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Diff for /OpenXM/src/asir-doc/parts/groebner.texi between version 1.15 and 1.19

version 1.15, 2004/09/14 02:28:20

version 1.19, 2016/08/29 04:56:58

Line 1

@comment $OpenXM: OpenXM/src/asir-doc/parts/groebner.texi,v 1.14 2004/09/14 01:32:34 noro Exp $

@comment $OpenXM: OpenXM/src/asir-doc/parts/groebner.texi,v 1.18 2016/03/24 20:58:50 noro Exp $

\BJP

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

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

Line 201 In an @b{Asir} session, it is displayed in the form li

\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,C19`<0(B (head monomial)

@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 monomial

@itemx head coefficient

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

Line 1464 Computation of the global b function is implemented as

* tolexm minipolym::

* 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_weyl_f4_main dp_weyl_f4_mod_main::

* nd_gr nd_gr_trace nd_f4 nd_weyl_gr nd_weyl_gr_trace::

* nd_gr nd_gr_trace nd_f4 nd_f4_trace nd_weyl_gr nd_weyl_gr_trace::

* dp_gr_flags dp_gr_print::

* dp_ord::

* dp_set_weight dp_set_top_weight dp_weyl_set_weight::

* dp_ptod::

* dp_dtop::

* dp_mod dp_rat::

* dp_homo dp_dehomo::

* dp_ptozp dp_prim::

* dp_nf dp_nf_mod dp_true_nf dp_true_nf_mod::

* dp_nf dp_nf_mod dp_true_nf dp_true_nf_mod dp_weyl_nf dp_weyl_nf_mod::

* dp_hm dp_ht dp_hc dp_rest::

* dp_td dp_sugar::

* dp_lcm::

Line 1541 Computation of the global b function is implemented as

Line 1542 Computation of the global b function is implemented as

strategy $B$K$h$k7W;;(B, @code{hgr()} $B$O(B trace-lifting $B$*$h$S(B

$B@F<!2=$K$h$k(B $B6:@5$5$l$?(B sugar strategy $B$K$h$k7W;;$r9T$&(B.

@item

@code{dgr()} $B$O(B, @code{gr()}, @code{dgr()} $B$r(B

@code{dgr()} $B$O(B, @code{gr()}, @code{hgr()} $B$r(B

$B;R%W%m%;%9%j%9%H(B @var{procs} $B$N(B 2 $B$D$N%W%m%;%9$K$h$jF1;~$K7W;;$5$;(B,

$B@h$K7k2L$rJV$7$?J}$N7k2L$rJV$9(B. $B7k2L$OF10l$G$"$k$,(B, $B$I$A$i$NJ}K!$,(B

$B9bB.$+0lHL$K$OITL@$N$?$a(B, $B<B:]$N7P2a;~4V$rC;=L$9$k$N$KM-8z$G$"$k(B.

Line 2304 except for lack of the argument for controlling homoge

Line 2305 except for lack of the argument for controlling homoge

\EG @fref{Controlling Groebner basis computations}

@end table

\JP @node nd_gr nd_gr_trace nd_f4 nd_weyl_gr nd_weyl_gr_trace,,, $B%0%l%V%J4pDl$K4X$9$kH!?t(B

\JP @node nd_gr nd_gr_trace nd_f4 nd_f4_trace nd_weyl_gr nd_weyl_gr_trace,,, $B%0%l%V%J4pDl$K4X$9$kH!?t(B

\EG @node nd_gr nd_gr_trace nd_f4 nd_weyl_gr nd_weyl_gr_trace,,, Functions for Groebner basis computation

\EG @node nd_gr nd_gr_trace nd_f4 nd_f4_trace nd_weyl_gr nd_weyl_gr_trace,,, Functions for Groebner basis computation

@subsection @code{nd_gr}, @code{nd_gr_trace}, @code{nd_f4}, @code{nd_weyl_gr}, @code{nd_weyl_gr_trace}

@subsection @code{nd_gr}, @code{nd_gr_trace}, @code{nd_f4}, @code{nd_f4_trace}, @code{nd_weyl_gr}, @code{nd_weyl_gr_trace}

