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-version 1.1, 1999/12/08 05:47:44
+version 1.23, 2019/09/13 09:31:00
 Line 1
 Line 1
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+ @comment $OpenXM: OpenXM/src/asir-doc/parts/groebner.texi,v 1.22 2019/03/29 04:54:25 noro Exp $
+ \BJP
  @node $B%0%l%V%J4pDl$N7W;;(B,,, Top
  @chapter $B%0%l%V%J4pDl$N7W;;(B
+ \E
+ \BEG
+ @node Groebner basis computation,,, Top
+ @chapter Groebner basis computation
+ \E
  @menu
+ \BJP
  * $BJ,;6I=8=B?9`<0(B::
  * $B%U%!%$%k$NFI$_9~$_(B::
  * $B4pK\E*$JH!?t(B::
  * $B7W;;$*$h$SI=<($N@)8f(B::
  * $B9`=g=x$N@_Dj(B::
+ * Weight::
  * $BM-M}<0$r78?t$H$9$k%0%l%V%J4pDl7W;;(B::
  * $B4pDlJQ49(B::
+ * Weyl $BBe?t(B::
+ * $BB?9`<04D>e$N2C72(B::
  * $B%0%l%V%J4pDl$K4X$9$kH!?t(B::
+ \E
+ \BEG
+ * Distributed polynomial::
+ * Reading files::
+ * Fundamental functions::
+ * Controlling Groebner basis computations::
+ * Setting term orderings::
+ * Weight::
+ * Groebner basis computation with rational function coefficients::
+ * Change of ordering::
+ * Weyl algebra::
+ * Module over a polynomial ring::
+ * Functions for Groebner basis computation::
+ \E
  @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
+ \E
+ \BEG
+ @node Distributed polynomial,,, Groebner basis computation
+ @section Distributed polynomial
+ \E
  @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.
+ \E
+ \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.'
+ \E
  @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 83  $(x+y+z)^2 = 1 \cdot x^2 + (2 \cdot y + (2 \cdot z)) \
 Line 35  $(x+y+z)^2 = 1 \cdot x^2 + (2 \cdot y + (2 \cdot z)) \
 Line 83  $(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.
+ \E
+ \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.)
+ \E
  @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 111  $(x+y+z)^2 = 1 \cdot x^2 + 2 \cdot xy + 2 \cdot xz + 1
 Line 50  $(x+y+z)^2 = 1 \cdot x^2 + 2 \cdot xy + 2 \cdot xz + 1
 Line 111  $(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.
+ \E
+ \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.
+ \E
  @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$*$$$F$O(B,
+ \E
+ \BEG
+ @item term
+ The power product of variables, i.e., a monomial with coefficient 1.
+ In an @b{Asir} session, it is displayed in the form like
+ \E
  @example
  <<0,1,2,3,4>>
  @end example
+ \BJP
  $B$H$$$&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.
+ \E
+ \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.
+ \E
+ \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.
+ \E
+ \BEG
+ @item term order
+ Terms are ordered according to a total order with the following properties.
+ \E
  @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.
+ \E
+ \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).
+ \E
+ \BJP
  @item $BC19`<0(B (monomial)
  $B9`$H78?t$N@Q(B.
+ \E
+ \BEG
+ @item monomial
+ The product of a term and a coefficient.
+ In an @b{Asir} session, it is displayed in the form like
+ \E
  @example
 *<<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.
- @itemx $BF,C19`<0(B (head monomial)
+ \BJP
  @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.
+ \E
+ \BEG
+ @item head term
+ @itemx head monomial
+ @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.
+ \E
  @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).
+ \E
+ \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).
+ \E
+ @item
+.15 algnum.c,
+.53 ctrl.c,
+.66 dp-supp.c,
+.105 dp.c,
+.73 gr.c,
+.4 reduct.c,
+.16 _distm.c,
+.17 dalg.c,
+.52 dist.c,
+.20 distm.c,
+.8  gmpq.c,
+.238 engine/nd.c,
+.102  ca.h,
+.411  version.h,
+.28 cpexpr.c,
+.42 pexpr.c,
+.20 pexpr_body.c,
+.40 spexpr.c,
+.27 arith.c,
+.77 eval.c,
+.56 parse.h,
+.37 parse.y,
+.8 stdio.c,
+.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
+ \E
+ \BEG
+ @node Reading files,,, Groebner basis computation
+ @section Reading files
+ \E
  @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.
+ \E
+ \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}.
+ \E
  @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
+ \E
+ \BEG
+ @node Fundamental functions,,, Groebner basis computation
+ @section Fundamental functions
+ \E
  @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.
+ \E
+ \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.
+ \E
  @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.
+ \E
+ \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.
+ \E
  @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?$$(B.
+ \E
+ \BEG
+ After homogenizing the input polynomials a candidate of the \gr basis
+ is computed by trace-lifting. Then the candidate is dehomogenized and
+ checked whether it is indeed a Groebner basis of the input.  Sugar
+ strategy often causes intermediate coefficient swells.  It is
+ empirically known that the combination of homogenization and supresses
+ the swells for such cases.
+ \E
  @item gr_mod(@var{plist},@var{vlist},@var{order},@var{p})
+ \BJP
  Gebauer-Moeller $B$K$h$k(B useless pair elimination criteria, sugar
  strategy $B$*$h$S(B Buchberger $B%"%k%4%j%:%`$K$h$k(B GF(p) $B78?t%0%l%V%J4pDl7W(B
  $B;;H!?t(B.
+ \E
+ \BEG
+ Function that computes Groebner bases over GF(@var{p}). The same
+ algorithm as @code{gr()} is used.
+ \E
  @end table
+ \BJP
  @node $B7W;;$*$h$SI=<($N@)8f(B,,, $B%0%l%V%J4pDl$N7W;;(B
  @section $B7W;;$*$h$SI=<($N@)8f(B
+ \E
+ \BEG
+ @node Controlling Groebner basis computations,,, Groebner basis computation
+ @section Controlling Groebner basis computations
+ \E
  @noindent
+ \BJP
  $B%0%l%V%J4pDl$N7W;;$K$*$$$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$$$k%Q%i%a%?$,(B, $BL>A0$HCM$N%j%9%H$GJV$5$l$k(B.
+ \E
+ \BEG
+ One can cotrol a Groebner basis computation by setting various parameters.
+ These parameters can be set and examined by a built-in function
+ @code{dp_gr_flags()}. Without argument it returns the current settings.
+ \E
  @example
  [100] dp_gr_flags();
- [Demand,0,NoSugar,0,NoCriB,0,NoGC,0,NoMC,0,NoRA,0,NoGCD,0,Top,0,ShowMag,1,
+ [Demand,0,NoSugar,0,NoCriB,0,NoGC,0,NoMC,0,NoRA,0,NoGCD,0,Top,0,
- Print,1,Stat,0,Reverse,0,InterReduce,0,Multiple,0]
+ ShowMag,1,Print,1,Stat,0,Reverse,0,InterReduce,0,Multiple,0]
  [101]
  @end example
+ \BJP
  $B0J2<$G(B, $B3F%Q%i%a%?$N0UL#$r@bL@$9$k(B. on $B$N>l9g$H$O(B, $B%Q%i%a%?$,(B 0 $B$G$J$$>l9g$r(B
  $B$$$&(B. $B$3$l$i$N%Q%i%a%?$N=i4|CM$OA4$F(B 0 (off) $B$G$"$k(B.
+ \E
+ \BEG
+ The return value is a list which contains the names of parameters and their
+ values. The meaning of the parameters are as follows. `on' means that the
+ parameter is not zero.
+ \E
  @table @code
  @item NoSugar
+ \BJP
  on $B$N>l9g(B, sugar strategy $B$NBe$o$j$K(B Buchberger$B$N(B normal strategy $B$,MQ(B
  $B$$$i$l$k(B.
+ \E
+ \BEG
+ If `on', Buchberger's normal strategy is used instead of sugar strategy.
+ \E
  @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.
+ \E
  @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$$(B.
+ \E
+ \BEG
+ If `on', the check that the resulting polynomials generates the same ideal as
+ the ideal generated by the input, is not executed.
+ \E
  @item NoRA
+ \BJP
  on $B$N>l9g(B, $B7k2L$r(B reduced $B%0%l%V%J4pDl$K$9$k$?$a$N(B
  interreduce $B$r9T$o$J$$(B.
+ \E
+ \BEG
+ If `on', the interreduction, which makes the Groebner basis reduced, is not
+ executed.
+ \E
  @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.
+ \E
+ \BEG
+ If `on', content removals are not executed during a Groebner basis computation
+ over a rational function field.
+ \E
  @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.
+ \E
+ \BEG
+ If `on', the selection strategy of reducer in a normal form computation
+ is such that a newer reducer is used first.
+ \E
  @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.
+ \E
+ @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.
+ \E
  @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.
+ \E
+ \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'.
+ \E
  @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.
+ \E
+ \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.
+ \E
- @item Multiple
+ @item Content
-$B$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$K(B
+$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.
+ \E
+ \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.
+ \E
  @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$$r;}$C$F>C5n$9$kI,MW$,$"$k(B.
+ \E
+ \BEG
+ If the value (a character string) is a valid directory name, then
+ generated basis elements are put in the directory and are loaded on
+ demand during normal form computations.  Each elements is saved in the
+ binary form and its name coincides with the index internally used in
+ the computation. These binary files are not removed automatically
+ and one should remove them by hand.
+ \E
  @end table
  @noindent
- @code{Print} $B$,(B 0 $B$G$J$$>l9g<!$N$h$&$J%G!<%?$,I=<($5$l$k(B.
+ \JP @code{Print} $B$,(B 0 $B$G$J$$>l9g<!$N$h$&$J%G!<%?$,I=<($5$l$k(B.
+ \EG If @code{Print} is `on', the following informations are shown.
  @example
  [93] gr(cyclic(4),[c0,c1,c2,c3],0)$
-Line 249  membercheck
+Line 585  membercheck
 Line 249  membercheck
 Line 585  membercheck
  (0,0)(0,0)(0,0)(0,0)
  gbcheck total 8 pairs
  ........
- UP=(0,0)SP=(0,0)SPM=(0,0)NF=(0,0)NFM=(0.010002,0)ZNFM=(0.010002,0)PZ=(0,0)
+ UP=(0,0)SP=(0,0)SPM=(0,0)NF=(0,0)NFM=(0.010002,0)ZNFM=(0.010002,0)
- NP=(0,0)MP=(0,0)RA=(0,0)MC=(0,0)GC=(0,0)T=40,B=0 M=8 F=6 D=12 ZR=5 NZR=6
+ PZ=(0,0)NP=(0,0)MP=(0,0)RA=(0,0)MC=(0,0)GC=(0,0)T=40,B=0 M=8 F=6
- Max_mag=6
+ D=12 ZR=5 NZR=6 Max_mag=6
  [94]
  @end example
  @noindent
+ \BJP
  $B:G=i$KI=<($5$l$k(B @code{mod}, @code{eval} $B$O(B, trace-lifting $B$GMQ$$$i$l$kK!(B
  $B$G$"$k(B. @code{mod} $B$OAG?t(B, @code{eval} $B$OM-M}<078?t$N>l9g$KMQ$$$i$l$k(B
  $B?t$N%j%9%H$G$"$k(B.
+ \E
+ \BEG
+ In this example @code{mod} and @code{eval} indicate moduli used in
+ trace-lifting. @code{mod} is a prime and @code{eval} is a list of integers
+ used for evaluation when the ground field is a field of rational functions.
+ \E
  @noindent
- $B7W;;ESCf$GB?9`<0$,@8@.$5$l$kKh$K<!$N7A$N%G!<%?$,I=<($5$l$k(B.
+ \JP $B7W;;ESCf$GB?9`<0$,@8@.$5$l$kKh$K<!$N7A$N%G!<%?$,I=<($5$l$k(B.
+ \EG The following information is shown after every normal form computation.
  @example
  (TNF)(TCONT)HT(INDEX),nb=NB,nab=NAB,rp=RP,sugar=S,mag=M
  @end example
  @noindent
- $B$=$l$i$N0UL#$O<!$NDL$j(B.
+ \JP $B$=$l$i$N0UL#$O<!$NDL$j(B.
+ \EG Meaning of each component is as follows.
  @table @code
+ \BJP
  @item TNF
  normal form $B7W;;;~4V(B ($BIC(B)
  @item TCONT
  content $B7W;;;~4V(B ($BIC(B)
  @item HT
  $B@8@.$5$l$?B?9`<0$NF,9`(B
  @item INDEX
  S-$BB?9`<0$r9=@.$9$kB?9`<0$N%$%s%G%C%/%9$N%Z%"(B
  @item NB
  $B8=:_$N(B, $B>iD9@-$r=|$$$?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. )
+ \E
+ \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'.)
+ \E
  @end table
  @noindent
+ \BJP
  $B:G8e$K(B, $B=87W%G!<%?$,I=<($5$l$k(B. $B0UL#$O<!$NDL$j(B.
  ($B;~4V$NI=<($K$*$$$F(B, $B?t;z$,(B 2 $B$D$"$k$b$N$O(B, $B7W;;;~4V$H(B GC $B;~4V$N%Z%"$G$"$k(B.)
+ \E
+ \BEG
+ The summary of the informations shown after a Groebner basis
+ computation is as follows.  If a component shows timings and it
+ contains two numbers, they are a pair of time for computation and time
+ for garbage colection.
+ \E
  @table @code
+ \BJP
  @item UP
  $B%Z%"$N%j%9%H$NA`:n$K$+$+$C$?;~4V(B
  @item SP
  $BM-M}?t>e$N(B S-$BB?9`<07W;;;~4V(B
  @item SPM
  $BM-8BBN>e$N(B S-$BB?9`<07W;;;~4V(B
  @item NF
  $BM-M}?t>e$N(B normal form $B7W;;;~4V(B
  @item NFM
  $BM-8BBN>e$N(B normal form $B7W;;;~4V(B
  @item ZNFM
  @code{NFM} $B$NFb(B, 0 $B$X$N(B reduction $B$K$+$+$C$?;~4V(B
  @item PZ
  content $B7W;;;~4V(B
  @item NP
  $BM-M}?t78?tB?9`<0$N78?t$KBP$9$k>jM>1i;;$N7W;;;~4V(B
  @item MP
  S-$BB?9`<0$r@8@.$9$k%Z%"$NA*Br$K$+$+$C$?;~4V(B
  @item RA
  interreduce $B7W;;;~4V(B
  @item MC
  trace-lifting $B$K$*$1$k(B, $BF~NOB?9`<0$N%a%s%P%7%C%W7W;;;~4V(B
  @item GC
  $B7k2L$N%0%l%V%J4pDl8uJd$N%0%l%V%J4pDl%A%'%C%/;~4V(B
  @item T
  $B@8@.$5$l$?%Z%"$N?t(B
  @item B, M, F, D
  $B3F(B criterion $B$K$h$j=|$+$l$?%Z%"$N?t(B
  @item ZR
 $B$K(B reduce $B$5$l$?%Z%"$N?t(B
  @item NZR
 $B$G$J$$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
+ \E
+ \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
+ \E
  @end table
+ \BJP
  @node $B9`=g=x$N@_Dj(B,,, $B%0%l%V%J4pDl$N7W;;(B
  @section $B9`=g=x$N@_Dj(B
+ \E
+ \BEG
+ @node Setting term orderings,,, Groebner basis computation
+ @section Setting term orderings
+ \E
  @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$$$O(B
  $B9TNs$GI=8=$5$l$k(B.
