===================================================================
RCS file: /home/cvs/OpenXM/src/asir-doc/parts/groebner.texi,v
retrieving revision 1.3
retrieving revision 1.18
diff -u -p -r1.3 -r1.18
--- OpenXM/src/asir-doc/parts/groebner.texi	1999/12/24 04:38:04	1.3
+++ OpenXM/src/asir-doc/parts/groebner.texi	2016/03/24 20:58:50	1.18
@@ -1,4 +1,4 @@
-@comment $OpenXM: OpenXM/src/asir-doc/parts/groebner.texi,v 1.2 1999/12/21 02:47:31 noro Exp $
+@comment $OpenXM: OpenXM/src/asir-doc/parts/groebner.texi,v 1.17 2006/09/06 23:53:31 noro Exp $
 \BJP
 @node $B%0%l%V%J4pDl$N7W;;(B,,, Top
 @chapter $B%0%l%V%J4pDl$N7W;;(B
@@ -15,8 +15,10 @@
 * $B4pK\E*$JH!?t(B::
 * $B7W;;$*$h$SI=<($N@)8f(B::
 * $B9`=g=x$N@_Dj(B::
+* Weight::
 * $BM-M}<0$r78?t$H$9$k%0%l%V%J4pDl7W;;(B::
 * $B4pDlJQ49(B::
+* Weyl $BBe?t(B::
 * $B%0%l%V%J4pDl$K4X$9$kH!?t(B::
 \E
 \BEG
@@ -25,8 +27,10 @@
 * Fundamental functions::
 * Controlling Groebner basis computations::
 * Setting term orderings::
+* Weight::
 * Groebner basis computation with rational function coefficients::
 * Change of ordering::
+* Weyl algebra::
 * Functions for Groebner basis computation::
 \E
 @end menu
@@ -228,23 +232,23 @@ the head term and the head coefficient respectively.
 @noindent
 \BJP
 $B%0%l%V%J4pDl$r7W;;$9$k$?$a$N4pK\E*$JH!?t$O(B @code{dp_gr_main()} $B$*$h$S(B
-@code{dp_gr_mod_main()} $B$J$k(B 2 $B$D$NAH$_9~$_H!?t$G$"$k$,(B, $BDL>o$O(B, $B%Q%i%a%?(B
+@code{dp_gr_mod_main()}, @code{dp_gr_f_main()}
+ $B$J$k(B 3 $B$D$NAH$_9~$_H!?t$G$"$k$,(B, $BDL>o$O(B, $B%Q%i%a%?(B
 $B@_Dj$J$I$r9T$C$?$N$A$3$l$i$r8F$S=P$9%f!<%6H!?t$rMQ$$$k$N$,JXMx$G$"$k(B. 
 $B$3$l$i$N%f!<%6H!?t$O(B, $B%U%!%$%k(B @samp{gr} $B$r(B @code{load()} $B$K$h$jFI(B
 $B$_9~$`$3$H$K$h$j;HMQ2DG=$H$J$k(B. @samp{gr} $B$O(B, @b{Asir} $B$NI8=`(B
-$B%i%$%V%i%j%G%#%l%/%H%j$KCV$+$l$F$$$k(B. $B$h$C$F(B, $B4D6-JQ?t(B @code{ASIR_LIBDIR}
-$B$rFC$K0[$J$k%Q%9$K@_Dj$7$J$$8B$j(B, $B%U%!%$%kL>$N$_$GFI$_9~$`$3$H$,$G$-$k(B. 
+$B%i%$%V%i%j%G%#%l%/%H%j$KCV$+$l$F$$$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}.
+Facilities for computing Groebner bases are 
+@code{dp_gr_main()}, @code{dp_gr_mod_main()}and @code{dp_gr_f_main()}.
+To call these functions, 
+it is necessary to set several parameters correctly and it is convenient
+to use a set of interface functions provided in the library file
+@samp{gr}.
 The facilities will be ready to use after you load the package by
 @code{load()}.  The package @samp{gr} is placed in the standard library
-directory of @b{Asir}.  Therefore, it is loaded simply by specifying
-its file name, unless the environment variable @code{ASIR_LIBDIR}
-is set to a non-standard one.
+directory of @b{Asir}.  
 \E
 
 @example
@@ -350,8 +354,8 @@ These parameters can be set and examined by a built-in
 
 @example
 [100] dp_gr_flags();
-[Demand,0,NoSugar,0,NoCriB,0,NoGC,0,NoMC,0,NoRA,0,NoGCD,0,Top,0,ShowMag,1,
-Print,1,Stat,0,Reverse,0,InterReduce,0,Multiple,0]
+[Demand,0,NoSugar,0,NoCriB,0,NoGC,0,NoMC,0,NoRA,0,NoGCD,0,Top,0,
+ShowMag,1,Print,1,Stat,0,Reverse,0,InterReduce,0,Multiple,0]
 [101]
 @end example
 
@@ -447,6 +451,13 @@ If `on', various informations during a Groebner basis 
 displayed.
 \E
 