@findex nd_gr

@findex nd_gr_trace

@findex nd_f4

@findex nd_f4_trace

@findex nd_weyl_gr

@findex nd_weyl_gr_trace

Line 2317 except for lack of the argument for controlling homoge

Line 2319 except for lack of the argument for controlling homoge

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

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

@itemx nd_f4(@var{plist},@var{vlist},@var{modular},@var{order})

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

@itemx nd_f4_trace(@var{plist},@var{vlist},@var{homo},@var{p},@var{order})

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

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

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

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

Line 2348 except for lack of the argument for controlling homoge

Line 2351 except for lack of the argument for controlling homoge

@item @code{nd_gr} $B$O(B, @code{p} $B$,(B 0 $B$N$H$-M-M}?tBN>e$N(B Buchberger

$B%"%k%4%j%:%`$r<B9T$9$k(B. @code{p} $B$,(B 2 $B0J>e$N<+A3?t$N$H$-(B, GF(p) $B>e$N(B

Buchberger $B%"%k%4%j%:%`$r<B9T$9$k(B.

@item @code{nd_gr_trace} $B$OM-M}?tBN>e$G(B trace $B%"%k%4%j%:%`$r<B9T$9$k(B.

@item @code{nd_gr_trace} $B$*$h$S(B @code{nd_f4_trace}

@code{p} $B$,(B 0 $B$^$?$O(B 1 $B$N$H$-(B, $B<+F0E*$KA*$P$l$?AG?t$rMQ$$$F(B, $B@.8y$9$k(B

$B$OM-M}?tBN>e$G(B trace $B%"%k%4%j%:%`$r<B9T$9$k(B.

@var{p} $B$,(B 0 $B$^$?$O(B 1 $B$N$H$-(B, $B<+F0E*$KA*$P$l$?AG?t$rMQ$$$F(B, $B@.8y$9$k(B

$B$^$G(B trace $B%"%k%4%j%:%`$r<B9T$9$k(B.

@code{p} $B$,(B 2 $B0J>e$N$H$-(B, trace $B$O(BGF(p) $B>e$G7W;;$5$l$k(B. trace $B%"%k%4%j%:%`(B

@var{p} $B$,(B 2 $B0J>e$N$H$-(B, trace $B$O(BGF(p) $B>e$G7W;;$5$l$k(B. trace $B%"%k%4%j%:%`(B

$B$,<:GT$7$?>l9g(B 0 $B$,JV$5$l$k(B. @code{p} $B$,Ii$N>l9g(B, $B%0%l%V%J4pDl%A%'%C%/$O(B

$B$,<:GT$7$?>l9g(B 0 $B$,JV$5$l$k(B. @var{p} $B$,Ii$N>l9g(B, $B%0%l%V%J4pDl%A%'%C%/$O(B

$B9T$o$J$$(B. $B$3$N>l9g(B, @code{p} $B$,(B -1 $B$J$i$P<+F0E*$KA*$P$l$?AG?t$,(B,

$B9T$o$J$$(B. $B$3$N>l9g(B, @var{p} $B$,(B -1 $B$J$i$P<+F0E*$KA*$P$l$?AG?t$,(B,

$B$=$l0J30$O;XDj$5$l$?AG?t$rMQ$$$F%0%l%V%J4pDl8uJd$N7W;;$,9T$o$l$k(B.

@code{nd_f4_trace} $B$O(B, $B3FA4<!?t$K$D$$$F(B, $B$"$kM-8BBN>e$G(B F4 $B%"%k%4%j%:%`(B

$B$G9T$C$?7k2L$r$b$H$K(B, $B$=$NM-8BBN>e$G(B 0 $B$G$J$$4pDl$rM?$($k(B S-$BB?9`<0$N$_$r(B

$BMQ$$$F9TNs@8@.$r9T$$(B, $B$=$NA4<!?t$K$*$1$k4pDl$r@8@.$9$kJ}K!$G$"$k(B. $BF@$i$l$k(B

$BB?9`<0=89g$O$d$O$j%0%l%V%J4pDl8uJd$G$"$j(B, @code{nd_gr_trace} $B$HF1MM$N(B

$B%A%'%C%/$,9T$o$l$k(B.