+ \E
+ \BEG
+ A term is internally represented as an integer vector whose components
+ are exponents with respect to variables. A variable list specifies the
+ correspondences between variables and components. A type of term ordering
+ specifies a total order for integer vectors. A type of term ordering is
+ represented by an integer, a list of integer or matrices.
+ \E
  @noindent
- $B4pK\E*$J9`=g=x7?$H$7$F<!$N(B 3 $B$D$,$"$k(B.
+ \JP $B4pK\E*$J9`=g=x7?$H$7$F<!$N(B 3 $B$D$,$"$k(B.
+ \EG There are following three fundamental types.
  @table @code
- @item 0 (DegRevLex; @b{$BA4<!?t5U<-=q<0=g=x(B})
+ \JP @item 0 (DegRevLex; @b{$BA4<!?t5U<-=q<0=g=x(B})
+ \EG @item 0 (DegRevLex; @b{total degree reverse lexicographic ordering})
+ \BJP
  $B0lHL$K(B, $B$3$N=g=x$K$h$k%0%l%V%J4pDl7W;;$,:G$b9bB.$G$"$k(B. $B$?$@$7(B,
  $BJ}Dx<0$r2r$/$H$$$&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$$$i$l$k(B.
+ \E
+ \BEG
+ In general, computation by this ordering shows the fastest speed
+ in most Groebner basis computations.
+ However, for the purpose to solve polynomial equations, this type
+ of ordering is, in general, not so suitable.
+ The Groebner bases obtained by this ordering is used for computing
+ the number of solutions, solving ideal membership problem and seeds
+ for conversion to other Groebner bases under different ordering.
+ \E
- @item 1 (DegLex; @b{$BA4<!?t<-=q<0=g=x(B})
+ \JP @item 1 (DegLex; @b{$BA4<!?t<-=q<0=g=x(B})
+ \EG @item 1 (DegLex; @b{total degree lexicographic ordering})
+ \BJP
  $B$3$N=g=x$b(B, $B<-=q<0=g=x$KHf$Y$F9bB.$K%0%l%V%J4pDl$r5a$a$k$3$H$,$G$-$k$,(B,
  @code{DegRevLex} $B$HF1MMD>@\$=$N7k2L$rMQ$$$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$$$k(B.
+ \E
+ \BEG
+ By this type term ordering, Groebner bases are obtained fairly faster
+ than Lex (lexicographic) ordering, too.
+ Alike the @code{DegRevLex} ordering, the result, in general, cannot directly
+ be used for solving polynomial equations.
+ It is used, however, in such a way
+ that a Groebner basis is computed in this ordering after homogenization
+ to obtain the final lexicographic Groebner basis.
+ \E
- @item 2 (Lex; @b{$B<-=q<0=g=x(B})
+ \JP @item 2 (Lex; @b{$B<-=q<0=g=x(B})
+ \EG @item 2 (Lex; @b{lexicographic ordering})
+ \BJP
  $B$3$N=g=x$K$h$k%0%l%V%J4pDl$O(B, $BJ}Dx<0$r2r$/>l9g$K:GE,$N7A$N4pDl$rM?$($k$,(B
  $B7W;;;~4V$,$+$+$j2a$.$k$N$,FqE@$G$"$k(B. $BFC$K(B, $B2r$,M-8B8D$N>l9g(B, $B7k2L$N(B
  $B78?t$,6K$a$FD9Bg$JB?G\D9?t$K$J$k>l9g$,B?$$(B. $B$3$N>l9g(B, @code{gr()},
  @code{hgr()} $B$K$h$k7W;;$,6K$a$FM-8z$K$J$k>l9g$,B?$$(B.
+ \E
+ \BEG
+ Groebner bases computed by this ordering give the most convenient
+ Groebner bases for solving the polynomial equations.
+ The only and serious shortcoming is the enormously long computation
+ time.
+ It is often observed that the number coefficients of the result becomes
+ very very long integers, especially if the ideal is 0-dimensional.
+ For such a case, it is empirically true for many cases
+ that i.e., computation by
+ @code{gr()} and/or @code{hgr()} may be quite effective.
+ \E
  @end table
  @noindent
+ \BJP
  $B$3$l$i$rAH$_9g$o$;$F%j%9%H$G;XDj$9$k$3$H$K$h$j(B, $BMM!9$J>C5n=g=x$,;XDj$G$-$k(B.
  $B$3$l$O(B,
+ \E
+ \BEG
+ By combining these fundamental orderingl into a list, one can make
+ various term ordering called elimination orderings.
+ \E
  @code{[[O1,L1],[O2,L2],...]}
  @noindent
+ \BJP
  $B$G;XDj$5$l$k(B. @code{Oi} $B$O(B 0, 1, 2 $B$N$$$:$l$+$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.
+ \E
+ \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.
+ \E
  @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.
+ \E
+ \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.
+ \E
  @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}.
+ \E
  @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.
+ \E
+ \BEG
+ For all non-negative integer vectors @code{v} the first non-zero component
+ of @code{Mv} is non-negative.
+ \E
  @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$$:G=i$N@.J,$,HsIi(B,
  $B$GDj5A$9$k$3$H$K$h$j9`=g=x$,Dj5A$G$-$k(B.
+ \E
+ \BEG
+ Then we can define a term ordering such that, for two vectors
+ @code{t}, @code{s}, @code{t>s} means that the first non-zero component
+ of @code{M(t-s)} is non-negative.
+ \E
  @noindent
+ \BJP
  $B9`=g=x7?$O(B, @code{gr()} $B$J$I$N0z?t$H$7$F;XDj$5$l$kB>(B, $BAH$_9~$_H!?t(B
  @code{dp_ord()} $B$G;XDj$5$l(B, $B$5$^$6$^$JH!?t$N<B9T$N:]$K;2>H$5$l$k(B.
+ \E
+ \BEG
+ Types of term orderings are used as arguments of functions such as
+ @code{gr()}. It is also set internally by @code{dp_ord()} and is used
+ during executions of various functions.
+ \E
  @noindent
+ \BJP
  $B$3$l$i$N=g=x$N6qBNE*$JDj5A$*$h$S%0%l%V%J4pDl$K4X$9$k99$K>\$7$$2r@b$O(B
  @code{[Becker,Weispfenning]} $B$J$I$r;2>H$N$3$H(B.
+ \E
+ \BEG
+ For concrete definitions of term ordering and more information
+ about Groebner basis, refer to, for example, the book
+ @code{[Becker,Weispfenning]}.
+ \E
  @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$rM?$($k(B.
+ \BEG
+ Note that the variable ordering have strong effects on the computation
+ time as well as the choice of types of term orderings.
+ \E
  @example
  [90] B=[x^10-t,x^8-z,x^31-x^6-x-y]$
-Line 443  trace-lifting $B$K$*$1$k(B, $BF~NOB?9`<0$N%a%s%P%7%
+Line 1047  trace-lifting $B$K$*$1$k(B, $BF~NOB?9`<0$N%a%s%P%7%
 Line 443  trace-lifting $B$K$*$1$k(B, $BF~NOB?9`<0$N%a%s%P%7%
 Line 1047  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
-*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,
-*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,
+*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$$$($P(B, $B<-=q<0=g=x$G%0%l%V%J4pDl$r5a$a$k(B
-Line 474  z^5-t^4]
+Line 1080  z^5-t^4]
 Line 474  z^5-t^4]
 Line 1080  z^5-t^4]
  @code{x} $B$GI=$5$l$F$$$k$3$H$+$i$3$N$h$&$J6KC<$J7k2L$H$J$C$?$o$1$G$"$k(B.
  $B<B:]$K8=$l$k7W;;$K$*$$$F$O(B, $B$3$N$h$&$KA*$V$Y$-JQ?t=g=x$,L@$i$+$G$"$k(B
  $B$3$H$O>/$J$/(B, $B;n9T:x8m$,I,MW$J>l9g$b$"$k(B.
+ \E
+ \BEG
+ As you see in the above example, the Groebner base under variable
+ ordering @code{[x,y,z,t]} has a lot of bases and each base itself is
+ large.  Under variable ordering @code{[t,z,y,x]}, however, @code{B} itself
+ is already the Groebner basis.
+ Roughly speaking, to obtain a Groebner base under the lexicographic
+ ordering is to express the variables on the left (having higher order)
+ in terms of variables on the right (having lower order).
+ In the example, variables @code{t},  @code{z}, and @code{y} are already
+ expressed by variable @code{x}, and the above explanation justifies
+ such a drastic experimental results.
+ In practice, however, optimum ordering for variables may not known
+ beforehand, and some heuristic trial may be inevitable.
+ \E
+ \BJP
+ @node Weight ,,, $B%0%l%V%J4pDl$N7W;;(B
+ @section Weight
+ \E
+ \BEG
+ @node Weight,,, Groebner basis computation
+ @section Weight
+ \E
+ \BJP
+ $BA0@a$G>R2p$7$?9`=g=x$O(B, $B3FJQ?t$K(B weight ($B=E$_(B) $B$r@_Dj$9$k$3$H$G(B
+ $B$h$j0lHLE*$J$b$N$H$J$k(B.
+ \E
+ \BEG
+ Term orderings introduced in the previous section can be generalized
+ by setting a weight for each variable.
+ \E
+ @example
+ [0] dp_td(<<1,1,1>>);
+ [1] dp_set_weight([1,2,3])$
+ [2] dp_td(<<1,1,1>>);
+ @end example
+ \BJP
+ $BC19`<0$NA4<!?t$r7W;;$9$k:](B, $B%G%U%)%k%H$G$O(B
+ $B3FJQ?t$N;X?t$NOB$rA4<!?t$H$9$k(B. $B$3$l$O3FJQ?t$N(B weight $B$r(B 1 $B$H(B
+ $B9M$($F$$$k$3$H$KAjEv$9$k(B. $B$3$NNc$G$O(B, $BBh0l(B, $BBhFs(B, $BBh;0JQ?t$N(B
+ weight $B$r$=$l$>$l(B 1,2,3 $B$H;XDj$7$F$$$k(B. $B$3$N$?$a(B, @code{<<1,1,1>>}
+ $B$NA4<!?t(B ($B0J2<$G$O$3$l$rC19`<0$N(B weight $B$H8F$V(B) $B$,(B @code{1*1+1*2+1*3=6} $B$H$J$k(B.
+ weight $B$r@_Dj$9$k$3$H$G(B, $BF1$89`=g=x7?$N$b$H$G0[$J$k9`=g=x$,Dj5A$G$-$k(B.
+ $BNc$($P(B, weight $B$r$&$^$/@_Dj$9$k$3$H$G(B, $BB?9`<0$r(B weighted homogeneous
+ $B$K$9$k$3$H$,$G$-$k>l9g$,$"$k(B.
+ \E
+ \BEG
+ By default, the total degree of a monomial is equal to
+ the sum of all exponents. This means that the weight for each variable
+ is set to 1.
+ In this example, the weights for the first, the second and the third
+ variable are set to 1, 2 and 3 respectively.
+ Therefore the total degree of @code{<<1,1,1>>} under this weight,
+ which is called the weight of the monomial, is @code{1*1+1*2+1*3=6}.
+ By setting weights, different term orderings can be set under a type of
+ term ordeing. In some case a polynomial can
+ be made weighted homogeneous by setting an appropriate weight.
+ \E
+ \BJP
+ $B3FJQ?t$KBP$9$k(B weight $B$r$^$H$a$?$b$N$r(B weight vector $B$H8F$V(B.
+ $B$9$Y$F$N@.J,$,@5$G$"$j(B, $B%0%l%V%J4pDl7W;;$K$*$$$F(B, $BA4<!?t$N(B
+ $BBe$o$j$KMQ$$$i$l$k$b$N$rFC$K(B sugar weight $B$H8F$V$3$H$K$9$k(B.
+ sugar strategy $B$K$*$$$F(B, $BA4<!?t$NBe$o$j$K;H$o$l$k$+$i$G$"$k(B.
+ $B0lJ}$G(B, $B3F@.J,$,I,$:$7$b@5$H$O8B$i$J$$(B weight vector $B$O(B,
+ sugar weight $B$H$7$F@_Dj$9$k$3$H$O$G$-$J$$$,(B, $B9`=g=x$N0lHL2=$K$O(B
+ $BM-MQ$G$"$k(B. $B$3$l$i$O(B, $B9TNs$K$h$k9`=g=x$N@_Dj$K$9$G$K8=$l$F(B
+ $B$$$k(B. $B$9$J$o$A(B, $B9`=g=x$rDj5A$9$k9TNs$N3F9T$,(B, $B0l$D$N(B weight vector
+ $B$H8+$J$5$l$k(B. $B$^$?(B, $B%V%m%C%/=g=x$O(B, $B3F%V%m%C%/$N(B
+ $BJQ?t$KBP1~$9$k@.J,$N$_(B 1 $B$GB>$O(B 0 $B$N(B weight vector $B$K$h$kHf3S$r(B
+ $B:G=i$K9T$C$F$+$i(B, $B3F%V%m%C%/Kh$N(B tie breaking $B$r9T$&$3$H$KAjEv$9$k(B.
+ \E
+ \BEG
+ A list of weights for all variables is called a weight vector.
+ A weight vector is called a sugar weight vector if
+ its elements are all positive and it is used for computing
+ a weighted total degree of a monomial, because such a weight
+ is used instead of total degree in sugar strategy.
+ On the other hand, a weight vector whose elements are not necessarily
+ positive cannot be set as a sugar weight, but it is useful for
+ generalizing term order. In fact, such a weight vector already
+ appeared in a matrix order. That is, each row of a matrix defining
+ a term order is regarded as a weight vector. A block order
+ is also considered as a refinement of comparison by weight vectors.
+ It compares two terms by using a weight vector whose elements
+ corresponding to variables in a block is 1 and 0 otherwise,
+ then it applies a tie breaker.