+@item PrintShort
+\JP on $B$G!"(BPrint $B$,(B off $B$N>l9g(B, $B%0%l%V%J4pDl7W;;$NESCf$N>pJs$rC;=L7A$GI=<($9$k(B. 
+\BEG
+If `on' and Print is `off', short information during a Groebner basis computation is
+displayed.
+\E
+
 @item Stat
 \BJP
 on $B$G(B @code{Print} $B$,(B off $B$J$i$P(B, @code{Print} $B$,(B on $B$N$H$-I=<($5(B
@@ -469,24 +480,28 @@ is shown after every normal computation.  After comlet
 computation the maximal value among the sums is shown.
 \E
 
-@item Multiple
+@item Content
+@itemx 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
+0 $B$G$J$$M-M}?t$N;~(B, $BM-M}?t>e$N@55,7A7W;;$K$*$$$F(B, $B78?t$N%S%C%HD9$NOB$,(B
+@code{Content} $BG\$K$J$k$4$H$K78?tA4BN$N(B GCD $B$,7W;;$5$l(B, $B$=$N(B GCD $B$G(B
+$B3d$C$?B?9`<0$r4JLs$9$k(B. @code{Content} $B$,(B 1 $B$J$i$P(B, $B4JLs$9$k$4$H$K(B
+GCD $B7W;;$,9T$o$l0lHL$K$O8zN($,0-$/$J$k$,(B, @code{Content} $B$r(B 2 $BDxEY(B
 $B$H$9$k$H(B, $B5pBg$J@0?t$,78?t$K8=$l$k>l9g(B, $B8zN($,NI$/$J$k>l9g$,$"$k(B. 
+backward compatibility $B$N$?$a!"(B@code{Multiple} $B$G@0?tCM$r;XDj$G$-$k(B.
 \E
 \BEG
-If a non-zero integer, in a normal form computation
+If a non-zero rational number, in a normal form computation
 over the rationals, the integer content of the polynomial being
-reduced is removed when its magnitude becomes @code{Multiple} times
+reduced is removed when its magnitude becomes @code{Content} times
 larger than a registered value, which is set to the magnitude of the
 input polynomial. After each content removal the registered value is
-set to the magnitude of the resulting polynomial. @code{Multiple} is
+set to the magnitude of the resulting polynomial. @code{Content} is
 equal to 1, the simiplification is done after every normal form computation.
-It is empirically known that it is often efficient to set @code{Multiple} to 2
+It is empirically known that it is often efficient to set @code{Content} to 2
 for the case where large integers appear during the computation.
+An integer value can be set by the keyword @code{Multiple} for
+backward compatibility.
 \E
 
 @item Demand
@@ -530,9 +545,9 @@ membercheck
 (0,0)(0,0)(0,0)(0,0)
 gbcheck total 8 pairs
 ........
-UP=(0,0)SP=(0,0)SPM=(0,0)NF=(0,0)NFM=(0.010002,0)ZNFM=(0.010002,0)PZ=(0,0)
-NP=(0,0)MP=(0,0)RA=(0,0)MC=(0,0)GC=(0,0)T=40,B=0 M=8 F=6 D=12 ZR=5 NZR=6
-Max_mag=6
+UP=(0,0)SP=(0,0)SPM=(0,0)NF=(0,0)NFM=(0.010002,0)ZNFM=(0.010002,0)
+PZ=(0,0)NP=(0,0)MP=(0,0)RA=(0,0)MC=(0,0)GC=(0,0)T=40,B=0 M=8 F=6
+D=12 ZR=5 NZR=6 Max_mag=6
 [94]
 @end example
 
