===================================================================
RCS file: /home/cvs/OpenXM/src/asir-doc/parts/groebner.texi,v
retrieving revision 1.1.1.1
retrieving revision 1.4
diff -u -p -r1.1.1.1 -r1.4
--- OpenXM/src/asir-doc/parts/groebner.texi	1999/12/08 05:47:44	1.1.1.1
+++ OpenXM/src/asir-doc/parts/groebner.texi	2003/04/19 15:44:56	1.4
@@ -1,7 +1,15 @@
+@comment $OpenXM: OpenXM/src/asir-doc/parts/groebner.texi,v 1.3 1999/12/24 04:38:04 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::
@@ -10,22 +18,56 @@
 * $BM-M}<0$r78?t$H$9$k%0%l%V%J4pDl7W;;(B::
 * $B4pDlJQ49(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::
+* Groebner basis computation with rational function coefficients::
+* Change of ordering::
+* 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
@@ -35,12 +77,25 @@ $(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
@@ -50,59 +105,128 @@ $(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
-@code{tu} > @code{su}
+\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}
+\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
 2*<<0,1,2,3,4>>
 @end example
 
-$B$H$$$&7A$GI=<($5$l(B, $B$^$?(B, $B$3$N7A$GF~NO2DG=$G$"$k(B. 
+\JP $B$H$$$&7A$GI=<($5$l(B, $B$^$?(B, $B$3$N7A$GF~NO2DG=$G$"$k(B. 
+\EG and also can be input in such a form.
 
+\BJP
 @itemx $BF,C19`<0(B (head monomial)
 @item $BF,9`(B (head term)
 @itemx $BF,78?t(B (head coefficient)
 $BJ,;6I=8=B?9`<0$K$*$1$k3FC19`<0$O(B, $B9`=g=x$K$h$j@0Ns$5$l$k(B. $B$3$N;~=g(B
 $B=x:GBg$NC19`<0$rF,C19`<0(B, $B$=$l$K8=$l$k9`(B, $B78?t$r$=$l$>$lF,9`(B, $BF,78?t(B
 $B$H8F$V(B. 
+\E
+\BEG
+@itemx head monomial
+@item head term
+@itemx head coefficient
+      
+Monomials in a distributed polynomial is sorted by a total order.
+In such representation, we call the monomial that is maximum
+with respect to the order the head monomial, and its term and coefficient
+the head term and the head coefficient respectively.
+\E
 @end table
 
+\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
 $B@_Dj$J$I$r9T$C$?$N$A$3$l$i$r8F$S=P$9%f!<%6H!?t$rMQ$$$k$N$,JXMx$G$"$k(B. 
@@ -110,52 +234,119 @@ $(x+y+z)^2 = 1 \cdot x^2 + 2 \cdot xy + 2 \cdot xz + 1
 $B$_9~$`$3$H$K$h$j;HMQ2DG=$H$J$k(B. @samp{gr} $B$O(B, @b{Asir} $B$NI8=`(B
 $B%i%$%V%i%j%G%#%l%/%H%j$KCV$+$l$F$$$k(B. $B$h$C$F(B, $B4D6-JQ?t(B @code{ASIR_LIBDIR}
 $B$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 provided not by built-in
+functions but by a set of user functions written in @b{Asir}.
+The set of functions is provided as a file (sometimes called package),
+named @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}.  Therefore, it is loaded simply by specifying
+its file name, unless the environment variable @code{ASIR_LIBDIR}
+is set to a non-standard one.
+\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();
@@ -164,73 +355,163 @@ 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
-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$K(B
-$B$h$k(B normal form $B$GCV$-49$($k(B.
+@comment @item Interreduce
+@comment \BJP
+@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 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
+\BJP
 0 $B$G$J$$@0?t$N;~(B, $BM-M}?t>e$N@55,7A7W;;$K$*$$$F(B, $B78?t$N%S%C%HD9$NOB$,(B
 @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
 $B3d$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
 GCD $B7W;;$,9T$o$l0lHL$K$O8zN($,0-$/$J$k$,(B, @code{Multiple} $B$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. 
+\E
+\BEG
+If a non-zero integer, in a normal form computation
+over the rationals, the integer content of the polynomial being
+reduced is removed when its magnitude becomes @code{Multiple} 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{Multiple} 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{Multiple} to 2
+for the case where large integers appear during the computation.
+\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)$
@@ -256,185 +537,453 @@ Max_mag=6
 @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
+
 0 $B$K(B reduce $B$5$l$?%Z%"$N?t(B
 
