NO$O(B, $B$"$i$+$8$a(B, @code{simp_ff()} $B$J$I$G(B,

$B9M$($kM-8BBN>e$K<M1F$5$l$F$$$kI,MW$,$"$k(B.

@item

$B%U%i%0(B @var{homo} $B$,(B 0 $B$G$J$$;~(B, $BF~NO$r@F<!2=$7$F$+$i(B Buchberger $B%"%k%4%j%:%`(B

$B$r<B9T$9$k(B.

@item

Line 1906 z^32+11405*z^31+20868*z^30+21602*z^29+...

Line 2049 z^32+11405*z^31+20868*z^30+21602*z^29+...

@item

These functions are fundamental built-in functions for Groebner basis

computation and @code{gr()},@code{hgr()} and @code{gr_mod()}

are all interfaces to these functions.

are all interfaces to these functions. Functions whose names

contain weyl are those for computation in Weyl algebra.

@item

@code{dp_gr_f_main()} and @code{dp_weyl_gr_f_main()}

are functions for Groebner basis computation

over various finite fields. Coefficients of input polynomials

must be converted to elements of a finite field

currently specified by @code{setmod_ff()}.

@item

If @var{homo} is not equal to 0, homogenization is applied before entering

Buchberger algorithm

@item

Line 1945 Actual computation is controlled by various parameters

Line 2095 Actual computation is controlled by various parameters

@fref{dp_ord},

@fref{dp_gr_flags dp_gr_print},

@fref{gr hgr gr_mod},

@fref{setmod_ff},

\JP @fref{$B7W;;$*$h$SI=<($N@)8f(B}.

\EG @fref{Controlling Groebner basis computations}

@end table

\JP @node dp_f4_main dp_f4_mod_main,,, $B%0%l%V%J4pDl$K4X$9$kH!?t(B

\JP @node dp_f4_main dp_f4_mod_main dp_weyl_f4_main dp_weyl_f4_mod_main,,, $B%0%l%V%J4pDl$K4X$9$kH!?t(B

\EG @node dp_f4_main dp_f4_mod_main,,, Functions for Groebner basis computation

\EG @node dp_f4_main dp_f4_mod_main dp_weyl_f4_main dp_weyl_f4_mod_main,,, Functions for Groebner basis computation

@subsection @code{dp_f4_main}, @code{dp_f4_mod_main}

@subsection @code{dp_f4_main}, @code{dp_f4_mod_main}, @code{dp_weyl_f4_main}, @code{dp_weyl_f4_mod_main}

@findex dp_f4_main

@findex dp_f4_mod_main

@findex dp_weyl_f4_main

@findex dp_weyl_f4_mod_main

@table @t

@item dp_f4_main(@var{plist},@var{vlist},@var{order})

@itemx dp_f4_mod_main(@var{plist},@var{vlist},@var{order})

@itemx dp_weyl_f4_main(@var{plist},@var{vlist},@var{order})

@itemx dp_weyl_f4_mod_main(@var{plist},@var{vlist},@var{order})

\JP :: F4 $B%"%k%4%j%:%`$K$h$k%0%l%V%J4pDl$N7W;;(B ($BAH$_9~$_H!?t(B)

\EG :: Groebner basis computation by F4 algorithm (built-in functions)

@end table

Line 1966 Actual computation is controlled by various parameters

Line 2121 Actual computation is controlled by various parameters

@item return

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

\EG list

@item plist, vlist

@item plist vlist

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

\EG list

@item order

Line 1983 F4 $B%"%k%4%j%:%`$O(B, J.C. Faugere $B$K$h$jDs>'$5$

Line 2138 F4 $B%"%k%4%j%:%`$O(B, J.C. Faugere $B$K$h$jDs>'$5$

$B;;K!$G$"$j(B, $BK\<BAu$O(B, $BCf9q>jM>DjM}$K$h$k@~7AJ}Dx<05a2r$rMQ$$$?(B

$B;n83E*$J<BAu$G$"$k(B.

@item

$B0z?t$*$h$SF0:n$O$=$l$>$l(B @code{dp_gr_main()}, @code{dp_gr_mod_main()}

$B@F<!2=$N0z?t$,$J$$$3$H$r=|$1$P(B, $B0z?t$*$h$SF0:n$O$=$l$>$l(B

@code{dp_gr_main()}, @code{dp_gr_mod_main()},

@code{dp_weyl_gr_main()}, @code{dp_weyl_gr_mod_main()}

$B$HF1MM$G$"$k(B.

