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version 1.1.1.1, 1999/12/08 05:47:44

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

Line 1

@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

\BEG

@node Groebner basis computation,,, Top

@chapter Groebner basis computation

@menu

\BJP

* $BJ,;6I=8=B?9`<0(B::

* $B%U%!%$%k$NFI$_9~$_(B::

* $B4pK\E*$JH!?t(B::

Line 9

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

* Distributed polynomial::

* Reading files::

* Fundamental functions::

* Controlling Groebner basis computations::

* Setting term orderings::

* Groebner basis computation with rational function coefficients::

* Change of ordering::

* Weyl algebra::

* Functions for Groebner basis computation::

@end menu

\BJP

@node $BJ,;6I=8=B?9`<0(B,,, $B%0%l%V%J4pDl$N7W;;(B

@section $BJ,;6I=8=B?9`<0(B

\BEG

@node Distributed polynomial,,, Groebner basis computation

@section Distributed polynomial

@noindent

\BJP

$BJ,;6I=8=B?9`<0$H$O(B, $BB?9`<0$NFbIt7A<0$N0l$D$G$"$k(B. $BDL>o$NB?9`<0(B

(@code{type} $B$,(B 2) $B$O(B, $B:F5"I=8=$H8F$P$l$k7A<0$GI=8=$5$l$F$$$k(B. $B$9$J$o(B

$B$A(B, $BFCDj$NJQ?t$r<gJQ?t$H$9$k(B 1 $BJQ?tB?9`<0$G(B, $B$=$NB>$NJQ?t$O(B, $B$=$N(B 1 $BJQ(B

$B?tB?9`<0$N78?t$K(B, $B<gJQ?t$r4^$^$J$$B?9`<0$H$7$F8=$l$k(B. $B$3$N78?t$,(B, $B$^$?(B,

$B$"$kJQ?t$r<gJQ?t$H$9$kB?9`<0$H$J$C$F$$$k$3$H$+$i:F5"I=8=$H8F$P$l$k(B.

\BEG

A distributed polynomial is a polynomial with a special internal

representation different from the ordinary one.

An ordinary polynomial (having @code{type} 2) is internally represented

in a format, called recursive representation.

In fact, it is represented as an uni-variate polynomial with respect to

a fixed variable, called main variable of that polynomial,

where the other variables appear in the coefficients which may again

polynomials in such variables other than the previous main variable.

A polynomial in the coefficients is again represented as

an uni-variate polynomial in a certain fixed variable,

the main variable. Thus, by this recursive structure of polynomial

representation, it is called the `recursive representation.'

@iftex

@tex

$(x+y+z)^2 = 1 \cdot x^2 + (2 \cdot y + (2 \cdot z)) \cdot x + ((2 \cdot z) \cdot y + (1 \cdot z^2 ))$

\JP $(x+y+z)^2 = 1 \cdot x^2 + (2 \cdot y + (2 \cdot z)) \cdot x + ((2 \cdot z) \cdot y + (1 \cdot z^2 ))$

\EG $(x+y+z)^2 = 1 \cdot x^2 + (2 \cdot y + (2 \cdot z)) \cdot x + ((2 \cdot z) \cdot y + (1 \cdot z^2 ))$

@end tex

@end iftex

@ifinfo

Line 35 $(x+y+z)^2 = 1 \cdot x^2 + (2 \cdot y + (2 \cdot z)) \

Line 79 $(x+y+z)^2 = 1 \cdot x^2 + (2 \cdot y + (2 \cdot z)) \

@end ifinfo

@noindent

\BJP

$B$3$l$KBP$7(B, $BB?9`<0$r(B, $BJQ?t$NQQ@Q$H78?t$N@Q$NOB$H$7$FI=8=$7$?$b$N$rJ,;6(B

$BI=8=$H8F$V(B.

\BEG

On the other hand,

we call a representation the distributed representation of a polynomial,

if a polynomial is represented, according to its original meaning,

as a sum of monomials,

where a monomial is the product of power product of variables

and a coefficient. We call a polynomial, represented in such an

internal format, a distributed polynomial. (This naming may sounds

something strange.)

@iftex

@tex

$(x+y+z)^2 = 1 \cdot x^2 + 2 \cdot xy + 2 \cdot xz + 1 \cdot y^2 + 2 \cdot yz +1 \cdot z^2$

\JP $(x+y+z)^2 = 1 \cdot x^2 + 2 \cdot xy + 2 \cdot xz + 1 \cdot y^2 + 2 \cdot yz +1 \cdot z^2$

\EG $(x+y+z)^2 = 1 \cdot x^2 + 2 \cdot xy + 2 \cdot xz + 1 \cdot y^2 + 2 \cdot yz +1 \cdot z^2$

@end tex

@end iftex

@ifinfo

Line 50 $(x+y+z)^2 = 1 \cdot x^2 + 2 \cdot xy + 2 \cdot xz + 1

Line 107 $(x+y+z)^2 = 1 \cdot x^2 + 2 \cdot xy + 2 \cdot xz + 1

@end ifinfo

@noindent

\BJP

$B%0%l%V%J4pDl7W;;$K$*$$$F$O(B, $BC19`<0$KCmL\$7$FA`:n$r9T$&$?$aB?9`<0$,J,;6I=8=(B

$B$5$l$F$$$kJ}$,$h$j8zN($N$h$$1i;;$,2DG=$K$J$k(B. $B$3$N$?$a(B, $BJ,;6I=8=B?9`<0$,(B,

$B<1JL;R(B 9 $B$N7?$H$7$F(B @b{Asir} $B$N%H%C%W%l%Y%k$+$iMxMQ2DG=$H$J$C$F$$$k(B.

$B$3$3$G(B, $B8e$N@bL@$N$?$a$K(B, $B$$$/$D$+$N8@MU$rDj5A$7$F$*$/(B.

\BEG

For computation of Groebner basis, efficient operation is expected if

polynomials are represented in a distributed representation,

because major operations for Groebner basis are performed with respect

to monomials.

From this view point, we provide the object type distributed polynomial

with its object identification number 9, and objects having such a type

are available by @b{Asir} language.

Here, we provide several definitions for the later description.

@table @b

\BJP

@item $B9`(B (term)

$BJQ?t$NQQ@Q(B. $B$9$J$o$A(B, $B78?t(B 1 $B$NC19`<0$N$3$H(B. @b{Asir} $B$K$*$$$F$O(B,

\BEG

@item term

The power product of variables, i.e., a monomial with coefficient 1.

In an @b{Asir} session, it is displayed in the form like

@example

<<0,1,2,3,4>>

@end example

\BJP

$B$H$$$&7A$GI=<($5$l(B, $B$^$?(B, $B$3$N7A$GF~NO2DG=$G$"$k(B. $B$3$NNc$O(B, 5 $BJQ?t$N9`(B

$B$r<($9(B. $B3FJQ?t$r(B @code{a}, @code{b}, @code{c}, @code{d}, @code{e} $B$H$9$k$H(B

$B$3$N9`$O(B @code{b*c^2*d^3*e^4} $B$rI=$9(B.

\BEG

and also can be input in such a form.

This example shows a term in 5 variables. If we assume the 5 variables

as @code{a}, @code{b}, @code{c}, @code{d}, and @code{e},

the term represents @code{b*c^2*d^3*e^4} in the ordinary expression.

\BJP

@item $B9`=g=x(B (term order)

$BJ,;6I=8=B?9`<0$K$*$1$k9`$O(B, $B<!$N@-<A$rK~$?$9A4=g=x$K$h$j@0Ns$5$l$k(B.

\BEG

@item term order

Terms are ordered according to a total order with the following properties.

@enumerate

@item

$BG$0U$N9`(B @code{t} $B$KBP$7(B @code{t} > 1

\JP $BG$0U$N9`(B @code{t} $B$KBP$7(B @code{t} > 1

\EG For all @code{t} @code{t} > 1.

@item

@code{t}, @code{s}, @code{u} $B$r9`$H$9$k;~(B, @code{t} > @code{s} $B$J$i$P(B

\JP @code{t}, @code{s}, @code{u} $B$r9`$H$9$k;~(B, @code{t} > @code{s} $B$J$i$P(B @code{tu} > @code{su}

@code{tu} > @code{su}

\EG For all @code{t}, @code{s}, @code{u} @code{t} > @code{s} implies @code{tu} > @code{su}.

@end enumerate

\BJP

$B$3$N@-<A$rK~$?$9A4=g=x$r9`=g=x$H8F$V(B. $B$3$N=g=x$OJQ?t=g=x(B ($BJQ?t$N%j%9%H(B)

$B$H9`=g=x7?(B ($B?t(B, $B%j%9%H$^$?$O9TNs(B) $B$K$h$j;XDj$5$l$k(B.

\BEG

Such a total order is called a term ordering. A term ordering is specified

by a variable ordering (a list of variables) and a type of term ordering

(an integer, a list or a matrix).

\BJP

@item $BC19`<0(B (monomial)

$B9`$H78?t$N@Q(B.

\BEG

@item monomial

The product of a term and a coefficient.

In an @b{Asir} session, it is displayed in the form like

@example

2*<<0,1,2,3,4>>

@end example

$B$H$$$&7A$GI=<($5$l(B, $B$^$?(B, $B$3$N7A$GF~NO2DG=$G$"$k(B.

\JP $B$H$$$&7A$GI=<($5$l(B, $B$^$?(B, $B$3$N7A$GF~NO2DG=$G$"$k(B.

\EG and also can be input in such a form.

\BJP

@itemx $BF,C19`<0(B (head monomial)

@item $BF,9`(B (head term)

@itemx $BF,78?t(B (head coefficient)

$BJ,;6I=8=B?9`<0$K$*$1$k3FC19`<0$O(B, $B9`=g=x$K$h$j@0Ns$5$l$k(B. $B$3$N;~=g(B

$B=x:GBg$NC19`<0$rF,C19`<0(B, $B$=$l$K8=$l$k9`(B, $B78?t$r$=$l$>$lF,9`(B, $BF,78?t(B

$B$H8F$V(B.

\BEG

@itemx head monomial

@item head term

@itemx head coefficient

Monomials in a distributed polynomial is sorted by a total order.

In such representation, we call the monomial that is maximum

with respect to the order the head monomial, and its term and coefficient

the head term and the head coefficient respectively.

@end table

\BJP

@node $B%U%!%$%k$NFI$_9~$_(B,,, $B%0%l%V%J4pDl$N7W;;(B

@section $B%U%!%$%k$NFI$_9~$_(B

\BEG

@node Reading files,,, Groebner basis computation

@section Reading files

@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

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

To call these functions,

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}.

@example

[0] load("gr")$

@end example

\BJP

@node $B4pK\E*$JH!?t(B,,, $B%0%l%V%J4pDl$N7W;;(B

@section $B4pK\E*$JH!?t(B

\BEG

@node Fundamental functions,,, Groebner basis computation

@section Fundamental functions

@noindent

\BJP

@samp{gr} $B$G$O?tB?$/$NH!?t$,Dj5A$5$l$F$$$k$,(B, $BD>@\(B

$B%0%l%V%J4pDl$r7W;;$9$k$?$a$N%H%C%W%l%Y%k$O<!$N(B 3 $B$D$G$"$k(B.

$B0J2<$G(B, @var{plist} $B$OB?9`<0$N%j%9%H(B, @var{vlist} $B$OJQ?t(B ($BITDj85(B) $B$N%j%9%H(B,

@var{order} $B$OJQ?t=g=x7?(B, @var{p} $B$O(B @code{2^27} $BL$K~$NAG?t$G$"$k(B.

\BEG

There are many functions and options defined in the package @samp{gr}.

Usually not so many of them are used. Top level functions for Groebner

basis computation are the following three functions.

In the following description, @var{plist}, @var{vlist}, @var{order}

and @var{p} stand for a list of polynomials, a list of variables

(indeterminates), a type of term ordering and a prime less than

@code{2^27} respectively.

@table @code

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

\BJP

Gebauer-Moeller $B$K$h$k(B useless pair elimination criteria, sugar

strategy $B$*$h$S(B Traverso $B$K$h$k(B trace-lifting $B$rMQ$$$?(B Buchberger $B%"%k(B

$B%4%j%:%`$K$h$kM-M}?t78?t%0%l%V%J4pDl7W;;H!?t(B. $B0lHL$K$O$3$NH!?t$rMQ$$$k(B.

\BEG

Function that computes Groebner bases over the rationals. The

algorithm is Buchberger algorithm with useless pair elimination

criteria by Gebauer-Moeller, sugar strategy and trace-lifting by

Traverso. For ordinary computation, this function is used.

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

\BJP

$BF~NOB?9`<0$r@F<!2=$7$?8e(B @code{gr()} $B$N%0%l%V%J4pDl8uJd@8@.It$K$h$j8u(B

$BJd@8@.$7(B, $BHs@F<!2=(B, interreduce $B$7$?$b$N$r(B @code{gr()} $B$N%0%l%V%J4pDl(B

$B%A%'%C%/It$G%A%'%C%/$9$k(B. 0 $B<!85%7%9%F%`(B ($B2r$N8D?t$,M-8B8D$NJ}Dx<07O(B)

$B$N>l9g(B, sugar strategy $B$,78?tKDD%$r0z$-5/$3$9>l9g$,$"$k(B. $B$3$N$h$&$J>l(B

$B9g(B, strategy $B$r@F<!2=$K$h$k(B strategy $B$KCV$-49$($k$3$H$K$h$j78?tKDD%$r(B

$BM^@)$9$k$3$H$,$G$-$k>l9g$,B?$$(B.

\BEG

After homogenizing the input polynomials a candidate of the \gr basis

is computed by trace-lifting. Then the candidate is dehomogenized and

checked whether it is indeed a Groebner basis of the input. Sugar

strategy often causes intermediate coefficient swells. It is

empirically known that the combination of homogenization and supresses

the swells for such cases.

@item gr_mod(@var{plist},@var{vlist},@var{order},@var{p})

\BJP

Gebauer-Moeller $B$K$h$k(B useless pair elimination criteria, sugar

strategy $B$*$h$S(B Buchberger $B%"%k%4%j%:%`$K$h$k(B GF(p) $B78?t%0%l%V%J4pDl7W(B

$B;;H!?t(B.

\BEG

Function that computes Groebner bases over GF(@var{p}). The same

algorithm as @code{gr()} is used.

@end table

\BJP

@node $B7W;;$*$h$SI=<($N@)8f(B,,, $B%0%l%V%J4pDl$N7W;;(B

@section $B7W;;$*$h$SI=<($N@)8f(B

\BEG

@node Controlling Groebner basis computations,,, Groebner basis computation

@section Controlling Groebner basis computations

@noindent

\BJP

$B%0%l%V%J4pDl$N7W;;$K$*$$$F(B, $B$5$^$6$^$J%Q%i%a%?@_Dj$r9T$&$3$H$K$h$j7W;;(B,

$BI=<($r@)8f$9$k$3$H$,$G$-$k(B. $B$3$l$i$O(B, $BAH$_9~$_H!?t(B @code{dp_gr_flags()}

$B$K$h$j@_Dj;2>H$9$k$3$H$,$G$-$k(B. $BL50z?t$G(B @code{dp_gr_flags()} $B$r<B9T$9$k(B

$B$H(B, $B8=:_@_Dj$5$l$F$$$k%Q%i%a%?$,(B, $BL>A0$HCM$N%j%9%H$GJV$5$l$k(B.

\BEG

One can cotrol a Groebner basis computation by setting various parameters.

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

@code{dp_gr_flags()}. Without argument it returns the current settings.

@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

\BJP

$B0J2<$G(B, $B3F%Q%i%a%?$N0UL#$r@bL@$9$k(B. on $B$N>l9g$H$O(B, $B%Q%i%a%?$,(B 0 $B$G$J$$>l9g$r(B

$B$$$&(B. $B$3$l$i$N%Q%i%a%?$N=i4|CM$OA4$F(B 0 (off) $B$G$"$k(B.

\BEG

The return value is a list which contains the names of parameters and their

values. The meaning of the parameters are as follows. `on' means that the

parameter is not zero.

@table @code

@item NoSugar

\BJP

on $B$N>l9g(B, sugar strategy $B$NBe$o$j$K(B Buchberger$B$N(B normal strategy $B$,MQ(B

$B$$$i$l$k(B.

\BEG

If `on', Buchberger's normal strategy is used instead of sugar strategy.