 @item NZR
+
 0 $B$G$J$$B?9`<0$K(B reduce $B$5$l$?%Z%"$N?t(B
 
 @item Max_mag
+
 $B@8@.$5$l$?B?9`<0$N(B, $B78?t$N%S%C%HD9$NOB$N:GBgCM(B
+\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]$
@@ -466,6 +1015,7 @@ z^5-t^4]
 @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$$$($P(B, $B<-=q<0=g=x$G%0%l%V%J4pDl$r5a$a$k(B
@@ -474,15 +1024,44 @@ z^5-t^4]
 @code{x} $B$GI=$5$l$F$$$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 $BM-M}<0$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);
@@ -490,6 +1069,7 @@ 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
@@ -501,11 +1081,36 @@ 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$$>l9g(B
@@ -516,16 +1121,44 @@ z^5-t^4]
 $B0J2<$N(B 2 $B$D$NH!?t$O(B, $BJQ?t=g=x(B @var{vlist1}, $B9`=g=x7?(B @var{order} $B$G(B
 $B4{$K%0%l%V%J4pDl$H$J$C$F$$$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, 
@@ -536,16 +1169,44 @@ 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 $B%0%l%V%J4pDl$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::
@@ -578,9 +1239,11 @@ 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::
 @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
@@ -592,21 +1255,27 @@ 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
-$B%j%9%H(B
-@item plist, vlist, procs
-$B%j%9%H(B
+\JP $B%j%9%H(B
+\EG list
+@item plist  vlist  procs
+\JP $B%j%9%H(B
+\EG list
 @item order
-$B?t(B, $B%j%9%H$^$?$O9TNs(B
+\JP $B?t(B, $B%j%9%H$^$?$O9TNs(B
+\EG number, list or matrix
 @item p
-2^27 $BL$K~$NAG?t(B
+\JP 2^27 $BL$K~$NAG?t(B
+\EG prime less than 2^27
 @end table
 
 @itemize @bullet
+\BJP
 @item
 $BI8=`%i%$%V%i%j$N(B @samp{gr} $B$GDj5A$5$l$F$$$k(B. 
 @item
@@ -628,6 +1297,35 @@ strategy $B$K$h$k7W;;(B, @code{hgr()} $B$O(B trace
 @item
 @code{dgr()} $B$GI=<($5$l$k;~4V$O(B, $B$3$NH!?t$,<B9T$5$l$F$$$k%W%m%;%9$G$N(B
 CPU $B;~4V$G$"$j(B, $B$3$NH!?t$N>l9g$O$[$H$s$IDL?.$N$?$a$N;~4V$G$"$k(B. 
+\E
+\BEG
+@item 
+These functions are defined in @samp{gr} in the standard library
+directory.
+@item 
+They compute a Groebner basis of a polynomial list @var{plist} with
+respect to the variable order @var{vlist} and the order type @var{order}.
+@code{gr()} and @code{hgr()} compute a Groebner basis over the rationals
+and @code{gr_mod} computes over GF(@var{p}).
+@item 
+Variables not included in @var{vlist} are regarded as
+included in the ground field.
+@item 
+@code{gr()} uses trace-lifting (an improvement by modular computation)
+ and sugar strategy.
+@code{hgr()} uses trace-lifting and a cured sugar strategy
+by using homogenization.
+@item 
+@code{dgr()} executes @code{gr()}, @code{dgr()} simultaneously on
+two process in a child process list @var{procs} and returns
+the result obtained first. The results returned from both the process
+should be equal, but it is not known in advance which method is faster.
+Therefore this function is useful to reduce the actual elapsed time.
+@item 
+The CPU time shown after an exection of @code{dgr()} indicates
+that of the master process, and most of the time corresponds to the time
+for communication.
+\E
 @end itemize
 
 @example
@@ -642,13 +1340,15 @@ 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
+\EG @item References
 @comment @fref{dp_gr_main dp_gr_mod_main},
 @fref{dp_gr_main dp_gr_mod_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
@@ -659,25 +1359,32 @@ 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
-$B%j%9%H(B
-@item plist, vlist1, vlist2, procs
-$B%j%9%H(B
+\JP $B%j%9%H(B
+\EG list
+@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$$$k(B. 
 @item
@@ -695,7 +1402,6 @@ CPU $B;~4V$G$"$j(B, $B$3$NH!?t$N>l9g$O$[$H$s$IDL?.$
 @item
 @code{lex_hensel()}, @code{lex_tl()} $B$K$*$$$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. 
@@ -750,6 +1456,87 @@ 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
@@ -771,12 +1558,15 @@ 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 $B;2>H(B
+\EG @item References
 @fref{dp_gr_main dp_gr_mod_main},
-@fref{dp_ord}, @fref{$BJ,;67W;;(B}
+\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
@@ -784,24 +1574,31 @@ CPU $B;~4V$G$"$j(B, $B$3$NH!?t$N>l9g$O$[$H$s$IDL?.$
 