+ \E
+ \BJP
+ weight vector $B$N@_Dj$O(B @code{dp_set_weight()} $B$G9T$&$3$H$,$G$-$k(B
+ $B$,(B, $B9`=g=x$r;XDj$9$k:]$NB>$N%Q%i%a%?(B ($B9`=g=x7?(B, $BJQ?t=g=x(B) $B$H(B
+ $B$^$H$a$F@_Dj$G$-$k$3$H$,K>$^$7$$(B. $B$3$N$?$a(B, $B<!$N$h$&$J7A$G$b(B
+ $B9`=g=x$,;XDj$G$-$k(B.
+ \E
+ \BEG
+ A weight vector can be set by using @code{dp_set_weight()}.
+ However it is more preferable if a weight vector can be set
+ together with other parapmeters such as a type of term ordering
+ and a variable order. This is realized as follows.
+ \E
+ @example
+ [64] B=[x+y+z-6,x*y+y*z+z*x-11,x*y*z-6]$
+ [65] dp_gr_main(B|v=[x,y,z],sugarweight=[3,2,1],order=0);
+ [z^3-6*z^2+11*z-6,x+y+z-6,-y^2+(-z+6)*y-z^2+6*z-11]
+ [66] dp_gr_main(B|v=[y,z,x],order=[[1,1,0],[0,1,0],[0,0,1]]);
+ [x^3-6*x^2+11*x-6,x+y+z-6,-x^2+(-y+6)*x-y^2+6*y-11]
+ [67] dp_gr_main(B|v=[y,z,x],order=[[x,1,y,2,z,3]]);
+ [x+y+z-6,x^3-6*x^2+11*x-6,-x^2+(-y+6)*x-y^2+6*y-11]
+ @end example
+ \BJP
+ $B$$$:$l$NNc$K$*$$$F$b(B, $B9`=g=x$O(B option $B$H$7$F;XDj$5$l$F$$$k(B.
+ $B:G=i$NNc$G$O(B @code{v} $B$K$h$jJQ?t=g=x$r(B, @code{sugarweight} $B$K$h$j(B
+ sugar weight vector $B$r(B, @code{order}$B$K$h$j9`=g=x7?$r;XDj$7$F$$$k(B.
+ $BFs$DL\$NNc$K$*$1$k(B @code{order} $B$N;XDj$O(B matrix order $B$HF1MM$G$"$k(B.
+ $B$9$J$o$A(B, $B;XDj$5$l$?(B weight vector $B$r:8$+$i=g$K;H$C$F(B weight $B$NHf3S(B
+ $B$r9T$&(B. $B;0$DL\$NNc$bF1MM$G$"$k$,(B, $B$3$3$G$O(B weight vector $B$NMWAG$r(B
+ $BJQ?tKh$K;XDj$7$F$$$k(B. $B;XDj$,$J$$$b$N$O(B 0 $B$H$J$k(B. $B;0$DL\$NNc$G$O(B,
+ @code{order} $B$K$h$k;XDj$G$O9`=g=x$,7hDj$7$J$$(B. $B$3$N>l9g$K$O(B,
+ tie breaker $B$H$7$FA4<!?t5U<-=q<0=g=x$,<+F0E*$K@_Dj$5$l$k(B.
+ $B$3$N;XDjJ}K!$O(B, @code{dp_gr_main}, @code{dp_gr_mod_main} $B$J$I(B
+ $B$NAH$_9~$_4X?t$G$N$_2DG=$G$"$j(B, @code{gr} $B$J$I$N%f!<%6Dj5A4X?t(B
+ $B$G$OL$BP1~$G$"$k(B.
+ \E
+ \BEG
+ In each example, a term ordering is specified as options.
+ In the first example, a variable order, a sugar weight vector
+ and a type of term ordering are specified by options @code{v},
+ @code{sugarweight} and @code{order} respectively.
+ In the second example, an option @code{order} is used
+ to set a matrix ordering. That is, the specified weight vectors
+ are used from left to right for comparing terms.
+ The third example shows a variant of specifying a weight vector,
+ where each component of a weight vector is specified variable by variable,
+ and unspecified components are set to zero. In this example,
+ a term order is not determined only by the specified weight vector.
+ In such a case a tie breaker by the graded reverse lexicographic ordering
+ is set automatically.
+ This type of a term ordering specification can be applied only to builtin
+ functions such as @code{dp_gr_main()}, @code{dp_gr_mod_main()}, not to
+ user defined functions such as @code{gr()}.
+ \E
+ \BJP
  @node $BM-M}<0$r78?t$H$9$k%0%l%V%J4pDl7W;;(B,,, $B%0%l%V%J4pDl$N7W;;(B
  @section $BM-M}<0$r78?t$H$9$k%0%l%V%J4pDl7W;;(B
+ \E
+ \BEG
+ @node Groebner basis computation with rational function coefficients,,, Groebner basis computation
+ @section Groebner basis computation with rational function coefficients
+ \E
  @noindent
+ \BJP
  @code{gr()} $B$J$I$N%H%C%W%l%Y%kH!?t$O(B, $B$$$:$l$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.
+ \E
+ \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()}.
+ \E
  @example
  [64] gr([a*x+b*y-c,d*x+e*y-f],[x,y],2);
-Line 490  z^5-t^4]
+Line 1258  z^5-t^4]
 Line 490  z^5-t^4]
 Line 1258  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 1270  z^5-t^4]
 Line 501  z^5-t^4]
 Line 1270  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.
+ \E
+ \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.
+ \E
+ \BJP
  @node $B4pDlJQ49(B,,, $B%0%l%V%J4pDl$N7W;;(B
  @section $B4pDlJQ49(B
+ \E
+ \BEG
+ @node Change of ordering,,, Groebner basis computation
+ @section Change of orderng
+ \E
  @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$$>l9g(B
-Line 516  z^5-t^4]
+Line 1310  z^5-t^4]
 Line 516  z^5-t^4]
 Line 1310  z^5-t^4]
  $B0J2<$N(B 2 $B$D$NH!?t$O(B, $BJQ?t=g=x(B @var{vlist1}, $B9`=g=x7?(B @var{order} $B$G(B
  $B4{$K%0%l%V%J4pDl$H$J$C$F$$$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.
+ \E
+ \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}.
+ \E
  @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.
+ \E
+ \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.
+ \E
  @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, $B
+Line 1358  trace-lifting$B$K$*$1$k%0%l%V%J4pDl8uJd$NF,9`(B, $B
 Line 536  trace-lifting$B$K$*$1$k%0%l%V%J4pDl8uJd$NF,9`(B, $B
 Line 1358  trace-lifting$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.
+ \E
+ \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.
+ \E
  @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.
+ \E
+ \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.
+ \E
+ \BJP
+ @node Weyl $BBe?t(B,,, $B%0%l%V%J4pDl$N7W;;(B
+ @section Weyl $BBe?t(B
+ \E
+ \BEG
+ @node Weyl algebra,,, Groebner basis computation
+ @section Weyl algebra
+ \E
+ @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
+ \E
+ \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:
+ \E
+ @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$$$&4X?t(B
+ $B$K$h$j<B9T$9$k(B.
+ \E
+ \BEG
+ @code{D} is the ring of differential operators whose coefficients
+ are polynomials in @code{K[x1,@dots{},xn]} and
+ @code{Di} denotes the differentiation with respect to  @code{xi}.
+ According to the commutation relation,
+ elements of @code{D} can be represented as a @code{K}-linear combination
+ of monomials @code{x1^i1*@dots{}*xn^in*D1^j1*@dots{}*Dn^jn}.
+ In Risa/Asir, this type of monomial is represented
+ by @code{<<i1,@dots{},in,j1,@dots{},jn>>} as in the case of commutative
+ polynomial.
+ That is, elements of @code{D} are represented by distributed polynomials.
+ Addition and subtraction can be done by @code{+}, @code{-},
+ but multiplication is done by calling @code{dp_weyl_mul()} because of
+ the non-commutativity of @code{D}.
+ \E
+ @example
+ [0] A=<<1,2,2,1>>;
+ (1)*<<1,2,2,1>>
+ [1] B=<<2,1,1,2>>;
+ (1)*<<2,1,1,2>>
+ [2] A*B;
+ (1)*<<3,3,3,3>>
+ [3] dp_weyl_mul(A,B);
+ (1)*<<3,3,3,3>>+(1)*<<3,2,3,2>>+(4)*<<2,3,2,3>>+(4)*<<2,2,2,2>>
+ +(2)*<<1,3,1,3>>+(2)*<<1,2,1,2>>
+ @end example
+ \BJP
+ $B%0%l%V%J4pDl7W;;$K$D$$$F$b(B, Weyl $BBe?t@lMQ$N4X?t$H$7$F(B,
+ $B<!$N4X?t$,MQ0U$7$F$"$k(B.
+ \E
+ \BEG
+ The following functions are avilable for Groebner basis computation
+ in Weyl algebra:
+ \E
+ @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$$$k(B.
+ \E
+ \BEG
+ Computation of the global b function is implemented as an application.
+ \E
+ \BJP
+ @node $BB?9`<04D>e$N2C72(B,,, $B%0%l%V%J4pDl$N7W;;(B
+ @section $BB?9`<04D>e$N2C72(B
+ \E
+ \BEG
+ @node Module over a polynomial ring,,, Groebner basis computation
+ @section Module over a polynomial ring
+ \E
+ @noindent
+ \BJP
+ $BB?9`<04D>e$N<+M32C72$N85$O(B, $B2C72C19`<0(B te_i $B$N@~7?OB$H$7$FFbItI=8=$5$l$k(B.
+ $B$3$3$G(B t $B$OB?9`<04D$NC19`<0(B, e_i $B$O<+M32C72$NI8=`4pDl$G$"$k(B. $B2C72C19`<0$O(B, $BB?9`<04D$NC19`<0(B
+ $B$K0LCV(B i $B$rDI2C$7$?(B @code{<<a,b,...,c:i>>} $B$GI=$9(B. $B2C72B?9`<0(B, $B$9$J$o$A2C72C19`<0$N@~7?OB$O(B,
+ $B@_Dj$5$l$F$$$k2C729`=g=x$K$7$?$,$C$F9_=g$K@0Ns$5$l$k(B. $B2C729`=g=x$K$O0J2<$N(B3$B<oN`$,$"$k(B.
+ @table @code
+ @item TOP $B=g=x(B
+ $B$3$l$O(B, te_i > se_j $B$H$J$k$N$O(B t>s $B$^$?$O(B (t=s $B$+$D(B i<j) $B$H$J$k$h$&$J9`=g=x$G$"$k(B. $B$3$3$G(B,
+ t, s $B$NHf3S$OB?9`<04D$K@_Dj$5$l$F$$$k=g=x$G9T$&(B.
+ $B$3$N7?$N=g=x$O(B, @code{dp_ord([0,Ord])} $B$K(B
+ $B$h$j@_Dj$9$k(B. $B$3$3$G(B, @code{Ord} $B$OB?9`<04D$N=g=x7?$G$"$k(B.
+ @item POT $B=g=x(B
+ $B$3$l$O(B, te_i > se_j $B$H$J$k$N$O(B i<j $B$^$?$O(B (i=j $B$+$D(B t>s) $B$H$J$k$h$&$J9`=g=x$G$"$k(B. $B$3$3$G(B,
+ t, s $B$NHf3S$OB?9`<04D$K@_Dj$5$l$F$$$k=g=x$G9T$&(B.
+ $B$3$N7?$N=g=x$O(B, @code{dp_ord([1,Ord])} $B$K(B
+ $B$h$j@_Dj$9$k(B. $B$3$3$G(B, @code{Ord} $B$OB?9`<04D$N=g=x7?$G$"$k(B.
+ @item Schreyer $B7?=g=x(B
+ $B3FI8=`4pDl(B e_i $B$KBP$7(B, $BJL$N<+M32C72$N2C72C19`<0(B T_i $B$,M?$($i$l$F$$$F(B, te_i > se_j $B$H$J$k$N$O(B
+ tT_i > sT_j $B$^$?$O(B (tT_i=sT_j $B$+$D(B i<j) $B$H$J$k$h$&$J9`=g=x$G$"$k(B. $B$3$3$G(B tT_i, sT_j $B$N(B
+ $BHf3S$O(B, $B$3$l$i$,=jB0$9$k<+M32C72$K@_Dj$5$l$F$$$k=g=x$G9T$&(B.
+ $B$3$N7?$N=g=x$O(B, $BDL>o:F5"E*$K@_Dj$5$l$k(B. $B$9$J$o$A(B, T_i $B$,=jB0$9$k<+M32C72$N=g=x$b(B Schreyer $B7?(B
+ $B$G$"$k$+(B, $B$^$?$O%\%H%`$H$J$k(B TOP, POT $B$J$I$N9`=g=x$H$J$k(B.
+ $B$3$N7?$N=g=x$O(B @code{dpm_set_schreyer([H_1,H_2,...])} $B$K$h$j;XDj$9$k(B. $B$3$3$G(B,
+ @code{H_i=[T_1,T_2,...]} $B$O2C72C19`<0$N%j%9%H$G(B, @code{[H_2,...]} $B$GDj5A$5$l$k(B Schreyer $B7?9`=g=x$r(B
+ @code{tT_i} $B$i$KE,MQ$9$k$H$$$&0UL#$G$"$k(B.
+ @end table
+ $B2C72B?9`<0$rF~NO$9$kJ}K!$H$7$F$O(B, @code{<<a,b,...:i>>} $B$J$k7A<0$GD>@\F~NO$9$kB>$K(B,
+ $BB?9`<0%j%9%H$r:n$j(B, @code{dpm_ltod()} $B$K$h$jJQ49$9$kJ}K!$b$"$k(B.
+ \E
+ \BEG
+ not yet
+ \E
+ \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
+ \E
+ \BEG
+ @node Functions for Groebner basis computation,,, Groebner basis computation
+ @section Functions for Groebner basis computation
+ \E
  @menu
  * gr hgr gr_mod::
-Line 553  trace-lifting$B$K$*$1$k%0%l%V%J4pDl8uJd$NF,9`(B, $B
+Line 1553  trace-lifting$B$K$*$1$k%0%l%V%J4pDl8uJd$NF,9`(B, $B
 Line 553  trace-lifting$B$K$*$1$k%0%l%V%J4pDl8uJd$NF,9`(B, $B
 Line 1553  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::
+ * 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::
  * 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::
+ * dpm_hm dpm_ht dpm_hc dpm_hp dpm_rest::
  * dp_td dp_sugar::
  * dp_lcm::
  * dp_redble::
-Line 578  trace-lifting$B$K$*$1$k%0%l%V%J4pDl8uJd$NF,9`(B, $B
+Line 1582  trace-lifting$B$K$*$1$k%0%l%V%J4pDl8uJd$NF,9`(B, $B
 Line 578  trace-lifting$B$K$*$1$k%0%l%V%J4pDl8uJd$NF,9`(B, $B
 Line 1582  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 bfct generic_bfct ann ann0::
  @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 1600  trace-lifting$B$K$*$1$k%0%l%V%J4pDl8uJd$NF,9`(B, $B
 Line 592  trace-lifting$B$K$*$1$k%0%l%V%J4pDl8uJd$NF,9`(B, $B
 Line 1600  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
-^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
+ 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$$$:$l$b(B, $BB?9`<0%j%9%H(B @var{plist} $B$N(B, $BJQ?t=g=x(B @var{vlist}, $B9`=g=x7?(B
  @var{order} $B$K4X$9$k%0%l%V%J4pDl$r5a$a$k(B. @code{gr()}, @code{hgr()}
  $B$O(B $BM-M}?t78?t(B, @code{gr_mod()} $B$O(B GF(@var{p}) $B78?t$H$7$F7W;;$9$k(B.
