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Diff for /OpenXM/src/asir-doc/parts/groebner.texi between version 1.18 and 1.22

version 1.18, 2016/03/24 20:58:50

version 1.22, 2019/03/29 04:54:25

Line 1

@comment $OpenXM: OpenXM/src/asir-doc/parts/groebner.texi,v 1.17 2006/09/06 23:53:31 noro Exp $

@comment $OpenXM: OpenXM/src/asir-doc/parts/groebner.texi,v 1.21 2018/09/06 05:42:43 takayama 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 220 the head term and the head coefficient respectively.

@end table

@noindent

ChangeLog

@itemize @bullet

\BJP

@item $BJ,;6I=8=B?9`<0$OG$0U$N%*%V%8%'%/%H$r78?t$K$b$F$k$h$&$K$J$C$?(B.

$B$^$?2C72$N(Bk$B@.J,$NMWAG$r<!$N7A<0(B <<d0,d1,...:k>> $B$GI=8=$9$k$h$&$K$J$C$?(B (2017-08-31).

\BEG

@item Distributed polynomials accept objects as coefficients.

The k-th element of a free module is expressed as <<d0,d1,...:k>> (2017-08-31).

@item

1.15 algnum.c,

1.53 ctrl.c,

1.66 dp-supp.c,

1.105 dp.c,

1.73 gr.c,

1.4 reduct.c,

1.16 _distm.c,

1.17 dalg.c,

1.52 dist.c,

1.20 distm.c,

1.8 gmpq.c,

1.238 engine/nd.c,

1.102 ca.h,

1.411 version.h,

1.28 cpexpr.c,

1.42 pexpr.c,

1.20 pexpr_body.c,

1.40 spexpr.c,

1.27 arith.c,

1.77 eval.c,

1.56 parse.h,

1.37 parse.y,

1.8 stdio.c,

1.31 plotf.c

@end itemize

\BJP

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

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

Line 1465 Computation of the global b function is implemented as

Line 1503 Computation of the global b function is implemented as

* 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_f4_trace nd_weyl_gr nd_weyl_gr_trace::

* nd_gr_postproc nd_weyl_gr_postproc::

* dp_gr_flags dp_gr_print::

* dp_ord::

* dp_set_weight dp_set_top_weight dp_weyl_set_weight::

Line 1531 Computation of the global b function is implemented as

Line 1570 Computation of the global b function is implemented as

@item

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

@item

gr $B$rL>A0$K4^$`4X?t$O8=:_%a%s%F$5$l$F$$$J$$(B. @code{nd_gr}$B7O$N4X?t$rBe$o$j$KMxMQ$9$Y$-$G$"$k(B(@fref{nd_gr nd_gr_trace nd_f4 nd_f4_trace nd_weyl_gr nd_weyl_gr_trace}).

@item

$B$$$:$l$b(B, $BB?9`<0%j%9%H(B @var{plist} $B$N(B, $BJQ?t=g=x(B @var{vlist}, $B9`=g=x7?(B

@var{order} $B$K4X$9$k%0%l%V%J4pDl$r5a$a$k(B. @code{gr()}, @code{hgr()}

$B$O(B $BM-M}?t78?t(B, @code{gr_mod()} $B$O(B GF(@var{p}) $B78?t$H$7$F7W;;$9$k(B.

Line 1562 CPU $B;~4V$G$"$j(B, $B$3$NH!?t$N>l9g$O$[$H$s$IDL?.$

Line 1603 CPU $B;~4V$G$"$j(B, $B$3$NH!?t$N>l9g$O$[$H$s$IDL?.$

@item

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

directory.

@item

Functions of which names contains gr are obsolted.

Functions of @code{nd_gr} families should be used (@fref{nd_gr nd_gr_trace nd_f4 nd_f4_trace nd_weyl_gr nd_weyl_gr_trace}).

@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}.

Line 2320 except for lack of the argument for controlling homoge

Line 2364 except for lack of the argument for controlling homoge

@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})

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

@item nd_weyl_gr(@var{plist},@var{vlist},@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 2451 ndv_alloc=1477188

Line 2495 ndv_alloc=1477188

\EG @fref{Controlling Groebner basis computations}

@end table

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

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

@subsection @code{nd_gr_postproc}, @code{nd_weyl_gr_postproc}

@findex nd_gr_postproc

@findex nd_weyl_gr_postproc

@table @t

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

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

\JP :: $B%0%l%V%J4pDl8uJd$N%A%'%C%/$*$h$SAj8_4JLs(B

\EG :: Check of Groebner basis candidate and inter-reduction

@end table

@table @var

@item return

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

\EG list or 0

@item plist vlist

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

\EG list

@item p

\JP $BAG?t$^$?$O(B 0

\EG prime or 0

@item order

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

\EG number, list or matrix

@item check

\JP 0 $B$^$?$O(B 1

\EG 0 or 1

@end table

@itemize @bullet

\BJP

@item

$B%0%l%V%J4pDl(B($B8uJd(B)$B$NAj8_4JLs$r9T$&(B.

@item

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

@item

@var{check=1} $B$N>l9g(B, @var{plist} $B$,(B, @var{vlist}, @var{p}, @var{order} $B$G;XDj$5$l$kB?9`<04D(B, $B9`=g=x$G%0%l%V%J!<4pDl$K$J$C$F$$$k$+(B

$B$N%A%'%C%/$b9T$&(B.

@item

$B@F<!2=$7$F7W;;$7$?%0%l%V%J!<4pDl$rHs@F<!2=$7$?$b$N$rAj8_4JLs$r9T$&(B, CRT $B$G7W;;$7$?%0%l%V%J!<4pDl8uJd$N%A%'%C%/$r9T$&$J$I$N>l9g$KMQ$$$k(B.

\BEG

@item

Perform the inter-reduction for a Groebner basis (candidate).

@item

@code{nd_weyl_gr_postproc} is for Weyl algebra.

@item

If @var{check=1} then the check whether @var{plist} is a Groebner basis with respect to a term order in a polynomial ring

or Weyl algebra specified by @var{vlist}, @var{p} and @var{order}.

@item

This function is used for inter-reduction of a non-reduced Groebner basis that is obtained by dehomogenizing a Groebner basis

computed via homogenization, or Groebner basis check of a Groebner basis candidate computed by CRT.

@end itemize

@example

afo

@end example

\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}

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

Line 3078 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 3900 refer to @code{dp_true_nf()} and @code{dp_true_nf_mod(

Line 4005 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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