 @table @t
 @item lex_hensel_gsl(@var{plist},@var{vlist1},@var{order},@var{vlist2},@var{homo})
-:: GSL $B7A<0$N%$%G%"%k4pDl$N7W;;(B
+\JP :: GSL $B7A<0$N%$%G%"%k4pDl$N7W;;(B
+\EG ::Computation of an GSL form ideal basis
 @item tolex_gsl(@var{plist},@var{vlist1},@var{order},@var{vlist2},@var{homo})
 @itemx tolex_gsl_d(@var{plist},@var{vlist1},@var{order},@var{vlist2},@var{homo},@var{procs})
-:: $B%0%l%V%J4pDl$rF~NO$H$9$k(B, GSL $B7A<0$N%$%G%"%k4pDl$N7W;;(B
+\JP :: $B%0%l%V%J4pDl$rF~NO$H$9$k(B, GSL $B7A<0$N%$%G%"%k4pDl$N7W;;(B
+\EG :: Computation of an GSL form ideal basis stating from a Groebner basis
 @end table
 
 @table @var
 @item return
-$B%j%9%H(B
-@item plist, vlist1, vlist2, procs
-$B%j%9%H(B
+\JP $B%j%9%H(B
+\EG list
+@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. 
@@ -813,7 +1610,7 @@ 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$$>l9g(B, @code{tolex()} $B$K(B
@@ -823,6 +1620,36 @@ GSL $B7A<0$K$h$jI=$5$l$k4pDl$O%0%l%V%J4pDl$G$O$J$$$,
 $B$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
@@ -839,39 +1666,51 @@ GSL $B7A<0$K$h$jI=$5$l$k4pDl$O%0%l%V%J4pDl$G$O$J$$$,
 @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
-$BB?9`<0(B
-@item plist, vlist
-$B%j%9%H(B
+\JP $BB?9`<0(B
+\EG polynomial
+@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$$(B, @code{minipoly()} $B$O(B
 $BF~NO$r%0%l%V%J4pDl$H$_$J$9(B. 
@@ -890,6 +1729,30 @@ K[@var{v}] $B$N85(B f(@var{v}) $B$K(B f(@var{p}) m
 @item
 @code{gr_minipoly()} $B$K;XDj$9$k9`=g=x$H$7$F$O(B, $BDL>oA4<!?t5U<-=q<0=g=x$r(B
 $BMQ$$$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
@@ -902,34 +1765,43 @@ 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()} : $BB?9`<0(B
-@item plist, vlist1, vlist2
-$B%j%9%H(B
+\JP @code{tolexm()} : $B%j%9%H(B, @code{minipolym()} : $BB?9`<0(B
+\EG @code{tolexm()} : list, @code{minipolym()} : polynomial
+@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. 
@@ -938,6 +1810,17 @@ 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
@@ -948,12 +1831,14 @@ 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,,, $B%0%l%V%J4pDl$K4X$9$kH!?t(B
+\EG @node dp_gr_main dp_gr_mod_main,,, Functions for Groebner basis computation
 @subsection @code{dp_gr_main}, @code{dp_gr_mod_main}
 @findex dp_gr_main
 @findex dp_gr_mod_main
@@ -961,23 +1846,30 @@ z^32+11405*z^31+20868*z^30+21602*z^29+...
 @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)
+\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
-$B%j%9%H(B
-@item plist, vlist
-$B%j%9%H(B
+\JP $B%j%9%H(B
+\EG list
+@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
@@ -1009,17 +1901,56 @@ 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.
+@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}.
+\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,,, $B%0%l%V%J4pDl$K4X$9$kH!?t(B
+\EG @node dp_f4_main dp_f4_mod_main,,, Functions for Groebner basis computation
 @subsection @code{dp_f4_main}, @code{dp_f4_mod_main}
 @findex dp_f4_main
 @findex dp_f4_mod_main
@@ -1027,19 +1958,24 @@ z^32+11405*z^31+20868*z^30+21602*z^29+...
 @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)
+\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
-$B%j%9%H(B
-@item plist, vlist
-$B%j%9%H(B
+\JP $B%j%9%H(B
+\EG list
+@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
@@ -1049,17 +1985,32 @@ F4 $B%"%k%4%j%:%`$O(B, J.C. Faugere $B$K$h$jDs>'$5$
 @item
 $B0z?t$*$h$SF0:n$O$=$l$>$l(B @code{dp_gr_main()}, @code{dp_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()}.
+\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 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
@@ -1067,17 +2018,23 @@ F4 $B%"%k%4%j%:%`$O(B, J.C. Faugere $B$K$h$jDs>'$5$
 @table @t
 @item dp_gr_flags([@var{list}])
 @itemx dp_gr_print([@var{0|1}])
-:: $B7W;;$*$h$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
 @end table
 