-Line 621  trace-lifting$B$K$*$1$k%0%l%V%J4pDl8uJd$NF,9`(B, $B
+Line 1637  trace-lifting$B$K$*$1$k%0%l%V%J4pDl8uJd$NF,9`(B, $B
 Line 621  trace-lifting$B$K$*$1$k%0%l%V%J4pDl8uJd$NF,9`(B, $B
 Line 1637  trace-lifting$B$K$*$1$k%0%l%V%J4pDl8uJd$NF,9`(B, $B
  strategy $B$K$h$k7W;;(B, @code{hgr()} $B$O(B trace-lifting $B$*$h$S(B
  $B@F<!2=$K$h$k(B $B6:@5$5$l$?(B sugar strategy $B$K$h$k7W;;$r9T$&(B.
  @item
- @code{dgr()} $B$O(B, @code{gr()}, @code{dgr()} $B$r(B
+ @code{dgr()} $B$O(B, @code{gr()}, @code{hgr()} $B$r(B
  $B;R%W%m%;%9%j%9%H(B @var{procs} $B$N(B 2 $B$D$N%W%m%;%9$K$h$jF1;~$K7W;;$5$;(B,
  $B@h$K7k2L$rJV$7$?J}$N7k2L$rJV$9(B. $B7k2L$OF10l$G$"$k$,(B, $B$I$A$i$NJ}K!$,(B
  $B9bB.$+0lHL$K$OITL@$N$?$a(B, $B<B:]$N7P2a;~4V$rC;=L$9$k$N$KM-8z$G$"$k(B.
  @item
  @code{dgr()} $B$GI=<($5$l$k;~4V$O(B, $B$3$NH!?t$,<B9T$5$l$F$$$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.
+ @item
+ $BB?9`<0%j%9%H(B @var{plist} $B$NMWAG$,J,;6I=8=B?9`<0$N>l9g$O(B
+ $B7k2L$bJ,;6I=8=B?9`<0$N%j%9%H$G$"$k(B.
+ $B$3$N>l9g(B, $B0z?t$NJ,;6B?9`<0$OM?$($i$l$?=g=x$K=>$$(B @code{dp_sort} $B$G(B
+ $B%=!<%H$5$l$F$+$i7W;;$5$l$k(B.
+ $BB?9`<0%j%9%H$NMWAG$,J,;6I=8=B?9`<0$N>l9g$b(B
+ $BJQ?t$N?tJ,$NITDj85$N%j%9%H$r(B @var{vlist} $B0z?t$H$7$FM?$($J$$$H$$$1$J$$(B
+ ($B%@%_!<(B).
+ \E
+ \BEG
+ @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}.
+ @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.
+ @item
+ When the elements of @var{plist} are distributed polynomials,
+ the result is also a list of distributed polynomials.
+ In this case, firstly  the elements of @var{plist} is sorted by @code{dp_sort}
+ and the Grobner basis computation is started.
+ Variables must be given in @var{vlist} even in this case
+ (these variables are dummy).
+ \E
  @end itemize
  @example
-Line 642  CPU $B;~4V$G$"$j(B, $B$3$NH!?t$N>l9g$O$[$H$s$IDL?.$
+Line 1705  CPU $B;~4V$G$"$j(B, $B$3$NH!?t$N>l9g$O$[$H$s$IDL?.$
 Line 642  CPU $B;~4V$G$"$j(B, $B$3$NH!?t$N>l9g$O$[$H$s$IDL?.$
 Line 1705  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 1723  CPU $B;~4V$G$"$j(B, $B$3$NH!?t$N>l9g$O$[$H$s$IDL?.$
 Line 659  CPU $B;~4V$G$"$j(B, $B$3$NH!?t$N>l9g$O$[$H$s$IDL?.$
 Line 1723  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$$$k(B.
  @item
-Line 695  CPU $B;~4V$G$"$j(B, $B$3$NH!?t$N>l9g$O$[$H$s$IDL?.$
+Line 1766  CPU $B;~4V$G$"$j(B, $B$3$NH!?t$N>l9g$O$[$H$s$IDL?.$
 Line 695  CPU $B;~4V$G$"$j(B, $B$3$NH!?t$N>l9g$O$[$H$s$IDL?.$
 Line 1766  CPU $B;~4V$G$"$j(B, $B$3$NH!?t$N>l9g$O$[$H$s$IDL?.$
  @item
  @code{lex_hensel()}, @code{lex_tl()} $B$K$*$$$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 1820  CPU $B;~4V$G$"$j(B, $B$3$NH!?t$N>l9g$O$[$H$s$IDL?.$
 Line 750  CPU $B;~4V$G$"$j(B, $B$3$NH!?t$N>l9g$O$[$H$s$IDL?.$
 Line 1820  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.
+ \E
+ \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.
+ \E
  @end itemize
  @example
-Line 771  CPU $B;~4V$G$"$j(B, $B$3$NH!?t$N>l9g$O$[$H$s$IDL?.$
+Line 1922  CPU $B;~4V$G$"$j(B, $B$3$NH!?t$N>l9g$O$[$H$s$IDL?.$
 Line 771  CPU $B;~4V$G$"$j(B, $B$3$NH!?t$N>l9g$O$[$H$s$IDL?.$
 Line 1922  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 1938  CPU $B;~4V$G$"$j(B, $B$3$NH!?t$N>l9g$O$[$H$s$IDL?.$
 Line 784  CPU $B;~4V$G$"$j(B, $B$3$NH!?t$N>l9g$O$[$H$s$IDL?.$
 Line 1938  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
- @item tolex_gsl(@var{plist},@var{vlist1},@var{order},@var{vlist2},@var{homo})
+ \EG ::Computation of an GSL form ideal basis
- @itemx tolex_gsl_d(@var{plist},@var{vlist1},@var{order},@var{vlist2},@var{homo},@var{procs})
+ @item tolex_gsl(@var{plist},@var{vlist1},@var{order},@var{vlist2})
- :: $B%0%l%V%J4pDl$rF~NO$H$9$k(B, GSL $B7A<0$N%$%G%"%k4pDl$N7W;;(B
+ @itemx tolex_gsl_d(@var{plist},@var{vlist1},@var{order},@var{vlist2},@var{procs})
+ \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 1974  CPU $B;~4V$G$"$j(B, $B$3$NH!?t$N>l9g$O$[$H$s$IDL?.$
 Line 813  CPU $B;~4V$G$"$j(B, $B$3$NH!?t$N>l9g$O$[$H$s$IDL?.$
 Line 1974  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 1984  GSL $B7A<0$K$h$jI=$5$l$k4pDl$O%0%l%V%J4pDl$G$O$J$$$,
 Line 823  GSL $B7A<0$K$h$jI=$5$l$k4pDl$O%0%l%V%J4pDl$G$O$J$$$,
 Line 1984  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$$(B.
+ \E
+ \BEG
+ @item
+ @code{lex_hensel_gsl()} and @code{lex_hensel()} are variants of
+ @code{tolex_gsl()} and @code{tolex()} respectively. The results are
+ Groebner basis or a kind of ideal basis, called GSL form.
+ @code{tolex_gsl_d()} does basis computations in parallel on child
+ processes specified in @code{procs}.
+ @item
+ If the input is zero-dimensional and a lex order Groebner basis has
+ the form @code{[f0,x1-f1,...,xn-fn]} (@code{f0},...,@code{fn} are
+ univariate polynomials of @code{x0}; SL form), then this these
+ functions return a list such as
+ @code{[[x1,g1,d1],...,[xn,gn,dn],[x0,f0,f0']]} (GSL form).  In this list
+ @code{gi} is a univariate polynomial of @code{x0} such that
+ @code{di*f0'*fi-gi} divides @code{f0} and the roots of the input ideal is
+ @code{[x1=g1/(d1*f0'),...,xn=gn/(dn*f0')]} for @code{x0}
+ such that @code{f0(x0)=0}.
+ If the lex order Groebner basis does not have the above form,
+ these functions return
+ a lex order Groebner basis computed by @code{tolex()}.
+ @item
+ Though an ideal basis represented as GSL form is not a Groebner basis
+ we can expect that the coefficients are much smaller than those in a Groebner
+ basis and that the computation is efficient.
+ The CPU time shown after an execution of @code{tolex_gsl_d()} indicates
+ that of the master process, and it does not include the time in child
+ processes.
+ \E
  @end itemize
  @example
-Line 835  GSL $B7A<0$K$h$jI=$5$l$k4pDl$O%0%l%V%J4pDl$G$O$J$$$,
+Line 2026  GSL $B7A<0$K$h$jI=$5$l$k4pDl$O%0%l%V%J4pDl$G$O$J$$$,
 Line 835  GSL $B7A<0$K$h$jI=$5$l$k4pDl$O%0%l%V%J4pDl$G$O$J$$$,
 Line 2026  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 2094  K[@var{v}] $B$N85(B f(@var{v}) $B$K(B f(@var{p}) m
 Line 890  K[@var{v}] $B$N85(B f(@var{v}) $B$K(B f(@var{p}) m
 Line 2094  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.
+ \E
+ \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()}.
+ \E
  @end itemize
  @example
-Line 902  K[@var{v}] $B$N85(B f(@var{v}) $B$K(B f(@var{p}) m
+Line 2130  K[@var{v}] $B$N85(B f(@var{v}) $B$K(B f(@var{p}) m
 Line 902  K[@var{v}] $B$N85(B f(@var{v}) $B$K(B f(@var{p}) m
 Line 2130  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 2175  K[@var{v}] $B$N85(B f(@var{v}) $B$K(B f(@var{p}) m
 Line 938  K[@var{v}] $B$N85(B f(@var{v}) $B$K(B f(@var{p}) m
 Line 2175  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.
+ \E
+ \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.
+ \E
  @end itemize
  @example
-Line 948  z^32+11405*z^31+20868*z^30+21602*z^29+...
+Line 2196  z^32+11405*z^31+20868*z^30+21602*z^29+...
 Line 948  z^32+11405*z^31+20868*z^30+21602*z^29+...
 Line 2196  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 2279  z^32+11405*z^31+20868*z^30+21602*z^29+...
 Line 1009  z^32+11405*z^31+20868*z^30+21602*z^29+...
 Line 2279  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.
+ \E
+ \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}.
+ \E
  @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 2373  F4 $B%"%k%4%j%:%`$O(B, J.C. Faugere $B$K$h$jDs>'$5$
 Line 1047  F4 $B%"%k%4%j%:%`$O(B, J.C. Faugere $B$K$h$jDs>'$5$
 Line 2373  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.
+ \E
+ \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.
+ \E
  @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 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_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_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
+ @table @t
+ @item nd_gr(@var{plist},@var{vlist},@var{p},@var{order}[|@var{option=value,...}])
+ @itemx nd_gr_trace(@var{plist},@var{vlist},@var{homo},@var{p},@var{order}[|@var{option=value,...}])
+ @itemx nd_f4(@var{plist},@var{vlist},@var{modular},@var{order}[|@var{option=value,...}])
+ @itemx nd_f4_trace(@var{plist},@var{vlist},@var{homo},@var{p},@var{order}[|@var{option=value,...}])
+ @itemx nd_weyl_gr(@var{plist},@var{vlist},@var{p},@var{order}[|@var{option=value,...}])
+ @itemx nd_weyl_gr_trace(@var{plist},@var{vlist},@var{homo},@var{p},@var{order}[|@var{option=value,...}])
+ \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
+ \JP $B%j%9%H(B
+ \EG list
+ @item plist  vlist
+ \JP $B%j%9%H(B
+ \EG list
+ @item order
+ \JP $B?t(B, $B%j%9%H$^$?$O9TNs(B
+ \EG number, list or matrix
+ @item homo
+ \JP $B%U%i%0(B
+ \EG flag
+ @item modular
+ \JP $B%U%i%0$^$?$OAG?t(B
+ \EG flag or prime
+ @end table
+ \BJP
+ @itemize @bullet
+ @item
+ $B$3$l$i$NH!?t$O(B, $B%0%l%V%J4pDl7W;;AH$_9~$_4X?t$N?7<BAu$G$"$k(B.
+ @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$*$h$S(B @code{nd_f4_trace}
+ $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.
+ @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. @var{p} $B$,Ii$N>l9g(B, $B%0%l%V%J4pDl%A%'%C%/$O(B
+ $B9T$o$J$$(B. $B$3$N>l9g(B, @var{p} $B$,(B -1 $B$J$i$P<+F0E*$KA*$P$l$?AG?t$,(B,
+ $B$=$l0J30$O;XDj$5$l$?AG?t$rMQ$$$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 @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),
+$B$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
+ @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.
+ @item
+ $B0J2<$N%*%W%7%g%s$,;XDj$G$-$k(B.
+ @table @code
+ @item homo
+$B$N$H$-(B, $B@F<!2=$r7PM3$7$F7W;;$9$k(B. (@code{nd_gr}, @code{nd_f4} $B$N$_(B)
+ @item dp
+$B$N$H$-(B, $BJ,;6I=8=B?9`<0(B ($B2C72$N>l9g$K$O2C72B?9`<0(B) $B$r7k2L$H$7$FJV$9(B.
+ @item nora
+$B$N$H$-(B, $B7k2L$NAj8_4JLs$r9T$o$J$$(B.
+ @end table
+ @end itemize
+ \E
+ \BEG
+ @itemize @bullet
+ @item
+ These functions are new implementations for computing Groebner bases.
+ @item @code{nd_gr} executes Buchberger algorithm over the rationals
+ if  @code{p} is 0, and that over GF(p) if @code{p} is a prime.
+ @item @code{nd_gr_trace} executes the trace algorithm over the rationals.
+ If @code{p} is 0 or 1, the trace algorithm is executed until it succeeds
+ by using automatically chosen primes.
+ If @code{p} a positive prime,
+ the trace is comuted over GF(p).