@item NoCriB

on $B$N>l9g(B, $BITI,MWBP8!=P5,=`$N$&$A(B, $B5,=`(B B $B$rE,MQ$7$J$$(B.

\JP on $B$N>l9g(B, $BITI,MWBP8!=P5,=`$N$&$A(B, $B5,=`(B B $B$rE,MQ$7$J$$(B.

\EG If `on', criterion B among the Gebauer-Moeller's criteria is not applied.

@item NoGC

on $B$N>l9g(B, $B7k2L$,%0%l%V%J4pDl$K$J$C$F$$$k$+$I$&$+$N%A%'%C%/$r9T$o$J$$(B.

\JP on $B$N>l9g(B, $B7k2L$,%0%l%V%J4pDl$K$J$C$F$$$k$+$I$&$+$N%A%'%C%/$r9T$o$J$$(B.

\BEG

If `on', the check that a Groebner basis candidate is indeed a Groebner basis,

is not executed.

@item NoMC

\BJP

on $B$N>l9g(B, $B7k2L$,F~NO%$%G%"%k$HF1Ey$N%$%G%"%k$G$"$k$+$I$&$+$N%A%'%C%/(B

$B$r9T$o$J$$(B.

\BEG

If `on', the check that the resulting polynomials generates the same ideal as

the ideal generated by the input, is not executed.

@item NoRA

\BJP

on $B$N>l9g(B, $B7k2L$r(B reduced $B%0%l%V%J4pDl$K$9$k$?$a$N(B

interreduce $B$r9T$o$J$$(B.

\BEG

If `on', the interreduction, which makes the Groebner basis reduced, is not

executed.

@item NoGCD

\BJP

on $B$N>l9g(B, $BM-M}<078?t$N%0%l%V%J4pDl7W;;$K$*$$$F(B, $B@8@.$5$l$?B?9`<0$N(B,

$B78?t$N(B content $B$r$H$i$J$$(B.

\BEG

If `on', content removals are not executed during a Groebner basis computation

over a rational function field.

@item Top

on $B$N>l9g(B, normal form $B7W;;$K$*$$$FF,9`>C5n$N$_$r9T$&(B.

\JP on $B$N>l9g(B, normal form $B7W;;$K$*$$$FF,9`>C5n$N$_$r9T$&(B.

\EG If `on', Only the head term of the polynomial being reduced is reduced.

@item Interreduce

@comment @item Interreduce

on $B$N>l9g(B, $BB?9`<0$r@8@.$9$kKh$K(B, $B$=$l$^$G@8@.$5$l$?4pDl$r$=$NB?9`<0$K(B

@comment \BJP

$B$h$k(B normal form $B$GCV$-49$($k(B.

@comment on $B$N>l9g(B, $BB?9`<0$r@8@.$9$kKh$K(B, $B$=$l$^$G@8@.$5$l$?4pDl$r$=$NB?9`<0$K(B

@comment $B$h$k(B normal form $B$GCV$-49$($k(B.

@comment \E

@comment \BEG

@comment If `on', intermediate basis elements are reduced by using a newly generated

@comment basis element.

@comment \E

@item Reverse

\BJP

on $B$N>l9g(B, normal form $B7W;;$N:]$N(B reducer $B$r(B, $B?7$7$/@8@.$5$l$?$b$N$rM%(B

$B@h$7$FA*$V(B.

\BEG

If `on', the selection strategy of reducer in a normal form computation

is such that a newer reducer is used first.

@item Print

on $B$N>l9g(B, $B%0%l%V%J4pDl7W;;$NESCf$K$*$1$k$5$^$6$^$J>pJs$rI=<($9$k(B.

\JP on $B$N>l9g(B, $B%0%l%V%J4pDl7W;;$NESCf$K$*$1$k$5$^$6$^$J>pJs$rI=<($9$k(B.

\BEG

If `on', various informations during a Groebner basis computation is

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

$B$l$k%G!<%?$NFb(B, $B=87W%G!<%?$N$_$,I=<($5$l$k(B.

\BEG

If `on', a summary of informations is shown after a Groebner basis

computation. Note that the summary is always shown if @code{Print} is `on'.

@item ShowMag

\BJP

on $B$G(B @code{Print} $B$,(B on $B$J$i$P(B, $B@8@.$,@8@.$5$l$kKh$K(B, $B$=$NB?9`<0$N(B

$B78?t$N%S%C%HD9$NOB$rI=<($7(B, $B:G8e$K(B, $B$=$l$i$NOB$N:GBgCM$rI=<($9$k(B.

\BEG

If `on' and @code{Print} is `on', the sum of bit length of

coefficients of a generated basis element, which we call @var{magnitude},

is shown after every normal computation. After comleting the

computation the maximal value among the sums is shown.

@item Multiple

@item Content

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

@itemx Multiple

@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

\BJP

$B3d$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

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

GCD $B7W;;$,9T$o$l0lHL$K$O8zN($,0-$/$J$k$,(B, @code{Multiple} $B$r(B 2 $BDxEY(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{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{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 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{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{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{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

\BJP

$B@5Ev$J%G%#%l%/%H%jL>(B ($BJ8;zNs(B) $B$rCM$K;}$D$H$-(B, $B@8@.$5$l$?B?9`<0$O%a%b%j(B

$BCf$K$*$+$l$:(B, $B$=$N%G%#%l%/%H%jCf$K%P%$%J%j%G!<%?$H$7$FCV$+$l(B, $B$=$NB?9`(B

$B<0$rMQ$$$k(B normal form $B7W;;$N:](B, $B<+F0E*$K%a%b%jCf$K%m!<%I$5$l$k(B. $B3FB?(B

$B9`<0$O(B, $BFbIt$G$N%$%s%G%C%/%9$r%U%!%$%kL>$K;}$D%U%!%$%k$K3JG<$5$l$k(B.

$B$3$3$G;XDj$5$l$?%G%#%l%/%H%j$K=q$+$l$?%U%!%$%k$O<+F0E*$K$O>C5n$5$l$J$$(B

$B$?$a(B, $B%f!<%6$,@UG$$r;}$C$F>C5n$9$kI,MW$,$"$k(B.

\BEG

If the value (a character string) is a valid directory name, then

generated basis elements are put in the directory and are loaded on

demand during normal form computations. Each elements is saved in the

binary form and its name coincides with the index internally used in

the computation. These binary files are not removed automatically

and one should remove them by hand.

@end table

@noindent

@code{Print} $B$,(B 0 $B$G$J$$>l9g<!$N$h$&$J%G!<%?$,I=<($5$l$k(B.

\JP @code{Print} $B$,(B 0 $B$G$J$$>l9g<!$N$h$&$J%G!<%?$,I=<($5$l$k(B.

\EG If @code{Print} is `on', the following informations are shown.

@example

[93] gr(cyclic(4),[c0,c1,c2,c3],0)$

Line 249 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

@noindent

\BJP

$B:G=i$KI=<($5$l$k(B @code{mod}, @code{eval} $B$O(B, trace-lifting $B$GMQ$$$i$l$kK!(B

$B$G$"$k(B. @code{mod} $B$OAG?t(B, @code{eval} $B$OM-M}<078?t$N>l9g$KMQ$$$i$l$k(B

$B?t$N%j%9%H$G$"$k(B.

\BEG

In this example @code{mod} and @code{eval} indicate moduli used in

trace-lifting. @code{mod} is a prime and @code{eval} is a list of integers

used for evaluation when the ground field is a field of rational functions.

@noindent

$B7W;;ESCf$GB?9`<0$,@8@.$5$l$kKh$K<!$N7A$N%G!<%?$,I=<($5$l$k(B.

\JP $B7W;;ESCf$GB?9`<0$,@8@.$5$l$kKh$K<!$N7A$N%G!<%?$,I=<($5$l$k(B.

\EG The following information is shown after every normal form computation.

@example

(TNF)(TCONT)HT(INDEX),nb=NB,nab=NAB,rp=RP,sugar=S,mag=M

@end example

@noindent

$B$=$l$i$N0UL#$O<!$NDL$j(B.

\JP $B$=$l$i$N0UL#$O<!$NDL$j(B.

\EG Meaning of each component is as follows.

@table @code

\BJP

@item TNF

normal form $B7W;;;~4V(B ($BIC(B)

@item TCONT

content $B7W;;;~4V(B ($BIC(B)

@item HT

$B@8@.$5$l$?B?9`<0$NF,9`(B

@item INDEX

S-$BB?9`<0$r9=@.$9$kB?9`<0$N%$%s%G%C%/%9$N%Z%"(B

@item NB

$B8=:_$N(B, $B>iD9@-$r=|$$$?4pDl$N?t(B

@item NAB

$B8=:_$^$G$K@8@.$5$l$?4pDl$N?t(B

@item RP

$B;D$j$N%Z%"$N?t(B

@item S

$B@8@.$5$l$?B?9`<0$N(B sugar $B$NCM(B

@item M

$B@8@.$5$l$?B?9`<0$N78?t$N%S%C%HD9$NOB(B (@code{ShowMag} $B$,(B on $B$N;~$KI=<($5$l$k(B. )

\BEG

@item TNF

CPU time for normal form computation (second)

@item TCONT

CPU time for content removal(second)

@item HT

Head term of the generated basis element

@item INDEX

Pair of indices which corresponds to the reduced S-polynomial

@item NB

Number of basis elements after removing redundancy

@item NAB

Number of all the basis elements

@item RP

Number of remaining pairs

@item S

Sugar of the generated basis element

@item M

Magnitude of the genrated basis element (shown if @code{ShowMag} is `on'.)

@end table

@noindent

\BJP

$B:G8e$K(B, $B=87W%G!<%?$,I=<($5$l$k(B. $B0UL#$O<!$NDL$j(B.

($B;~4V$NI=<($K$*$$$F(B, $B?t;z$,(B 2 $B$D$"$k$b$N$O(B, $B7W;;;~4V$H(B GC $B;~4V$N%Z%"$G$"$k(B.)

\BEG

The summary of the informations shown after a Groebner basis

computation is as follows. If a component shows timings and it

contains two numbers, they are a pair of time for computation and time

for garbage colection.

@table @code

\BJP

@item UP

$B%Z%"$N%j%9%H$NA`:n$K$+$+$C$?;~4V(B

@item SP

$BM-M}?t>e$N(B S-$BB?9`<07W;;;~4V(B

@item SPM

$BM-8BBN>e$N(B S-$BB?9`<07W;;;~4V(B

@item NF

$BM-M}?t>e$N(B normal form $B7W;;;~4V(B

@item NFM

$BM-8BBN>e$N(B normal form $B7W;;;~4V(B

@item ZNFM

@code{NFM} $B$NFb(B, 0 $B$X$N(B reduction $B$K$+$+$C$?;~4V(B

@item PZ

content $B7W;;;~4V(B

@item NP

$BM-M}?t78?tB?9`<0$N78?t$KBP$9$k>jM>1i;;$N7W;;;~4V(B

@item MP

S-$BB?9`<0$r@8@.$9$k%Z%"$NA*Br$K$+$+$C$?;~4V(B

@item RA

interreduce $B7W;;;~4V(B

@item MC

trace-lifting $B$K$*$1$k(B, $BF~NOB?9`<0$N%a%s%P%7%C%W7W;;;~4V(B

@item GC

$B7k2L$N%0%l%V%J4pDl8uJd$N%0%l%V%J4pDl%A%'%C%/;~4V(B

@item T

$B@8@.$5$l$?%Z%"$N?t(B

@item B, M, F, D

$B3F(B criterion $B$K$h$j=|$+$l$?%Z%"$N?t(B

@item ZR

0 $B$K(B reduce $B$5$l$?%Z%"$N?t(B

@item NZR

0 $B$G$J$$B?9`<0$K(B reduce $B$5$l$?%Z%"$N?t(B

@item Max_mag

$B@8@.$5$l$?B?9`<0$N(B, $B78?t$N%S%C%HD9$NOB$N:GBgCM(B

\BEG

@item UP

Time to manipulate the list of critical pairs

@item SP

Time to compute S-polynomials over the rationals

@item SPM

Time to compute S-polynomials over a finite field

@item NF

Time to compute normal forms over the rationals

@item NFM

Time to compute normal forms over a finite field

@item ZNFM

Time for zero reductions in @code{NFM}

@item PZ

Time to remove integer contets

@item NP

Time to compute remainders for coefficients of polynomials with coeffieints

in the rationals

@item MP

Time to select pairs from which S-polynomials are computed

@item RA

Time to interreduce the Groebner basis candidate

@item MC

Time to check that each input polynomial is a member of the ideal

generated by the Groebner basis candidate.

@item GC

Time to check that the Groebner basis candidate is a Groebner basis

@item T

Number of critical pairs generated

@item B, M, F, D

Number of critical pairs removed by using each criterion

@item ZR

Number of S-polynomials reduced to 0

@item NZR

Number of S-polynomials reduced to non-zero results

@item Max_mag

Maximal magnitude among all the generated polynomials

@end table

\BJP

@node $B9`=g=x$N@_Dj(B,,, $B%0%l%V%J4pDl$N7W;;(B

@section $B9`=g=x$N@_Dj(B

\BEG

@node Setting term orderings,,, Groebner basis computation

@section Setting term orderings

@noindent

\BJP

$B9`$OFbIt$G$O(B, $B3FJQ?t$K4X$9$k;X?t$r@.J,$H$9$k@0?t%Y%/%H%k$H$7$FI=8=$5$l(B

$B$k(B. $BB?9`<0$rJ,;6I=8=B?9`<0$KJQ49$9$k:](B, $B3FJQ?t$,$I$N@.J,$KBP1~$9$k$+$r(B

$B;XDj$9$k$N$,(B, $BJQ?t%j%9%H$G$"$k(B. $B$5$i$K(B, $B$=$l$i@0?t%Y%/%H%k$NA4=g=x$r(B

$B;XDj$9$k$N$,9`=g=x$N7?$G$"$k(B. $B9`=g=x7?$O(B, $B?t(B, $B?t$N%j%9%H$"$k$$$O(B

$B9TNs$GI=8=$5$l$k(B.

\BEG

A term is internally represented as an integer vector whose components

are exponents with respect to variables. A variable list specifies the

correspondences between variables and components. A type of term ordering

specifies a total order for integer vectors. A type of term ordering is

represented by an integer, a list of integer or matrices.

@noindent

$B4pK\E*$J9`=g=x7?$H$7$F<!$N(B 3 $B$D$,$"$k(B.

\JP $B4pK\E*$J9`=g=x7?$H$7$F<!$N(B 3 $B$D$,$"$k(B.

\EG There are following three fundamental types.

@table @code

@item 0 (DegRevLex; @b{$BA4<!?t5U<-=q<0=g=x(B})

\JP @item 0 (DegRevLex; @b{$BA4<!?t5U<-=q<0=g=x(B})

\EG @item 0 (DegRevLex; @b{total degree reverse lexicographic ordering})

\BJP

$B0lHL$K(B, $B$3$N=g=x$K$h$k%0%l%V%J4pDl7W;;$,:G$b9bB.$G$"$k(B. $B$?$@$7(B,

$BJ}Dx<0$r2r$/$H$$$&L\E*$KMQ$$$k$3$H$O(B, $B0lHL$K$O$G$-$J$$(B. $B$3$N(B

$B=g=x$K$h$k%0%l%V%J4pDl$O(B, $B2r$N8D?t$N7W;;(B, $B%$%G%"%k$N%a%s%P%7%C%W$d(B,

$BB>$NJQ?t=g=x$X$N4pDlJQ49$N$?$a$N%=!<%9$H$7$FMQ$$$i$l$k(B.

\BEG

In general, computation by this ordering shows the fastest speed

in most Groebner basis computations.

However, for the purpose to solve polynomial equations, this type

of ordering is, in general, not so suitable.

The Groebner bases obtained by this ordering is used for computing

the number of solutions, solving ideal membership problem and seeds

for conversion to other Groebner bases under different ordering.

@item 1 (DegLex; @b{$BA4<!?t<-=q<0=g=x(B})

\JP @item 1 (DegLex; @b{$BA4<!?t<-=q<0=g=x(B})

\EG @item 1 (DegLex; @b{total degree lexicographic ordering})

\BJP

$B$3$N=g=x$b(B, $B<-=q<0=g=x$KHf$Y$F9bB.$K%0%l%V%J4pDl$r5a$a$k$3$H$,$G$-$k$,(B,

@code{DegRevLex} $B$HF1MMD>@\$=$N7k2L$rMQ$$$k$3$H$O:$Fq$G$"$k(B. $B$7$+$7(B,

$B<-=q<0=g=x$N%0%l%V%J4pDl$r5a$a$k:]$K(B, $B@F<!2=8e$K$3$N=g=x$G%0%l%V%J4pDl(B

$B$r5a$a$F$$$k(B.

