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Diff for /OpenXM/src/asir-doc/parts/groebner.texi between version 1.2 and 1.7

version 1.2, 1999/12/21 02:47:31

version 1.7, 2003/04/21 03:07:32

Line 1

@comment $OpenXM$

@comment $OpenXM: OpenXM/src/asir-doc/parts/groebner.texi,v 1.6 2003/04/20 09:55:18 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 1239 Refer to the sections for each functions.

Line 1352 Refer to the sections for each functions.

* katsura hkatsura cyclic hcyclic::

* dp_vtoe dp_etov::

* lex_hensel_gsl tolex_gsl tolex_gsl_d::

* primadec primedec::

* primedec_mod::

* bfunction generic_bfct::

@end menu

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

Line 1262 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 1341 for communication.

Line 1457 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 1371 for communication.

Line 1486 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 1559 processes.

Line 1674 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 1585 processes.

Line 1700 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 1661 processes.

Line 1776 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 1691 processes.

Line 1807 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 1788 for @code{gr_minipoly()}.

Line 1904 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 1836 z^32+11405*z^31+20868*z^30+21602*z^29+...

Line 1952 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 1853 z^32+11405*z^31+20868*z^30+21602*z^29+...

Line 1977 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 1872 z^32+11405*z^31+20868*z^30+21602*z^29+...

Line 1996 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 1905 z^32+11405*z^31+20868*z^30+21602*z^29+...

Line 2034 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 1944 Actual computation is controlled by various parameters

Line 2080 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 1965 Actual computation is controlled by various parameters

Line 2106 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 1982 F4 $B%"%k%4%j%:%`$O(B, J.C. Faugere $B$K$h$jDs>'$5$

Line 2123 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 1994 invented by J.C. Faugere. The current implementation o

Line 2137 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 2016 Arguments and actions are the same as those of

Line 2161 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 2030 and showing informations.

Line 2175 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 2043 and showing informations.

Line 2191 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 2060 Arguments must be specified as a list such as

Line 2217 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 2211 the coefficient field.

Line 2377 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 2263 variables of @var{dpoly}.

Line 2430 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 2616 For single computation @code{p_nf} and @code{p_true_nf

Line 2784 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 2790 selection strategy of critical pairs in Groebner basis

Line 2961 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 2833 two polynomials, where coefficient is always set to 1.

Line 3004 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 2888 Used for finding candidate terms at reduction of polyn

Line 3059 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 3112 values of @code{dp_mag()} for intermediate basis eleme

Line 3283 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 3136 values of @code{dp_mag()} for intermediate basis eleme

Line 3307 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 3155 the divisibility of the head term of @var{dpoly2} by t

Line 3326 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 3169 The result is a list @code{[@var{a dpoly1},@var{a dpol

Line 3340 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 3196 The result is a list @code{[@var{a dpoly1},@var{a dpol

Line 3367 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 3272 as a form of @code{[numerator, denominator]})

Line 3443 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 3408 exists.

Line 3579 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 3427 u0^6,u0^5,u0^4,u0^3,u0^2,u0,1]

Line 3598 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 3518 Polynomial set @code{cyclic} is sometimes called by ot

Line 3689 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 3545 u0^2-u0+2*u4^2+2*u3^2+2*u2^2+2*u1^2+2*u5^2]

Line 3716 u0^2-u0+2*u4^2+2*u3^2+2*u2^2+2*u1^2+2*u5^2]

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

\EG @item References

@fref{dp_dtop}.

@end table

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

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

@subsection @code{primadec}, @code{primedec}

@findex primadec

@findex primedec

@table @t

@item primadec(@var{plist},@var{vlist})

@item primedec(@var{plist},@var{vlist})

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

\EG :: Computes decompositions of ideals.

@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

@end table

@itemize @bullet

\BJP

@item

@code{primadec()}, @code{primedec} $B$O(B @samp{primdec} $B$GDj5A$5$l$F$$$k(B.