 @itemize @bullet
+\BJP
 @item
 @code{dp_gr_main()}, @code{dp_gr_mod_main()} $B<B9T;~$K$*$1$k$5$^$6$^(B
 $B$J%Q%i%a%?$r@_Dj(B, $B;2>H$9$k(B. 
@@ -1091,30 +2048,54 @@ F4 $B%"%k%4%j%:%`$O(B, J.C. Faugere $B$K$h$jDs>'$5$
 $B$G$-$k(B. $B$3$l$O(B, @code{dp_gr_main()} $B$J$I$r%5%V%k!<%A%s$H$7$FMQ$$$k%f!<%6(B
 $BH!?t$K$*$$$F(B, @code{Print} $B$NCM$r8+$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}. This functions is prepared to get quickly the value of
+@code{Print} 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 $B;2>H(B
-@fref{$B7W;;$*$h$SI=<($N@)8f(B}
+\JP @item $B;2>H(B
+\EG @item References
+\JP @fref{$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. 
@@ -1137,6 +2118,34 @@ 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. 
+\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.
+\E
 @end itemize
 
 @example
@@ -1149,33 +2158,51 @@ F4 $B%"%k%4%j%:%`$O(B, J.C. Faugere $B$K$h$jDs>'$5$
 @end example
 
 @table @t
-@item $B;2>H(B
-@fref{$B9`=g=x$N@_Dj(B}
+\JP @item $B;2>H(B
+\EG @item References
+\JP @fref{$B9`=g=x$N@_Dj(B}
+\EG @fref{Setting term orderings}
 @end table
 
-@node dp_ptod,,, $B%0%l%V%J4pDl$K4X$9$kH!?t(B
+\JP @node dp_ptod,,, $B%0%l%V%J4pDl$K4X$9$kH!?t(B
+\EG @node dp_ptod,,, Functions for Groebner basis computation
 @subsection @code{dp_ptod}
 @findex dp_ptod
 
 @table @t
 @item dp_ptod(@var{poly},@var{vlist})
-:: $BB?9`<0$rJ,;6I=8=B?9`<0$KJQ49$9$k(B. 
+\JP :: $BB?9`<0$rJ,;6I=8=B?9`<0$KJQ49$9$k(B. 
+\EG :: Converts an ordinary polynomial into a distributed polynomial.
 @end table
 
 @table @var
 @item return
-$BJ,;6I=8=B?9`<0(B
+\JP $BJ,;6I=8=B?9`<0(B
+\EG distributed polynomial
 @item poly
-$BB?9`<0(B
+\JP $BB?9`<0(B
+\EG polynomial
 @item vlist
-$B%j%9%H(B
+\JP $B%j%9%H(B
+\EG list
 @end table
 
 @itemize @bullet
+\BJP
 @item
 $BJQ?t=g=x(B @var{vlist} $B$*$h$S8=:_$NJQ?t=g=x7?$K=>$C$FJ,;6I=8=B?9`<0$KJQ49$9$k(B. 
 @item
 @var{vlist} $B$K4^$^$l$J$$ITDj85$O(B, $B78?tBN$KB0$9$k$H$7$FJQ49$5$l$k(B. 
+\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
@@ -1189,34 +2216,50 @@ 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
+\EG @item References
 @fref{dp_dtop},
 @fref{dp_ord}.
 @end table
 
-@node dp_dtop,,, $B%0%l%V%J4pDl$K4X$9$kH!?t(B
+\JP @node dp_dtop,,, $B%0%l%V%J4pDl$K4X$9$kH!?t(B
+\EG @node dp_dtop,,, Functions for Groebner basis computation
 @subsection @code{dp_dtop}
 @findex dp_dtop
 
 @table @t
 @item dp_dtop(@var{dpoly},@var{vlist})
-:: $BJ,;6I=8=B?9`<0$rB?9`<0$KJQ49$9$k(B. 
+\JP :: $BJ,;6I=8=B?9`<0$rB?9`<0$KJQ49$9$k(B. 
+\EG :: Converts a distributed polynomial into an ordinary polynomial.
 @end table
 