\BEG

Line 1995 invented by J.C. Faugere. The current implementation o

Line 2152 invented by J.C. Faugere. The current implementation o

uses Chinese Remainder theorem and not highly optimized.

@item

Arguments and actions are the same as those of

@code{dp_gr_main()}, @code{dp_gr_mod_main()}.

@code{dp_gr_main()}, @code{dp_gr_mod_main()},

@code{dp_weyl_gr_main()}, @code{dp_weyl_gr_mod_main()},

except for lack of the argument for controlling homogenization.

@end itemize

Line 2017 Arguments and actions are the same as those of

Line 2176 Arguments and actions are the same as those of

@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 2031 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

\BJP

@item

@code{dp_gr_main()}, @code{dp_gr_mod_main()} $B<B9T;~$K$*$1$k$5$^$6$^(B

@code{dp_gr_main()}, @code{dp_gr_mod_main()}, @code{dp_gr_f_main()} $B<B9T;~$K$*$1$k$5$^$6$^(B

$B$J%Q%i%a%?$r@_Dj(B, $B;2>H$9$k(B.

@item

$B0z?t$,$J$$>l9g(B, $B8=:_$N@_Dj$,JV$5$l$k(B.

Line 2044 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 2061 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 2212 the coefficient field.

Line 2392 the coefficient field.

(1)*<<2,0,0>>+(2)*<<1,1,0>>+(1)*<<0,2,0>>+(2)*<<1,0,1>>+(2)*<<0,1,1>>

+(1)*<<0,0,2>>

[52] dp_ptod((x+y+z)^2,[x,y]);

(1)*<<2,0>>+(2)*<<1,1>>+(1)*<<0,2>>+(2*z)*<<1,0>>+(2*z)*<<0,1>>+(z^2)*<<0,0>>

(1)*<<2,0>>+(2)*<<1,1>>+(1)*<<0,2>>+(2*z)*<<1,0>>+(2*z)*<<0,1>>

+(z^2)*<<0,0>>

@end example

@table @t

Line 2264 variables of @var{dpoly}.

Line 2445 variables of @var{dpoly}.

@example

[53] T=dp_ptod((x+y+z)^2,[x,y]);

(1)*<<2,0>>+(2)*<<1,1>>+(1)*<<0,2>>+(2*z)*<<1,0>>+(2*z)*<<0,1>>+(z^2)*<<0,0>>

(1)*<<2,0>>+(2)*<<1,1>>+(1)*<<0,2>>+(2*z)*<<1,0>>+(2*z)*<<0,1>>

+(z^2)*<<0,0>>

[54] P=dp_dtop(T,[a,b]);

z^2+(2*a+2*b)*z+a^2+2*b*a+b^2

@end example

Line 2617 For single computation @code{p_nf} and @code{p_true_nf

Line 2799 For single computation @code{p_nf} and @code{p_true_nf

[74] DP2=newvect(length(G),map(dp_ptod,G,V))$

[75] T=dp_ptod((u0-u1+u2-u3+u4)^2,V)$

[76] dp_dtop(dp_nf([0,1,2,3,4],T,DP1,1),V);

u4^2+(6*u3+2*u2+6*u1-2)*u4+9*u3^2+(6*u2+18*u1-6)*u3+u2^2+(6*u1-2)*u2+9*u1^2-6*u1+1

u4^2+(6*u3+2*u2+6*u1-2)*u4+9*u3^2+(6*u2+18*u1-6)*u3+u2^2

+(6*u1-2)*u2+9*u1^2-6*u1+1

[77] dp_dtop(dp_nf([4,3,2,1,0],T,DP1,1),V);

-5*u4^2+(-4*u3-4*u2-4*u1)*u4-u3^2-3*u3-u2^2+(2*u1-1)*u2-2*u1^2-3*u1+1

[78] dp_dtop(dp_nf([0,1,2,3,4],T,DP2,1),V);

-1138087976845165778088612297273078520347097001020471455633353049221045677593

-11380879768451657780886122972730785203470970010204714556333530492210

0005716505560062087150928400876150217079820311439477560587583488*u4^15+...

456775930005716505560062087150928400876150217079820311439477560587583

488*u4^15+...