+ If the trace algorithm fails 0 is returned.
+ If @code{p} is negative,
+ the Groebner basis check and ideal-membership check are omitted.
+ 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 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
+ 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.
+ @item
+ The fallowing options can be specified.
+ @table @code
+ @item homo
+ If set to 1, the computation is done via homogenization. (only for @code{nd_gr} and @code{nd_f4})
+ @item dp
+ If set to 1, the functions return a list of distributed polynomials (a list of
+ module polynomials when the input is a sub-module).
+ @item nora
+ If set to 1, the inter-reduction is not performed.
+ @end table
+ @end itemize
+ \E
+ @example
+ [38] load("cyclic")$
+ [49] C=cyclic(7)$
+ [50] V=vars(C)$
+ [51] cputime(1)$
+ [52] dp_gr_mod_main(C,V,0,31991,0)$
+.06sec + gc : 0.313sec(26.4sec)
+ [53] nd_gr(C,V,31991,0)$
+ ndv_alloc=1477188
+.737sec + gc : 0.1837sec(5.921sec)
+ [54] dp_f4_mod_main(C,V,31991,0)$
+.51sec + gc : 0.7109sec(4.221sec)
+ [55] nd_f4(C,V,31991,0)$
+.906sec + gc : 0.126sec(2.032sec)
+ @end example
+ @table @t
+ \JP @item $B;2>H(B
+ \EG @item References
+ @fref{dp_ord},
+ @fref{dp_gr_flags dp_gr_print},
+ \JP @fref{$B7W;;$*$h$SI=<($N@)8f(B}.
+ \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$$$k(B.
+ \E
+ \BEG
+ @item
+ Perform the inter-reduction for a Groebner basis (candidate).
+ @item
+ @code{nd_weyl_gr_postproc} is for Weyl algebra.
+ @item
+ If @var{check=1} then the check whether @var{plist} is a Groebner basis with respect to a term order in a polynomial ring
+ or Weyl algebra specified by @var{vlist}, @var{p} and @var{order}.
+ @item
+ This function is used for inter-reduction of a non-reduced Groebner basis that is obtained by dehomogenizing a Groebner basis
+ computed via homogenization, or Groebner basis check of a Groebner basis candidate computed by CRT.
+ \E
+ @end itemize
+ @example
+ afo
+ @end example
+ \JP @node dp_gr_flags dp_gr_print,,, $B%0%l%V%J4pDl$K4X$9$kH!?t(B
+ \EG @node dp_gr_flags dp_gr_print,,, Functions for Groebner basis computation
  @subsection @code{dp_gr_flags}, @code{dp_gr_print}
  @findex dp_gr_flags
  @findex dp_gr_print
  @table @t
  @item dp_gr_flags([@var{list}])
- @itemx dp_gr_print([@var{0|1}])
+ @itemx dp_gr_print([@var{i}])
- :: $B7W;;$*$h$SI=<(MQ%Q%i%a%?$N@_Dj(B, $B;2>H(B
+ \JP :: $B7W;;$*$h$SI=<(MQ%Q%i%a%?$N@_Dj(B, $B;2>H(B
+ \BEG :: Set and show various parameters for cotrolling computations
+ and showing informations.
+ \E
  @end table
  @table @var
  @item return
- $B@_DjCM(B
+ \JP $B@_DjCM(B
+ \EG value currently set
  @item list
- $B%j%9%H(B
+ \JP $B%j%9%H(B
+ \EG list
+ @item i
+ \JP $B@0?t(B
+ \EG integer
  @end table
  @itemize @bullet
+ \BJP
  @item
- @code{dp_gr_main()}, @code{dp_gr_mod_main()} $B<B9T;~$K$*$1$k$5$^$6$^(B
+ @code{dp_gr_main()}, @code{dp_gr_mod_main()}, @code{dp_gr_f_main()}  $B<B9T;~$K$*$1$k$5$^$6$^(B
  $B$J%Q%i%a%?$r@_Dj(B, $B;2>H$9$k(B.
  @item
  $B0z?t$,$J$$>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 2669  F4 $B%"%k%4%j%:%`$O(B, J.C. Faugere $B$K$h$jDs>'$5$
 Line 1087  F4 $B%"%k%4%j%:%`$O(B, J.C. Faugere $B$K$h$jDs>'$5$
 Line 2669  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$$$k(B.
+ \E
+ \BEG
+ @item
+ @code{dp_gr_flags()} sets and shows various parameters for Groebner basis
+  computation.
+ @item
+ If no argument is specified the current settings are returned.
+ @item
+ Arguments must be specified as a list such as
+  @code{["Print",1,"NoSugar",1,...]}. Names of parameters must be character
+ strings.
+ @item
+ @code{dp_gr_print()} is used to set and show the value of a parameter
+ @code{Print} and @code{PrintShort}.
+ @table @var
+ @item i=0
+ @code{Print=0}, @code{PrintShort=0}
+ @item i=1
+ @code{Print=1}, @code{PrintShort=0}
+ @item i=2
+ @code{Print=0}, @code{PrintShort=1}
+ @end table
+ This functions is prepared to get quickly the value
+ when a user defined function calling @code{dp_gr_main()} etc.
+ uses the value as a flag for showing intermediate informations.
+ \E
  @end itemize
  @table @t
- @item $B;2>H(B
+ \JP @item $B;2>H(B
- @fref{$B7W;;$*$h$SI=<($N@)8f(B}
+ \EG @item References
+ \JP @fref{$B7W;;$*$h$SI=<($N@)8f(B}
+ \EG @fref{Controlling Groebner basis computations}
  @end table
- @node dp_ord,,, $B%0%l%V%J4pDl$K4X$9$kH!?t(B
+ \JP @node dp_ord,,, $B%0%l%V%J4pDl$K4X$9$kH!?t(B
+ \EG @node dp_ord,,, Functions for Groebner basis computation
  @subsection @code{dp_ord}
  @findex dp_ord
  @table @t
  @item dp_ord([@var{order}])
- :: $BJQ?t=g=x7?$N@_Dj(B, $B;2>H(B
+ \JP :: $BJQ?t=g=x7?$N@_Dj(B, $B;2>H(B
+ \EG :: Set and show the ordering type.
  @end table
  @table @var
  @item return
- $BJQ?t=g=x7?(B ($B?t(B, $B%j%9%H$^$?$O9TNs(B)
+ \JP $BJQ?t=g=x7?(B ($B?t(B, $B%j%9%H$^$?$O9TNs(B)
+ \EG ordering type (number, list or matrix)
  @item order
- $B?t(B, $B%j%9%H$^$?$O9TNs(B
+ \JP $B?t(B, $B%j%9%H$^$?$O9TNs(B
+ \EG number, list or matrix
  @end table
  @itemize @bullet
+ \BJP
  @item
  $B0z?t$,$"$k;~(B, $BJQ?t=g=x7?$r(B @var{order} $B$K@_Dj$9$k(B. $B0z?t$,$J$$;~(B,
  $B8=:_@_Dj$5$l$F$$$kJQ?t=g=x7?$rJV$9(B.
-Line 1137  F4 $B%"%k%4%j%:%`$O(B, J.C. Faugere $B$K$h$jDs>'$5$
+Line 2761  F4 $B%"%k%4%j%:%`$O(B, J.C. Faugere $B$K$h$jDs>'$5$
 Line 1137  F4 $B%"%k%4%j%:%`$O(B, J.C. Faugere $B$K$h$jDs>'$5$
 Line 2761  F4 $B%"%k%4%j%:%`$O(B, J.C. Faugere $B$K$h$jDs>'$5$
  @item
  $B%H%C%W%l%Y%kH!?t0J30$NH!?t$rD>@\8F$S=P$9>l9g$K$O(B, $B$3$NH!?t$K$h$j(B
  $BJQ?t=g=x7?$r@5$7$/@_Dj$7$J$1$l$P$J$i$J$$(B.
+ @item
+ $B0z?t$,%j%9%H$N>l9g(B, $B<+M32C72$K$*$1$k9`=g=x7?$r@_Dj$9$k(B. $B0z?t$,(B@code{[0,Ord]} $B$N>l9g(B,
+ $BB?9`<04D>e$G(B @code{Ord} $B$G;XDj$5$l$k9`=g=x$K4p$E$/(B TOP $B=g=x(B, $B0z?t$,(B @code{[1,Ord]} $B$N>l9g(B
+ OPT $B=g=x$r@_Dj$9$k(B.
+ \E
+ \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.
+ @item
+ If the argument is a list, then an ordering type in a free module is set.
+ If the argument is @code{[0,Ord]} then a TOP ordering based on the ordering type specified
+ by @code{Ord} is set.
+ If the argument is @code{[1,Ord]} then a POT ordering is set.
+ \E
  @end itemize
  @example
-Line 1149  F4 $B%"%k%4%j%:%`$O(B, J.C. Faugere $B$K$h$jDs>'$5$
+Line 2813  F4 $B%"%k%4%j%:%`$O(B, J.C. Faugere $B$K$h$jDs>'$5$
 Line 1149  F4 $B%"%k%4%j%:%`$O(B, J.C. Faugere $B$K$h$jDs>'$5$
 Line 2813  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_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.
+ \E
+ \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.
+ \E
+ @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}
  @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.
+ \E
+ \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.
+ \E
  @end itemize
  @example
-Line 1185  F4 $B%"%k%4%j%:%`$O(B, J.C. Faugere $B$K$h$jDs>'$5$
+Line 2940  F4 $B%"%k%4%j%:%`$O(B, J.C. Faugere $B$K$h$jDs>'$5$
 Line 1185  F4 $B%"%k%4%j%:%`$O(B, J.C. Faugere $B$K$h$jDs>'$5$
 Line 2940  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 dpm_dptodpm,,, $B%0%l%V%J4pDl$K4X$9$kH!?t(B
+ \EG @node dpm_dptodpm,,, Functions for Groebner basis computation
+ @subsection @code{dpm_dptodpm}
+ @findex dpm_dptodpm
+ @table @t
+ @item dpm_dptodpm(@var{dpoly},@var{pos})
+ \JP :: $BJ,;6I=8=B?9`<0$r2C72B?9`<0$KJQ49$9$k(B.
+ \EG :: Converts a distributed polynomial into a module polynomial.
+ @end table
+ @table @var
+ @item return
+ \JP $B2C72B?9`<0(B
+ \EG module polynomial
+ @item dpoly
+ \JP $BJ,;6I=8=B?9`<0(B
+ \EG distributed polynomial
+ @item pos
+ \JP $B@5@0?t(B
+ \EG positive integer
+ @end table
+ @itemize @bullet
+ \BJP
+ @item
+ $BJ,;6I=8=B?9`<0$r2C72B?9`<0$KJQ49$9$k(B.
+ @item
+ $B=PNO$O2C72B?9`<0(B @code{dpoly e_pos} $B$G$"$k(B.
+ \E
+ \BEG
+ @item
+ This function converts a distributed polynomial into a module polynomial.
+ @item
+ The output is @code{dpoly e_pos}.
+ \E
+ @end itemize
+ @example
+ [50] dp_ord([0,0])$
+ [51] D=dp_ptod((x+y+z)^2,[x,y,z]);
+ (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_dptodpm(D,2);
+ (1)*<<2,0,0:2>>+(2)*<<1,1,0:2>>+(1)*<<0,2,0:2>>+(2)*<<1,0,1:2>>
+ +(2)*<<0,1,1:2>>+(1)*<<0,0,2:2>>
+ @end example
+ @table @t
+ \JP @item $B;2>H(B
+ \EG @item References
+ @fref{dp_ptod},
+ @fref{dp_ord}.
+ @end table
+ \JP @node dpm_ltod,,, $B%0%l%V%J4pDl$K4X$9$kH!?t(B
+ \EG @node dpm_ltod,,, Functions for Groebner basis computation
+ @subsection @code{dpm_ltod}
+ @findex dpm_ltod
+ @table @t
+ @item dpm_dptodpm(@var{plist},@var{vlist})
+ \JP :: $BB?9`<0%j%9%H$r2C72B?9`<0$KJQ49$9$k(B.
+ \EG :: Converts a list of polynomials into a module polynomial.
+ @end table
+ @table @var
+ @item return
+ \JP $B2C72B?9`<0(B
+ \EG module polynomial
+ @item 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
+ $BB?9`<0%j%9%H$r2C72B?9`<0$KJQ49$9$k(B.
+ @item
+ @code{[p1,...,pm]} $B$O(B @code{p1 e1+...+pm em} $B$KJQ49$5$l$k(B.
+ \E
+ \BEG
+ @item
+ This function converts a list of polynomials into a module polynomial.
+ @item
+ @code{[p1,...,pm]} is converted into @code{p1 e1+...+pm em}.
+ \E
+ @end itemize
+ @example
+ [2126] dp_ord([0,0])$
+ [2127] dpm_ltod([x^2+y^2,x,y-z],[x,y,z]);
+ (1)*<<2,0,0:1>>+(1)*<<0,2,0:1>>+(1)*<<1,0,0:2>>+(1)*<<0,1,0:3>>
+ +(-1)*<<0,0,1:3>>
+ @end example
+ @table @t
+ \JP @item $B;2>H(B
+ \EG @item References
+ @fref{dpm_dtol},
+ @fref{dp_ord}.
+ @end table
+ \JP @node dpm_dtol,,, $B%0%l%V%J4pDl$K4X$9$kH!?t(B
+ \EG @node dpm_dtol,,, Functions for Groebner basis computation
+ @subsection @code{dpm_dtol}
+ @findex dpm_dtol
+ @table @t
+ @item dpm_dptodpm(@var{poly},@var{vlist})
+ \JP :: $B2C72B?9`<0$rB?9`<0%j%9%H$KJQ49$9$k(B.
+ \EG :: Converts a module polynomial into a list of polynomials.
+ @end table
+ @table @var
+ @item return
+ \JP $BB?9`<0%j%9%H(B
+ \EG list of polynomials
+ @item poly
+ \JP $B2C72B?9`<0(B
+ \EG module polynomial
+ @item vlist
+ \JP $BJQ?t%j%9%H(B
+ \EG list of variables
+ @end table
+ @itemize @bullet
+ \BJP
+ @item
+ $B2C72B?9`<0$rB?9`<0%j%9%H$KJQ49$9$k(B.
+ @item
+ @code{p1 e1+...+pm em} $B$O(B @code{[p1,...,pm]} $B$KJQ49$5$l$k(B.
+ @item
+ $B=PNO%j%9%H$ND9$5$O(B, @code{poly} $B$K4^$^$l$kI8=`4pDl$N:GBg%$%s%G%C%/%9$H$J$k(B.
+ \E
+ \BEG
+ @item
+ This function converts a module polynomial into a list of polynomials.