\BEG

By this type term ordering, Groebner bases are obtained fairly faster

than Lex (lexicographic) ordering, too.

Alike the @code{DegRevLex} ordering, the result, in general, cannot directly

be used for solving polynomial equations.

It is used, however, in such a way

that a Groebner basis is computed in this ordering after homogenization

to obtain the final lexicographic Groebner basis.

@item 2 (Lex; @b{$B<-=q<0=g=x(B})

\JP @item 2 (Lex; @b{$B<-=q<0=g=x(B})

\EG @item 2 (Lex; @b{lexicographic ordering})

\BJP

$B$3$N=g=x$K$h$k%0%l%V%J4pDl$O(B, $BJ}Dx<0$r2r$/>l9g$K:GE,$N7A$N4pDl$rM?$($k$,(B

$B7W;;;~4V$,$+$+$j2a$.$k$N$,FqE@$G$"$k(B. $BFC$K(B, $B2r$,M-8B8D$N>l9g(B, $B7k2L$N(B

$B78?t$,6K$a$FD9Bg$JB?G\D9?t$K$J$k>l9g$,B?$$(B. $B$3$N>l9g(B, @code{gr()},

@code{hgr()} $B$K$h$k7W;;$,6K$a$FM-8z$K$J$k>l9g$,B?$$(B.

\BEG

Groebner bases computed by this ordering give the most convenient

Groebner bases for solving the polynomial equations.

The only and serious shortcoming is the enormously long computation

time.

It is often observed that the number coefficients of the result becomes

very very long integers, especially if the ideal is 0-dimensional.

For such a case, it is empirically true for many cases

that i.e., computation by

@code{gr()} and/or @code{hgr()} may be quite effective.

@end table

@noindent

\BJP

$B$3$l$i$rAH$_9g$o$;$F%j%9%H$G;XDj$9$k$3$H$K$h$j(B, $BMM!9$J>C5n=g=x$,;XDj$G$-$k(B.

$B$3$l$O(B,

\BEG

By combining these fundamental orderingl into a list, one can make

various term ordering called elimination orderings.

@code{[[O1,L1],[O2,L2],...]}

@noindent

\BJP

$B$G;XDj$5$l$k(B. @code{Oi} $B$O(B 0, 1, 2 $B$N$$$:$l$+$G(B, @code{Li} $B$OJQ?t$N8D(B

$B?t$rI=$9(B. $B$3$N;XDj$O(B, $BJQ?t$r@hF,$+$i(B @code{L1}, @code{L2} , ...$B8D(B

$B$:$D$NAH$KJ,$1(B, $B$=$l$>$l$NJQ?t$K4X$7(B, $B=g$K(B @code{O1}, @code{O2},

...$B$N9`=g=x7?$GBg>.$,7hDj$9$k$^$GHf3S$9$k$3$H$r0UL#$9$k(B. $B$3$N7?$N(B

$B=g=x$O0lHL$K>C5n=g=x$H8F$P$l$k(B.

\BEG

In this example @code{Oi} indicates 0, 1 or 2 and @code{Li} indicates

the number of variables subject to the correspoinding orderings.

This specification means the following.

The variable list is separated into sub lists from left to right where

the @code{i}-th list contains @code{Li} members and it corresponds to

the ordering of type @code{Oi}. The result of a comparison is equal

to that for the leftmost different sub components. This type of ordering

is called an elimination ordering.

@noindent

\BJP

$B$5$i$K(B, $B9TNs$K$h$j9`=g=x$r;XDj$9$k$3$H$,$G$-$k(B. $B0lHL$K(B, @code{n} $B9T(B

@code{m} $BNs$N<B?t9TNs(B @code{M} $B$,<!$N@-<A$r;}$D$H$9$k(B.

\BEG

Furthermore one can specify a term ordering by a matix.

Suppose that a real @code{n}, @code{m} matrix @code{M} has the

following properties.

@enumerate

@item

$BD9$5(B @code{m} $B$N@0?t%Y%/%H%k(B @code{v} $B$KBP$7(B @code{Mv=0} $B$H(B @code{v=0} $B$OF1CM(B.

\JP $BD9$5(B @code{m} $B$N@0?t%Y%/%H%k(B @code{v} $B$KBP$7(B @code{Mv=0} $B$H(B @code{v=0} $B$OF1CM(B.

\BEG

For all integer verctors @code{v} of length @code{m} @code{Mv=0} is equivalent

to @code{v=0}.

@item

\BJP

$BHsIi@.J,$r;}$DD9$5(B @code{m} $B$N(B 0 $B$G$J$$@0?t%Y%/%H%k(B @code{v} $B$KBP$7(B,

@code{Mv} $B$N(B 0 $B$G$J$$:G=i$N@.J,$OHsIi(B.

\BEG

For all non-negative integer vectors @code{v} the first non-zero component

of @code{Mv} is non-negative.

@end enumerate

@noindent

\BJP

$B$3$N;~(B, 2 $B$D$N%Y%/%H%k(B @code{t}, @code{s} $B$KBP$7(B,

@code{t>s} $B$r(B, @code{M(t-s)} $B$N(B 0 $B$G$J$$:G=i$N@.J,$,HsIi(B,

$B$GDj5A$9$k$3$H$K$h$j9`=g=x$,Dj5A$G$-$k(B.

\BEG

Then we can define a term ordering such that, for two vectors

@code{t}, @code{s}, @code{t>s} means that the first non-zero component

of @code{M(t-s)} is non-negative.

@noindent

\BJP

$B9`=g=x7?$O(B, @code{gr()} $B$J$I$N0z?t$H$7$F;XDj$5$l$kB>(B, $BAH$_9~$_H!?t(B

@code{dp_ord()} $B$G;XDj$5$l(B, $B$5$^$6$^$JH!?t$N<B9T$N:]$K;2>H$5$l$k(B.

\BEG

Types of term orderings are used as arguments of functions such as

@code{gr()}. It is also set internally by @code{dp_ord()} and is used

during executions of various functions.

@noindent

\BJP

$B$3$l$i$N=g=x$N6qBNE*$JDj5A$*$h$S%0%l%V%J4pDl$K4X$9$k99$K>\$7$$2r@b$O(B

@code{[Becker,Weispfenning]} $B$J$I$r;2>H$N$3$H(B.

\BEG

For concrete definitions of term ordering and more information

about Groebner basis, refer to, for example, the book

@code{[Becker,Weispfenning]}.

@noindent

$B9`=g=x7?$N@_Dj$NB>$K(B, $BJQ?t$N=g=x<+BN$b7W;;;~4V$KBg$-$J1F6A$rM?$($k(B.

\JP $B9`=g=x7?$N@_Dj$NB>$K(B, $BJQ?t$N=g=x<+BN$b7W;;;~4V$KBg$-$J1F6A$rM?$($k(B.

\BEG

Note that the variable ordering have strong effects on the computation

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

@example

[90] B=[x^10-t,x^8-z,x^31-x^6-x-y]$

Line 443 trace-lifting $B$K$*$1$k(B, $BF~NOB?9`<0$N%a%s%P%7%

Line 1005 trace-lifting $B$K$*$1$k(B, $BF~NOB?9`<0$N%a%s%P%7%

-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

@noindent

\BJP

$BJQ?t=g=x(B @code{[x,y,z,t]} $B$K$*$1$k%0%l%V%J4pDl$O(B, $B4pDl$N?t$bB?$/(B, $B$=$l$>$l$N(B

$B<0$bBg$-$$(B. $B$7$+$7(B, $B=g=x(B @code{[t,z,y,x]} $B$K$b$H$G$O(B, @code{B} $B$,$9$G$K(B

$B%0%l%V%J4pDl$H$J$C$F$$$k(B. $BBg;(GD$K$$$($P(B, $B<-=q<0=g=x$G%0%l%V%J4pDl$r5a$a$k(B

Line 474 z^5-t^4]

Line 1038 z^5-t^4]

@code{x} $B$GI=$5$l$F$$$k$3$H$+$i$3$N$h$&$J6KC<$J7k2L$H$J$C$?$o$1$G$"$k(B.

$B<B:]$K8=$l$k7W;;$K$*$$$F$O(B, $B$3$N$h$&$KA*$V$Y$-JQ?t=g=x$,L@$i$+$G$"$k(B

$B$3$H$O>/$J$/(B, $B;n9T:x8m$,I,MW$J>l9g$b$"$k(B.

\BEG

As you see in the above example, the Groebner base under variable

ordering @code{[x,y,z,t]} has a lot of bases and each base itself is

large. Under variable ordering @code{[t,z,y,x]}, however, @code{B} itself

is already the Groebner basis.

Roughly speaking, to obtain a Groebner base under the lexicographic

ordering is to express the variables on the left (having higher order)

in terms of variables on the right (having lower order).

In the example, variables @code{t}, @code{z}, and @code{y} are already

expressed by variable @code{x}, and the above explanation justifies

such a drastic experimental results.

In practice, however, optimum ordering for variables may not known

beforehand, and some heuristic trial may be inevitable.

\BJP

@node $BM-M}<0$r78?t$H$9$k%0%l%V%J4pDl7W;;(B,,, $B%0%l%V%J4pDl$N7W;;(B

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

\BEG

@node Groebner basis computation with rational function coefficients,,, Groebner basis computation

@section Groebner basis computation with rational function coefficients

@noindent

\BJP

@code{gr()} $B$J$I$N%H%C%W%l%Y%kH!?t$O(B, $B$$$:$l$b(B, $BF~NOB?9`<0%j%9%H$K(B

$B8=$l$kJQ?t(B ($BITDj85(B) $B$H(B, $BJQ?t%j%9%H$K8=$l$kJQ?t$rHf3S$7$F(B, $BJQ?t%j%9%H$K(B

$B$J$$JQ?t$,F~NOB?9`<0$K8=$l$F$$$k>l9g$K$O(B, $B<+F0E*$K(B, $B$=$NJQ?t$r(B, $B78?t(B

$BBN$N85$H$7$F07$&(B.

\BEG

Such variables that appear within the input polynomials but

not appearing in the input variable list are automatically treated

as elements in the coefficient field

by top level functions, such as @code{gr()}.

@example

[64] gr([a*x+b*y-c,d*x+e*y-f],[x,y],2);

Line 490 z^5-t^4]

Line 1083 z^5-t^4]

@end example

@noindent

\BJP

$B$3$NNc$G$O(B, @code{a}, @code{b}, @code{c}, @code{d} $B$,78?tBN$N85$H$7$F(B

$B07$o$l$k(B. $B$9$J$o$A(B, $BM-M}H!?tBN(B

@b{F} = @b{Q}(@code{a},@code{b},@code{c},@code{d}) $B>e$N(B 2 $BJQ?tB?9`<04D(B

Line 501 z^5-t^4]

Line 1095 z^5-t^4]

$B$K$O0[$J$k(B. $B$^$?(B, $B<g$H$7$F7W;;8zN(>e$NLdBj$N$?$a(B, $BJ,;6I=8=B?9`<0(B

$B$N78?t$H$7$F<B:]$K5v$5$l$k$N$OB?9`<0$^$G$G$"$k(B. $B$9$J$o$A(B, $BJ,Jl$r(B

$B;}$DM-M}<0$OJ,;6I=8=B?9`<0$N78?t$H$7$F$O5v$5$l$J$$(B.

\BEG

In this example, variables @code{a}, @code{b}, @code{c}, and @code{d}

are treated as elements in the coefficient field.

In this case, a Groebner basis is computed

on a bi-variate polynomial ring

@b{F}[@code{x},@code{y}]

over rational function field

@b{F} = @b{Q}(@code{a},@code{b},@code{c},@code{d}).

Notice that coefficients are considered as a member in a field.

As a consequence, polynomial factors common to the coefficients

are removed so that the result, in general, is different from

the result that would be obtained when the problem is considered

as a computation of Groebner basis over a polynomial ring

with rational function coefficients.

And note that coefficients of a distributed polynomial are limited

to numbers and polynomials because of efficiency.

\BJP

@node $B4pDlJQ49(B,,, $B%0%l%V%J4pDl$N7W;;(B

@section $B4pDlJQ49(B

\BEG

@node Change of ordering,,, Groebner basis computation

@section Change of orderng

@noindent

\BJP

$B<-=q<0=g=x$N%0%l%V%J4pDl$r5a$a$k>l9g(B, $BD>@\(B @code{gr()} $B$J$I$r5/F0$9$k(B

$B$h$j(B, $B0lC6B>$N=g=x(B ($BNc$($PA4<!?t5U<-=q<0=g=x(B) $B$N%0%l%V%J4pDl$r7W;;$7$F(B,

$B$=$l$rF~NO$H$7$F<-=q<0=g=x$N%0%l%V%J4pDl$r7W;;$9$kJ}$,8zN($,$h$$>l9g(B

Line 516 z^5-t^4]

Line 1135 z^5-t^4]

$B0J2<$N(B 2 $B$D$NH!?t$O(B, $BJQ?t=g=x(B @var{vlist1}, $B9`=g=x7?(B @var{order} $B$G(B

$B4{$K%0%l%V%J4pDl$H$J$C$F$$$kB?9`<0%j%9%H(B @var{gbase} $B$r(B, $BJQ?t=g=x(B

@var{vlist2} $B$K$*$1$k<-=q<0=g=x$N%0%l%V%J4pDl$KJQ49$9$kH!?t$G$"$k(B.

\BEG

When we compute a lex order Groebner basis, it is often efficient to

compute it via Groebner basis with respect to another order such as

degree reverse lex order, rather than to compute it directory by

@code{gr()} etc. If we know that an input is a Groebner basis with

respect to an order, we can apply special methods called change of

ordering for a Groebner basis computation with respect to another

order, without using Buchberger algorithm. The following two functions

are ones for change of ordering such that they convert a Groebner

basis @var{gbase} with respect to the variable order @var{vlist1} and

the order type @var{order} into a lex Groebner basis with respect

to the variable order @var{vlist2}.

@table @code

@item tolex(@var{gbase},@var{vlist1},@var{order},@var{vlist2})

\BJP

$B$3$NH!?t$O(B, @var{gbase} $B$,M-M}?tBN>e$N%7%9%F%`$N>l9g$K$N$_;HMQ2DG=$G$"$k(B.

$B$3$NH!?t$O(B, $B<-=q<0=g=x$N%0%l%V%J4pDl$r(B, $BM-8BBN>e$G7W;;$5$l$?%0%l%V%J4pDl(B

$B$r?w7?$H$7$F(B, $BL$Dj78?tK!$*$h$S(B Hensel $B9=@.$K$h$j5a$a$k$b$N$G$"$k(B.

\BEG

This function can be used only when @var{gbase} is an ideal over the

rationals. The input @var{gbase} must be a Groebner basis with respect

to the variable order @var{vlist1} and the order type @var{order}. Moreover

the ideal generated by @var{gbase} must be zero-dimensional.

This computes the lex Groebner basis of @var{gbase}

by using the modular change of ordering algorithm. The algorithm first

computes the lex Groebner basis over a finite field. Then each element

in the lex Groebner basis over the rationals is computed with undetermined

coefficient method and linear equation solving by Hensel lifting.

@item tolex_tl(@var{gbase},@var{vlist1},@var{order},@var{vlist2},@var{homo})

\BJP

$B$3$NH!?t$O(B, $B<-=q<0=g=x$N%0%l%V%J4pDl$r(B Buchberger $B%"%k%4%j%:%`$K$h$j5a(B

$B$a$k$b$N$G$"$k$,(B, $BF~NO$,$"$k=g=x$K$*$1$k%0%l%V%J4pDl$G$"$k>l9g$N(B

trace-lifting$B$K$*$1$k%0%l%V%J4pDl8uJd$NF,9`(B, $BF,78?t$N@-<A$rMxMQ$7$F(B,

Line 536 trace-lifting$B$K$*$1$k%0%l%V%J4pDl8uJd$NF,9`(B, $B

Line 1183 trace-lifting$B$K$*$1$k%0%l%V%J4pDl8uJd$NF,9`(B, $B

$BD>@\<-=q<0=g=x$N7W;;$r9T$&$h$j8zN($,$h$$(B. ($B$b$A$m$sNc30$"$j(B. )

$B0z?t(B @var{homo} $B$,(B 0 $B$G$J$$;~(B, @code{hgr()} $B$HF1MM$K@F<!2=$r7PM3$7$F(B

$B7W;;$r9T$&(B.