 @table @var
 @item return
-$BB?9`<0(B
+\JP $BB?9`<0(B
+\EG polynomial
 @item dpoly
-$BJ,;6I=8=B?9`<0(B
+\JP $BJ,;6I=8=B?9`<0(B
+\EG distributed polynomial
 @item vlist
-$B%j%9%H(B
+\JP $B%j%9%H(B
+\EG list
 @end table
 
 @itemize @bullet
+\BJP
 @item
 $BJ,;6I=8=B?9`<0$r(B, $BM?$($i$l$?ITDj85%j%9%H$rMQ$$$FB?9`<0$KJQ49$9$k(B. 
 @item
 $BITDj85%j%9%H$O(B, $BD9$5J,;6I=8=B?9`<0$NJQ?t$N8D?t$H0lCW$7$F$$$l$P2?$G$b$h$$(B. 
+\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
@@ -1226,30 +2269,41 @@ F4 $B%"%k%4%j%:%`$O(B, J.C. Faugere $B$K$h$jDs>'$5$
 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
@@ -1263,38 +2317,64 @@ 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
+\EG @item References
 @fref{dp_nf dp_nf_mod dp_true_nf dp_true_nf_mod},
 @fref{subst psubst},
 @fref{setmod}.
 @end table
 
-@node dp_homo dp_dehomo,,, $B%0%l%V%J4pDl$K4X$9$kH!?t(B
+\JP @node dp_homo dp_dehomo,,, $B%0%l%V%J4pDl$K4X$9$kH!?t(B
+\EG @node dp_homo dp_dehomo,,, Functions for Groebner basis computation
 @subsection @code{dp_homo}, @code{dp_dehomo}
 @findex dp_homo
 @findex dp_dehomo
 
 @table @t
 @item dp_homo(@var{dpoly})
-:: $BJ,;6I=8=B?9`<0$N@F<!2=(B
+\JP :: $BJ,;6I=8=B?9`<0$N@F<!2=(B
+\EG :: Homogenize a distributed polynomial
 @item dp_dehomo(@var{dpoly})
-:: $B@F<!J,;6I=8=B?9`<0$NHs@F<!2=(B
+\JP :: $B@F<!J,;6I=8=B?9`<0$NHs@F<!2=(B
+\EG :: Dehomogenize a homogenious distributed polynomial
 @end table
 
 @table @var
 @item return
-$BJ,;6I=8=B?9`<0(B
+\JP $BJ,;6I=8=B?9`<0(B
+\EG distributed polynomial
 @item dpoly
-$BJ,;6I=8=B?9`<0(B
+\JP $BJ,;6I=8=B?9`<0(B
+\EG distributed polynomial
 @end table
 
 @itemize @bullet
+\BJP
 @item
 @code{dp_homo()} $B$O(B, @var{dpoly} $B$N(B $B3F9`(B @var{t} $B$K$D$$$F(B, $B;X?t%Y%/%H%k$ND9$5$r(B
 1 $B?-$P$7(B, $B:G8e$N@.J,$NCM$r(B @var{d}-@code{deg(@var{t})} 
@@ -1307,6 +2387,23 @@ z^2+(2*a+2*b)*z+a^2+2*b*a+b^2
 $B@5$7$/@_Dj$9$kI,MW$,$"$k(B. 
 @item
 @code{hgr()} $B$J$I$K$*$$$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
@@ -1319,30 +2416,45 @@ z^2+(2*a+2*b)*z+a^2+2*b*a+b^2
 @end example
 
 @table @t
-@item $B;2>H(B
+\JP @item $B;2>H(B
+\EG @item References
 @fref{gr hgr gr_mod}.
 @end table
 
-@node dp_ptozp dp_prim,,, $B%0%l%V%J4pDl$K4X$9$kH!?t(B
+\JP @node dp_ptozp dp_prim,,, $B%0%l%V%J4pDl$K4X$9$kH!?t(B
+\EG @node dp_ptozp dp_prim,,, Functions for Groebner basis computation
 @subsection @code{dp_ptozp}, @code{dp_prim}
 @findex dp_ptozp
 @findex dp_prim
 