@item

@code{primadec()}, @code{primedec()} $B$O$=$l$>$lM-M}?tBN>e$G$N%$%G%"%k$N(B

$B=`AGJ,2r(B, $B:,4p$NAG%$%G%"%kJ,2r$r9T$&(B.

@item

$B0z?t$OB?9`<0%j%9%H$*$h$SJQ?t%j%9%H$G$"$k(B. $BB?9`<0$OM-M}?t78?t$N$_$,5v$5$l$k(B.

@item

@code{primadec} $B$O(B @code{[$B=`AG@.J,(B, $BIUB0AG%$%G%"%k(B]} $B$N%j%9%H$rJV$9(B.

@item

@code{primadec} $B$O(B $BAG0x;R$N%j%9%H$rJV$9(B.

@item

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

$B%0%l%V%J4pDl$G$"$k(B. $BBP1~$9$k9`=g=x$O(B, $B$=$l$>$l(B

$BJQ?t(B @code{PRIMAORD}, @code{PRIMEORD} $B$K3JG<$5$l$F$$$k(B.

@item

@code{primadec} $B$O(B @code{[Shimoyama,Yokoyama]} $B$N=`AGJ,2r%"%k%4%j%:%`(B

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

@item

$B$b$7AG0x;R$N$_$r5a$a$?$$$J$i(B, @code{primedec} $B$r;H$&J}$,$h$$(B.

$B$3$l$O(B, $BF~NO%$%G%"%k$,:,4p%$%G%"%k$G$J$$>l9g$K(B, @code{primadec}

$B$N7W;;$KM>J,$J%3%9%H$,I,MW$H$J$k>l9g$,$"$k$+$i$G$"$k(B.

\BEG

@item

Function @code{primadec()} and @code{primedec} are defined in @samp{primdec}.

@item

@code{primadec()}, @code{primedec()} are the function for primary

ideal decomposition and prime decomposition of the radical over the

rationals respectively.

@item

The arguments are a list of polynomials and a list of variables.

These functions accept ideals with rational function coefficients only.

@item

@code{primadec} returns the list of pair lists consisting a primary component

and its associated prime.

@item

@code{primedec} returns the list of prime components.

@item

Each component is a Groebner basis and the corresponding term order

is indicated by the global variables @code{PRIMAORD}, @code{PRIMEORD}

respectively.

@item

@code{primadec} implements the primary decompostion algorithm

in @code{[Shimoyama,Yokoyama]}.

@item

If one only wants to know the prime components of an ideal, then

use @code{primedec} because @code{primadec} may need additional costs

if an input ideal is not radical.

@end itemize

@example

[84] load("primdec")$

[102] primedec([p*q*x-q^2*y^2+q^2*y,-p^2*x^2+p^2*x+p*q*y,

(q^3*y^4-2*q^3*y^3+q^3*y^2)*x-q^3*y^4+q^3*y^3,

-q^3*y^4+2*q^3*y^3+(-q^3+p*q^2)*y^2],[p,q,x,y]);

[[y,x],[y,p],[x,q],[q,p],[x-1,q],[y-1,p],[(y-1)*x-y,q*y^2-2*q*y-p+q]]

[103] primadec([x,z*y,w*y^2,w^2*y-z^3,y^3],[x,y,z,w]);

[[[x,z*y,y^2,w^2*y-z^3],[z,y,x]],[[w,x,z*y,z^3,y^3],[w,z,y,x]]]

@end example

@table @t

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

\EG @item References

@fref{fctr sqfr},

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

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

@subsection @code{bfunction}, @code{generic_bfct}

@findex bfunction

@findex generic_bfct

@table @t

@item bfunction(@var{f})

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

\JP :: b $B4X?t$N7W;;(B

\EG :: Computes the global b function of a polynomial or an ideal

@end table

@table @var

@item return

@itemx 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})} $B$OB?9`<0(B @var{f} $B$N(B global 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.

@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, [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

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}

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 [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

@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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