\BEG

This function computes the lex Groebner basis of @var{gbase}. The

input @var{gbase} must be a Groebner basis with respect to the

variable order @var{vlist1} and the order type @var{order}.

Buchberger algorithm with trace lifting is used to compute the lex

Groebner basis, however the Groebner basis check and the ideal

membership check can be omitted by using several properties derived

from the fact that the input is a Groebner basis. So it is more

efficient than simple repetition of Buchberger algorithm. If the input

is zero-dimensional, this function inserts automatically a computation

of Groebner basis with respect to an elimination order, which makes

the whole computation more efficient for many cases. If @var{homo} is

not equal to 0, homogenization is used in each step.

@end table

@noindent

\BJP

$B$=$NB>(B, 0 $B<!85%7%9%F%`$KBP$7(B, $BM?$($i$l$?B?9`<0$N:G>.B?9`<0$r5a$a$k(B

$BH!?t(B, 0 $B<!85%7%9%F%`$N2r$r(B, $B$h$j%3%s%Q%/%H$KI=8=$9$k$?$a$NH!?t$J$I$,(B

@samp{gr} $B$GDj5A$5$l$F$$$k(B. $B$3$l$i$K$D$$$F$O8D!9$NH!?t$N@bL@$r;2>H$N$3$H(B.

\BEG

For zero-dimensional systems, there are several fuctions to

compute the minimal polynomial of a polynomial and or a more compact

representation for zeros of the system. They are all defined in @samp{gr}.

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

\BEG

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

@section Functions for Groebner basis computation

@menu

* gr hgr gr_mod::

Line 553 trace-lifting$B$K$*$1$k%0%l%V%J4pDl8uJd$NF,9`(B, $B

Line 1327 trace-lifting$B$K$*$1$k%0%l%V%J4pDl8uJd$NF,9`(B, $B

* 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 578 trace-lifting$B$K$*$1$k%0%l%V%J4pDl8uJd$NF,9`(B, $B

Line 1352 trace-lifting$B$K$*$1$k%0%l%V%J4pDl8uJd$NF,9`(B, $B

* 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

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

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

\EG @node gr hgr gr_mod,,, Functions for Groebner basis computation

@subsection @code{gr}, @code{hgr}, @code{gr_mod}, @code{dgr}

@findex gr

@findex hgr

Line 592 trace-lifting$B$K$*$1$k%0%l%V%J4pDl8uJd$NF,9`(B, $B

Line 1370 trace-lifting$B$K$*$1$k%0%l%V%J4pDl8uJd$NF,9`(B, $B

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

@itemx gr_mod(@var{plist},@var{vlist},@var{order},@var{p})

@itemx dgr(@var{plist},@var{vlist},@var{order},@var{procs})

:: $B%0%l%V%J4pDl$N7W;;(B

\JP :: $B%0%l%V%J4pDl$N7W;;(B

\EG :: Groebner basis computation

@end table

@table @var

@item return

$B%j%9%H(B

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

@item plist, vlist, procs

\EG list

$B%j%9%H(B

@item plist vlist procs

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

\EG list

@item order

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

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

\EG number, list or matrix

@item p

2^27 $BL$K~$NAG?t(B

\JP 2^27 $BL$K~$NAG?t(B

\EG prime less than 2^27

@end table

@itemize @bullet

\BJP

@item

$BI8=`%i%$%V%i%j$N(B @samp{gr} $B$GDj5A$5$l$F$$$k(B.

@item

Line 628 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.

\BEG

@item

These functions are defined in @samp{gr} in the standard library

directory.

@item

They compute a Groebner basis of a polynomial list @var{plist} with

respect to the variable order @var{vlist} and the order type @var{order}.

@code{gr()} and @code{hgr()} compute a Groebner basis over the rationals

and @code{gr_mod} computes over GF(@var{p}).

@item

Variables not included in @var{vlist} are regarded as

included in the ground field.

@item

@code{gr()} uses trace-lifting (an improvement by modular computation)

and sugar strategy.

@code{hgr()} uses trace-lifting and a cured sugar strategy

by using homogenization.

@item

@code{dgr()} executes @code{gr()}, @code{dgr()} simultaneously on

two process in a child process list @var{procs} and returns

the result obtained first. The results returned from both the process

should be equal, but it is not known in advance which method is faster.

Therefore this function is useful to reduce the actual elapsed time.

@item

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.

@end itemize

@example

Line 642 CPU $B;~4V$G$"$j(B, $B$3$NH!?t$N>l9g$O$[$H$s$IDL?.$

Line 1455 CPU $B;~4V$G$"$j(B, $B$3$NH!?t$N>l9g$O$[$H$s$IDL?.$

@end example

@table @t

@item $B;2>H(B

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

@comment @fref{dp_gr_main dp_gr_mod_main},

\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},

@fref{dp_ord}.

@end table

@node lex_hensel lex_tl tolex tolex_d tolex_tl,,, $B%0%l%V%J4pDl$K4X$9$kH!?t(B

\JP @node lex_hensel lex_tl tolex tolex_d tolex_tl,,, $B%0%l%V%J4pDl$K4X$9$kH!?t(B

\EG @node lex_hensel lex_tl tolex tolex_d tolex_tl,,, Functions for Groebner basis computation

@subsection @code{lex_hensel}, @code{lex_tl}, @code{tolex}, @code{tolex_d}, @code{tolex_tl}

@findex lex_hensel

@findex lex_tl

Line 659 CPU $B;~4V$G$"$j(B, $B$3$NH!?t$N>l9g$O$[$H$s$IDL?.$

Line 1473 CPU $B;~4V$G$"$j(B, $B$3$NH!?t$N>l9g$O$[$H$s$IDL?.$

@table @t

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

@itemx lex_tl(@var{plist},@var{vlist1},@var{order},@var{vlist2},@var{homo})

:: $B4pDlJQ49$K$h$k<-=q<0=g=x%0%l%V%J4pDl$N7W;;(B

\JP :: $B4pDlJQ49$K$h$k<-=q<0=g=x%0%l%V%J4pDl$N7W;;(B

\EG:: Groebner basis computation with respect to a lex order by change of ordering

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

@itemx tolex_d(@var{plist},@var{vlist1},@var{order},@var{vlist2},@var{procs})

@itemx tolex_tl(@var{plist},@var{vlist1},@var{order},@var{vlist2},@var{homo})

:: $B%0%l%V%J4pDl$rF~NO$H$9$k(B, $B4pDlJQ49$K$h$k<-=q<0=g=x%0%l%V%J4pDl$N7W;;(B

\JP :: $B%0%l%V%J4pDl$rF~NO$H$9$k(B, $B4pDlJQ49$K$h$k<-=q<0=g=x%0%l%V%J4pDl$N7W;;(B

\EG :: Groebner basis computation with respect to a lex order by change of ordering, starting from a Groebner basis

@end table

@table @var

@item return

$B%j%9%H(B

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

@item plist, vlist1, vlist2, procs

\EG list

$B%j%9%H(B

@item plist vlist1 vlist2 procs

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

\EG list

@item order

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

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

\EG number, list or matrix

@item homo

$B%U%i%0(B

\JP $B%U%i%0(B

\EG flag

@end table

@itemize @bullet

\BJP

@item

$BI8=`%i%$%V%i%j$N(B @samp{gr} $B$GDj5A$5$l$F$$$k(B.

@item

Line 695 CPU $B;~4V$G$"$j(B, $B$3$NH!?t$N>l9g$O$[$H$s$IDL?.$

Line 1516 CPU $B;~4V$G$"$j(B, $B$3$NH!?t$N>l9g$O$[$H$s$IDL?.$

@item

@code{lex_hensel()}, @code{lex_tl()} $B$K$*$$$F$O(B, $B<-=q<0=g=x%0%l%V%J4pDl$N(B

$B7W;;$O<!$N$h$&$K9T$o$l$k(B. (@code{[Noro,Yokoyama]} $B;2>H(B.)

@enumerate

@item

@var{vlist1}, @var{order} $B$K4X$9$k%0%l%V%J4pDl(B @var{G0} $B$r7W;;$9$k(B.

Line 750 CPU $B;~4V$G$"$j(B, $B$3$NH!?t$N>l9g$O$[$H$s$IDL?.$

Line 1570 CPU $B;~4V$G$"$j(B, $B$3$NH!?t$N>l9g$O$[$H$s$IDL?.$

@item

@code{tolex_d()} $B$GI=<($5$l$k;~4V$O(B, $B$3$NH!?t$,<B9T$5$l$F$$$k%W%m%;%9$K(B

$B$*$$$F9T$o$l$?7W;;$KBP1~$7$F$$$F(B, $B;R%W%m%;%9$K$*$1$k;~4V$O4^$^$l$J$$(B.

\BEG

@item

These functions are defined in @samp{gr} in the standard library

directory.

@item

@code{lex_hensel()} and @code{lex_tl()} first compute a Groebner basis

with respect to the variable order @var{vlist1} and the order type @var{order}.

Then the Groebner basis is converted into a lex order Groebner basis

with respect to the varable order @var{vlist2}.

@item

@code{tolex()} and @code{tolex_tl()} convert a Groebner basis @var{plist}

with respect to the variable order @var{vlist1} and the order type @var{order}

into a lex order Groebner basis

with respect to the varable order @var{vlist2}.

@code{tolex_d()} does computations of basis elements in @code{tolex()}

in parallel on the processes in a child process list @var{procs}.

@item

In @code{lex_hensel()} and @code{tolex_hensel()} a lex order Groebner basis

is computed as follows.(Refer to @code{[Noro,Yokoyama]}.)

@enumerate

@item

Compute a Groebner basis @var{G0} with respect to @var{vlist1} and @var{order}.

(Only in @code{lex_hensel()}. )

@item

Choose a prime which does not divide head coefficients of elements in @var{G0}

with respect to @var{vlist1} and @var{order}. Then compute a lex order

Groebner basis @var{Gp} over GF(@var{p}) with respect to @var{vlist2}.

@item

Compute @var{NF}, the set of all the normal forms with respect to

@var{G0} of terms appearing in @var{Gp}.

@item

For each element @var{f} in @var{Gp}, replace coefficients and terms in @var{f}

with undetermined coefficients and the corresponding polynomials in @var{NF}

respectively, and generate a system of liear equation @var{Lf} by equating

the coefficients of terms in the replaced polynomial with 0.

@item

Solve @var{Lf} by Hensel lifting, starting from the unique mod @var{p}

solution.

@item

If all the linear equations generated from the elements in @var{Gp}

could be solved, then the set of solutions corresponds to a lex order

Groebner basis. Otherwise redo the whole process with another @var{p}.

@end enumerate

@item

In @code{lex_tl()} and @code{tolex_tl()} a lex order Groebner basis

is computed as follows.(Refer to @code{[Noro,Yokoyama]}.)

@enumerate

@item

Compute a Groebner basis @var{G0} with respect to @var{vlist1} and @var{order}.

(Only in @code{lex_tl()}. )

@item

If @var{G0} is not zero-dimensional, choose a prime which does not divide

head coefficients of elements in @var{G0} with respect to @var{vlist1} and

@var{order}. Then compute a candidate of a lex order Groebner basis

via trace lifting with @var{p}. If it succeeds the candidate is indeed

a lex order Groebner basis without any check. Otherwise redo the whole

process with another @var{p}.

@item

If @var{G0} is zero-dimensional, starting from @var{G0},

compute a Groebner basis @var{G1} with respect to an elimination order

to eliminate variables other than the last varibale in @var{vlist2}.

Then compute a lex order Groebner basis stating from @var{G1}. These

computations are done by trace lifting and the selection of a mudulus

@var{p} is the same as in non zero-dimensional cases.

@end enumerate

@item

Computations with rational function coefficients can be done only by

@code{lex_tl()} and @code{tolex_tl()}.

@item

If @code{homo} is not equal to 0, homogenization is used in Buchberger

algorithm.

@item

The CPU time shown after an execution of @code{tolex_d()} indicates

that of the master process, and it does not include the time in child

processes.

@end itemize

@example

Line 771 CPU $B;~4V$G$"$j(B, $B$3$NH!?t$N>l9g$O$[$H$s$IDL?.$

Line 1672 CPU $B;~4V$G$"$j(B, $B$3$NH!?t$N>l9g$O$[$H$s$IDL?.$

@end example

@table @t

@item $B;2>H(B

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

@fref{dp_gr_main dp_gr_mod_main},

\EG @item References

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

@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

@node lex_hensel_gsl tolex_gsl tolex_gsl_d,,, $B%0%l%V%J4pDl$K4X$9$kH!?t(B

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

\EG @node lex_hensel_gsl tolex_gsl tolex_gsl_d,,, Functions for Groebner basis computation

@subsection @code{lex_hensel_gsl}, @code{tolex_gsl}, @code{tolex_gsl_d}

@findex lex_hensel_gsl

@findex tolex_gsl

Line 784 CPU $B;~4V$G$"$j(B, $B$3$NH!?t$N>l9g$O$[$H$s$IDL?.$

Line 1688 CPU $B;~4V$G$"$j(B, $B$3$NH!?t$N>l9g$O$[$H$s$IDL?.$

@table @t

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

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

\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})

@itemx tolex_gsl_d(@var{plist},@var{vlist1},@var{order},@var{vlist2},@var{homo},@var{procs})

:: $B%0%l%V%J4pDl$rF~NO$H$9$k(B, GSL $B7A<0$N%$%G%"%k4pDl$N7W;;(B

\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

@table @var

@item return

$B%j%9%H(B

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

@item plist, vlist1, vlist2, procs

\EG list

$B%j%9%H(B

@item plist vlist1 vlist2 procs

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

\EG list

@item order

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

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

\EG number, list or matrix

@item homo

$B%U%i%0(B

\JP $B%U%i%0(B

\EG flag

@end table

@itemize @bullet

\BJP

@item

@code{lex_hensel_gsl()} $B$O(B @code{lex_hensel()} $B$N(B, @code{tolex_gsl()} $B$O(B

@code{tolex()} $B$NJQ<o$G(B, $B7k2L$N$_$,0[$J$k(B.

Line 813 CPU $B;~4V$G$"$j(B, $B$3$NH!?t$N>l9g$O$[$H$s$IDL?.$

Line 1724 CPU $B;~4V$G$"$j(B, $B$3$NH!?t$N>l9g$O$[$H$s$IDL?.$

@code{x0} $B$N(B 1 $BJQ?tB?9`<0(B) $B$J$k7A(B ($B$3$l$r(B SL $B7A<0$H8F$V(B) $B$r;}$D>l9g(B,

@code{[[x1,g1,d1],...,[xn,gn,dn],[x0,f0,f0']]} $B$J$k%j%9%H(B ($B$3$l$r(B GSL $B7A<0$H8F$V(B)

$B$rJV$9(B.

$B$3$3$G(B, @code{gi} $B$O(B, @code{f0'fi-gi} $B$,(B @code{f0} $B$G3d$j@Z$l$k$h$&$J(B

$B$3$3$G(B, @code{gi} $B$O(B, @code{di*f0'*fi-gi} $B$,(B @code{f0} $B$G3d$j@Z$l$k$h$&$J(B

@code{x0} $B$N(B1 $BJQ?tB?9`<0$G(B,

$B2r$O(B @code{f0(x0)=0} $B$J$k(B @code{x0} $B$KBP$7(B, @code{[x1=g1/(d1*f0'),...,xn=gn/(dn*f0')]}

$B$H$J$k(B. $B<-=q<0=g=x%0%l%V%J4pDl$,>e$N$h$&$J7A$G$J$$>l9g(B, @code{tolex()} $B$K(B

Line 823 GSL $B7A<0$K$h$jI=$5$l$k4pDl$O%0%l%V%J4pDl$G$O$J$$$,

Line 1734 GSL $B7A<0$K$h$jI=$5$l$k4pDl$O%0%l%V%J4pDl$G$O$J$$$,

$B$N%0%l%V%J4pDl$h$jHs>o$K>.$5$$$?$a7W;;$bB.$/(B, $B2r$b5a$a$d$9$$(B.

@code{tolex_gsl_d()} $B$GI=<($5$l$k;~4V$O(B, $B$3$NH!?t$,<B9T$5$l$F$$$k%W%m%;%9$K(B

$B$*$$$F9T$o$l$?7W;;$KBP1~$7$F$$$F(B, $B;R%W%m%;%9$K$*$1$k;~4V$O4^$^$l$J$$(B.