+ @item
+ @code{p1 e1+...+pm em} is converted into @code{[p1,...,pm]}.
+ @item
+ The length of the output list is equal to the largest index among those of the standard bases
+ containd in @code{poly}.
+ \E
+ @end itemize
+ @example
+ [2126] dp_ord([0,0])$
+ [2127] D=(1)*<<2,0,0:1>>+(1)*<<0,2,0:1>>+(1)*<<1,0,0:2>>+(1)*<<0,1,0:3>>
+ +(-1)*<<0,0,1:3>>$
+ [2128] dpm_dtol(D,[x,y,z]);
+ [x^2+y^2,x,y-z]
+ @end example
+ @table @t
+ \JP @item $B;2>H(B
+ \EG @item References
+ @fref{dpm_ltod},
+ @fref{dp_ord}.
+ @end table
+ \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.
+ \E
+ \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}.
+ \E
  @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.
+ \E
  @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 3212  z^2+(2*a+2*b)*z+a^2+2*b*a+b^2
 Line 1263  z^2+(2*a+2*b)*z+a^2+2*b*a+b^2
 Line 3212  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.
+ \E
+ \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.
+ \E
  @end itemize
  @example
  @end example
  @table @t
- @item $B;2>H(B
+ \JP @item $B;2>H(B
- @fref{dp_nf dp_nf_mod dp_true_nf dp_true_nf_mod},
+ \EG @item References
+ @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
- @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
 $B?-$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 3282  z^2+(2*a+2*b)*z+a^2+2*b*a+b^2
 Line 1307  z^2+(2*a+2*b)*z+a^2+2*b*a+b^2
 Line 3282  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.
+ \E
+ \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.
+ \E
  @end itemize
  @example
-Line 1319  z^2+(2*a+2*b)*z+a^2+2*b*a+b^2
+Line 3311  z^2+(2*a+2*b)*z+a^2+2*b*a+b^2
 Line 1319  z^2+(2*a+2*b)*z+a^2+2*b*a+b^2
 Line 3311  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.
- @itemx dp_prim(@var{dpoly})
+ \BEG
- :: $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.
+ :: Converts a distributed polynomial @var{poly} with rational coefficients
+ into an integral distributed polynomial such that GCD of all its coefficients
+ is 1.
+ \E
+ @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
+ coefficients into an integral distributed polynomial such that polynomial
+ GCD of all its coefficients is 1.
+ \E
  @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 3357  z^2+(2*a+2*b)*z+a^2+2*b*a+b^2
 Line 1350  z^2+(2*a+2*b)*z+a^2+2*b*a+b^2
 Line 3357  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.
+ \E
+ \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.
+ \E
  @end itemize
  @example
-Line 1362  z^2+(2*a+2*b)*z+a^2+2*b*a+b^2
+Line 3378  z^2+(2*a+2*b)*z+a^2+2*b*a+b^2
 Line 1362  z^2+(2*a+2*b)*z+a^2+2*b*a+b^2
 Line 3378  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 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 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})
- :: $BJ,;6I=8=B?9`<0$N@55,7A$r5a$a$k(B. ($B7k2L$ODj?tG\$5$l$F$$$k2DG=@-$"$j(B)
+ @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$$$k2DG=@-$"$j(B)
+ \BEG
+ :: Computes the normal form of a distributed polynomial.
+ (The result may be multiplied by a constant in the ground field.)
+ \E
  @item dp_true_nf(@var{indexlist},@var{dpoly},@var{dpolyarray},@var{fullreduce})
  @item dp_true_nf_mod(@var{indexlist},@var{dpoly},@var{dpolyarray},@var{fullreduce},@var{mod})
- :: $BJ,;6I=8=B?9`<0$N@55,7A$r5a$a$k(B. ($B??$N7k2L$r(B @code{[$BJ,;R(B, $BJ,Jl(B]} $B$N7A$GJV$9(B)
+ \JP :: $BJ,;6I=8=B?9`<0$N@55,7A$r5a$a$k(B. ($B??$N7k2L$r(B @code{[$BJ,;R(B, $BJ,Jl(B]} $B$N7A$GJV$9(B)
+ \BEG
+ :: Computes the normal form of a distributed polynomial. (The true result
+ is returned in such a list as @code{[numerator, denominator]})
+ \E
  @end table
  @table @var
  @item return
- @code{dp_nf()} : $BJ,;6I=8=B?9`<0(B, @code{dp_true_nf()} : $B%j%9%H(B
+ \JP @code{dp_nf()} : $BJ,;6I=8=B?9`<0(B, @code{dp_true_nf()} : $B%j%9%H(B
+ \EG @code{dp_nf()} : distributed polynomial, @code{dp_true_nf()} : list
  @item indexlist
- $B%j%9%H(B
+ \JP $B%j%9%H(B
+ \EG list
  @item dpoly
- $BJ,;6I=8=B?9`<0(B
+ \JP $BJ,;6I=8=B?9`<0(B
+ \EG distributed polynomial
  @item dpolyarray
- $BG[Ns(B
+ \JP $BG[Ns(B
+ \EG array of distributed polynomial
  @item fullreduce
- $B%U%i%0(B
+ \JP $B%U%i%0(B
+ \EG flag
  @item mod
- $BAG?t(B
+ \JP $BAG?t(B
+ \EG prime
  @end table
  @itemize @bullet
+ \BJP
  @item
  $BJ,;6I=8=B?9`<0(B @var{dpoly} $B$N@55,7A$r5a$a$k(B.
  @item
+ $BL>A0$K(B weyl $B$r4^$`4X?t$O%o%$%kBe?t$K$*$1$k@55,7A7W;;$r9T$&(B. $B0J2<$N@bL@$O(B weyl $B$r4^$`$b$N$KBP$7$F$bF1MM$K@.N)$9$k(B.
+ @item
  @code{dp_nf_mod()}, @code{dp_true_nf_mod()} $B$NF~NO$O(B, @code{dp_mod()} $B$J$I(B
  $B$K$h$j(B, $BM-8BBN>e$NJ,;6I=8=B?9`<0$K$J$C$F$$$J$1$l$P$J$i$J$$(B.
  @item
-Line 1429  z^2+(2*a+2*b)*z+a^2+2*b*a+b^2
+Line 3468  z^2+(2*a+2*b)*z+a^2+2*b*a+b^2
 Line 1429  z^2+(2*a+2*b)*z+a^2+2*b*a+b^2
 Line 3468  z^2+(2*a+2*b)*z+a^2+2*b*a+b^2
  $BJ,;6I=8=$G$J$$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.
+ \E
+ \BEG
+ @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
+ 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.
+ \E
  @end itemize
  @example
-Line 1443  z^2+(2*a+2*b)*z+a^2+2*b*a+b^2
+Line 3521  z^2+(2*a+2*b)*z+a^2+2*b*a+b^2
 Line 1443  z^2+(2*a+2*b)*z+a^2+2*b*a+b^2
 Line 3521  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
+*u4^15+...
  [79] dp_dtop(dp_nf([4,3,2,1,0],T,DP2,1),V);
- -1138087976845165778088612297273078520347097001020471455633353049221045677593
+ -11380879768451657780886122972730785203470970010204714556333530492210
- 0005716505560062087150928400876150217079820311439477560587583488*u4^15+...
+ 456775930005716505560062087150928400876150217079820311439477560587583
+*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 dpm_nf dpm_nf_and_quotient,,, $B%0%l%V%J4pDl$K4X$9$kH!?t(B
+ \EG @node dpm_nf dpm_nf_and_quotient,,, Functions for Groebner basis computation
+ @subsection @code{dpm_nf}, @code{dpm_nf_and_quotient}
+ @findex dpm_nf
+ @findex dpm_nf_and_quotient
+ @table @t
+ @item dpm_nf([@var{indexlist},]@var{dpoly},@var{dpolyarray},@var{fullreduce})
+ \JP :: $B2C72B?9`<0$N@55,7A$r5a$a$k(B. ($B7k2L$ODj?tG\$5$l$F$$$k2DG=@-$"$j(B)
+ \BEG
+ :: Computes the normal form of a module polynomial.
+ (The result may be multiplied by a constant in the ground field.)
+ \E
+ @item dpm_nf_and_quotient([@var{indexlist},]@var{dpoly},@var{dpolyarray})
+ \JP :: $B2C72B?9`<0$N@55,7A$H>&$r5a$a$k(B.
+ \BEG
+ :: Computes the normal form of a module polynomial and the quotient.
+ \E
+ @end table
+ @table @var
+ @item return
+ \JP @code{dpm_nf()} : $B2C72B?9`<0(B, @code{dpm_nf_and_quotient()} : $B%j%9%H(B
+ \EG @code{dpm_nf()} : module polynomial, @code{dpm_nf_and_quotient()} : list
+ @item indexlist
+ \JP $B%j%9%H(B
+ \EG list
+ @item dpoly
+ \JP $B2C72B?9`<0(B
+ \EG module polynomial
+ @item dpolyarray
+ \JP $BG[Ns(B
+ \EG array of module polynomial
+ @end table
+ @itemize @bullet
+ \BJP
+ @item
+ $B2C72B?9`<0(B @var{dpoly} $B$N@55,7A$r5a$a$k(B.
+ @item
+ $B7k2L$KM-M}?t(B, $BM-M}<0$,4^$^$l$k$N$rHr$1$k$?$a(B, @code{dpm_nf()} $B$O(B
+ $B??$NCM$NDj?tG\$NCM$rJV$9(B.
+ @item
+ @var{dpolyarray} $B$O2C72B?9`<0$rMWAG$H$9$k%Y%/%H%k(B,
+ @var{indexlist} $B$O@55,2=7W;;$KMQ$$$k(B @var{dpolyarray} $B$NMWAG$N%$%s%G%C%/%9(B
+ @item
+ @var{indexlist} $B$,M?$($i$l$F$$$k>l9g(B, @var{dpolyarray} $B$NCf$G(B, @var{indexlist} $B$G;XDj$5$l$?$b$N$N$_$,(B, $BA0$NJ}$+$iM%@hE*$K;H$o$l$k(B.
+ @var{indexlist} $B$,M?$($i$l$F$$$J$$>l9g$K$O(B, @var{dpolyarray} $B$NCf$NA4$F$NB?9`<0$,A0$NJ}$+$iM%@hE*$K;H$o$l$k(B.
+ @item
+ @code{dpm_nf_and_quotient()} $B$O(B,
+ @code{[@var{nm},@var{dn},@var{quo}]} $B$J$k7A$N%j%9%H$rJV$9(B.
+ $B$?$@$7(B, @var{nm} $B$O78?t$KJ,?t$r4^$^$J$$2C72B?9`<0(B, @var{dn} $B$O(B
+ $B?t$^$?$OB?9`<0$G(B @var{nm}/@var{dn} $B$,??$NCM$H$J$k(B.
+ @var{quo} $B$O=|;;$N>&$rI=$9G[Ns$G(B, @var{dn}@var{dpoly}=@var{nm}+@var{quo[0]dpolyarray[0]+...} $B$,@.$jN)$D(B.
+ $B$N%j%9%H(B.
+ @item
+ @var{fullreduce} $B$,(B 0 $B$G$J$$$H$-A4$F$N9`$KBP$7$F4JLs$r9T$&(B. @var{fullreduce}
+ $B$,(B 0 $B$N$H$-F,9`$N$_$KBP$7$F4JLs$r9T$&(B.
+ \E
+ \BEG
+ @item
+ Computes the normal form of a module polynomial.
+ @item
+ The result of @code{dpm_nf()} may be multiplied by a constant in the
+ ground field in order to make the result integral.
+ @item
+ @var{dpolyarray} is a vector whose components are module polynomials
+ and @var{indexlist} is a list of indices which is used for the normal form
+ computation.
+ @item
+ If @var{indexlist} is given, only the polynomials in @var{dpolyarray} specified in @var{indexlist}
+ is used in the division. An index placed at the preceding position has priority to be selected.
+ If @var{indexlist} is not given, all the polynomials in @var{dpolyarray} are used.
+ @item
+ @code{dpm_nf_and_quotient()} returns
+ such a list as @code{[@var{nm},@var{dn},@var{quo}]}.
+ Here @var{nm} is a module 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.
+ @var{quo} is an array containing the quotients of the division satisfying
+ @var{dn}@var{dpoly}=@var{nm}+@var{quo[0]dpolyarray[0]+...}.
+ @item
+ When argument @var{fullreduce} has non-zero value,
+ all terms are reduced. When it has value 0,
+ only the head term is reduced.
+ \E
+ @end itemize
+ @example
+ [2126] dp_ord([1,0])$
+ [2127] S=ltov([(1)*<<0,0,2,0:1>>+(1)*<<0,0,1,1:1>>+(1)*<<0,0,0,2:1>>
+ +(-1)*<<3,0,0,0:2>>+(-1)*<<0,0,2,1:2>>+(-1)*<<0,0,1,2:2>>
+ +(1)*<<3,0,1,0:3>>+(1)*<<3,0,0,1:3>>+(1)*<<0,0,2,2:3>>,
+ (-1)*<<0,1,0,0:1>>+(-1)*<<0,0,1,0:1>>+(-1)*<<0,0,0,1:1>>
+ +(-1)*<<3,0,0,0:3>>+(1)*<<0,1,1,1:3>>,(1)*<<0,1,0,0:2>>
+ +(1)*<<0,0,1,0:2>>+(1)*<<0,0,0,1:2>>+(-1)*<<0,1,1,0:3>>
+ +(-1)*<<0,1,0,1:3>>+(-1)*<<0,0,1,1:3>>])$
+ [2128] U=dpm_sp(S[0],S[1]);
+ (1)*<<0,0,3,0:1>>+(-1)*<<0,1,1,1:1>>+(1)*<<0,0,2,1:1>>
+ +(-1)*<<0,1,0,2:1>>+(1)*<<3,1,0,0:2>>+(1)*<<0,1,2,1:2>>
+ +(1)*<<0,1,1,2:2>>+(-1)*<<3,1,1,0:3>>+(1)*<<3,0,2,0:3>>
+ +(-1)*<<3,1,0,1:3>>+(-1)*<<0,1,3,1:3>>+(-1)*<<0,1,2,2:3>>
+ [2129] dpm_nf(U,S,1);
+ [2130] L=dpm_nf_and_quotient(U,S)$
+ [2131] Q=L[2]$
+ [2132] D=L[1]$
+ [2133] D*U-(Q[1]*S[1]+Q[2]*S[2]);
+ @end example
+ @table @t
+ \JP @item $B;2>H(B
+ \EG @item References
+ @fref{dpm_sp},
+ @fref{dp_ord}.