[79] dp_dtop(dp_nf([4,3,2,1,0],T,DP2,1),V);

-1138087976845165778088612297273078520347097001020471455633353049221045677593

-11380879768451657780886122972730785203470970010204714556333530492210

0005716505560062087150928400876150217079820311439477560587583488*u4^15+...

456775930005716505560062087150928400876150217079820311439477560587583

488*u4^15+...

[80] @@78==@@79;

@end example

Line 2791 selection strategy of critical pairs in Groebner basis

Line 2976 selection strategy of critical pairs in Groebner basis

@item return

\JP $BJ,;6I=8=B?9`<0(B

\EG distributed polynomial

@item dpoly1, dpoly2

@item dpoly1 dpoly2

\JP $BJ,;6I=8=B?9`<0(B

\EG distributed polynomial

@end table

Line 2834 two polynomials, where coefficient is always set to 1.

Line 3019 two polynomials, where coefficient is always set to 1.

@item return

\JP $B@0?t(B

\EG integer

@item dpoly1, dpoly2

@item dpoly1 dpoly2

\JP $BJ,;6I=8=B?9`<0(B

\EG distributed polynomial

@end table

Line 2889 Used for finding candidate terms at reduction of polyn

Line 3074 Used for finding candidate terms at reduction of polyn

@item return

\JP $BJ,;6I=8=B?9`<0(B

\EG distributed polynomial

@item dpoly1, dpoly2

@item dpoly1 dpoly2

\JP $BJ,;6I=8=B?9`<0(B

\EG distributed polynomial

@end table

Line 3113 values of @code{dp_mag()} for intermediate basis eleme

Line 3298 values of @code{dp_mag()} for intermediate basis eleme

@item return

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

\EG list

@item dpoly1, dpoly2, dpoly3

@item dpoly1 dpoly2 dpoly3

\JP $BJ,;6I=8=B?9`<0(B

\EG distributed polynomial

@item vlist

Line 3137 values of @code{dp_mag()} for intermediate basis eleme

Line 3322 values of @code{dp_mag()} for intermediate basis eleme

$B$J$i$J$$(B.

@item

$B0z?t$,@0?t78?t$N;~(B, $B4JLs$O(B, $BJ,?t$,8=$l$J$$$h$&(B, $B@0?t(B @var{a}, @var{b},

$B9`(B @var{t} $B$K$h$j(B @var{a(dpoly1 + dpoly2)-bt dpoly3} $B$H$7$F7W;;$5$l$k(B.

$B9`(B @var{t} $B$K$h$j(B @var{a}(@var{dpoly1} + @var{dpoly2})-@var{bt} @var{dpoly3} $B$H$7$F7W;;$5$l$k(B.

@item

$B7k2L$O(B, @code{[@var{a dpoly1},@var{a dpoly2 - bt dpoly3}]} $B$J$k%j%9%H$G$"$k(B.

Line 3156 the divisibility of the head term of @var{dpoly2} by t

Line 3341 the divisibility of the head term of @var{dpoly2} by t

When integral coefficients, computation is so carefully performed that

no rational operations appear in the reduction procedure.

It is computed for integers @var{a} and @var{b}, and a term @var{t} as:

@var{a(dpoly1 + dpoly2)-bt dpoly3}.

@var{a}(@var{dpoly1} + @var{dpoly2})-@var{bt} @var{dpoly3}.

@item

The result is a list @code{[@var{a dpoly1},@var{a dpoly2 - bt dpoly3}]}.

Line 3170 The result is a list @code{[@var{a dpoly1},@var{a dpol

Line 3355 The result is a list @code{[@var{a dpoly1},@var{a dpol

[159] C=12*<<1,1,1,0,0>>+(1)*<<0,1,1,1,0>>+(1)*<<1,1,0,0,1>>;

(12)*<<1,1,1,0,0>>+(1)*<<0,1,1,1,0>>+(1)*<<1,1,0,0,1>>

[160] dp_red(D,R,C);

[(6)*<<2,1,0,0,0>>+(6)*<<1,2,0,0,0>>+(2)*<<0,3,0,0,0>>,(-1)*<<0,1,1,1,0>>

[(6)*<<2,1,0,0,0>>+(6)*<<1,2,0,0,0>>+(2)*<<0,3,0,0,0>>,

+(-1)*<<1,1,0,0,1>>]

(-1)*<<0,1,1,1,0>>+(-1)*<<1,1,0,0,1>>]