@@ -992,24 +1007,25 @@ time as well as the choice of types of term orderings.
 -40*t^8+70*t^7+252*t^6+30*t^5-140*t^4-168*t^3+2*t^2-12*t+16)*z^2*y
 +(-12*t^16+72*t^13-28*t^11-180*t^10+112*t^8+240*t^7+28*t^6-127*t^5
 -167*t^4-55*t^3+30*t^2+58*t-15)*z^4,
-(y+t^2*z^2)*x+y^7+(20*t^2+6*t+1)*y^2+(-t^17+6*t^14-21*t^12-15*t^11+84*t^9
-+20*t^8-35*t^7-126*t^6-15*t^5+70*t^4+84*t^3-t^2+5*t-9)*z^2*y+(6*t^16-36*t^13
-+14*t^11+90*t^10-56*t^8-120*t^7-14*t^6+64*t^5+84*t^4+27*t^3-16*t^2-30*t+7)*z^4,
-(t^3-1)*x-y^6+(-6*t^13+24*t^10-20*t^8-36*t^7+40*t^5+24*t^4-6*t^3-20*t^2-6*t-1)*y
-+(t^17-6*t^14+9*t^12+15*t^11-36*t^9-20*t^8-5*t^7+54*t^6+15*t^5+10*t^4-36*t^3
--11*t^2-5*t+9)*z^2,
+(y+t^2*z^2)*x+y^7+(20*t^2+6*t+1)*y^2+(-t^17+6*t^14-21*t^12-15*t^11
++84*t^9+20*t^8-35*t^7-126*t^6-15*t^5+70*t^4+84*t^3-t^2+5*t-9)*z^2*y
++(6*t^16-36*t^13+14*t^11+90*t^10-56*t^8-120*t^7-14*t^6+64*t^5+84*t^4
++27*t^3-16*t^2-30*t+7)*z^4,
+(t^3-1)*x-y^6+(-6*t^13+24*t^10-20*t^8-36*t^7+40*t^5+24*t^4-6*t^3-20*t^2
+-6*t-1)*y+(t^17-6*t^14+9*t^12+15*t^11-36*t^9-20*t^8-5*t^7+54*t^6+15*t^5
++10*t^4-36*t^3-11*t^2-5*t+9)*z^2,
 -y^8-8*t*y^3+16*z^2*y^2+(-8*t^16+48*t^13-56*t^11-120*t^10+224*t^8+160*t^7
--56*t^6-336*t^5-112*t^4+112*t^3+224*t^2+24*t-56)*z^4*y+(t^24-8*t^21+20*t^19
-+28*t^18-120*t^16-56*t^15+14*t^14+300*t^13+70*t^12-56*t^11-400*t^10-84*t^9
-+84*t^8+268*t^7+84*t^6-56*t^5-63*t^4-36*t^3+46*t^2-12*t+1)*z,
-2*t*y^5+z*y^2+(-2*t^11+8*t^8-20*t^6-12*t^5+40*t^3+8*t^2-10*t-20)*z^3*y+8*t^14
--32*t^11+48*t^8-t^7-32*t^5-6*t^4+9*t^2-t,
+-56*t^6-336*t^5-112*t^4+112*t^3+224*t^2+24*t-56)*z^4*y+(t^24-8*t^21
++20*t^19+28*t^18-120*t^16-56*t^15+14*t^14+300*t^13+70*t^12-56*t^11
+-400*t^10-84*t^9+84*t^8+268*t^7+84*t^6-56*t^5-63*t^4-36*t^3+46*t^2
+-12*t+1)*z,2*t*y^5+z*y^2+(-2*t^11+8*t^8-20*t^6-12*t^5+40*t^3+8*t^2
+-10*t-20)*z^3*y+8*t^14-32*t^11+48*t^8-t^7-32*t^5-6*t^4+9*t^2-t,
 -z*y^3+(t^7-2*t^4+3*t^2+t)*y+(-2*t^6+4*t^3+2*t-2)*z^2,
-2*t^2*y^3+z^2*y^2+(-2*t^5+4*t^2-6)*z^4*y+(4*t^8-t^7-8*t^5+2*t^4-4*t^3+5*t^2-t)*z,
+2*t^2*y^3+z^2*y^2+(-2*t^5+4*t^2-6)*z^4*y
++(4*t^8-t^7-8*t^5+2*t^4-4*t^3+5*t^2-t)*z,
 z^3*y^2+2*t^3*y+(-t^7+2*t^4+t^2-t)*z^2,
 -t*z*y^2-2*z^3*y+t^8-2*t^5-t^3+t^2,
--t^3*y^2-2*t^2*z^2*y+(t^6-2*t^3-t+1)*z^4,
-z^5-t^4]
+-t^3*y^2-2*t^2*z^2*y+(t^6-2*t^3-t+1)*z^4,z^5-t^4]
 [93] gr(B,[t,z,y,x],2);
 [x^10-t,x^8-z,x^31-x^6-x-y]
 @end example
@@ -1041,6 +1057,139 @@ beforehand, and some heuristic trial may be inevitable
 \E
 