\BEG

@item

@code{lex_hensel_gsl()} and @code{lex_hensel()} are variants of

@code{tolex_gsl()} and @code{tolex()} respectively. The results are

Groebner basis or a kind of ideal basis, called GSL form.

@code{tolex_gsl_d()} does basis computations in parallel on child

processes specified in @code{procs}.

@item

If the input is zero-dimensional and a lex order Groebner basis has

the form @code{[f0,x1-f1,...,xn-fn]} (@code{f0},...,@code{fn} are

univariate polynomials of @code{x0}; SL form), then this these

functions return a list such as

@code{[[x1,g1,d1],...,[xn,gn,dn],[x0,f0,f0']]} (GSL form). In this list

@code{gi} is a univariate polynomial of @code{x0} such that

@code{di*f0'*fi-gi} divides @code{f0} and the roots of the input ideal is

@code{[x1=g1/(d1*f0'),...,xn=gn/(dn*f0')]} for @code{x0}

such that @code{f0(x0)=0}.

If the lex order Groebner basis does not have the above form,

these functions return

a lex order Groebner basis computed by @code{tolex()}.

@item

Though an ideal basis represented as GSL form is not a Groebner basis

we can expect that the coefficients are much smaller than those in a Groebner

basis and that the computation is efficient.

The CPU time shown after an execution of @code{tolex_gsl_d()} indicates

that of the master process, and it does not include the time in child

processes.

@end itemize

@example

Line 835 GSL $B7A<0$K$h$jI=$5$l$k4pDl$O%0%l%V%J4pDl$G$O$J$$$,

Line 1776 GSL $B7A<0$K$h$jI=$5$l$k4pDl$O%0%l%V%J4pDl$G$O$J$$$,

[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

@item $B;2>H(B

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

\EG @item References

@fref{lex_hensel lex_tl tolex tolex_d tolex_tl},

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

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

\EG @fref{Distributed computation}

@end table

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

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

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

@subsection @code{gr_minipoly}, @code{minipoly}

@findex gr_minipoly

@findex minipoly

@table @t

@item gr_minipoly(@var{plist},@var{vlist},@var{order},@var{poly},@var{v},@var{homo})

:: $BB?9`<0$N(B, $B%$%G%"%k$rK!$H$7$?:G>.B?9`<0$N7W;;(B

\JP :: $BB?9`<0$N(B, $B%$%G%"%k$rK!$H$7$?:G>.B?9`<0$N7W;;(B

\EG :: Computation of the minimal polynomial of a polynomial modulo an ideal

@item minipoly(@var{plist},@var{vlist},@var{order},@var{poly},@var{v})

:: $B%0%l%V%J4pDl$rF~NO$H$9$k(B, $BB?9`<0$N:G>.B?9`<0$N7W;;(B

\JP :: $B%0%l%V%J4pDl$rF~NO$H$9$k(B, $BB?9`<0$N:G>.B?9`<0$N7W;;(B

\EG :: Computation of the minimal polynomial of a polynomial modulo an ideal

@end table

@table @var

@item return

$BB?9`<0(B

\JP $BB?9`<0(B

@item plist, vlist

\EG polynomial

$B%j%9%H(B

@item plist vlist

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

\EG list

@item order

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

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

\EG number, list or matrix

@item poly

$BB?9`<0(B

\JP $BB?9`<0(B

\EG polynomial

@item v

$BITDj85(B

\JP $BITDj85(B

\EG indeterminate

@item homo

$B%U%i%0(B

\JP $B%U%i%0(B

\EG flag

@end table

@itemize @bullet

\BJP

@item

@code{gr_minipoly()} $B$O%0%l%V%J4pDl$N7W;;$+$i9T$$(B, @code{minipoly()} $B$O(B

$BF~NO$r%0%l%V%J4pDl$H$_$J$9(B.

Line 890 K[@var{v}] $B$N85(B f(@var{v}) $B$K(B f(@var{p}) m

Line 1844 K[@var{v}] $B$N85(B f(@var{v}) $B$K(B f(@var{p}) m

@item

@code{gr_minipoly()} $B$K;XDj$9$k9`=g=x$H$7$F$O(B, $BDL>oA4<!?t5U<-=q<0=g=x$r(B

$BMQ$$$k(B.

\BEG

@item

@code{gr_minipoly()} begins by computing a Groebner basis.

@code{minipoly()} regards an input as a Groebner basis with respect to

the variable order @var{vlist} and the order type @var{order}.

@item

Let K be a field. If an ideal @var{I} in K[X] is zero-dimensional, then, for

a polynomial @var{p} in K[X], the kernel of a homomorphism from

K[@var{v}] to K[X]/@var{I} which maps f(@var{v}) to f(@var{p}) mod @var{I}

is generated by a polynomial. The generator is called the minimal polynomial

of @var{p} modulo @var{I}.

@item

@code{gr_minipoly()} and @code{minipoly()} computes the minimal polynomial

of a polynomial @var{p} and returns it as a polynomial of @var{v}.

@item

The minimal polynomial can be computed as an element of a Groebner basis.

But if we are only interested in the minimal polynomial,

@code{minipoly()} and @code{gr_minipoly()} can compute it more efficiently

than methods using Groebner basis computation.

@item

It is recommended to use a degree reverse lex order as a term order

for @code{gr_minipoly()}.

@end itemize

@example

Line 902 K[@var{v}] $B$N85(B f(@var{v}) $B$K(B f(@var{p}) m

Line 1880 K[@var{v}] $B$N85(B f(@var{v}) $B$K(B f(@var{p}) m

@end example

@table @t

@item $B;2>H(B

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

\EG @item References

@fref{lex_hensel lex_tl tolex tolex_d tolex_tl}.

@end table

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

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

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

@subsection @code{tolexm}, @code{minipolym}

@findex tolexm

@findex minipolym

@table @t

@item tolexm(@var{plist},@var{vlist1},@var{order},@var{vlist2},@var{mod})

:: $BK!(B @var{mod} $B$G$N4pDlJQ49$K$h$k%0%l%V%J4pDl7W;;(B

\JP :: $BK!(B @var{mod} $B$G$N4pDlJQ49$K$h$k%0%l%V%J4pDl7W;;(B

\EG :: Groebner basis computation modulo @var{mod} by change of ordering.

@item minipolym(@var{plist},@var{vlist1},@var{order},@var{poly},@var{v},@var{mod})

:: $BK!(B @var{mod} $B$G$N%0%l%V%J4pDl$K$h$kB?9`<0$N:G>.B?9`<0$N7W;;(B

\JP :: $BK!(B @var{mod} $B$G$N%0%l%V%J4pDl$K$h$kB?9`<0$N:G>.B?9`<0$N7W;;(B

\EG :: Minimal polynomial computation modulo @var{mod} the same method as

@end table

@table @var

@item return

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

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

@item plist, vlist1, vlist2

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

$B%j%9%H(B

@item plist vlist1 vlist2

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

\EG list

@item order

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

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

\EG number, list or matrix

@item mod

$BAG?t(B

\JP $BAG?t(B

\EG prime

@end table

@itemize @bullet

\BJP

@item

$BF~NO(B @var{plist} $B$O$$$:$l$b(B $BJQ?t=g=x(B @var{vlist1}, $B9`=g=x7?(B @var{order},

$BK!(B @var{mod} $B$K$*$1$k%0%l%V%J4pDl$G$J$1$l$P$J$i$J$$(B.

Line 938 K[@var{v}] $B$N85(B f(@var{v}) $B$K(B f(@var{p}) m

Line 1925 K[@var{v}] $B$N85(B f(@var{v}) $B$K(B f(@var{p}) m

@item

@code{tolexm()} $B$O(B FGLM $BK!$K$h$k4pDlJQ49$K$h$j(B @var{vlist2},

$B<-=q<0=g=x$K$h$k%0%l%V%J4pDl$r7W;;$9$k(B.

\BEG

@item

An input @var{plist} must be a Groebner basis modulo @var{mod}

with respect to the variable order @var{vlist1} and the order type @var{order}.

@item

@code{minipolym()} executes the same computation as in @code{minipoly}.

@item

@code{tolexm()} computes a lex order Groebner basis modulo @var{mod}

with respect to the variable order @var{vlist2}, by using FGLM algorithm.

@end itemize

@example

Line 948 z^32+11405*z^31+20868*z^30+21602*z^29+...

Line 1946 z^32+11405*z^31+20868*z^30+21602*z^29+...

@end example

@table @t

@item $B;2>H(B

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

\EG @item References

@fref{lex_hensel lex_tl tolex tolex_d tolex_tl},

@fref{gr_minipoly minipoly}.

@end table

@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

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

\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}, @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})

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

@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

@table @var

@item return

$B%j%9%H(B

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

@item plist, vlist

\EG list

$B%j%9%H(B

@item plist vlist

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

\EG list

@item order

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

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

\EG number, list or matrix

@item homo

$B%U%i%0(B

\JP $B%U%i%0(B

\EG flag

@item modular

$B%U%i%0$^$?$OAG?t(B

\JP $B%U%i%0$^$?$OAG?t(B

\EG flag or prime

@end table

@itemize @bullet

\BJP

@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 1009 z^32+11405*z^31+20868*z^30+21602*z^29+...

Line 2029 z^32+11405*z^31+20868*z^30+21602*z^29+...

@item

@var{homo}, @var{modular} $B$NB>$K(B, @code{dp_gr_flags()} $B$G@_Dj$5$l$k(B

$B$5$^$6$^$J%U%i%0$K$h$j7W;;$,@)8f$5$l$k(B.

\BEG

@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. 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

For @code{dp_gr_mod_main()}, @var{modular} means a computation over

GF(@var{modular}).

For @code{dp_gr_main()}, @var{modular} has the following mean.

@enumerate

@item

If @var{modular} is 1 , trace lifting is used. Primes for trace lifting

are generated by @code{lprime()}, starting from @code{lprime(0)}, until

the computation succeeds.

@item

If @var{modular} is an integer greater than 1, the integer is regarded as a

prime and trace lifting is executed by using the prime. If the computation

fails then 0 is returned.

@item

If @var{modular} is negative, the above rule is applied for @var{-modular}

but the Groebner basis check and ideal-membership check are omitted in

the last stage of trace lifting.

@end enumerate

@item

@code{gr(P,V,O)}, @code{hgr(P,V,O)} and @code{gr_mod(P,V,O,M)} execute

@code{dp_gr_main(P,V,0,1,O)}, @code{dp_gr_main(P,V,1,1,O)}

and @code{dp_gr_mod_main(P,V,0,M,O)} respectively.

@item

Actual computation is controlled by various parameters set by

@code{dp_gr_flags()}, other then by @var{homo} and @var{modular}.

@end itemize

@table @t

@item $B;2>H(B

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

\EG @item References

@fref{dp_ord},

@fref{dp_gr_flags dp_gr_print},

@fref{gr hgr gr_mod},

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

@fref{setmod_ff},

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

\EG @fref{Controlling Groebner basis computations}

@end table

@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

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

\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}, @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})

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

@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

@table @var

@item return

$B%j%9%H(B

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

@item plist, vlist

\EG list

$B%j%9%H(B

@item plist vlist

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

\EG list

@item order

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

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

\EG number, list or matrix

@end table

@itemize @bullet

\BJP

@item

F4 $B%"%k%4%j%:%`$K$h$j%0%l%V%J4pDl$N7W;;$r9T$&(B.

@item

Line 1047 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

@item

These functions compute Groebner bases by F4 algorithm.

@item

F4 is a new generation algorithm for Groebner basis computation

invented by J.C. Faugere. The current implementation of @code{dp_f4_main()}

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

except for lack of the argument for controlling homogenization.

@end itemize

@table @t

@item $B;2>H(B

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

\EG @item References

@fref{dp_ord},

@fref{dp_gr_flags dp_gr_print},

@fref{gr hgr gr_mod},

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

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

\EG @fref{Controlling Groebner basis computations}

@end table

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

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

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

@subsection @code{dp_gr_flags}, @code{dp_gr_print}

@findex dp_gr_flags

@findex dp_gr_print

@table @t

@item dp_gr_flags([@var{list}])

@itemx dp_gr_print([@var{0|1}])

@itemx dp_gr_print([@var{i}])

:: $B7W;;$*$h$SI=<(MQ%Q%i%a%?$N@_Dj(B, $B;2>H(B

\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.

@end table

@table @var

@item return

$B@_DjCM(B

\JP $B@_DjCM(B

\EG value currently set

@item list

$B%j%9%H(B

\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 1087 F4 $B%"%k%4%j%:%`$O(B, J.C. Faugere $B$K$h$jDs>'$5$

Line 2191 F4 $B%"%k%4%j%:%`$O(B, J.C. Faugere $B$K$h$jDs>'$5$

$B0z?t$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

@item

@code{dp_gr_flags()} sets and shows various parameters for Groebner basis

computation.

@item

If no argument is specified the current settings are returned.

@item

Arguments must be specified as a list such as

@code{["Print",1,"NoSugar",1,...]}. Names of parameters must be character

strings.

@item

@code{dp_gr_print()} is used to set and show the value of a parameter

@code{Print} and @code{PrintShort}.

@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

@table @t

@item $B;2>H(B

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

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

\EG @item References

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

\EG @fref{Controlling Groebner basis computations}

@end table

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

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

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

@subsection @code{dp_ord}

@findex dp_ord

@table @t

@item dp_ord([@var{order}])

:: $BJQ?t=g=x7?$N@_Dj(B, $B;2>H(B

\JP :: $BJQ?t=g=x7?$N@_Dj(B, $B;2>H(B

\EG :: Set and show the ordering type.

@end table

@table @var

@item return

$BJQ?t=g=x7?(B ($B?t(B, $B%j%9%H$^$?$O9TNs(B)

\JP $BJQ?t=g=x7?(B ($B?t(B, $B%j%9%H$^$?$O9TNs(B)

\EG ordering type (number, list or matrix)

@item order

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

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

\EG number, list or matrix

@end table

@itemize @bullet

\BJP

@item

$B0z?t$,$"$k;~(B, $BJQ?t=g=x7?$r(B @var{order} $B$K@_Dj$9$k(B. $B0z?t$,$J$$;~(B,

$B8=:_@_Dj$5$l$F$$$kJQ?t=g=x7?$rJV$9(B.

Line 1137 F4 $B%"%k%4%j%:%`$O(B, J.C. Faugere $B$K$h$jDs>'$5$

Line 2283 F4 $B%"%k%4%j%:%`$O(B, J.C. Faugere $B$K$h$jDs>'$5$

@item

$B%H%C%W%l%Y%kH!?t0J30$NH!?t$rD>@\8F$S=P$9>l9g$K$O(B, $B$3$NH!?t$K$h$j(B

$BJQ?t=g=x7?$r@5$7$/@_Dj$7$J$1$l$P$J$i$J$$(B.

\BEG

@item

If an argument is specified, the function

sets the current ordering type to @var{order}.

If no argument is specified, the function returns the ordering

type currently set.

@item

There are two types of functions concerning distributed polynomial,

functions which take a ordering type and those which don't take it.

The latter ones use the current setting.

@item

Functions such as @code{gr()}, which need a ordering type as an argument,

call @code{dp_ord()} internally during the execution.

The setting remains after the execution.

Fundamental arithmetics for distributed polynomial also use the current

setting. Therefore, when such arithmetics for distributed polynomials

are done, the current setting must coincide with the ordering type

which was used upon the creation of the polynomials. It is assumed

that such polynomials were generated under the same ordering type.

@item

Type of term ordering must be correctly set by this function

when functions other than top level functions are called directly.

@end itemize

@example

Line 1149 F4 $B%"%k%4%j%:%`$O(B, J.C. Faugere $B$K$h$jDs>'$5$

Line 2323 F4 $B%"%k%4%j%:%`$O(B, J.C. Faugere $B$K$h$jDs>'$5$

@end example

@table @t

@item $B;2>H(B

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

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

\EG @item References

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

\EG @fref{Setting term orderings}

@end table

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

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

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

@subsection @code{dp_ptod}

@findex dp_ptod

@table @t

@item dp_ptod(@var{poly},@var{vlist})

:: $BB?9`<0$rJ,;6I=8=B?9`<0$KJQ49$9$k(B.

\JP :: $BB?9`<0$rJ,;6I=8=B?9`<0$KJQ49$9$k(B.

