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Diff for /OpenXM/src/asir-doc/parts/groebner.texi between version 1.15 and 1.17

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

version 1.17, 2006/09/06 23:53:31

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 1.16 2004/10/20 00:30:55 fujiwara Exp $

\BJP

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

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

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_ptod::

Line 1541 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

\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 2318 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})

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

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

Line 2348 except for lack of the argument for controlling homoge

Line 2350 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}

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

@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

$B$^$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

$B$,<: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

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

$B$=$l0J30$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

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

@item

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

Line 2393 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).

@item

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

@item

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