 \BJP
+@node Weight ,,, $B%0%l%V%J4pDl$N7W;;(B
+@section Weight
+\E
+\BEG
+@node Weight,,, Groebner basis computation
+@section Weight
+\E
+\BJP
+$BA0@a$G>R2p$7$?9`=g=x$O(B, $B3FJQ?t$K(B weight ($B=E$_(B) $B$r@_Dj$9$k$3$H$G(B
+$B$h$j0lHLE*$J$b$N$H$J$k(B. 
+\E
+\BEG
+Term orderings introduced in the previous section can be generalized
+by setting a weight for each variable.
+\E
+@example
+[0] dp_td(<<1,1,1>>);
+3
+[1] dp_set_weight([1,2,3])$
+[2] dp_td(<<1,1,1>>);
+6
+@end example
+\BJP
+$BC19`<0$NA4<!?t$r7W;;$9$k:](B, $B%G%U%)%k%H$G$O(B
+$B3FJQ?t$N;X?t$NOB$rA4<!?t$H$9$k(B. $B$3$l$O3FJQ?t$N(B weight $B$r(B 1 $B$H(B
+$B9M$($F$$$k$3$H$KAjEv$9$k(B. $B$3$NNc$G$O(B, $BBh0l(B, $BBhFs(B, $BBh;0JQ?t$N(B
+weight $B$r$=$l$>$l(B 1,2,3 $B$H;XDj$7$F$$$k(B. $B$3$N$?$a(B, @code{<<1,1,1>>}
+$B$NA4<!?t(B ($B0J2<$G$O$3$l$rC19`<0$N(B weight $B$H8F$V(B) $B$,(B @code{1*1+1*2+1*3=6} $B$H$J$k(B.
+weight $B$r@_Dj$9$k$3$H$G(B, $BF1$89`=g=x7?$N$b$H$G0[$J$k9`=g=x$,Dj5A$G$-$k(B.
+$BNc$($P(B, weight $B$r$&$^$/@_Dj$9$k$3$H$G(B, $BB?9`<0$r(B weighted homogeneous
+$B$K$9$k$3$H$,$G$-$k>l9g$,$"$k(B. 
+\E
+\BEG
+By default, the total degree of a monomial is equal to
+the sum of all exponents. This means that the weight for each variable
+is set to 1.
+In this example, the weights for the first, the second and the third
+variable are set to 1, 2 and 3 respectively.
+Therefore the total degree of @code{<<1,1,1>>} under this weight,
+which is called the weight of the monomial, is @code{1*1+1*2+1*3=6}.
+By setting weights, different term orderings can be set under a type of
+term ordeing. In some case a polynomial can 
+be made weighted homogeneous by setting an appropriate weight.
+\E
+
+\BJP
+$B3FJQ?t$KBP$9$k(B weight $B$r$^$H$a$?$b$N$r(B weight vector $B$H8F$V(B.
+$B$9$Y$F$N@.J,$,@5$G$"$j(B, $B%0%l%V%J4pDl7W;;$K$*$$$F(B, $BA4<!?t$N(B
+$BBe$o$j$KMQ$$$i$l$k$b$N$rFC$K(B sugar weight $B$H8F$V$3$H$K$9$k(B.
+sugar strategy $B$K$*$$$F(B, $BA4<!?t$NBe$o$j$K;H$o$l$k$+$i$G$"$k(B.
+$B0lJ}$G(B, $B3F@.J,$,I,$:$7$b@5$H$O8B$i$J$$(B weight vector $B$O(B,
+sugar weight $B$H$7$F@_Dj$9$k$3$H$O$G$-$J$$$,(B, $B9`=g=x$N0lHL2=$K$O(B
+$BM-MQ$G$"$k(B. $B$3$l$i$O(B, $B9TNs$K$h$k9`=g=x$N@_Dj$K$9$G$K8=$l$F(B
+$B$$$k(B. $B$9$J$o$A(B, $B9`=g=x$rDj5A$9$k9TNs$N3F9T$,(B, $B0l$D$N(B weight vector
+$B$H8+$J$5$l$k(B. $B$^$?(B, $B%V%m%C%/=g=x$O(B, $B3F%V%m%C%/$N(B
+$BJQ?t$KBP1~$9$k@.J,$N$_(B 1 $B$GB>$O(B 0 $B$N(B weight vector $B$K$h$kHf3S$r(B
+$B:G=i$K9T$C$F$+$i(B, $B3F%V%m%C%/Kh$N(B tie breaking $B$r9T$&$3$H$KAjEv$9$k(B.
+\E
+
+\BEG
+A list of weights for all variables is called a weight vector.
+A weight vector is called a sugar weight vector if
+its elements are all positive and it is used for computing
+a weighted total degree of a monomial, because such a weight
+is used instead of total degree in sugar strategy.
+On the other hand, a weight vector whose elements are not necessarily
+positive cannot be set as a sugar weight, but it is useful for
+generalizing term order. In fact, such a weight vector already
+appeared in a matrix order. That is, each row of a matrix defining
+a term order is regarded as a weight vector. A block order 
+is also considered as a refinement of comparison by weight vectors.