+ @end table
+ \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 3676  u4^2+(6*u3+2*u2+6*u1-2)*u4+9*u3^2+(6*u2+18*u1-6)*u3+u2
 Line 1473  u4^2+(6*u3+2*u2+6*u1-2)*u4+9*u3^2+(6*u2+18*u1-6)*u3+u2
 Line 3676  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
+ \E
+ \BEG
+ @item return
+ @code{dp_hm()}, @code{dp_ht()}, @code{dp_rest()} : distributed polynomial
+ @code{dp_hc()} : number or polynomial
+ @item dpoly
+ distributed polynomial
+ \E
  @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.
+ \E
+ \BEG
+ @item
+ These are used to get various parts of a distributed polynomial.
+ @item
+ The next equations hold for a distributed polynomial @var{p}.
+ \E
  @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 3740  u4^2+(6*u3+2*u2+6*u1-2)*u4+9*u3^2+(6*u2+18*u1-6)*u3+u2
 Line 1516  u4^2+(6*u3+2*u2+6*u1-2)*u4+9*u3^2+(6*u2+18*u1-6)*u3+u2
 Line 3740  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 dpm_hm dpm_ht dpm_hc dpm_hp dpm_rest,,, $B%0%l%V%J4pDl$K4X$9$kH!?t(B
+ \EG @node dpm_hm dpm_ht dpm_hc dpm_hp dpm_rest,,, Functions for Groebner basis computation
+ @subsection @code{dpm_hm}, @code{dpm_ht}, @code{dpm_hc}, @code{dpm_hp}, @code{dpm_rest}
+ @findex dpm_hm
+ @findex dpm_ht
+ @findex dpm_hc
+ @findex dpm_hp
+ @findex dpm_rest
+ @table @t
+ @item dpm_hm(@var{dpoly})
+ \JP :: $B2C72B?9`<0$NF,C19`<0$r<h$j=P$9(B.
+ \EG :: Gets the head monomial of a module polynomial.
+ @item dpm_ht(@var{dpoly})
+ \JP :: $B2C72B?9`<0$NF,9`$r<h$j=P$9(B.
+ \EG :: Gets the head term of a module polynomial.
+ @item dpm_hc(@var{dpoly})
+ \JP :: $B2C72B?9`<0$NF,78?t$r<h$j=P$9(B.
+ \EG :: Gets the head coefficient of a module polynomial.
+ @item dpm_hp(@var{dpoly})
+ \JP :: $B2C72B?9`<0$NF,0LCV$r<h$j=P$9(B.
+ \EG :: Gets the head position of a module polynomial.
+ @item dpm_rest(@var{dpoly})
+ \JP :: $B2C72B?9`<0$NF,C19`<0$r<h$j=|$$$?;D$j$rJV$9(B.
+ \EG :: Gets the remainder of a module polynomial where the head monomial is removed.
+ @end table
+ @table @var
+ \BJP
+ @item return
+ @code{dp_hm()}, @code{dp_ht()}, @code{dp_rest()} : $B2C72B?9`<0(B,
+ @code{dp_hc()} : $B?t$^$?$OB?9`<0(B
+ @item dpoly
+ $B2C72B?9`<0(B
+ \E
+ \BEG
+ @item return
+ @code{dpm_hm()}, @code{dpm_ht()}, @code{dpm_rest()} : module polynomial
+ @code{dpm_hc()} : monomial
+ @item dpoly
+ distributed polynomial
+ \E
+ @end table
+ @itemize @bullet
+ \BJP
+ @item
+ $B$3$l$i$O(B, $B2C72B?9`<0$N3FItJ,$r<h$j=P$9$?$a$NH!?t$G$"$k(B.
+ @item
+ @code{dpm_hc()} $B$O(B, @code{dpm_hm()} $B$N(B, $BI8=`4pDl$K4X$9$k78?t$G$"$kC19`<0$rJV$9(B.
+ $B%9%+%i!<78?t$r<h$j=P$9$K$O(B, $B$5$i$K(B @code{dp_hc()} $B$r<B9T$9$k(B.
+ @item
+ @code{dpm_hp()} $B$O(B, $BF,2C72C19`<0$K4^$^$l$kI8=`4pDl$N%$%s%G%C%/%9$rJV$9(B.
+ \E
+ \BEG
+ @item
+ These are used to get various parts of a module polynomial.
+ @item
+ @code{dpm_hc()} returns the monomial that is the coefficient of @code{dpm_hm()} with respect to the
+ standard base.
+ For getting its scalar coefficient apply @code{dp_hc()}.
+ @item
+ @code{dpm_hp()} returns the index of the standard base conteind in the head module monomial.
+ \E
+ @end itemize
+ @example
+ [2126] dp_ord([1,0]);
+ [1,0]
+ [2127] F=2*<<1,2,0:2>>-3*<<1,0,2:3>>+<<2,1,0:2>>;
+ (1)*<<2,1,0:2>>+(2)*<<1,2,0:2>>+(-3)*<<1,0,2:3>>
+ [2128] M=dpm_hm(F);
+ (1)*<<2,1,0:2>>
+ [2129] C=dpm_hc(F);
+ (1)*<<2,1,0>>
+ [2130] R=dpm_rest(F);
+ (2)*<<1,2,0:2>>+(-3)*<<1,0,2:3>>
+ [2131] dpm_hp(F);
+ @end example
+ \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 3859  u4^2+(6*u3+2*u2+6*u1-2)*u4+9*u3^2+(6*u2+18*u1-6)*u3+u2
 Line 1546  u4^2+(6*u3+2*u2+6*u1-2)*u4+9*u3^2+(6*u2+18*u1-6)*u3+u2
 Line 3859  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.
+ \E
+ \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.
+ \E
  @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 3885  u4^2+(6*u3+2*u2+6*u1-2)*u4+9*u3^2+(6*u2+18*u1-6)*u3+u2
 Line 1558  u4^2+(6*u3+2*u2+6*u1-2)*u4+9*u3^2+(6*u2+18*u1-6)*u3+u2
 Line 3885  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.
+ \E
+ \BEG
+ @item
+ Returns the least common multiple of the head terms of the given
+ two polynomials, where coefficient is always set to 1.
+ \E
  @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 3923  u4^2+(6*u3+2*u2+6*u1-2)*u4+9*u3^2+(6*u2+18*u1-6)*u3+u2
 Line 1585  u4^2+(6*u3+2*u2+6*u1-2)*u4+9*u3^2+(6*u2+18*u1-6)*u3+u2
 Line 3923  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
 $B$rJV$9(B.
  @item
  $BB?9`<0$N4JLs$r9T$&:](B, $B$I$N9`$r4JLs$G$-$k$+$rC5$9$N$KMQ$$$k(B.
+ \E
+ \BEG
+ @item
+ Returns 1 if the head term of @var{dpoly2} divides the head term of
+ @var{dpoly1}; otherwise 0.
+ @item
+ Used for finding candidate terms at reduction of polynomials.
+ \E
  @end itemize
  @example
-Line 1626  u4^2+(6*u3+2*u2+6*u1-2)*u4+9*u3^2+(6*u2+18*u1-6)*u3+u2
+Line 3978  u4^2+(6*u3+2*u2+6*u1-2)*u4+9*u3^2+(6*u2+18*u1-6)*u3+u2
 Line 1626  u4^2+(6*u3+2*u2+6*u1-2)*u4+9*u3^2+(6*u2+18*u1-6)*u3+u2
 Line 3978  u4^2+(6*u3+2*u2+6*u1-2)*u4+9*u3^2+(6*u2+18*u1-6)*u3+u2
  @end example
  @table @t
- @item $B;2>H(B
+ \JP @item $B;2>H(B
+ \EG @item References
  @fref{dp_red dp_red_mod}.
  @end table
- @node dp_subd,,, $B%0%l%V%J4pDl$K4X$9$kH!?t(B
+ \JP @node dpm_redble,,, $B%0%l%V%J4pDl$K4X$9$kH!?t(B
+ \EG @node dpm_redble,,, Functions for Groebner basis computation
+ @subsection @code{dpm_redble}
+ @findex dpm_redble
+ @table @t
+ @item dpm_redble(@var{dpoly1},@var{dpoly2})
+ \JP :: $BF,9`$I$&$7$,@0=|2DG=$+$I$&$+D4$Y$k(B.
+ \EG :: Checks whether one head term is divisible by the other head term.
+ @end table
+ @table @var
+ @item return
+ \JP $B@0?t(B
+ \EG integer
+ @item dpoly1  dpoly2
+ \JP $B2C72B?9`<0(B
+ \EG module polynomial
+ @end table
+ @itemize @bullet
+ \BJP
+ @item
+ @var{dpoly1} $B$NF,9`$,(B @var{dpoly2} $B$NF,9`$G3d$j@Z$l$l$P(B 1, $B3d$j@Z$l$J$1$l$P(B
+$B$rJV$9(B.
+ @item
+ $BB?9`<0$N4JLs$r9T$&:](B, $B$I$N9`$r4JLs$G$-$k$+$rC5$9$N$KMQ$$$k(B.
+ \E
+ \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.
+ \E
+ @end itemize
+ \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.
+ \E
+ \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.
+ \E
  @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 4063  u4^2+(6*u3+2*u2+6*u1-2)*u4+9*u3^2+(6*u2+18*u1-6)*u3+u2
 Line 1660  u4^2+(6*u3+2*u2+6*u1-2)*u4+9*u3^2+(6*u2+18*u1-6)*u3+u2
 Line 4063  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.
+ \E
+ \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}.
+ \E
  @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 4122  u4^2+(6*u3+2*u2+6*u1-2)*u4+9*u3^2+(6*u2+18*u1-6)*u3+u2
 Line 1703  u4^2+(6*u3+2*u2+6*u1-2)*u4+9*u3^2+(6*u2+18*u1-6)*u3+u2
 Line 4122  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$$$F(B,
-Line 1727  u4^2+(6*u3+2*u2+6*u1-2)*u4+9*u3^2+(6*u2+18*u1-6)*u3+u2
+Line 4151  u4^2+(6*u3+2*u2+6*u1-2)*u4+9*u3^2+(6*u2+18*u1-6)*u3+u2
 Line 1727  u4^2+(6*u3+2*u2+6*u1-2)*u4+9*u3^2+(6*u2+18*u1-6)*u3+u2
 Line 4151  u4^2+(6*u3+2*u2+6*u1-2)*u4+9*u3^2+(6*u2+18*u1-6)*u3+u2
  K $B>eM-8B<!85@~7A6u4V$G$"$k(B K[X]/I $B$N(B monomial $B$K$h$k4pDl$r5a$a$k(B.
  @item
  $BF@$i$l$?4pDl$N8D?t$,(B, K[X]/I $B$N(B K-$B@~7A6u4V$H$7$F$N<!85$KEy$7$$(B.
+ \E
+ \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.
+ \E
  @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 4177  u1*u2,u1^2,u4*u0,u3*u0,u2*u0,u1*u0,u0^2,u4,u3,u2,u1,u0
 Line 1741  u1*u2,u1^2,u4*u0,u3*u0,u2*u0,u1*u0,u0^2,u4,u3,u2,u1,u0
 Line 4177  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 4214  u1*u2,u1^2,u4*u0,u3*u0,u2*u0,u1*u0,u0^2,u4,u3,u2,u1,u0
 Line 1772  u1*u2,u1^2,u4*u0,u3*u0,u2*u0,u1*u0,u0^2,u4,u3,u2,u1,u0
 Line 4214  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.
+ \E
+ \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.
+ \E
  @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 4237  u1*u2,u1^2,u4*u0,u3*u0,u2*u0,u1*u0,u0^2,u4,u3,u2,u1,u0
 Line 1781  u1*u2,u1^2,u4*u0,u3*u0,u2*u0,u1*u0,u0^2,u4,u3,u2,u1,u0
 Line 4237  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 4251  u1*u2,u1^2,u4*u0,u3*u0,u2*u0,u1*u0,u0^2,u4,u3,u2,u1,u0
 Line 1793  u1*u2,u1^2,u4*u0,u3*u0,u2*u0,u1*u0,u0^2,u4,u3,u2,u1,u0
 Line 4251  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
 $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 4283  u1*u2,u1^2,u4*u0,u3*u0,u2*u0,u1*u0,u0^2,u4,u3,u2,u1,u0
 Line 1819  u1*u2,u1^2,u4*u0,u3*u0,u2*u0,u1*u0,u0^2,u4,u3,u2,u1,u0
 Line 4283  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.
+ \E
+ \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}]}.
+ \E
  @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 4316  u1*u2,u1^2,u4*u0,u3*u0,u2*u0,u1*u0,u0^2,u4,u3,u2,u1,u0
 Line 1832  u1*u2,u1^2,u4*u0,u3*u0,u2*u0,u1*u0,u0^2,u4,u3,u2,u1,u0
 Line 4316  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 4335  u1*u2,u1^2,u4*u0,u3*u0,u2*u0,u1*u0,u0^2,u4,u3,u2,u1,u0
 Line 1849  u1*u2,u1^2,u4*u0,u3*u0,u2*u0,u1*u0,u0^2,u4,u3,u2,u1,u0
 Line 4335  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 4360  u1*u2,u1^2,u4*u0,u3*u0,u2*u0,u1*u0,u0^2,u4,u3,u2,u1,u0
 Line 1869  u1*u2,u1^2,u4*u0,u3*u0,u2*u0,u1*u0,u0^2,u4,u3,u2,u1,u0
 Line 4360  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.
+ \E
+ \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.
+ \E
  @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 4383  u1*u2,u1^2,u4*u0,u3*u0,u2*u0,u1*u0,u0^2,u4,u3,u2,u1,u0
 Line 1881  u1*u2,u1^2,u4*u0,u3*u0,u2*u0,u1*u0,u0^2,u4,u3,u2,u1,u0
 Line 4383  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 dpm_sp,,, $B%0%l%V%J4pDl$K4X$9$kH!?t(B
+ \EG @node dmp_sp,,, Functions for Groebner basis computation
+ @subsection @code{dpm_sp}
+ @findex dpm_sp
+ @table @t
+ @item dpm_sp(@var{dpoly1},@var{dpoly2}[|coef=1])
+ \JP :: S-$BB?9`<0$N7W;;(B
+ \EG :: Computation of an S-polynomial
+ @end table
+ @table @var
+ @item return
+ \JP $B2C72B?9`<0$^$?$O%j%9%H(B
+ \EG module polynomial or list
+ @item dpoly1  dpoly2
+ \JP $B2C72B?9`<0(B
+ \EG module polynomial
+ \JP $BJ,;6I=8=B?9`<0(B
+ @end table
+ @itemize @bullet
+ \BJP
+ @item
+ @var{dpoly1}, @var{dpoly2} $B$N(B S-$BB?9`<0$r7W;;$9$k(B.
