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

Return to groebner.texi CVS log

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

Diff for /OpenXM/src/asir-doc/parts/groebner.texi between version 1.2 and 1.5

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

version 1.5, 2003/04/20 08:01:25

Line 1

@comment $OpenXM$

@comment $OpenXM: OpenXM/src/asir-doc/parts/groebner.texi,v 1.4 2003/04/19 15:44:56 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 530 membercheck

Line 532 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 994 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 1214 Refer to the sections for each functions.

Line 1217 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_f4_main dp_f4_mod_main::

* dp_gr_flags dp_gr_print::

* dp_ord::

Line 1239 Refer to the sections for each functions.

Line 1242 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::

@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 1267 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 1346 for communication.

@table @t

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

\EG @item References

@comment @fref{dp_gr_main dp_gr_mod_main},

@comment @fref{dp_gr_main dp_gr_mod_main dp_gr_f_main},

@fref{dp_gr_main dp_gr_mod_main},

@fref{dp_gr_main dp_gr_mod_main dp_gr_f_main},

@fref{dp_ord}.

@end table

Line 1371 for communication.

Line 1376 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 1564 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},

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

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

@end table

Line 1585 processes.

Line 1590 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 1666 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 1697 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 1794 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 1842 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,,, $B%0%l%V%J4pDl$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,,, 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}

@findex dp_gr_main

@findex dp_gr_mod_main

@findex dp_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})

\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 1861 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 1874 z^32+11405*z^31+20868*z^30+21602*z^29+...

Line 1882 z^32+11405*z^31+20868*z^30+21602*z^29+...

@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.

@item

@code{dp_gr_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 1907 These functions are fundamental built-in functions for

Line 1919 These functions are fundamental built-in functions for

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

are all interfaces to these functions.

@item

@code{dp_gr_f_main()} is a function 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 1961 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

Line 1965 Actual computation is controlled by various parameters

Line 1983 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 2035 and showing informations.

Line 2053 and showing informations.

@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 2211 the coefficient field.

Line 2229 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 2282 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 2636 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 2813 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 2856 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 2911 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 3135 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 3159 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 3178 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 3192 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 3219 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 3295 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 3431 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 3450 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 3541 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 3546 u0^2-u0+2*u4^2+2*u3^2+2*u2^2+2*u1^2+2*u5^2]

Line 3569 u0^2-u0+2*u4^2+2*u3^2+2*u2^2+2*u1^2+2*u5^2]

\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

\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.

\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.

\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.

@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},

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

\EG @fref{Setting term orderings}.

@end table

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