+It compares two terms by using a weight vector whose elements
+corresponding to variables in a block is 1 and 0 otherwise,
+then it applies a tie breaker.
+\E
+
+\BJP
+weight vector $B$N@_Dj$O(B @code{dp_set_weight()} $B$G9T$&$3$H$,$G$-$k(B
+$B$,(B, $B9`=g=x$r;XDj$9$k:]$NB>$N%Q%i%a%?(B ($B9`=g=x7?(B, $BJQ?t=g=x(B) $B$H(B
+$B$^$H$a$F@_Dj$G$-$k$3$H$,K>$^$7$$(B. $B$3$N$?$a(B, $B<!$N$h$&$J7A$G$b(B
+$B9`=g=x$,;XDj$G$-$k(B.
+\E
+\BEG
+A weight vector can be set by using @code{dp_set_weight()}.
+However it is more preferable if a weight vector can be set
+together with other parapmeters such as a type of term ordering
+and a variable order. This is realized as follows.
+\E
+
+@example
+[64] B=[x+y+z-6,x*y+y*z+z*x-11,x*y*z-6]$
+[65] dp_gr_main(B|v=[x,y,z],sugarweight=[3,2,1],order=0);
+[z^3-6*z^2+11*z-6,x+y+z-6,-y^2+(-z+6)*y-z^2+6*z-11]
+[66] dp_gr_main(B|v=[y,z,x],order=[[1,1,0],[0,1,0],[0,0,1]]);
+[x^3-6*x^2+11*x-6,x+y+z-6,-x^2+(-y+6)*x-y^2+6*y-11]
+[67] dp_gr_main(B|v=[y,z,x],order=[[x,1,y,2,z,3]]);
+[x+y+z-6,x^3-6*x^2+11*x-6,-x^2+(-y+6)*x-y^2+6*y-11]
+@end example
+
+\BJP
+$B$$$:$l$NNc$K$*$$$F$b(B, $B9`=g=x$O(B option $B$H$7$F;XDj$5$l$F$$$k(B.
+$B:G=i$NNc$G$O(B @code{v} $B$K$h$jJQ?t=g=x$r(B, @code{sugarweight} $B$K$h$j(B
+sugar weight vector $B$r(B, @code{order}$B$K$h$j9`=g=x7?$r;XDj$7$F$$$k(B.
+$BFs$DL\$NNc$K$*$1$k(B @code{order} $B$N;XDj$O(B matrix order $B$HF1MM$G$"$k(B.
+$B$9$J$o$A(B, $B;XDj$5$l$?(B weight vector $B$r:8$+$i=g$K;H$C$F(B weight $B$NHf3S(B
+$B$r9T$&(B. $B;0$DL\$NNc$bF1MM$G$"$k$,(B, $B$3$3$G$O(B weight vector $B$NMWAG$r(B
+$BJQ?tKh$K;XDj$7$F$$$k(B. $B;XDj$,$J$$$b$N$O(B 0 $B$H$J$k(B. $B;0$DL\$NNc$G$O(B,
+@code{order} $B$K$h$k;XDj$G$O9`=g=x$,7hDj$7$J$$(B. $B$3$N>l9g$K$O(B,
+tie breaker $B$H$7$FA4<!?t5U<-=q<0=g=x$,<+F0E*$K@_Dj$5$l$k(B.
+$B$3$N;XDjJ}K!$O(B, @code{dp_gr_main}, @code{dp_gr_mod_main} $B$J$I(B
+$B$NAH$_9~$_4X?t$G$N$_2DG=$G$"$j(B, @code{gr} $B$J$I$N%f!<%6Dj5A4X?t(B
+$B$G$OL$BP1~$G$"$k(B.
+\E
+\BEG
+In each example, a term ordering is specified as options.
+In the first example, a variable order, a sugar weight vector
+and a type of term ordering are specified by options @code{v}, 
+@code{sugarweight} and @code{order} respectively.
+In the second example, an option @code{order} is used
+to set a matrix ordering. That is, the specified weight vectors
+are used from left to right for comparing terms.
+The third example shows a variant of specifying a weight vector,
+where each component of a weight vector is specified variable by variable,
+and unspecified components are set to zero. In this example,
+a term order is not determined only by the specified weight vector.
+In such a case a tie breaker by the graded reverse lexicographic ordering
+is set automatically.
+This type of a term ordering specification can be applied only to builtin
+functions such as @code{dp_gr_main()}, @code{dp_gr_mod_main()}, not to
+user defined functions such as @code{gr()}.
+\E
+
+\BJP
 @node $BM-M}<0$r78?t$H$9$k%0%l%V%J4pDl7W;;(B,,, $B%0%l%V%J4pDl$N7W;;(B
 @section $BM-M}<0$r78?t$H$9$k%0%l%V%J4pDl7W;;(B
 \E
@@ -1200,6 +1349,105 @@ Refer to the sections for each functions.
 \E
 