@end example

@table @t

Line 3197 The result is a list @code{[@var{a dpoly1},@var{a dpol

Line 3382 The result is a list @code{[@var{a dpoly1},@var{a dpol

@item return

\JP $BJ,;6I=8=B?9`<0(B

\EG distributed polynomial

@item dpoly1, dpoly2

@item dpoly1 dpoly2

\JP $BJ,;6I=8=B?9`<0(B

\EG distributed polynomial

@item mod

Line 3273 as a form of @code{[numerator, denominator]})

Line 3458 as a form of @code{[numerator, denominator]})

@item poly

\JP $BB?9`<0(B

\EG polynomial

@item plist,vlist

@item plist vlist

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

\EG list

@item order

Line 3409 exists.

Line 3594 exists.

@example

[233] G=gr(katsura(5),[u5,u4,u3,u2,u1,u0],2)$

[234] p_terms(G[0],[u5,u4,u3,u2,u1,u0],2);

[u5,u0^31,u0^30,u0^29,u0^28,u0^27,u0^26,u0^25,u0^24,u0^23,u0^22,u0^21,u0^20,

[u5,u0^31,u0^30,u0^29,u0^28,u0^27,u0^26,u0^25,u0^24,u0^23,u0^22,

u0^19,u0^18,u0^17,u0^16,u0^15,u0^14,u0^13,u0^12,u0^11,u0^10,u0^9,u0^8,u0^7,

u0^21,u0^20,u0^19,u0^18,u0^17,u0^16,u0^15,u0^14,u0^13,u0^12,u0^11,

u0^6,u0^5,u0^4,u0^3,u0^2,u0,1]

u0^10,u0^9,u0^8,u0^7,u0^6,u0^5,u0^4,u0^3,u0^2,u0,1]

@end example

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

Line 3428 u0^6,u0^5,u0^4,u0^3,u0^2,u0,1]

Line 3613 u0^6,u0^5,u0^4,u0^3,u0^2,u0,1]

@table @var

\JP @item return 0 $B$^$?$O(B 1

\EG @item return 0 or 1

@item plist1, plist2

@item plist1 plist2

@end table

@itemize @bullet

Line 3519 Polynomial set @code{cyclic} is sometimes called by ot

Line 3704 Polynomial set @code{cyclic} is sometimes called by ot

[79] load("cyclic")$

[89] katsura(5);

[u0+2*u4+2*u3+2*u2+2*u1+2*u5-1,2*u4*u0-u4+2*u1*u3+u2^2+2*u5*u1,

2*u3*u0+2*u1*u4-u3+(2*u1+2*u5)*u2,2*u2*u0+2*u2*u4+(2*u1+2*u5)*u3-u2+u1^2,

2*u3*u0+2*u1*u4-u3+(2*u1+2*u5)*u2,2*u2*u0+2*u2*u4+(2*u1+2*u5)*u3

2*u1*u0+(2*u3+2*u5)*u4+2*u2*u3+2*u1*u2-u1,

-u2+u1^2,2*u1*u0+(2*u3+2*u5)*u4+2*u2*u3+2*u1*u2-u1,

u0^2-u0+2*u4^2+2*u3^2+2*u2^2+2*u1^2+2*u5^2]

[90] hkatsura(5);

[-t+u0+2*u4+2*u3+2*u2+2*u1+2*u5,

Line 3642 if an input ideal is not radical.

Line 3827 if an input ideal is not radical.

\JP @fref{$B9`=g=x$N@_Dj(B}.

\EG @fref{Setting term orderings}.

@end table

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

\EG @node primedec_mod,,, Functions for Groebner basis computation

@subsection @code{primedec_mod}

@findex primedec_mod

@table @t

@item primedec_mod(@var{plist},@var{vlist},@var{ord},@var{mod},@var{strategy})

\JP :: $B%$%G%"%k$NJ,2r(B

\EG :: Computes decompositions of ideals over small finite fields.

@end table

@table @var

@item return

@itemx plist

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

\EG list of polynomials

@item vlist

\JP $BJQ?t%j%9%H(B

\EG list of variables

@item ord

\JP $B?t(B, $B%j%9%H$^$?$O9TNs(B

\EG number, list or matrix

@item mod

\JP $B@5@0?t(B

\EG positive integer

@item strategy

\JP $B@0?t(B

\EG integer

@end table

@itemize @bullet

\BJP

@item

@code{primedec_mod()} $B$O(B @samp{primdec_mod}

$B$GDj5A$5$l$F$$$k(B. @code{[Yokoyama]} $B$NAG%$%G%"%kJ,2r%"%k%4%j%:%`(B

$B$r<BAu$7$F$$$k(B.