@item

@code{nd_f4} $B$O(B, $BM-8BBN>e$N(B F4 $B%"%k%4%j%:%`$r<B9T$9$k(B.

@code{nd_f4} $B$O(B @code{modular} $B$,(B 0 $B$N$H$-M-M}?tBN>e$N(B, @code{modular} $B$,(B

$B%^%7%s%5%$%:AG?t$N$H$-M-8BBN>e$N(B F4 $B%"%k%4%j%:%`$r<B9T$9$k(B.

@item

@var{plist} $B$,B?9`<0%j%9%H$N>l9g(B, @var{plist}$B$G@8@.$5$l$k%$%G%"%k$N%0%l%V%J!<4pDl$,(B

$B7W;;$5$l$k(B. @var{plist} $B$,B?9`<0%j%9%H$N%j%9%H$N>l9g(B, $B3FMWAG$OB?9`<04D>e$N<+M32C72$N85$H8+$J$5$l(B,

$B$3$l$i$,@8@.$9$kItJ,2C72$N%0%l%V%J!<4pDl$,7W;;$5$l$k(B. $B8e<T$N>l9g(B, $B9`=g=x$O2C72$KBP$9$k9`=g=x$r(B

$B;XDj$9$kI,MW$,$"$k(B. $B$3$l$O(B @var{[s,ord]} $B$N7A$G;XDj$9$k(B. @var{s} $B$,(B 0 $B$J$i$P(B TOP (Term Over Position),

1 $B$J$i$P(B POT (Position Over Term) $B$r0UL#$7(B, @var{ord} $B$OB?9`<04D$NC19`<0$KBP$9$k9`=g=x$G$"$k(B.

@item

@code{nd_weyl_gr}, @code{nd_weyl_gr_trace} $B$O(B Weyl $BBe?tMQ$G$"$k(B.

@item

$B$$$:$l$N4X?t$b(B, $BM-M}4X?tBN>e$N7W;;$OL$BP1~$G$"$k(B.

@code{f4} $B7O4X?t0J30$O$9$Y$FM-M}4X?t78?t$N7W;;$,2DG=$G$"$k(B.

@item

$B0lHL$K(B @code{dp_gr_main}, @code{dp_gr_mod_main} $B$h$j9bB.$G$"$k$,(B,

$BFC$KM-8BBN>e$N>l9g82Cx$G$"$k(B.

Line 2384 the Groebner basis check and ideal-membership check ar

Line 2400 the Groebner basis check and ideal-membership check ar

In this case, an automatically chosen prime if @code{p} is 1,

otherwise the specified prime is used to compute a Groebner basis

candidate.

Execution of @code{nd_f4_trace} is done as follows:

For each total degree, an F4-reduction of S-polynomials over a finite field

is done, and S-polynomials which give non-zero basis elements are gathered.

Then F4-reduction over Q is done for the gathered S-polynomials.

The obtained polynomial set is a Groebner basis candidate and the same

check procedure as in the case of @code{nd_gr_trace} is done.

@item

@code{nd_f4} executes F4 algorithm over a finite field.

@code{nd_f4} executes F4 algorithm over Q if @code{modular} is equal to 0,

or over a finite field GF(@code{modular})

if @code{modular} is a prime number of machine size (<2^29).

If @var{plist} is a list of polynomials, then a Groebner basis of the ideal generated by @var{plist}

is computed. If @var{plist} is a list of lists of polynomials, then each list of polynomials are regarded

as an element of a free module over a polynomial ring and a Groebner basis of the sub-module generated by @var{plist}

in the free module. In the latter case a term order in the free module should be specified.

This is specified by @var{[s,ord]}. If @var{s} is 0 then it means TOP (Term Over Position).

If @var{s} is 1 then it means POT 1 (Position Over Term). @var{ord} is a term order in the base polynomial ring.

@item

@code{nd_weyl_gr}, @code{nd_weyl_gr_trace} are for Weyl algebra computation.

@item

Each function cannot handle rational function coefficient cases.

Functions except for F4 related ones can handle rational coeffient cases.