\EG :: Converts an ordinary polynomial into a distributed polynomial.

@end table

@table @var

@item return

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

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

\EG distributed polynomial

@item poly

$BB?9`<0(B

\JP $BB?9`<0(B

\EG polynomial

@item vlist

$B%j%9%H(B

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

\EG list

@end table

@itemize @bullet

\BJP

@item

$BJQ?t=g=x(B @var{vlist} $B$*$h$S8=:_$NJQ?t=g=x7?$K=>$C$FJ,;6I=8=B?9`<0$KJQ49$9$k(B.

@item

@var{vlist} $B$K4^$^$l$J$$ITDj85$O(B, $B78?tBN$KB0$9$k$H$7$FJQ49$5$l$k(B.

\BEG

@item

According to the variable ordering @var{vlist} and current

type of term ordering, this function converts an ordinary

polynomial into a distributed polynomial.

@item

Indeterminates not included in @var{vlist} are regarded to belong to

the coefficient field.

@end itemize

@example

Line 1185 F4 $B%"%k%4%j%:%`$O(B, J.C. Faugere $B$K$h$jDs>'$5$

Line 2377 F4 $B%"%k%4%j%:%`$O(B, J.C. Faugere $B$K$h$jDs>'$5$

(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

@item $B;2>H(B

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

\EG @item References

@fref{dp_dtop},

@fref{dp_ord}.

@end table

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

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

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

@subsection @code{dp_dtop}

@findex dp_dtop

@table @t

@item dp_dtop(@var{dpoly},@var{vlist})

:: $BJ,;6I=8=B?9`<0$rB?9`<0$KJQ49$9$k(B.

\JP :: $BJ,;6I=8=B?9`<0$rB?9`<0$KJQ49$9$k(B.

\EG :: Converts a distributed polynomial into an ordinary polynomial.

@end table

@table @var

@item return

$BB?9`<0(B

\JP $BB?9`<0(B

\EG polynomial

@item dpoly

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

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

\EG distributed polynomial

@item vlist

$B%j%9%H(B

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

\EG list

@end table

@itemize @bullet

\BJP

@item

$BJ,;6I=8=B?9`<0$r(B, $BM?$($i$l$?ITDj85%j%9%H$rMQ$$$FB?9`<0$KJQ49$9$k(B.

@item

$BITDj85%j%9%H$O(B, $BD9$5J,;6I=8=B?9`<0$NJQ?t$N8D?t$H0lCW$7$F$$$l$P2?$G$b$h$$(B.

\BEG

@item

This function converts a distributed polynomial into an ordinary polynomial

according to a list of indeterminates @var{vlist}.

@item

@var{vlist} is such a list that its length coincides with the number of

variables of @var{dpoly}.

@end itemize

@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

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

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

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

@subsection @code{dp_mod}, @code{dp_rat}

@findex dp_mod

@findex dp_rat

@table @t

@item dp_mod(@var{p},@var{mod},@var{subst})

:: $BM-M}?t78?tJ,;6I=8=B?9`<0$NM-8BBN78?t$X$NJQ49(B

\JP :: $BM-M}?t78?tJ,;6I=8=B?9`<0$NM-8BBN78?t$X$NJQ49(B

\EG :: Converts a disributed polynomial into one with coefficients in a finite field.

@item dp_rat(@var{p})

:: $BM-8BBN78?tJ,;6I=8=B?9`<0$NM-M}?t78?t$X$NJQ49(B

\JP :: $BM-8BBN78?tJ,;6I=8=B?9`<0$NM-M}?t78?t$X$NJQ49(B

\BEG

:: Converts a distributed polynomial with coefficients in a finite field into

one with coefficients in the rationals.

@end table

@table @var

@item return

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

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

\EG distributed polynomial

@item p

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

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

\EG distributed polynomial

@item mod

$BAG?t(B

\JP $BAG?t(B

\EG prime

@item subst

$B%j%9%H(B

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

\EG list

@end table

@itemize @bullet

\BJP

@item

@code{dp_nf_mod()}, @code{dp_true_nf_mod()} $B$O(B, $BF~NO$H$7$FM-8BBN78?t$N(B

$BJ,;6I=8=B?9`<0$rI,MW$H$9$k(B. $B$3$N$h$&$J>l9g(B, @code{dp_mod()} $B$K$h$j(B

Line 1263 z^2+(2*a+2*b)*z+a^2+2*b*a+b^2

Line 2484 z^2+(2*a+2*b)*z+a^2+2*b*a+b^2

@var{subst} $B$O(B, $B78?t$,M-M}<0$N>l9g(B, $B$=$NM-M}<0$NJQ?t$K$"$i$+$8$a?t$rBeF~(B

$B$7$?8eM-8BBN78?t$KJQ49$9$k$H$$$&A`:n$r9T$&:]$N(B, $BBeF~CM$r;XDj$9$k$b$N$G(B,

@code{[[@var{var},@var{value}],...]} $B$N7A$N%j%9%H$G$"$k(B.

\BEG

@item

@code{dp_nf_mod()} and @code{dp_true_nf_mod()} require

distributed polynomials with coefficients in a finite field as arguments.

@code{dp_mod()} is used to convert distributed polynomials with rational

number coefficients into appropriate ones.

Polynomials with coefficients in a finite field

cannot be used as inputs of operations with polynomials

with rational number coefficients. @code{dp_rat()} is used for such cases.

@item

The ground finite field must be set in advance by using @code{setmod()}.

@item

@var{subst} is such a list as @code{[[@var{var},@var{value}],...]}.

This is valid when the ground field of the input polynomial is a

rational function field. @var{var}'s are variables in the ground field and

the list means that @var{value} is substituted for @var{var} before

converting the coefficients into elements of a finite field.

@end itemize

@example

@end example

@table @t

@item $B;2>H(B

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

\EG @item References

@fref{dp_nf dp_nf_mod dp_true_nf dp_true_nf_mod},

@fref{subst psubst},

@fref{setmod}.

@end table

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

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

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

@subsection @code{dp_homo}, @code{dp_dehomo}

@findex dp_homo

@findex dp_dehomo

@table @t

@item dp_homo(@var{dpoly})

:: $BJ,;6I=8=B?9`<0$N@F<!2=(B

\JP :: $BJ,;6I=8=B?9`<0$N@F<!2=(B

\EG :: Homogenize a distributed polynomial

@item dp_dehomo(@var{dpoly})

:: $B@F<!J,;6I=8=B?9`<0$NHs@F<!2=(B

\JP :: $B@F<!J,;6I=8=B?9`<0$NHs@F<!2=(B

\EG :: Dehomogenize a homogenious distributed polynomial

@end table

@table @var

@item return

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

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

\EG distributed polynomial

@item dpoly

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

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

\EG distributed polynomial

@end table

@itemize @bullet

\BJP

@item

@code{dp_homo()} $B$O(B, @var{dpoly} $B$N(B $B3F9`(B @var{t} $B$K$D$$$F(B, $B;X?t%Y%/%H%k$ND9$5$r(B

1 $B?-$P$7(B, $B:G8e$N@.J,$NCM$r(B @var{d}-@code{deg(@var{t})}

Line 1307 z^2+(2*a+2*b)*z+a^2+2*b*a+b^2

Line 2554 z^2+(2*a+2*b)*z+a^2+2*b*a+b^2

$B@5$7$/@_Dj$9$kI,MW$,$"$k(B.

@item

@code{hgr()} $B$J$I$K$*$$$F(B, $BFbItE*$KMQ$$$i$l$F$$$k(B.

\BEG

@item

@code{dp_homo()} makes a copy of @var{dpoly}, extends

the length of the exponent vector of each term @var{t} in the copy by 1,

and sets the value of the newly appended

component to @var{d}-@code{deg(@var{t})}, where @var{d} is the total

degree of @var{dpoly}.

@item

@code{dp_dehomo()} make a copy of @var{dpoly} and removes the last component

of each terms in the copy.

@item

Appropriate term orderings must be set when the results are used as inputs

of some operations.

@item

These are used internally in @code{hgr()} etc.

@end itemize

@example

Line 1319 z^2+(2*a+2*b)*z+a^2+2*b*a+b^2

Line 2583 z^2+(2*a+2*b)*z+a^2+2*b*a+b^2

@end example

@table @t

@item $B;2>H(B

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

\EG @item References

@fref{gr hgr gr_mod}.

@end table

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

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

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

@subsection @code{dp_ptozp}, @code{dp_prim}

@findex dp_ptozp

@findex dp_prim

@table @t

@item dp_ptozp(@var{dpoly})

:: $BDj?tG\$7$F78?t$r@0?t78?t$+$D78?t$N@0?t(B GCD $B$r(B 1 $B$K$9$k(B.

\JP :: $BDj?tG\$7$F78?t$r@0?t78?t$+$D78?t$N@0?t(B GCD $B$r(B 1 $B$K$9$k(B.

\BEG

:: Converts a distributed polynomial @var{poly} with rational coefficients

into an integral distributed polynomial such that GCD of all its coefficients

is 1.

@itemx dp_prim(@var{dpoly})

:: $BM-M}<0G\$7$F78?t$r@0?t78?tB?9`<078?t$+$D78?t$NB?9`<0(B GCD $B$r(B 1 $B$K$9$k(B.

\JP :: $BM-M}<0G\$7$F78?t$r@0?t78?tB?9`<078?t$+$D78?t$NB?9`<0(B GCD $B$r(B 1 $B$K$9$k(B.

\BEG

:: Converts a distributed polynomial @var{poly} with rational function

coefficients into an integral distributed polynomial such that polynomial

GCD of all its coefficients is 1.

@end table

@table @var

@item return

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

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

\EG distributed polynomial

@item dpoly

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

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

\EG distributed polynomial

@end table

@itemize @bullet

\BJP

@item

@code{dp_ptozp()} $B$O(B, @code{ptozp()} $B$KAjEv$9$kA`:n$rJ,;6I=8=B?9`<0$K(B

$BBP$7$F9T$&(B. $B78?t$,B?9`<0$r4^$`>l9g(B, $B78?t$K4^$^$l$kB?9`<06&DL0x;R$O(B

Line 1350 z^2+(2*a+2*b)*z+a^2+2*b*a+b^2

Line 2629 z^2+(2*a+2*b)*z+a^2+2*b*a+b^2

@item

@code{dp_prim()} $B$O(B, $B78?t$,B?9`<0$r4^$`>l9g(B, $B78?t$K4^$^$l$kB?9`<06&DL0x;R(B

$B$r<h$j=|$/(B.

\BEG

@item

@code{dp_ptozp()} executes the same operation as @code{ptozp()} for

a distributed polynomial. If the coefficients include polynomials,

polynomial contents included in the coefficients are not removed.

@item

@code{dp_prim()} removes polynomial contents.

@end itemize

@example

Line 1362 z^2+(2*a+2*b)*z+a^2+2*b*a+b^2

Line 2650 z^2+(2*a+2*b)*z+a^2+2*b*a+b^2

@end example

@table @t

@item $B;2>H(B

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

\EG @item References

@fref{ptozp}.

@end table

@node dp_nf dp_nf_mod dp_true_nf dp_true_nf_mod,,, $B%0%l%V%J4pDl$K4X$9$kH!?t(B

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

\EG @node dp_nf dp_nf_mod dp_true_nf dp_true_nf_mod,,, Functions for Groebner basis computation

@subsection @code{dp_nf}, @code{dp_nf_mod}, @code{dp_true_nf}, @code{dp_true_nf_mod}

@findex dp_nf

@findex dp_true_nf

Line 1376 z^2+(2*a+2*b)*z+a^2+2*b*a+b^2

Line 2666 z^2+(2*a+2*b)*z+a^2+2*b*a+b^2

@table @t

@item dp_nf(@var{indexlist},@var{dpoly},@var{dpolyarray},@var{fullreduce})

@item dp_nf_mod(@var{indexlist},@var{dpoly},@var{dpolyarray},@var{fullreduce},@var{mod})

:: $BJ,;6I=8=B?9`<0$N@55,7A$r5a$a$k(B. ($B7k2L$ODj?tG\$5$l$F$$$k2DG=@-$"$j(B)

\JP :: $BJ,;6I=8=B?9`<0$N@55,7A$r5a$a$k(B. ($B7k2L$ODj?tG\$5$l$F$$$k2DG=@-$"$j(B)

\BEG

:: Computes the normal form of a distributed polynomial.

(The result may be multiplied by a constant in the ground field.)

@item dp_true_nf(@var{indexlist},@var{dpoly},@var{dpolyarray},@var{fullreduce})

@item dp_true_nf_mod(@var{indexlist},@var{dpoly},@var{dpolyarray},@var{fullreduce},@var{mod})

:: $BJ,;6I=8=B?9`<0$N@55,7A$r5a$a$k(B. ($B??$N7k2L$r(B @code{[$BJ,;R(B, $BJ,Jl(B]} $B$N7A$GJV$9(B)

\JP :: $BJ,;6I=8=B?9`<0$N@55,7A$r5a$a$k(B. ($B??$N7k2L$r(B @code{[$BJ,;R(B, $BJ,Jl(B]} $B$N7A$GJV$9(B)

\BEG

:: Computes the normal form of a distributed polynomial. (The true result

is returned in such a list as @code{[numerator, denominator]})

@end table

@table @var

@item return

@code{dp_nf()} : $BJ,;6I=8=B?9`<0(B, @code{dp_true_nf()} : $B%j%9%H(B

\JP @code{dp_nf()} : $BJ,;6I=8=B?9`<0(B, @code{dp_true_nf()} : $B%j%9%H(B

\EG @code{dp_nf()} : distributed polynomial, @code{dp_true_nf()} : list

@item indexlist

$B%j%9%H(B

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

\EG list

@item dpoly

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

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

\EG distributed polynomial

@item dpolyarray

$BG[Ns(B

\JP $BG[Ns(B

\EG array of distributed polynomial

@item fullreduce

$B%U%i%0(B

\JP $B%U%i%0(B

\EG flag

@item mod

$BAG?t(B

\JP $BAG?t(B

\EG prime

@end table

@itemize @bullet

\BJP

@item

$BJ,;6I=8=B?9`<0(B @var{dpoly} $B$N@55,7A$r5a$a$k(B.

@item

Line 1429 z^2+(2*a+2*b)*z+a^2+2*b*a+b^2

Line 2734 z^2+(2*a+2*b)*z+a^2+2*b*a+b^2

$BJ,;6I=8=$G$J$$8GDj$5$l$?B?9`<0=89g$K$h$k@55,7A$rB??t5a$a$kI,MW$,$"$k>l9g(B

$B$KJXMx$G$"$k(B. $BC10l$N1i;;$K4X$7$F$O(B, @code{p_nf}, @code{p_true_nf} $B$r(B

$BMQ$$$k$H$h$$(B.

\BEG

@item

Computes the normal form of a distributed polynomial.

@item

@code{dp_nf_mod()} and @code{dp_true_nf_mod()} require

distributed polynomials with coefficients in a finite field as arguments.

@item

The result of @code{dp_nf()} may be multiplied by a constant in the

ground field in order to make the result integral. The same is true

for @code{dp_nf_mod()}, but it returns the true normal form if

the ground field is a finite field.

@item

@code{dp_true_nf()} and @code{dp_true_nf_mod()} return

such a list as @code{[@var{nm},@var{dn}]}.

Here @var{nm} is a distributed polynomial whose coefficients are integral

in the ground field, @var{dn} is an integral element in the ground

field and @var{nm}/@var{dn} is the true normal form.

@item

@var{dpolyarray} is a vector whose components are distributed polynomials

and @var{indexlist} is a list of indices which is used for the normal form

computation.

@item

When argument @var{fullreduce} has non-zero value,

all terms are reduced. When it has value 0,

only the head term is reduced.

@item

As for the polynomials specified by @var{indexlist}, one specified by

an index placed at the preceding position has priority to be selected.

@item

In general, the result of the function may be different depending on

@var{indexlist}. However, the result is unique for Groebner bases.

@item

These functions are useful when a fixed non-distributed polynomial set

is used as a set of reducers to compute normal forms of many polynomials.

For single computation @code{p_nf} and @code{p_true_nf} are sufficient.

@end itemize

@example

Line 1443 z^2+(2*a+2*b)*z+a^2+2*b*a+b^2

Line 2784 z^2+(2*a+2*b)*z+a^2+2*b*a+b^2

[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

@table @t

@item $B;2>H(B

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

\EG @item References

@fref{dp_dtop},

@fref{dp_ord},

@fref{dp_mod dp_rat},

@fref{p_nf p_nf_mod p_true_nf p_true_nf_mod}.