 @table @t
 @item dp_ptozp(@var{dpoly})
-:: $BDj?tG\$7$F78?t$r@0?t78?t$+$D78?t$N@0?t(B GCD $B$r(B 1 $B$K$9$k(B. 
+\JP :: $BDj?tG\$7$F78?t$r@0?t78?t$+$D78?t$N@0?t(B GCD $B$r(B 1 $B$K$9$k(B. 
+\BEG
+:: Converts a distributed polynomial @var{poly} with rational coefficients
+into an integral distributed polynomial such that GCD of all its coefficients
+is 1.
+\E
 @itemx dp_prim(@var{dpoly})
-:: $BM-M}<0G\$7$F78?t$r@0?t78?tB?9`<078?t$+$D78?t$NB?9`<0(B GCD $B$r(B 1 $B$K$9$k(B. 
+\JP :: $BM-M}<0G\$7$F78?t$r@0?t78?tB?9`<078?t$+$D78?t$NB?9`<0(B GCD $B$r(B 1 $B$K$9$k(B. 
+\BEG
+:: Converts a distributed polynomial @var{poly} with rational function
+coefficients into an integral distributed polynomial such that polynomial 
+GCD of all its coefficients is 1.
+\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
@@ -1350,6 +2462,15 @@ 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
@@ -1362,11 +2483,13 @@ z^2+(2*a+2*b)*z+a^2+2*b*a+b^2
 @end example
 
 @table @t
-@item $B;2>H(B
+\JP @item $B;2>H(B
+\EG @item References
 @fref{ptozp}.
 @end table
 
-@node dp_nf dp_nf_mod dp_true_nf dp_true_nf_mod,,, $B%0%l%V%J4pDl$K4X$9$kH!?t(B
+\JP @node dp_nf dp_nf_mod dp_true_nf dp_true_nf_mod,,, $B%0%l%V%J4pDl$K4X$9$kH!?t(B
+\EG @node dp_nf dp_nf_mod dp_true_nf dp_true_nf_mod,,, Functions for Groebner basis computation
 @subsection @code{dp_nf}, @code{dp_nf_mod}, @code{dp_true_nf}, @code{dp_true_nf_mod}
 @findex dp_nf
 @findex  dp_true_nf
@@ -1376,29 +2499,44 @@ z^2+(2*a+2*b)*z+a^2+2*b*a+b^2
 @table @t
 @item dp_nf(@var{indexlist},@var{dpoly},@var{dpolyarray},@var{fullreduce})
 @item dp_nf_mod(@var{indexlist},@var{dpoly},@var{dpolyarray},@var{fullreduce},@var{mod})
-:: $BJ,;6I=8=B?9`<0$N@55,7A$r5a$a$k(B. ($B7k2L$ODj?tG\$5$l$F$$$k2DG=@-$"$j(B)
+\JP :: $BJ,;6I=8=B?9`<0$N@55,7A$r5a$a$k(B. ($B7k2L$ODj?tG\$5$l$F$$$k2DG=@-$"$j(B)
 
+\BEG
+:: Computes the normal form of a distributed polynomial.
+(The result may be multiplied by a constant in the ground field.)
+\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
@@ -1429,7 +2567,43 @@ 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
+@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
@@ -1457,14 +2631,16 @@ 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_dtop},
 @fref{dp_ord},
 @fref{dp_mod dp_rat},
 @fref{p_nf p_nf_mod p_true_nf p_true_nf_mod}.
 @end table
 
-@node dp_hm dp_ht dp_hc dp_rest,,, $B%0%l%V%J4pDl$K4X$9$kH!?t(B
+\JP @node dp_hm dp_ht dp_hc dp_rest,,, $B%0%l%V%J4pDl$K4X$9$kH!?t(B
+\EG @node dp_hm dp_ht dp_hc dp_rest,,, Functions for Groebner basis computation
 @subsection @code{dp_hm}, @code{dp_ht}, @code{dp_hc}, @code{dp_rest}
 @findex dp_hm
 @findex dp_ht
@@ -1473,28 +2649,49 @@ 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})
@@ -1516,28 +2713,35 @@ u4^2+(6*u3+2*u2+6*u1-2)*u4+9*u3^2+(6*u2+18*u1-6)*u3+u2
 +(-490)*<<0,0,0>>
 @end example
 
-@node dp_td dp_sugar,,, $B%0%l%V%J4pDl$K4X$9$kH!?t(B
+\JP @node dp_td dp_sugar,,, $B%0%l%V%J4pDl$K4X$9$kH!?t(B
+\EG @node dp_td dp_sugar,,, Functions for Groebner basis computation
 @subsection @code{dp_td}, @code{dp_sugar}
 @findex dp_td
 @findex dp_sugar
 
 @table @t
 @item dp_td(@var{dpoly})
-:: $BF,9`$NA4<!?t$rJV$9(B. 
+\JP :: $BF,9`$NA4<!?t$rJV$9(B. 
+\EG :: Gets the total degree of the head term.
 @item dp_sugar(@var{dpoly})
-:: $BB?9`<0$N(B @code{sugar} $B$rJV$9(B. 
+\JP :: $BB?9`<0$N(B @code{sugar} $B$rJV$9(B. 
+\EG :: Gets the @code{sugar} of a polynomial.
 @end table
 