@item

In general these functions are more efficient than

@code{dp_gr_main}, @code{dp_gr_mod_main}, especially over finite fields.

Line 2597 when functions other than top level functions are call

Line 2627 when functions other than top level functions are call

\EG @fref{Setting term orderings}

@end table

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

\EG @node dp_set_weight dp_set_top_weight dp_weyl_set_weight,,, Functions for Groebner basis computation

@subsection @code{dp_set_weight}, @code{dp_set_top_weight}, @code{dp_weyl_set_weight}

@findex dp_set_weight

@findex dp_set_top_weight

@findex dp_weyl_set_weight

@table @t

@item dp_set_weight([@var{weight}])

\JP :: sugar weight $B$N@_Dj(B, $B;2>H(B

\EG :: Set and show the sugar weight.

@item dp_set_top_weight([@var{weight}])

\JP :: top weight $B$N@_Dj(B, $B;2>H(B

\EG :: Set and show the top weight.

@item dp_weyl_set_weight([@var{weight}])

\JP :: weyl weight $B$N@_Dj(B, $B;2>H(B

\EG :: Set and show the weyl weight.

@end table

@table @var

@item return

\JP $B%Y%/%H%k(B

\EG a vector

@item weight

\JP $B@0?t$N%j%9%H$^$?$O%Y%/%H%k(B

\EG a list or vector of integers

@end table

@itemize @bullet

\BJP

@item

@code{dp_set_weight} $B$O(B sugar weight $B$r(B @var{weight} $B$K@_Dj$9$k(B. $B0z?t$,$J$$;~(B,

$B8=:_@_Dj$5$l$F$$$k(B sugar weight $B$rJV$9(B. sugar weight $B$O@5@0?t$r@.J,$H$9$k%Y%/%H%k$G(B,

$B3FJQ?t$N=E$_$rI=$9(B. $B<!?t$D$-=g=x$K$*$$$F(B, $BC19`<0$N<!?t$r7W;;$9$k:]$KMQ$$$i$l$k(B.

$B@F<!2=JQ?tMQ$K(B, $BKvHx$K(B 1 $B$rIU$12C$($F$*$/$H0BA4$G$"$k(B.

@item

@code{dp_set_top_weight} $B$O(B top weight $B$r(B @var{weight} $B$K@_Dj$9$k(B. $B0z?t$,$J$$;~(B,

$B8=:_@_Dj$5$l$F$$$k(B top weight $B$rJV$9(B. top weight $B$,@_Dj$5$l$F$$$k$H$-(B,

$B$^$:(B top weight $B$K$h$kC19`<0Hf3S$r@h$K9T$&(B. tie breaker $B$H$7$F8=:_@_Dj$5$l$F$$$k(B

$B9`=g=x$,MQ$$$i$l$k$,(B, $B$3$NHf3S$K$O(B top weight $B$OMQ$$$i$l$J$$(B.

@item

@code{dp_weyl_set_weight} $B$O(B weyl weight $B$r(B @var{weight} $B$K@_Dj$9$k(B. $B0z?t$,$J$$;~(B,

$B8=:_@_Dj$5$l$F$$$k(B weyl weight $B$rJV$9(B. weyl weight w $B$r@_Dj$9$k$H(B,

$B9`=g=x7?(B 11 $B$G$N7W;;$K$*$$$F(B, (-w,w) $B$r(B top weight, tie breaker $B$r(B graded reverse lex

$B$H$7$?9`=g=x$,@_Dj$5$l$k(B.

\BEG

@item

@code{dp_set_weight} sets the sugar weight=@var{weight}. It returns the current sugar weight.

A sugar weight is a vector with positive integer components and it represents the weights of variables.

It is used for computing the weight of a monomial in a graded ordering.

It is recommended to append a component 1 at the end of the weight vector for a homogenizing variable.

@item

@code{dp_set_top_weight} sets the top weight=@var{weight}. It returns the current top weight.

It a top weight is set, the weights of monomials under the top weight are firstly compared.

If the the weights are equal then the current term ordering is applied as a tie breaker, but

the top weight is not used in the tie breaker.