@item

@code{primedec_mod()} $B$OM-8BBN>e$G$N%$%G%"%k$N(B

$B:,4p$NAG%$%G%"%kJ,2r$r9T$$(B, $BAG%$%G%"%k$N%j%9%H$rJV$9(B.

@item

@code{primedec_mod()} $B$O(B, GF(@var{mod}) $B>e$G$NJ,2r$rM?$($k(B.

$B7k2L$N3F@.J,$N@8@.85$O(B, $B@0?t78?tB?9`<0$G$"$k(B.

@item

$B7k2L$K$*$$$F(B, $BB?9`<0%j%9%H$H$7$FI=<($5$l$F$$$k3F%$%G%"%k$OA4$F(B

[@var{vlist},@var{ord}] $B$G;XDj$5$l$k9`=g=x$K4X$9$k%0%l%V%J4pDl$G$"$k(B.

@item

@var{strategy} $B$,(B 0 $B$G$J$$$H$-(B, incremental $B$K(B component $B$N6&DL(B

$BItJ,$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

Function @code{primedec_mod()}

is defined in @samp{primdec_mod} and implements the prime decomposition

algorithm in @code{[Yokoyama]}.

@item

@code{primedec_mod()}

is the function for prime ideal decomposition

of the radical of a polynomial ideal over small finite field,

and they return a list of prime ideals, which are associated primes

of the input ideal.

@item

@code{primedec_mod()} gives the decomposition over GF(@var{mod}).

The generators of each resulting component consists of integral polynomials.

@item

Each resulting component is a Groebner basis with respect to

a term order specified by [@var{vlist},@var{ord}].

@item

If @var{strategy} is non zero, then the early termination strategy

is tried by computing the intersection of obtained components

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

@example

[0] load("primdec_mod")$

[246] PP444=[x^8+x^2+t,y^8+y^2+t,z^8+z^2+t]$

[247] primedec_mod(PP444,[x,y,z,t],0,2,1);

[[y+z,x+z,z^8+z^2+t],[x+y,y^2+y+z^2+z+1,z^8+z^2+t],

[y+z+1,x+z+1,z^8+z^2+t],[x+z,y^2+y+z^2+z+1,z^8+z^2+t],

[y+z,x^2+x+z^2+z+1,z^8+z^2+t],[y+z+1,x^2+x+z^2+z+1,z^8+z^2+t],

[x+z+1,y^2+y+z^2+z+1,z^8+z^2+t],[y+z+1,x+z,z^8+z^2+t],

[x+y+1,y^2+y+z^2+z+1,z^8+z^2+t],[y+z,x+z+1,z^8+z^2+t]]

[248]

@end example

@table @t

\JP @item $B;2>H(B

\EG @item References

@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},

@fref{dp_gr_flags dp_gr_print}.

@end table

\JP @node bfunction bfct generic_bfct ann ann0,,, $B%0%l%V%J4pDl$K4X$9$kH!?t(B

\EG @node bfunction bfct generic_bfct ann ann0,,, Functions for Groebner basis computation

@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})

@itemx bfct(@var{f})

@itemx generic_bfct(@var{plist},@var{vlist},@var{dvlist},@var{weight})

\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

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

\EG polynomial or list

@item f

\JP $BB?9`<0(B

\EG polynomial

@item plist

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

\EG list of polynomials

@item vlist dvlist

\JP $BJQ?t%j%9%H(B

\EG list of variables

@end table

@itemize @bullet

\BJP

@item @samp{bfct} $B$GDj5A$5$l$F$$$k(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 @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 @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})} 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 @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 @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

@example

[0] load("bfct")$

[216] bfunction(x^3+y^3+z^3+x^2*y^2*z^2+x*y*z);

-9*s^5-63*s^4-173*s^3-233*s^2-154*s-40

[217] fctr(@@);

[[-1,1],[s+2,1],[3*s+4,1],[3*s+5,1],[s+1,2]]

[218] F = [4*x^3*dt+y*z*dt+dx,x*z*dt+4*y^3*dt+dy,

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

\JP @item $B;2>H(B

\EG @item References

\JP @fref{Weyl $BBe?t(B}.

\EG @fref{Weyl algebra}.

@end table

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