 @table @var
 @item return
-$B<+A3?t(B
+\JP $B<+A3?t(B
+\EG non-negative integer
 @item dpoly
-$BJ,;6I=8=B?9`<0(B
+\JP $BJ,;6I=8=B?9`<0(B
+\EG distributed polynomial
 @item onoff
-$B%U%i%0(B
+\JP $B%U%i%0(B
+\EG flag
 @end table
 
 @itemize @bullet
+\BJP
 @item
 @code{dp_td()} $B$O(B, $BF,9`$NA4<!?t(B, $B$9$J$o$A3FJQ?t$N;X?t$NOB$rJV$9(B. 
 @item
@@ -1546,6 +2750,20 @@ 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
@@ -1558,25 +2776,36 @@ u4^2+(6*u3+2*u2+6*u1-2)*u4+9*u3^2+(6*u2+18*u1-6)*u3+u2
 3
 @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
-$BJ,;6I=8=B?9`<0(B
-@item dpoly1, dpoly2
-$BJ,;6I=8=B?9`<0(B
+\JP $BJ,;6I=8=B?9`<0(B
+\EG distributed polynomial
+@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
@@ -1585,32 +2814,46 @@ 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
-$B@0?t(B
-@item dpoly1, dpoly2
-$BJ,;6I=8=B?9`<0(B
+\JP $B@0?t(B
+\EG integer
+@item dpoly1  dpoly2
+\JP $BJ,;6I=8=B?9`<0(B
+\EG distributed polynomial
 @end table
 
 @itemize @bullet
+\BJP
 @item
 @var{dpoly1} $B$NF,9`$,(B @var{dpoly2} $B$NF,9`$G3d$j@Z$l$l$P(B 1, $B3d$j@Z$l$J$1$l$P(B
 0 $B$rJV$9(B. 
 @item
 $BB?9`<0$N4JLs$r9T$&:](B, $B$I$N9`$r4JLs$G$-$k$+$rC5$9$N$KMQ$$$k(B. 
+\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
@@ -1626,32 +2869,46 @@ u4^2+(6*u3+2*u2+6*u1-2)*u4+9*u3^2+(6*u2+18*u1-6)*u3+u2
 @end example
 
 @table @t
-@item $B;2>H(B
+\JP @item $B;2>H(B
+\EG @item References
 @fref{dp_red dp_red_mod}.
 @end table
 
-@node dp_subd,,, $B%0%l%V%J4pDl$K4X$9$kH!?t(B
+\JP @node dp_subd,,, $B%0%l%V%J4pDl$K4X$9$kH!?t(B
+\EG @node dp_subd,,, Functions for Groebner basis computation
 @subsection @code{dp_subd}
 @findex dp_subd
 
 @table @t
 @item dp_subd(@var{dpoly1},@var{dpoly2})
-:: $BF,9`$N>&C19`<0$rJV$9(B. 
+\JP :: $BF,9`$N>&C19`<0$rJV$9(B. 
+\EG :: Returns the quotient monomial of the head terms.
 @end table
 
 @table @var
 @item return
-$BJ,;6I=8=B?9`<0(B
-@item dpoly1, dpoly2
-$BJ,;6I=8=B?9`<0(B
+\JP $BJ,;6I=8=B?9`<0(B
+\EG distributed polynomial
+@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
@@ -1660,37 +2917,53 @@ 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
@@ -1703,23 +2976,28 @@ 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, 
@@ -1727,6 +3005,18 @@ 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
@@ -1741,27 +3031,33 @@ 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. 
@@ -1772,6 +3068,20 @@ 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
@@ -1781,11 +3091,13 @@ 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
@@ -1793,21 +3105,27 @@ 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
-$B%j%9%H(B
-@item dpoly1, dpoly2, dpoly3
-$BJ,;6I=8=B?9`<0(B
+\JP $B%j%9%H(B
+\EG list
+@item dpoly1  dpoly2  dpoly3
+\JP $BJ,;6I=8=B?9`<0(B
+\EG distributed polynomial
 @item vlist
-$B%j%9%H(B
+\JP $B%j%9%H(B
+\EG list
 @item mod
-$BAG?t(B
+\JP $BAG?t(B
+\EG prime
 @end table
 