@end table

@node dp_hm dp_ht dp_hc dp_rest,,, $B%0%l%V%J4pDl$K4X$9$kH!?t(B

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

\EG @node dp_hm dp_ht dp_hc dp_rest,,, Functions for Groebner basis computation

@subsection @code{dp_hm}, @code{dp_ht}, @code{dp_hc}, @code{dp_rest}

@findex dp_hm

@findex dp_ht

Line 1473 u4^2+(6*u3+2*u2+6*u1-2)*u4+9*u3^2+(6*u2+18*u1-6)*u3+u2

Line 2819 u4^2+(6*u3+2*u2+6*u1-2)*u4+9*u3^2+(6*u2+18*u1-6)*u3+u2

@table @t

@item dp_hm(@var{dpoly})

:: $BF,C19`<0$r<h$j=P$9(B.

\JP :: $BF,C19`<0$r<h$j=P$9(B.

\EG :: Gets the head monomial.

@item dp_ht(@var{dpoly})

:: $BF,9`$r<h$j=P$9(B.

\JP :: $BF,9`$r<h$j=P$9(B.

\EG :: Gets the head term.

@item dp_hc(@var{dpoly})

:: $BF,78?t$r<h$j=P$9(B.

\JP :: $BF,78?t$r<h$j=P$9(B.

\EG :: Gets the head coefficient.

@item dp_rest(@var{dpoly})

:: $BF,C19`<0$r<h$j=|$$$?;D$j$rJV$9(B.

\JP :: $BF,C19`<0$r<h$j=|$$$?;D$j$rJV$9(B.

\EG :: Gets the remainder of the polynomial where the head monomial is removed.

@end table

@table @var

\BJP

@item return

@code{dp_hm()}, @code{dp_ht()}, @code{dp_rest()} : $BJ,;6I=8=B?9`<0(B,

@code{dp_hc()} : $B?t$^$?$OB?9`<0(B

@item dpoly

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

\BEG

@item return

@code{dp_hm()}, @code{dp_ht()}, @code{dp_rest()} : distributed polynomial

@code{dp_hc()} : number or polynomial

@item dpoly

distributed polynomial

@end table

@itemize @bullet

\BJP

@item

$B$3$l$i$O(B, $BJ,;6I=8=B?9`<0$N3FItJ,$r<h$j=P$9$?$a$NH!?t$G$"$k(B.

@item

$BJ,;6I=8=B?9`<0(B @var{p} $B$KBP$7<!$,@.$jN)$D(B.

\BEG

@item

These are used to get various parts of a distributed polynomial.

@item

The next equations hold for a distributed polynomial @var{p}.

@table @code

@item @var{p} = dp_hm(@var{p}) + dp_rest(@var{p})

@item dp_hm(@var{p}) = dp_hc(@var{p}) dp_ht(@var{p})

Line 1516 u4^2+(6*u3+2*u2+6*u1-2)*u4+9*u3^2+(6*u2+18*u1-6)*u3+u2

Line 2883 u4^2+(6*u3+2*u2+6*u1-2)*u4+9*u3^2+(6*u2+18*u1-6)*u3+u2

+(-490)*<<0,0,0>>

@end example

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

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

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

@subsection @code{dp_td}, @code{dp_sugar}

@findex dp_td

@findex dp_sugar

@table @t

@item dp_td(@var{dpoly})

:: $BF,9`$NA4<!?t$rJV$9(B.

\JP :: $BF,9`$NA4<!?t$rJV$9(B.

\EG :: Gets the total degree of the head term.

@item dp_sugar(@var{dpoly})

:: $BB?9`<0$N(B @code{sugar} $B$rJV$9(B.

\JP :: $BB?9`<0$N(B @code{sugar} $B$rJV$9(B.

\EG :: Gets the @code{sugar} of a polynomial.

@end table

@table @var

@item return

$B<+A3?t(B

\JP $B<+A3?t(B

\EG non-negative integer

@item dpoly

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

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

\EG distributed polynomial

@item onoff

$B%U%i%0(B

\JP $B%U%i%0(B

\EG flag

@end table

@itemize @bullet

\BJP

@item

@code{dp_td()} $B$O(B, $BF,9`$NA4<!?t(B, $B$9$J$o$A3FJQ?t$N;X?t$NOB$rJV$9(B.

@item

Line 1546 u4^2+(6*u3+2*u2+6*u1-2)*u4+9*u3^2+(6*u2+18*u1-6)*u3+u2

Line 2920 u4^2+(6*u3+2*u2+6*u1-2)*u4+9*u3^2+(6*u2+18*u1-6)*u3+u2

@item

@code{sugar} $B$O(B, $B%0%l%V%J4pDl7W;;$K$*$1$k@55,2=BP$NA*Br$N%9%H%i%F%8$r(B

$B7hDj$9$k$?$a$N=EMW$J;X?K$H$J$k(B.

\BEG

@item

Function @code{dp_td()} returns the total degree of the head term,

i.e., the sum of all exponent of variables in that term.

@item

Upon creation of a distributed polynomial, an integer called @code{sugar}

is associated. This value is

the total degree of the virtually homogenized one of the original

polynomial.

@item

The quantity @code{sugar} is an important guide to determine the

selection strategy of critical pairs in Groebner basis computation.

@end itemize

@example

Line 1558 u4^2+(6*u3+2*u2+6*u1-2)*u4+9*u3^2+(6*u2+18*u1-6)*u3+u2

Line 2946 u4^2+(6*u3+2*u2+6*u1-2)*u4+9*u3^2+(6*u2+18*u1-6)*u3+u2

@end example

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

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

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

@subsection @code{dp_lcm}

@findex dp_lcm

@table @t

@item dp_lcm(@var{dpoly1},@var{dpoly2})

:: $B:G>.8xG\9`$rJV$9(B.

\JP :: $B:G>.8xG\9`$rJV$9(B.

\EG :: Returns the least common multiple of the head terms of the given two polynomials.

@end table

@table @var

@item return

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

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

@item dpoly1, dpoly2

\EG distributed polynomial

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

@item dpoly1 dpoly2

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

\EG distributed polynomial

@end table

@itemize @bullet

\BJP

@item

$B$=$l$>$l$N0z?t$NF,9`$N:G>.8xG\9`$rJV$9(B. $B78?t$O(B 1 $B$G$"$k(B.

\BEG

@item

Returns the least common multiple of the head terms of the given

two polynomials, where coefficient is always set to 1.

@end itemize

@example

Line 1585 u4^2+(6*u3+2*u2+6*u1-2)*u4+9*u3^2+(6*u2+18*u1-6)*u3+u2

Line 2984 u4^2+(6*u3+2*u2+6*u1-2)*u4+9*u3^2+(6*u2+18*u1-6)*u3+u2

@end example

@table @t

@item $B;2>H(B

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

\EG @item References

@fref{p_nf p_nf_mod p_true_nf p_true_nf_mod}.

@end table

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

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

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

@subsection @code{dp_redble}

@findex dp_redble

@table @t

@item dp_redble(@var{dpoly1},@var{dpoly2})

:: $BF,9`$I$&$7$,@0=|2DG=$+$I$&$+D4$Y$k(B.

\JP :: $BF,9`$I$&$7$,@0=|2DG=$+$I$&$+D4$Y$k(B.

\EG :: Checks whether one head term is divisible by the other head term.

@end table

@table @var

@item return

$B@0?t(B

\JP $B@0?t(B

@item dpoly1, dpoly2

\EG integer

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

@item dpoly1 dpoly2

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

\EG distributed polynomial

@end table

@itemize @bullet

\BJP

@item

@var{dpoly1} $B$NF,9`$,(B @var{dpoly2} $B$NF,9`$G3d$j@Z$l$l$P(B 1, $B3d$j@Z$l$J$1$l$P(B

0 $B$rJV$9(B.

@item

$BB?9`<0$N4JLs$r9T$&:](B, $B$I$N9`$r4JLs$G$-$k$+$rC5$9$N$KMQ$$$k(B.

\BEG

@item

Returns 1 if the head term of @var{dpoly2} divides the head term of

@var{dpoly1}; otherwise 0.

@item

Used for finding candidate terms at reduction of polynomials.

@end itemize

@example

Line 1626 u4^2+(6*u3+2*u2+6*u1-2)*u4+9*u3^2+(6*u2+18*u1-6)*u3+u2

Line 3039 u4^2+(6*u3+2*u2+6*u1-2)*u4+9*u3^2+(6*u2+18*u1-6)*u3+u2

@end example

@table @t

@item $B;2>H(B

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

\EG @item References

@fref{dp_red dp_red_mod}.

@end table

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

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

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

@subsection @code{dp_subd}

@findex dp_subd

@table @t

@item dp_subd(@var{dpoly1},@var{dpoly2})

:: $BF,9`$N>&C19`<0$rJV$9(B.

\JP :: $BF,9`$N>&C19`<0$rJV$9(B.

\EG :: Returns the quotient monomial of the head terms.

@end table

@table @var

@item return

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

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

@item dpoly1, dpoly2

\EG distributed polynomial

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

@item dpoly1 dpoly2

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

\EG distributed polynomial

@end table

@itemize @bullet

\BJP

@item

@code{dp_ht(@var{dpoly1})/dp_ht(@var{dpoly2})} $B$r5a$a$k(B. $B7k2L$N78?t$O(B 1

$B$G$"$k(B.

@item

$B3d$j@Z$l$k$3$H$,$"$i$+$8$a$o$+$C$F$$$kI,MW$,$"$k(B.

\BEG

@item

Gets @code{dp_ht(@var{dpoly1})/dp_ht(@var{dpoly2})}.

The coefficient of the result is always set to 1.

@item

Divisibility assumed.

@end itemize

@example

Line 1660 u4^2+(6*u3+2*u2+6*u1-2)*u4+9*u3^2+(6*u2+18*u1-6)*u3+u2

Line 3087 u4^2+(6*u3+2*u2+6*u1-2)*u4+9*u3^2+(6*u2+18*u1-6)*u3+u2

@end example

@table @t

@item $B;2>H(B

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

\EG @item References

@fref{dp_red dp_red_mod}.

@end table

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

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

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

@subsection @code{dp_vtoe}, @code{dp_etov}

@findex dp_vtoe

@findex dp_etov

@table @t

@item dp_vtoe(@var{vect})

:: $B;X?t%Y%/%H%k$r9`$KJQ49(B

\JP :: $B;X?t%Y%/%H%k$r9`$KJQ49(B

\EG :: Converts an exponent vector into a term.

@item dp_etov(@var{dpoly})

:: $BF,9`$r;X?t%Y%/%H%k$KJQ49(B

\JP :: $BF,9`$r;X?t%Y%/%H%k$KJQ49(B

\EG :: Convert the head term of a distributed polynomial into an exponent vector.

@end table

@table @var

@item return

@code{dp_vtoe} : $BJ,;6I=8=B?9`<0(B, @code{dp_etov} : $B%Y%/%H%k(B

\JP @code{dp_vtoe} : $BJ,;6I=8=B?9`<0(B, @code{dp_etov} : $B%Y%/%H%k(B

\EG @code{dp_vtoe} : distributed polynomial, @code{dp_etov} : vector

@item vect

$B%Y%/%H%k(B

\JP $B%Y%/%H%k(B

\EG vector

@item dpoly

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

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

\EG distributed polynomial

@end table

@itemize @bullet

\BJP

@item

@code{dp_vtoe()} $B$O(B, $B%Y%/%H%k(B @var{vect} $B$r;X?t%Y%/%H%k$H$9$k9`$r@8@.$9$k(B.

@item

@code{dp_etov()} $B$O(B, $BJ,;6I=8=B?9`<0(B @code{dpoly} $B$NF,9`$N;X?t%Y%/%H%k$r(B

$B%Y%/%H%k$KJQ49$9$k(B.

\BEG

@item

@code{dp_vtoe()} generates a term whose exponent vector is @var{vect}.

@item

@code{dp_etov()} generates a vector which is the exponent vector of the

head term of @code{dpoly}.

@end itemize

@example

Line 1703 u4^2+(6*u3+2*u2+6*u1-2)*u4+9*u3^2+(6*u2+18*u1-6)*u3+u2

Line 3146 u4^2+(6*u3+2*u2+6*u1-2)*u4+9*u3^2+(6*u2+18*u1-6)*u3+u2

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

@end example

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

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

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

@subsection @code{dp_mbase}

@findex dp_mbase

@table @t

@item dp_mbase(@var{dplist})

:: monomial $B4pDl$N7W;;(B

\JP :: monomial $B4pDl$N7W;;(B

\EG :: Computes the monomial basis

@end table

@table @var

@item return

$BJ,;6I=8=B?9`<0$N%j%9%H(B

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

\EG list of distributed polynomial

@item dplist

$BJ,;6I=8=B?9`<0$N%j%9%H(B

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

\EG list of distributed polynomial

@end table

@itemize @bullet

\BJP

@item

$B$"$k=g=x$G%0%l%V%J4pDl$H$J$C$F$$$kB?9`<0=89g$N(B, $B$=$N=g=x$K4X$9$kJ,;6I=8=(B

$B$G$"$k(B @var{dplist} $B$K$D$$$F(B,

Line 1727 u4^2+(6*u3+2*u2+6*u1-2)*u4+9*u3^2+(6*u2+18*u1-6)*u3+u2

Line 3175 u4^2+(6*u3+2*u2+6*u1-2)*u4+9*u3^2+(6*u2+18*u1-6)*u3+u2

K $B>eM-8B<!85@~7A6u4V$G$"$k(B K[X]/I $B$N(B monomial $B$K$h$k4pDl$r5a$a$k(B.

@item

$BF@$i$l$?4pDl$N8D?t$,(B, K[X]/I $B$N(B K-$B@~7A6u4V$H$7$F$N<!85$KEy$7$$(B.

\BEG

@item

Assuming that @var{dplist} is a list of distributed polynomials which

is a Groebner basis with respect to the current ordering type and

that the ideal @var{I} generated by @var{dplist} in K[X] is zero-dimensional,

this function computes the monomial basis of a finite dimenstional K-vector

space K[X]/I.

@item

The number of elements in the monomial basis is equal to the

K-dimenstion of K[X]/I.

@end itemize

@example

Line 1741 u1*u2,u1^2,u4*u0,u3*u0,u2*u0,u1*u0,u0^2,u4,u3,u2,u1,u0

Line 3201 u1*u2,u1^2,u4*u0,u3*u0,u2*u0,u1*u0,u0^2,u4,u3,u2,u1,u0

@end example

@table @t

@item $B;2>H(B

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

\EG @item References

@fref{gr hgr gr_mod}.

@end table

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

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

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

@subsection @code{dp_mag}

@findex dp_mag

@table @t

@item dp_mag(@var{p})

:: $B78?t$N%S%C%HD9$NOB$rJV$9(B

\JP :: $B78?t$N%S%C%HD9$NOB$rJV$9(B

\EG :: Computes the sum of bit lengths of coefficients of a distributed polynomial.

@end table

@table @var

@item return

$B?t(B

\JP $B?t(B

\EG integer

@item p

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

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

\EG distributed polynomial

@end table

@itemize @bullet

\BJP

@item

$BJ,;6I=8=B?9`<0$N78?t$K8=$l$kM-M}?t$K$D$-(B, $B$=$NJ,JlJ,;R(B ($B@0?t$N>l9g$OJ,;R(B)

$B$N%S%C%HD9$NAmOB$rJV$9(B.

Line 1772 u1*u2,u1^2,u4*u0,u3*u0,u2*u0,u1*u0,u0^2,u4,u3,u2,u1,u0

Line 3238 u1*u2,u1^2,u4*u0,u3*u0,u2*u0,u1*u0,u0^2,u4,u3,u2,u1,u0

@item

@code{dp_gr_flags()} $B$G(B, @code{ShowMag}, @code{Print} $B$r(B on $B$K$9$k$3$H$K$h$j(B

$BESCf@8@.$5$l$kB?9`<0$K$?$$$9$k(B @code{dp_mag()} $B$NCM$r8+$k$3$H$,$G$-$k(B.

\BEG

@item

This function computes the sum of bit lengths of coefficients of a

distributed polynomial @var{p}. If a coefficient is non integral,

the sum of bit lengths of the numerator and the denominator is taken.