 \BJP
+@node Weyl $BBe?t(B,,, $B%0%l%V%J4pDl$N7W;;(B
+@section Weyl $BBe?t(B
+\E
+\BEG
+@node Weyl algebra,,, Groebner basis computation
+@section Weyl algebra
+\E
+
+@noindent
+
+\BJP
+$B$3$l$^$G$O(B, $BDL>o$N2D49$JB?9`<04D$K$*$1$k%0%l%V%J4pDl7W;;$K$D$$$F(B
+$B=R$Y$F$-$?$,(B, $B%0%l%V%J4pDl$NM}O@$O(B, $B$"$k>r7o$rK~$?$9Hs2D49$J(B
+$B4D$K$b3HD%$G$-$k(B. $B$3$N$h$&$J4D$NCf$G(B, $B1~MQ>e$b=EMW$J(B, 
+Weyl $BBe?t(B, $B$9$J$o$AB?9`<04D>e$NHyJ,:nMQAG4D$N1i;;$*$h$S(B
+$B%0%l%V%J4pDl7W;;$,(B Risa/Asir $B$K<BAu$5$l$F$$$k(B. 
+
+$BBN(B @code{K} $B>e$N(B @code{n} $B<!85(B Weyl $BBe?t(B 
+@code{D=K<x1,@dots{},xn,D1,@dots{},Dn>} $B$O(B
+\E
+
+\BEG
+So far we have explained Groebner basis computation in
+commutative polynomial rings. However Groebner basis can be
+considered in more general non-commutative rings.
+Weyl algebra is one of such rings and 
+Risa/Asir implements fundamental operations
+in Weyl algebra and Groebner basis computation in Weyl algebra.
+
+The @code{n} dimensional Weyl algebra over a field @code{K},
+@code{D=K<x1,@dots{},xn,D1,@dots{},Dn>} is a non-commutative
+algebra which has the following fundamental relations:
+\E
+
+@code{xi*xj-xj*xi=0}, @code{Di*Dj-Dj*Di=0}, @code{Di*xj-xj*Di=0} (@code{i!=j}),
+@code{Di*xi-xi*Di=1}
+
+\BJP
+$B$H$$$&4pK\4X78$r;}$D4D$G$"$k(B. @code{D} $B$O(B $BB?9`<04D(B @code{K[x1,@dots{},xn]} $B$r78?t(B
+$B$H$9$kHyJ,:nMQAG4D$G(B,  @code{Di} $B$O(B @code{xi} $B$K$h$kHyJ,$rI=$9(B. $B8r494X78$K$h$j(B, 
+@code{D} $B$N85$O(B, @code{x1^i1*@dots{}*xn^in*D1^j1*@dots{}*Dn^jn} $B$J$kC19`(B
+$B<0$N(B @code{K} $B@~7A7k9g$H$7$F=q$-I=$9$3$H$,$G$-$k(B. 
+Risa/Asir $B$K$*$$$F$O(B, $B$3$NC19`<0$r(B, $B2D49$JB?9`<0$HF1MM$K(B
+@code{<<i1,@dots{},in,j1,@dots{},jn>>} $B$GI=$9(B. $B$9$J$o$A(B, @code{D} $B$N85$b(B
+$BJ,;6I=8=B?9`<0$H$7$FI=$5$l$k(B. $B2C8:;;$O(B, $B2D49$N>l9g$HF1MM$K(B, @code{+}, @code{-}
+$B$K$h$j(B
+$B<B9T$G$-$k$,(B, $B>h;;$O(B, $BHs2D49@-$r9MN8$7$F(B @code{dp_weyl_mul()} $B$H$$$&4X?t(B
+$B$K$h$j<B9T$9$k(B.
+\E
+
+\BEG
+@code{D} is the ring of differential operators whose coefficients
+are polynomials in @code{K[x1,@dots{},xn]} and
+@code{Di} denotes the differentiation with respect to  @code{xi}.
+According to the commutation relation,
+elements of @code{D} can be represented as a @code{K}-linear combination 
+of monomials @code{x1^i1*@dots{}*xn^in*D1^j1*@dots{}*Dn^jn}.
+In Risa/Asir, this type of monomial is represented 
+by @code{<<i1,@dots{},in,j1,@dots{},jn>>} as in the case of commutative
+polynomial.
+That is, elements of @code{D} are represented by distributed polynomials.
+Addition and subtraction can be done by @code{+}, @code{-},
+but multiplication is done by calling @code{dp_weyl_mul()} because of
+the non-commutativity of @code{D}.
+\E
+
+@example
+[0] A=<<1,2,2,1>>;
+(1)*<<1,2,2,1>>
+[1] B=<<2,1,1,2>>;
+(1)*<<2,1,1,2>>
+[2] A*B;
+(1)*<<3,3,3,3>>
+[3] dp_weyl_mul(A,B);
+(1)*<<3,3,3,3>>+(1)*<<3,2,3,2>>+(4)*<<2,3,2,3>>+(4)*<<2,2,2,2>>
++(2)*<<1,3,1,3>>+(2)*<<1,2,1,2>>
+@end example
+
+\BJP
+$B%0%l%V%J4pDl7W;;$K$D$$$F$b(B, Weyl $BBe?t@lMQ$N4X?t$H$7$F(B, 
+$B<!$N4X?t$,MQ0U$7$F$"$k(B. 
+\E
+\BEG
+The following functions are avilable for Groebner basis computation
+in Weyl algebra:
+\E
+@code{dp_weyl_gr_main()},
+@code{dp_weyl_gr_mod_main()},
+@code{dp_weyl_gr_f_main()},
+@code{dp_weyl_f4_main()},
+@code{dp_weyl_f4_mod_main()}.
+\BJP
+$B$^$?(B, $B1~MQ$H$7$F(B, global b $B4X?t$N7W;;$,<BAu$5$l$F$$$k(B. 
+\E
+\BEG
+Computation of the global b function is implemented as an application.
+\E
+
+\BJP
 @node $B%0%l%V%J4pDl$K4X$9$kH!?t(B,,, $B%0%l%V%J4pDl$N7W;;(B
 @section $B%0%l%V%J4pDl$K4X$9$kH!?t(B
 \E
@@ -1214,16 +1462,18 @@ Refer to the sections for each functions.
 * lex_hensel_gsl tolex_gsl tolex_gsl_d::
 * gr_minipoly minipoly::
 * tolexm minipolym::
-* dp_gr_main dp_gr_mod_main::
-* dp_f4_main dp_f4_mod_main::
+* dp_gr_main dp_gr_mod_main dp_gr_f_main dp_weyl_gr_main dp_weyl_gr_mod_main dp_weyl_gr_f_main::
+* dp_f4_main dp_f4_mod_main dp_weyl_f4_main dp_weyl_f4_mod_main::
+* nd_gr nd_gr_trace nd_f4 nd_f4_trace nd_weyl_gr nd_weyl_gr_trace::
 * dp_gr_flags dp_gr_print::
 * dp_ord::
+* dp_set_weight dp_set_top_weight dp_weyl_set_weight::
 * dp_ptod::
 * dp_dtop::
 * dp_mod dp_rat::
 * dp_homo dp_dehomo::
 * dp_ptozp dp_prim::
-* dp_nf dp_nf_mod dp_true_nf dp_true_nf_mod::
+* dp_nf dp_nf_mod dp_true_nf dp_true_nf_mod dp_weyl_nf dp_weyl_nf_mod::
 * dp_hm dp_ht dp_hc dp_rest::
 * dp_td dp_sugar::
 * dp_lcm::
@@ -1240,6 +1490,8 @@ Refer to the sections for each functions.
 * dp_vtoe dp_etov::
 * lex_hensel_gsl tolex_gsl tolex_gsl_d::
 * primadec primedec::
+* primedec_mod::
+* bfunction bfct generic_bfct ann ann0::
 @end menu
 