 @itemize @bullet
+\BJP
 @item
 @var{dpoly1} + @var{dpoly2} $B$J$kJ,;6I=8=B?9`<0$r(B @var{dpoly3} $B$G(B
 1 $B2s4JLs$9$k(B.
@@ -1819,9 +3137,29 @@ 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
@@ -1837,11 +3175,13 @@ 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 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
@@ -1849,19 +3189,24 @@ 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
-$BJ,;6I=8=B?9`<0(B
-@item dpoly1, dpoly2
-$BJ,;6I=8=B?9`<0(B
+\JP $BJ,;6I=8=B?9`<0(B
+\EG distributed polynomial
+@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
@@ -1869,6 +3214,17 @@ 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
@@ -1881,10 +3237,12 @@ u1*u2,u1^2,u4*u0,u3*u0,u2*u0,u1*u0,u0^2,u4,u3,u2,u1,u0
 @end example
 
 @table @t
-@item $B;2>H(B
+\JP @item $B;2>H(B
+\EG @item References
 @fref{dp_mod dp_rat}.
 @end table
-@node p_nf p_nf_mod p_true_nf p_true_nf_mod,,, $B%0%l%V%J4pDl$K4X$9$kH!?t(B
+\JP @node p_nf p_nf_mod p_true_nf p_true_nf_mod,,, $B%0%l%V%J4pDl$K4X$9$kH!?t(B
+\EG @node p_nf p_nf_mod p_true_nf p_true_nf_mod,,, Functions for Groebner basis computation
 @subsection @code{p_nf}, @code{p_nf_mod}, @code{p_true_nf}, @code{p_true_nf_mod}
 @findex p_nf
 @findex p_nf_mod
@@ -1894,26 +3252,40 @@ 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
-@item plist,vlist
-$B%j%9%H(B
+\JP $BB?9`<0(B
+\EG polynomial
+@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
@@ -1934,6 +3306,30 @@ 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
@@ -1949,34 +3345,42 @@ u1*u2,u1^2,u4*u0,u3*u0,u2*u0,u1*u0,u0^2,u4,u3,u2,u1,u0
 @end example
 
 @table @t
-@item $B;2>H(B
+\JP @item $B;2>H(B
+\EG @item References
 @fref{dp_ptod},
 @fref{dp_dtop},
 @fref{dp_ord},
 @fref{dp_nf dp_nf_mod dp_true_nf dp_true_nf_mod}.
 @end table
 
-@node p_terms,,, $B%0%l%V%J4pDl$K4X$9$kH!?t(B
+\JP @node p_terms,,, $B%0%l%V%J4pDl$K4X$9$kH!?t(B
+\EG @node p_terms,,, Functions for Groebner basis computation
 @subsection @code{p_terms}
 @findex p_terms
 
 @table @t
 @item p_terms(@var{poly},@var{vlist},@var{order})
-:: $BB?9`<0$K$"$i$o$l$kC19`$r%j%9%H$K$9$k(B. 
+\JP :: $BB?9`<0$K$"$i$o$l$kC19`$r%j%9%H$K$9$k(B. 
+\EG :: Monomials appearing in the given polynomial is collected into a list.
 @end table
 
 @table @var
 @item return
-$B%j%9%H(B
+\JP $B%j%9%H(B
+\EG list
 @item poly
-$BB?9`<0(B
+\JP $BB?9`<0(B
+\EG polynomial
 @item vlist
-$B%j%9%H(B
+\JP $B%j%9%H(B
+\EG list
 @item order
-$B?t(B, $B%j%9%H$^$?$O9TNs(B
+\JP $B?t(B, $B%j%9%H$^$?$O9TNs(B
+\EG number, list or matrix
 @end table
 
 @itemize @bullet
+\BJP
 @item
 @samp{gr} $B$GDj5A$5$l$F$$$k(B. 
 @item
@@ -1986,6 +3390,20 @@ u1*u2,u1^2,u4*u0,u3*u0,u2*u0,u1*u0,u0^2,u4,u3,u2,u1,u0
 @item
 $B%0%l%V%J4pDl$O$7$P$7$P78?t$,5pBg$K$J$k$?$a(B, $B<B:]$K$I$N9`$,8=$l$F(B
 $B$$$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
@@ -1996,27 +3414,42 @@ 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]
 @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
-@item plist1, plist2
+\JP @item return 0 $B$^$?$O(B 1
+\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
@@ -2029,7 +3462,8 @@ u0^6,u0^5,u0^4,u0^3,u0^2,u0,1]
 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
@@ -2041,17 +3475,21 @@ 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. 
@@ -2061,6 +3499,19 @@ 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
@@ -2092,7 +3543,102 @@ 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