@item

@code{dp_weyl_set_weight} sets the weyl weigh=@var{weight}. It returns the current weyl weight.

If a weyl weight w is set, in the comparsion by the term order type 11, a term order with

the top weight=(-w,w) and the tie breaker=graded reverse lex is applied.

@end itemize

@table @t

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

\EG @item References

@fref{Weight}

@end table

\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}

Line 2779 converting the coefficients into elements of a finite

Line 2882 converting the coefficients into elements of a finite

@table @t

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

\EG @item References

@fref{dp_nf dp_nf_mod dp_true_nf dp_true_nf_mod},

@fref{dp_nf dp_nf_mod dp_true_nf dp_true_nf_mod dp_weyl_nf dp_weyl_nf_mod},

@fref{subst psubst},

@fref{setmod}.

@end table

Line 2870 These are used internally in @code{hgr()} etc.

Line 2973 These are used internally in @code{hgr()} etc.

into an integral distributed polynomial such that GCD of all its coefficients

is 1.

@itemx dp_prim(@var{dpoly})

@item dp_prim(@var{dpoly})

\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

Line 2923 polynomial contents included in the coefficients are n

Line 3026 polynomial contents included in the coefficients are n

@fref{ptozp}.

@end table

\JP @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 dp_weyl_nf dp_weyl_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

\EG @node dp_nf dp_nf_mod dp_true_nf dp_true_nf_mod dp_weyl_nf dp_weyl_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

@findex dp_nf_mod

@findex dp_true_nf_mod

@findex dp_weyl_nf

@findex dp_weyl_nf_mod

@table @t

@item dp_nf(@var{indexlist},@var{dpoly},@var{dpolyarray},@var{fullreduce})

@item dp_weyl_nf(@var{indexlist},@var{dpoly},@var{dpolyarray},@var{fullreduce})

@item dp_nf_mod(@var{indexlist},@var{dpoly},@var{dpolyarray},@var{fullreduce},@var{mod})

@item dp_weyl_nf_mod(@var{indexlist},@var{dpoly},@var{dpolyarray},@var{fullreduce},@var{mod})

\JP :: $BJ,;6I=8=B?9`<0$N@55,7A$r5a$a$k(B. ($B7k2L$ODj?tG\$5$l$F$$$k2DG=@-$"$j(B)

\BEG

Line 2975 is returned in such a list as @code{[numerator, denomi

Line 3082 is returned in such a list as @code{[numerator, denomi

@item

$BJ,;6I=8=B?9`<0(B @var{dpoly} $B$N@55,7A$r5a$a$k(B.

@item

$BL>A0$K(B weyl $B$r4^$`4X?t$O%o%$%kBe?t$K$*$1$k@55,7A7W;;$r9T$&(B. $B0J2<$N@bL@$O(B weyl $B$r4^$`$b$N$KBP$7$F$bF1MM$K@.N)$9$k(B.

@item

@code{dp_nf_mod()}, @code{dp_true_nf_mod()} $B$NF~NO$O(B, @code{dp_mod()} $B$J$I(B

$B$K$h$j(B, $BM-8BBN>e$NJ,;6I=8=B?9`<0$K$J$C$F$$$J$1$l$P$J$i$J$$(B.

@item

Line 3007 is returned in such a list as @code{[numerator, denomi

Line 3116 is returned in such a list as @code{[numerator, denomi

@item

Computes the normal form of a distributed polynomial.

@item

Functions whose name contain @code{weyl} compute normal forms in Weyl algebra. The description below also applies to

the functions for Weyl algebra.

@item

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

distributed polynomials with coefficients in a finite field as arguments.

@item

Line 3788 refer to @code{dp_true_nf()} and @code{dp_true_nf_mod(

Line 3900 refer to @code{dp_true_nf()} and @code{dp_true_nf_mod(

@fref{dp_ptod},

@fref{dp_dtop},

@fref{dp_ord},

@fref{dp_nf dp_nf_mod dp_true_nf dp_true_nf_mod}.

@fref{dp_nf dp_nf_mod dp_true_nf dp_true_nf_mod dp_weyl_nf dp_weyl_nf_mod}.

@end table

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

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