 \JP @node gr hgr gr_mod,,, $B%0%l%V%J4pDl$K4X$9$kH!?t(B
@@ -1263,7 +1515,7 @@ Refer to the sections for each functions.
 @item return
 \JP $B%j%9%H(B
 \EG list
-@item plist, vlist, procs
+@item plist  vlist  procs
 \JP $B%j%9%H(B
 \EG list
 @item order
@@ -1290,13 +1542,21 @@ Refer to the sections for each functions.
 strategy $B$K$h$k7W;;(B, @code{hgr()} $B$O(B trace-lifting $B$*$h$S(B
 $B@F<!2=$K$h$k(B $B6:@5$5$l$?(B sugar strategy $B$K$h$k7W;;$r9T$&(B. 
 @item
-@code{dgr()} $B$O(B, @code{gr()}, @code{dgr()} $B$r(B
+@code{dgr()} $B$O(B, @code{gr()}, @code{hgr()} $B$r(B
 $B;R%W%m%;%9%j%9%H(B @var{procs} $B$N(B 2 $B$D$N%W%m%;%9$K$h$jF1;~$K7W;;$5$;(B, 
 $B@h$K7k2L$rJV$7$?J}$N7k2L$rJV$9(B. $B7k2L$OF10l$G$"$k$,(B, $B$I$A$i$NJ}K!$,(B
 $B9bB.$+0lHL$K$OITL@$N$?$a(B, $B<B:]$N7P2a;~4V$rC;=L$9$k$N$KM-8z$G$"$k(B. 
 @item
 @code{dgr()} $B$GI=<($5$l$k;~4V$O(B, $B$3$NH!?t$,<B9T$5$l$F$$$k%W%m%;%9$G$N(B
 CPU $B;~4V$G$"$j(B, $B$3$NH!?t$N>l9g$O$[$H$s$IDL?.$N$?$a$N;~4V$G$"$k(B. 
+@item
+$BB?9`<0%j%9%H(B @var{plist} $B$NMWAG$,J,;6I=8=B?9`<0$N>l9g$O(B
+$B7k2L$bJ,;6I=8=B?9`<0$N%j%9%H$G$"$k(B.
+$B$3$N>l9g(B, $B0z?t$NJ,;6B?9`<0$OM?$($i$l$?=g=x$K=>$$(B @code{dp_sort} $B$G(B
+$B%=!<%H$5$l$F$+$i7W;;$5$l$k(B.
+$BB?9`<0%j%9%H$NMWAG$,J,;6I=8=B?9`<0$N>l9g$b(B
+$BJQ?t$N?tJ,$NITDj85$N%j%9%H$r(B @var{vlist} $B0z?t$H$7$FM?$($J$$$H$$$1$J$$(B
+($B%@%_!<(B).
 \E
 \BEG
 @item 
@@ -1325,6 +1585,13 @@ Therefore this function is useful to reduce the actual
 The CPU time shown after an exection of @code{dgr()} indicates
 that of the master process, and most of the time corresponds to the time
 for communication.
+@item
+When the elements of @var{plist} are distributed polynomials,
+the result is also a list of distributed polynomials.
+In this case, firstly  the elements of @var{plist} is sorted by @code{dp_sort}
+and the Grobner basis computation is started.
+Variables must be given in @var{vlist} even in this case
+(these variables are dummy).
 \E
 @end itemize
 