+ @item
+ $B%*%W%7%g%s(B @var{coef=1} $B$,;XDj$5$l$F$$$k>l9g(B, @code{[S,t1,t2]} $B$J$k%j%9%H$rJV$9(B.
+ $B$3$3$G(B, @code{t1}, @code{t2} $B$O(BS-$BB?9`<0$r:n$k:]$N78?tC19`<0$G(B @code{S=t1 dpoly1-t2 dpoly2}
+ $B$rK~$?$9(B.
+ \E
+ \BEG
+ @item
+ This function computes the S-polynomial of @var{dpoly1} and @var{dpoly2}.
+ @item
+ If an option @var{coef=1} is specified, it returns a list @code{[S,t1,t2]},
+ where @code{S} is the S-polynmial and @code{t1}, @code{t2} are monomials satisfying @code{S=t1 dpoly1-t2 dpoly2}.
+ \E
+ @end itemize
+ \JP @node p_nf p_nf_mod p_true_nf p_true_nf_mod,,, $B%0%l%V%J4pDl$K4X$9$kH!?t(B
+ \EG @node p_nf p_nf_mod p_true_nf p_true_nf_mod,,, Functions for Groebner basis computation
  @subsection @code{p_nf}, @code{p_nf_mod}, @code{p_true_nf}, @code{p_true_nf_mod}
  @findex p_nf
  @findex p_nf_mod
-Line 1894  u1*u2,u1^2,u4*u0,u3*u0,u2*u0,u1*u0,u0^2,u4,u3,u2,u1,u0
+Line 4438  u1*u2,u1^2,u4*u0,u3*u0,u2*u0,u1*u0,u0^2,u4,u3,u2,u1,u0
 Line 1894  u1*u2,u1^2,u4*u0,u3*u0,u2*u0,u1*u0,u0^2,u4,u3,u2,u1,u0
 Line 4438  u1*u2,u1^2,u4*u0,u3*u0,u2*u0,u1*u0,u0^2,u4,u3,u2,u1,u0
  @table @t
  @item p_nf(@var{poly},@var{plist},@var{vlist},@var{order})
  @itemx p_nf_mod(@var{poly},@var{plist},@var{vlist},@var{order},@var{mod})
- :: $BI=8=B?9`<0$N@55,7A$r5a$a$k(B. ($B7k2L$ODj?tG\$5$l$F$$$k2DG=@-$"$j(B)
+ \JP :: $BI=8=B?9`<0$N@55,7A$r5a$a$k(B. ($B7k2L$ODj?tG\$5$l$F$$$k2DG=@-$"$j(B)
+ \BEG
+ :: Computes the normal form of the given polynomial.
+ (The result may be multiplied by a constant.)
+ \E
  @item p_true_nf(@var{poly},@var{plist},@var{vlist},@var{order})
  @itemx p_true_nf_mod(@var{poly},@var{plist},@var{vlist},@var{order},@var{mod})
- :: $BI=8=B?9`<0$N@55,7A$r5a$a$k(B. ($B??$N7k2L$r(B @code{[$BJ,;R(B, $BJ,Jl(B]} $B$N7A$GJV$9(B)
+ \JP :: $BI=8=B?9`<0$N@55,7A$r5a$a$k(B. ($B??$N7k2L$r(B @code{[$BJ,;R(B, $BJ,Jl(B]} $B$N7A$GJV$9(B)
+ \BEG
+ :: Computes the normal form of the given polynomial. (The result is returned
+ as a form of @code{[numerator, denominator]})
+ \E
  @end table
  @table @var
  @item return
- @code{p_nf} : $BB?9`<0(B, @code{p_true_nf} : $B%j%9%H(B
+ \JP @code{p_nf} : $BB?9`<0(B, @code{p_true_nf} : $B%j%9%H(B
+ \EG @code{p_nf} : polynomial, @code{p_true_nf} : list
  @item poly
- $BB?9`<0(B
+ \JP $BB?9`<0(B
- @item plist,vlist
+ \EG polynomial
- $B%j%9%H(B
+ @item plist vlist
+ \JP $B%j%9%H(B
+ \EG list
  @item order
- $B?t(B, $B%j%9%H$^$?$O9TNs(B
+ \JP $B?t(B, $B%j%9%H$^$?$O9TNs(B
+ \EG number, list or matrix
  @item mod
- $BAG?t(B
+ \JP $BAG?t(B
+ \EG prime
  @end table
  @itemize @bullet
+ \BJP
  @item
  @samp{gr} $B$GDj5A$5$l$F$$$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 4492  u1*u2,u1^2,u4*u0,u3*u0,u2*u0,u1*u0,u0^2,u4,u3,u2,u1,u0
 Line 1934  u1*u2,u1^2,u4*u0,u3*u0,u2*u0,u1*u0,u0^2,u4,u3,u2,u1,u0
 Line 4492  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.
+ \E
+ \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()}.
+ \E
  @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 4531  u1*u2,u1^2,u4*u0,u3*u0,u2*u0,u1*u0,u0^2,u4,u3,u2,u1,u0
 Line 1949  u1*u2,u1^2,u4*u0,u3*u0,u2*u0,u1*u0,u0^2,u4,u3,u2,u1,u0
 Line 4531  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}.
+ @fref{dp_nf dp_nf_mod dp_true_nf dp_true_nf_mod dp_weyl_nf dp_weyl_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$$$k(B.
  @item
-Line 1986  u1*u2,u1^2,u4*u0,u3*u0,u2*u0,u1*u0,u0^2,u4,u3,u2,u1,u0
+Line 4576  u1*u2,u1^2,u4*u0,u3*u0,u2*u0,u1*u0,u0^2,u4,u3,u2,u1,u0
 Line 1986  u1*u2,u1^2,u4*u0,u3*u0,u2*u0,u1*u0,u0^2,u4,u3,u2,u1,u0
 Line 4576  u1*u2,u1^2,u4*u0,u3*u0,u2*u0,u1*u0,u0^2,u4,u3,u2,u1,u0
  @item
  $B%0%l%V%J4pDl$O$7$P$7$P78?t$,5pBg$K$J$k$?$a(B, $B<B:]$K$I$N9`$,8=$l$F(B
  $B$$$k$N$+$r8+$k$?$a$J$I$KMQ$$$k(B.
+ \E
+ \BEG
+ @item
+ Defined in the package @samp{gr}.
+ @item
+ This returns a list which contains all non-zero monomials in the given
+ polynomial.  The monomials are ordered according to the current
+ type of term ordering and @var{vlist}.
+ @item
+ Since polynomials in a Groebner base often have very large coefficients,
+ examining a polynomial as it is may sometimes be difficult to perform.
+ For such a case, this function enables to examine which term is really
+ exists.
+ \E
  @end itemize
  @example
  [233] G=gr(katsura(5),[u5,u4,u3,u2,u1,u0],2)$
  [234] p_terms(G[0],[u5,u4,u3,u2,u1,u0],2);
- [u5,u0^31,u0^30,u0^29,u0^28,u0^27,u0^26,u0^25,u0^24,u0^23,u0^22,u0^21,u0^20,
+ [u5,u0^31,u0^30,u0^29,u0^28,u0^27,u0^26,u0^25,u0^24,u0^23,u0^22,
- u0^19,u0^18,u0^17,u0^16,u0^15,u0^14,u0^13,u0^12,u0^11,u0^10,u0^9,u0^8,u0^7,
+ u0^21,u0^20,u0^19,u0^18,u0^17,u0^16,u0^15,u0^14,u0^13,u0^12,u0^11,
- u0^6,u0^5,u0^4,u0^3,u0^2,u0,1]
+ u0^10,u0^9,u0^8,u0^7,u0^6,u0^5,u0^4,u0^3,u0^2,u0,1]
  @end example
- @node gb_comp,,, $B%0%l%V%J4pDl$K4X$9$kH!?t(B
+ \JP @node gb_comp,,, $B%0%l%V%J4pDl$K4X$9$kH!?t(B
+ \EG @node gb_comp,,, Functions for Groebner basis computation
  @subsection @code{gb_comp}
  @findex gb_comp
  @table @t
  @item gb_comp(@var{plist1}, @var{plist2})
- :: $BB?9`<0%j%9%H$,(B, $BId9f$r=|$$$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.
+ \E
+ \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.
+ \E
  @end itemize
  @example
-Line 2029  u0^6,u0^5,u0^4,u0^3,u0^2,u0,1]
+Line 4648  u0^6,u0^5,u0^4,u0^3,u0^2,u0,1]
 Line 2029  u0^6,u0^5,u0^4,u0^3,u0^2,u0,1]
 Line 4648  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 4661  u0^6,u0^5,u0^4,u0^3,u0^2,u0,1]
 Line 2041  u0^6,u0^5,u0^4,u0^3,u0^2,u0,1]
 Line 4661  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 4685  u0^6,u0^5,u0^4,u0^3,u0^2,u0,1]
 Line 2061  u0^6,u0^5,u0^4,u0^3,u0^2,u0,1]
 Line 4685  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.
+ \E
+ \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}.
+ \E
  @end itemize
  @example
-Line 2068  u0^6,u0^5,u0^4,u0^3,u0^2,u0,1]
+Line 4705  u0^6,u0^5,u0^4,u0^3,u0^2,u0,1]
 Line 2068  u0^6,u0^5,u0^4,u0^3,u0^2,u0,1]
 Line 4705  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,
-*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,
+*u3*u0+2*u1*u4-u3+(2*u1+2*u5)*u2,2*u2*u0+2*u2*u4+(2*u1+2*u5)*u3
-*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 4729  u0^2-u0+2*u4^2+2*u3^2+2*u2^2+2*u1^2+2*u5^2]
 Line 2092  u0^2-u0+2*u4^2+2*u3^2+2*u2^2+2*u1^2+2*u5^2]
 Line 4729  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.
+ \E
+ \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.
+ \E
+ @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.
+ \E
+ \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.
+ \E
+ @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 bfct generic_bfct ann ann0,,, $B%0%l%V%J4pDl$K4X$9$kH!?t(B
+ \EG @node bfunction bfct generic_bfct ann ann0,,, Functions for Groebner basis computation
+ @subsection @code{bfunction}, @code{bfct}, @code{generic_bfct}, @code{ann}, @code{ann0}
+ @findex bfunction
+ @findex bfct
+ @findex generic_bfct
+ @findex ann
+ @findex ann0
+ @table @t
+ @item bfunction(@var{f})
+ @itemx bfct(@var{f})
+ @itemx generic_bfct(@var{plist},@var{vlist},@var{dvlist},@var{weight})
+ \JP :: @var{b} $B4X?t$N7W;;(B
+ \EG :: Computes the global @var{b} function of a polynomial or an ideal
+ @item ann(@var{f})
+ @itemx ann0(@var{f})
+ \JP :: $BB?9`<0$N%Y%-$N(B annihilator $B$N7W;;(B
+ \EG :: Computes the annihilator of a power of polynomial
+ @end table
+ @table @var
+ @item return
+ \JP $BB?9`<0$^$?$O%j%9%H(B
+ \EG polynomial or list
+ @item 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$$$k(B.
+ @item @code{bfunction(@var{f})}, @code{bfct(@var{f})} $B$OB?9`<0(B @var{f} $B$N(B global @var{b} $B4X?t(B @code{b(s)} $B$r(B
+ $B7W;;$9$k(B. @code{b(s)} $B$O(B, Weyl $BBe?t(B @code{D} $B>e$N0lJQ?tB?9`<04D(B @code{D[s]}
+ $B$N85(B @code{P(x,s)} $B$,B8:_$7$F(B, @code{P(x,s)f^(s+1)=b(s)f^s} $B$rK~$?$9$h$&$J(B
+ $BB?9`<0(B @code{b(s)} $B$NCf$G(B, $B<!?t$,:G$bDc$$$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 @var{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 @code{bfunction} $B$H(B @code{bfct} $B$G$OMQ$$$F$$$k%"%k%4%j%:%`$,(B
+ $B0[$J$k(B. $B$I$A$i$,9bB.$+$OF~NO$K$h$k(B.
+ @item @code{ann(@var{f})} $B$O(B, @code{@var{f}^s} $B$N(B annihilator ideal
+ $B$N@8@.7O$rJV$9(B. @code{ann(@var{f})} $B$O(B, @code{[@var{a},@var{list}]}
+ $B$J$k%j%9%H$rJV$9(B. $B$3$3$G(B, @var{a} $B$O(B @var{f} $B$N(B @var{b} $B4X?t$N:G>.@0?t:,(B,
+ @var{list} $B$O(B @code{ann(@var{f})} $B$N7k2L$N(B @code{s}$ $B$K(B, @var{a} $B$r(B
+ $BBeF~$7$?$b$N$G$"$k(B.
+ @item $B>\:Y$K$D$$$F$O(B, [Saito,Sturmfels,Takayama] $B$r8+$h(B.
+ \E
+ \BEG
+ @item These functions are defined in @samp{bfct}.
+ @item @code{bfunction(@var{f})} and @code{bfct(@var{f})} compute the global @var{b}-function @code{b(s)} of
+ a polynomial @var{f}.
+ @code{b(s)} is a polynomial of the minimal degree
+ such that there exists @code{P(x,s)} in D[s], which is a polynomial
+ ring over Weyl algebra @code{D}, and @code{P(x,s)f^(s+1)=b(s)f^s} holds.
+ @item @code{generic_bfct(@var{f},@var{vlist},@var{dvlist},@var{weight})}
+ computes the global @var{b}-function of a left ideal @code{I} in @code{D}
+ generated by @var{plist}, with respect to @var{weight}.
+ @var{vlist} is the list of @code{x}-variables,
+ @var{vlist} is the list of corresponding @code{D}-variables.
+ @item @code{bfunction(@var{f})} and @code{bfct(@var{f})} implement
+ different algorithms and the efficiency depends on inputs.
+ @item @code{ann(@var{f})} returns the generator set of the annihilator
+ ideal of @code{@var{f}^s}.
+ @code{ann(@var{f})} returns a list @code{[@var{a},@var{list}]},
+ where @var{a} is the minimal integral root of the global @var{b}-function
+ of @var{f}, and @var{list} is a list of polynomials obtained by
+ substituting @code{s} in @code{ann(@var{f})} with @var{a}.
+ @item See [Saito,Sturmfels,Takayama] for the details.
+ \E
+ @end itemize
+ @example
+ [0] load("bfct")$
+ [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]);
+*s^10-70000*s^9+101750*s^8-79375*s^7+35768*s^6-9277*s^5
+ +1278*s^4-72*s^3
+ [220] P=x^3-y^2$
+ [221] ann(P);
+ [2*dy*x+3*dx*y^2,-3*dx*x-2*dy*y+6*s]
+ [222] ann0(P);
+ [-1,[2*dy*x+3*dx*y^2,-3*dx*x-2*dy*y-6]]
+ @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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