@item

This is a measure of the size of a polynomial. Especially for

zero-dimensional system coefficient swells are often serious and

the returned value is useful to detect such swells.

@item

If @code{ShowMag} and @code{Print} for @code{dp_gr_flags()} are on,

values of @code{dp_mag()} for intermediate basis elements are shown.

@end itemize

@example

Line 1781 u1*u2,u1^2,u4*u0,u3*u0,u2*u0,u1*u0,u0^2,u4,u3,u2,u1,u0

Line 3261 u1*u2,u1^2,u4*u0,u3*u0,u2*u0,u1*u0,u0^2,u4,u3,u2,u1,u0

@end example

@table @t

@item $B;2>H(B

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

\EG @item References

@fref{dp_gr_flags dp_gr_print}.

@end table

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

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

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

@subsection @code{dp_red}, @code{dp_red_mod}

@findex dp_red

@findex dp_red_mod

Line 1793 u1*u2,u1^2,u4*u0,u3*u0,u2*u0,u1*u0,u0^2,u4,u3,u2,u1,u0

Line 3275 u1*u2,u1^2,u4*u0,u3*u0,u2*u0,u1*u0,u0^2,u4,u3,u2,u1,u0

@table @t

@item dp_red(@var{dpoly1},@var{dpoly2},@var{dpoly3})

@item dp_red_mod(@var{dpoly1},@var{dpoly2},@var{dpoly3},@var{mod})

:: $B0l2s$N4JLsA`:n(B

\JP :: $B0l2s$N4JLsA`:n(B

\EG :: Single reduction operation

@end table

@table @var

@item return

$B%j%9%H(B

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

@item dpoly1, dpoly2, dpoly3

\EG list

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

@item dpoly1 dpoly2 dpoly3

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

\EG distributed polynomial

@item vlist

$B%j%9%H(B

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

\EG list

@item mod

$BAG?t(B

\JP $BAG?t(B

\EG prime

@end table

@itemize @bullet

\BJP

@item

@var{dpoly1} + @var{dpoly2} $B$J$kJ,;6I=8=B?9`<0$r(B @var{dpoly3} $B$G(B

1 $B2s4JLs$9$k(B.

Line 1819 u1*u2,u1^2,u4*u0,u3*u0,u2*u0,u1*u0,u0^2,u4,u3,u2,u1,u0

Line 3307 u1*u2,u1^2,u4*u0,u3*u0,u2*u0,u1*u0,u0^2,u4,u3,u2,u1,u0

$B$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.

\BEG

@item

Reduces a distributed polynomial, @var{dpoly1} + @var{dpoly2},

by @var{dpoly3} for single time.

@item

An input for @code{dp_red_mod()} must be converted into a distributed

polynomial with coefficients in a finite field.

@item

This implies that

the divisibility of the head term of @var{dpoly2} by the head term of

@var{dpoly3} is assumed.

@item

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

@item

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

@end itemize

@example

Line 1832 u1*u2,u1^2,u4*u0,u3*u0,u2*u0,u1*u0,u0^2,u4,u3,u2,u1,u0

Line 3340 u1*u2,u1^2,u4*u0,u3*u0,u2*u0,u1*u0,u0^2,u4,u3,u2,u1,u0

[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

@item $B;2>H(B

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

\EG @item References

@fref{dp_mod dp_rat}.

@end table

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

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

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

@subsection @code{dp_sp}, @code{dp_sp_mod}

@findex dp_sp

@findex dp_sp_mod

Line 1849 u1*u2,u1^2,u4*u0,u3*u0,u2*u0,u1*u0,u0^2,u4,u3,u2,u1,u0

Line 3359 u1*u2,u1^2,u4*u0,u3*u0,u2*u0,u1*u0,u0^2,u4,u3,u2,u1,u0

@table @t

@item dp_sp(@var{dpoly1},@var{dpoly2})

@item dp_sp_mod(@var{dpoly1},@var{dpoly2},@var{mod})

:: S-$BB?9`<0$N7W;;(B

\JP :: S-$BB?9`<0$N7W;;(B

\EG :: Computation of an S-polynomial

@end table

@table @var

@item return

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

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

@item dpoly1, dpoly2

\EG distributed polynomial

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

@item dpoly1 dpoly2

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

\EG distributed polynomial

@item mod

$BAG?t(B

\JP $BAG?t(B

\EG prime

@end table

@itemize @bullet

\BJP

@item

@var{dpoly1}, @var{dpoly2} $B$N(B S-$BB?9`<0$r7W;;$9$k(B.

@item

Line 1869 u1*u2,u1^2,u4*u0,u3*u0,u2*u0,u1*u0,u0^2,u4,u3,u2,u1,u0

Line 3384 u1*u2,u1^2,u4*u0,u3*u0,u2*u0,u1*u0,u0^2,u4,u3,u2,u1,u0

@item

$B7k2L$KM-M}?t(B, $BM-M}<0$,F~$k$N$rHr$1$k$?$a(B, $B7k2L$,Dj?tG\(B, $B$"$k$$$OB?9`<0(B

$BG\$5$l$F$$$k2DG=@-$,$"$k(B.

\BEG

@item

This function computes the S-polynomial of @var{dpoly1} and @var{dpoly2}.

@item

Inputs of @code{dp_sp_mod()} must be polynomials with coefficients in a

finite field.

@item

The result may be multiplied by a constant in the ground field in order to

make the result integral.

@end itemize

@example

Line 1881 u1*u2,u1^2,u4*u0,u3*u0,u2*u0,u1*u0,u0^2,u4,u3,u2,u1,u0

Line 3407 u1*u2,u1^2,u4*u0,u3*u0,u2*u0,u1*u0,u0^2,u4,u3,u2,u1,u0

@end example

@table @t

@item $B;2>H(B

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

\EG @item References

@fref{dp_mod dp_rat}.

@end table

@node p_nf p_nf_mod p_true_nf p_true_nf_mod,,, $B%0%l%V%J4pDl$K4X$9$kH!?t(B

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

\EG @node p_nf p_nf_mod p_true_nf p_true_nf_mod,,, Functions for Groebner basis computation

@subsection @code{p_nf}, @code{p_nf_mod}, @code{p_true_nf}, @code{p_true_nf_mod}

@findex p_nf

@findex p_nf_mod

Line 1894 u1*u2,u1^2,u4*u0,u3*u0,u2*u0,u1*u0,u0^2,u4,u3,u2,u1,u0

Line 3422 u1*u2,u1^2,u4*u0,u3*u0,u2*u0,u1*u0,u0^2,u4,u3,u2,u1,u0

@table @t

@item p_nf(@var{poly},@var{plist},@var{vlist},@var{order})

@itemx p_nf_mod(@var{poly},@var{plist},@var{vlist},@var{order},@var{mod})

:: $BI=8=B?9`<0$N@55,7A$r5a$a$k(B. ($B7k2L$ODj?tG\$5$l$F$$$k2DG=@-$"$j(B)

\JP :: $BI=8=B?9`<0$N@55,7A$r5a$a$k(B. ($B7k2L$ODj?tG\$5$l$F$$$k2DG=@-$"$j(B)

\BEG

:: Computes the normal form of the given polynomial.

(The result may be multiplied by a constant.)

@item p_true_nf(@var{poly},@var{plist},@var{vlist},@var{order})

@itemx p_true_nf_mod(@var{poly},@var{plist},@var{vlist},@var{order},@var{mod})

:: $BI=8=B?9`<0$N@55,7A$r5a$a$k(B. ($B??$N7k2L$r(B @code{[$BJ,;R(B, $BJ,Jl(B]} $B$N7A$GJV$9(B)

\JP :: $BI=8=B?9`<0$N@55,7A$r5a$a$k(B. ($B??$N7k2L$r(B @code{[$BJ,;R(B, $BJ,Jl(B]} $B$N7A$GJV$9(B)

\BEG

:: Computes the normal form of the given polynomial. (The result is returned

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

@end table

@table @var

@item return

@code{p_nf} : $BB?9`<0(B, @code{p_true_nf} : $B%j%9%H(B

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

\EG @code{p_nf} : polynomial, @code{p_true_nf} : list

@item poly

$BB?9`<0(B

\JP $BB?9`<0(B

@item plist,vlist

\EG polynomial

$B%j%9%H(B

@item plist vlist

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

\EG list

@item order

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

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

\EG number, list or matrix

@item mod

$BAG?t(B

\JP $BAG?t(B

\EG prime

@end table

@itemize @bullet

\BJP

@item

@samp{gr} $B$GDj5A$5$l$F$$$k(B.

@item

Line 1934 u1*u2,u1^2,u4*u0,u3*u0,u2*u0,u1*u0,u0^2,u4,u3,u2,u1,u0

Line 3476 u1*u2,u1^2,u4*u0,u3*u0,u2*u0,u1*u0,u0^2,u4,u3,u2,u1,u0

@item

@code{p_true_nf()}, @code{p_true_nf_mod()} $B$N=PNO$K4X$7$F$O(B,

@code{dp_true_nf()}, @code{dp_true_nf_mod()} $B$N9`$r;2>H(B.

\BEG

@item

Defined in the package @samp{gr}.

@item

Obtains the normal form of a polynomial by a polynomial list.

@item

These are interfaces to @code{dp_nf()}, @code{dp_true_nf()}, @code{dp_nf_mod()},

@code{dp_true_nf_mod}

@item

The polynomial @var{poly} and the polynomials in @var{plist} is

converted, according to the variable ordering @var{vlist} and

type of term ordering @var{otype}, into their distributed polynomial

counterparts and passed to @code{dp_nf()}.

@item

@code{dp_nf()}, @code{dp_true_nf()}, @code{dp_nf_mod()} and

@code{dp_true_nf_mod()}

is called with value 1 for @var{fullreduce}.

@item

The result is converted back into an ordinary polynomial.

@item

As for @code{p_true_nf()}, @code{p_true_nf_mod()}

refer to @code{dp_true_nf()} and @code{dp_true_nf_mod()}.

@end itemize

@example

Line 1949 u1*u2,u1^2,u4*u0,u3*u0,u2*u0,u1*u0,u0^2,u4,u3,u2,u1,u0

Line 3515 u1*u2,u1^2,u4*u0,u3*u0,u2*u0,u1*u0,u0^2,u4,u3,u2,u1,u0

@end example

@table @t

@item $B;2>H(B

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

\EG @item References

@fref{dp_ptod},

@fref{dp_dtop},

@fref{dp_ord},

@fref{dp_nf dp_nf_mod dp_true_nf dp_true_nf_mod}.

@end table

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

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

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

@subsection @code{p_terms}

@findex p_terms

@table @t

@item p_terms(@var{poly},@var{vlist},@var{order})

:: $BB?9`<0$K$"$i$o$l$kC19`$r%j%9%H$K$9$k(B.

\JP :: $BB?9`<0$K$"$i$o$l$kC19`$r%j%9%H$K$9$k(B.

\EG :: Monomials appearing in the given polynomial is collected into a list.

@end table

@table @var

@item return

$B%j%9%H(B

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

\EG list

@item poly

$BB?9`<0(B

\JP $BB?9`<0(B

\EG polynomial

@item vlist

$B%j%9%H(B

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

\EG list

@item order

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

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

\EG number, list or matrix

@end table

@itemize @bullet

\BJP

@item

@samp{gr} $B$GDj5A$5$l$F$$$k(B.

@item

Line 1986 u1*u2,u1^2,u4*u0,u3*u0,u2*u0,u1*u0,u0^2,u4,u3,u2,u1,u0

Line 3560 u1*u2,u1^2,u4*u0,u3*u0,u2*u0,u1*u0,u0^2,u4,u3,u2,u1,u0

@item

$B%0%l%V%J4pDl$O$7$P$7$P78?t$,5pBg$K$J$k$?$a(B, $B<B:]$K$I$N9`$,8=$l$F(B

$B$$$k$N$+$r8+$k$?$a$J$I$KMQ$$$k(B.

\BEG

@item

Defined in the package @samp{gr}.

@item

This returns a list which contains all non-zero monomials in the given

polynomial. The monomials are ordered according to the current

type of term ordering and @var{vlist}.

@item

Since polynomials in a Groebner base often have very large coefficients,

examining a polynomial as it is may sometimes be difficult to perform.

For such a case, this function enables to examine which term is really

exists.

@end itemize

@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

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

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

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

@subsection @code{gb_comp}

@findex gb_comp

@table @t

@item gb_comp(@var{plist1}, @var{plist2})

:: $BB?9`<0%j%9%H$,(B, $BId9f$r=|$$$F=89g$H$7$FEy$7$$$+$I$&$+D4$Y$k(B.

\JP :: $BB?9`<0%j%9%H$,(B, $BId9f$r=|$$$F=89g$H$7$FEy$7$$$+$I$&$+D4$Y$k(B.

\EG :: Checks whether two polynomial lists are equal or not as a set

@end table

@table @var

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

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

@item plist1, plist2

\EG @item return 0 or 1

@item plist1 plist2

@end table

@itemize @bullet

\BJP

@item

@var{plist1}, @var{plist2} $B$K$D$$$F(B, $BId9f$r=|$$$F=89g$H$7$FEy$7$$$+$I$&$+(B

$BD4$Y$k(B.

@item

$B0[$J$kJ}K!$G5a$a$?%0%l%V%J4pDl$O(B, $B4pDl$N=g=x(B, $BId9f$,0[$J$k>l9g$,$"$j(B,

$B$=$l$i$,Ey$7$$$+$I$&$+$rD4$Y$k$?$a$KMQ$$$k(B.

\BEG

@item

This function checks whether @var{plist1} and @var{plist2} are equal or

not as a set .

@item

For the same input and the same term ordering different

functions for Groebner basis computations may produce different outputs

as lists. This function compares such lists whether they are equal

as a generating set of an ideal.

@end itemize

@example

Line 2029 u0^6,u0^5,u0^4,u0^3,u0^2,u0,1]

Line 3632 u0^6,u0^5,u0^4,u0^3,u0^2,u0,1]

@end example

@node katsura hkatsura cyclic hcyclic,,, $B%0%l%V%J4pDl$K4X$9$kH!?t(B

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

\EG @node katsura hkatsura cyclic hcyclic,,, Functions for Groebner basis computation

@subsection @code{katsura}, @code{hkatsura}, @code{cyclic}, @code{hcyclic}

@findex katsura

@findex hkatsura

Line 2041 u0^6,u0^5,u0^4,u0^3,u0^2,u0,1]

Line 3645 u0^6,u0^5,u0^4,u0^3,u0^2,u0,1]

@item hkatsura(@var{n})

@item cyclic(@var{n})

@item hcyclic(@var{n})

:: $BB?9`<0%j%9%H$N@8@.(B

\JP :: $BB?9`<0%j%9%H$N@8@.(B

\EG :: Generates a polynomial list of standard benchmark.

@end table

@table @var

@item return

$B%j%9%H(B

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

\EG list

@item n

$B@0?t(B

\JP $B@0?t(B

\EG integer

@end table

@itemize @bullet

\BJP

@item

@code{katsura()} $B$O(B @samp{katsura}, @code{cyclic()} $B$O(B @samp{cyclic}

$B$GDj5A$5$l$F$$$k(B.

Line 2061 u0^6,u0^5,u0^4,u0^3,u0^2,u0,1]

Line 3669 u0^6,u0^5,u0^4,u0^3,u0^2,u0,1]

@item

@code{cyclic} $B$O(B @code{Arnborg}, @code{Lazard}, @code{Davenport} $B$J$I$N(B

$BL>$G8F$P$l$k$3$H$b$"$k(B.

\BEG

@item

Function @code{katsura()} is defined in @samp{katsura}, and

function @code{cyclic()} in @samp{cyclic}.

@item

These functions generate a series of polynomial sets, respectively,

which are often used for testing and bench marking:

@code{katsura}, @code{cyclic} and their homogenized versions.

@item

Polynomial set @code{cyclic} is sometimes called by other name:

@code{Arnborg}, @code{Lazard}, and @code{Davenport}.

@end itemize

@example

Line 2068 u0^6,u0^5,u0^4,u0^3,u0^2,u0,1]

Line 3689 u0^6,u0^5,u0^4,u0^3,u0^2,u0,1]

[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 2092 u0^2-u0+2*u4^2+2*u3^2+2*u2^2+2*u1^2+2*u5^2]

Line 3713 u0^2-u0+2*u4^2+2*u3^2+2*u2^2+2*u1^2+2*u5^2]

@end example

@table @t

@item $B;2>H(B

\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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