@@ -1342,8 +1609,7 @@ for communication.
 @table @t
 \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_gr_main dp_gr_mod_main dp_gr_f_main dp_weyl_gr_main dp_weyl_gr_mod_main dp_weyl_gr_f_main},
 @fref{dp_ord}.
 @end table
 
@@ -1372,7 +1638,7 @@ for communication.
 @item return
 \JP $B%j%9%H(B
 \EG list
-@item plist, vlist1, vlist2, procs
+@item plist  vlist1  vlist2  procs
 \JP $B%j%9%H(B
 \EG list
 @item order
@@ -1560,7 +1826,7 @@ processes.
 @table @t
 \JP @item $B;2>H(B
 \EG @item References
-@fref{dp_gr_main dp_gr_mod_main},
+@fref{dp_gr_main dp_gr_mod_main dp_gr_f_main dp_weyl_gr_main dp_weyl_gr_mod_main dp_weyl_gr_f_main},
 \JP @fref{dp_ord}, @fref{$BJ,;67W;;(B}
 \EG @fref{dp_ord}, @fref{Distributed computation}
 @end table
@@ -1576,8 +1842,8 @@ processes.
 @item lex_hensel_gsl(@var{plist},@var{vlist1},@var{order},@var{vlist2},@var{homo})
 \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})
+@item tolex_gsl(@var{plist},@var{vlist1},@var{order},@var{vlist2})
+@itemx tolex_gsl_d(@var{plist},@var{vlist1},@var{order},@var{vlist2},@var{procs})
 \JP :: $B%0%l%V%J4pDl$rF~NO$H$9$k(B, GSL $B7A<0$N%$%G%"%k4pDl$N7W;;(B
 \EG :: Computation of an GSL form ideal basis stating from a Groebner basis
 @end table
@@ -1586,7 +1852,7 @@ processes.
 @item return
 \JP $B%j%9%H(B
 \EG list
-@item plist, vlist1, vlist2, procs
+@item plist  vlist1  vlist2  procs
 \JP $B%j%9%H(B
 \EG list
 @item order
@@ -1662,7 +1928,8 @@ processes.
 [108] GSL[1];
 [u2,10352277157007342793600000000*u0^31-...]
 [109] GSL[5];
-[u0,11771021876193064124640000000*u0^32-...,376672700038178051988480000000*u0^31-...]
+[u0,11771021876193064124640000000*u0^32-...,
+376672700038178051988480000000*u0^31-...]
 @end example
 
 @table @t
@@ -1692,7 +1959,7 @@ processes.
 @item return
 \JP $BB?9`<0(B
 \EG polynomial
-@item plist, vlist
+@item plist  vlist
 \JP $B%j%9%H(B
 \EG list
 @item order
@@ -1789,7 +2056,7 @@ for @code{gr_minipoly()}.
 @item return
 \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
+@item plist  vlist1  vlist2
 \JP $B%j%9%H(B
 \EG list
 @item order
@@ -1837,15 +2104,23 @@ z^32+11405*z^31+20868*z^30+21602*z^29+...
 @fref{gr_minipoly minipoly}.
 @end table
 
-\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}
+\JP @node dp_gr_main dp_gr_mod_main dp_gr_f_main dp_weyl_gr_main dp_weyl_gr_mod_main dp_weyl_gr_f_main,,, $B%0%l%V%J4pDl$K4X$9$kH!?t(B
+\EG @node dp_gr_main dp_gr_mod_main dp_gr_f_main dp_weyl_gr_main dp_weyl_gr_mod_main dp_weyl_gr_f_main,,, Functions for Groebner basis computation
+@subsection @code{dp_gr_main}, @code{dp_gr_mod_main}, @code{dp_gr_f_main}, @code{dp_weyl_gr_main}, @code{dp_weyl_gr_mod_main}, @code{dp_weyl_gr_f_main}
 @findex dp_gr_main
 @findex dp_gr_mod_main
+@findex dp_gr_f_main
+@findex dp_weyl_gr_main
+@findex dp_weyl_gr_mod_main
+@findex dp_weyl_gr_f_main
 
 @table @t
 @item dp_gr_main(@var{plist},@var{vlist},@var{homo},@var{modular},@var{order})
 @itemx dp_gr_mod_main(@var{plist},@var{vlist},@var{homo},@var{modular},@var{order})
+@itemx dp_gr_f_main(@var{plist},@var{vlist},@var{homo},@var{order})
+@itemx dp_weyl_gr_main(@var{plist},@var{vlist},@var{homo},@var{modular},@var{order})
+@itemx dp_weyl_gr_mod_main(@var{plist},@var{vlist},@var{homo},@var{modular},@var{order})
+@itemx dp_weyl_gr_f_main(@var{plist},@var{vlist},@var{homo},@var{order})
 \JP :: $B%0%l%V%J4pDl$N7W;;(B ($BAH$_9~$_H!?t(B)
 \EG :: Groebner basis computation (built-in functions)
 @end table
@@ -1854,7 +2129,7 @@ z^32+11405*z^31+20868*z^30+21602*z^29+...
 @item return
 \JP $B%j%9%H(B
 \EG list
-@item plist, vlist
+@item plist  vlist
 \JP $B%j%9%H(B
 \EG list
 @item order
@@ -1873,8 +2148,13 @@ z^32+11405*z^31+20868*z^30+21602*z^29+...
 @item
 $B$3$l$i$NH!?t$O(B, $B%0%l%V%J4pDl7W;;$N4pK\E*AH$_9~$_H!?t$G$"$j(B, @code{gr()},
 @code{hgr()}, @code{gr_mod()} $B$J$I$O$9$Y$F$3$l$i$NH!?t$r8F$S=P$7$F7W;;(B
-$B$r9T$C$F$$$k(B. 
+$B$r9T$C$F$$$k(B. $B4X?tL>$K(B weyl $B$,F~$C$F$$$k$b$N$O(B, Weyl $BBe?t>e$N7W;;(B
+$B$N$?$a$N4X?t$G$"$k(B. 
 @item
+@code{dp_gr_f_main()}, @code{dp_weyl_f_main()} $B$O(B, $B<o!9$NM-8BBN>e$N%0%l%V%J4pDl$r7W;;$9$k(B
+$B>l9g$KMQ$$$k(B. $BF~NO$O(B, $B$"$i$+$8$a(B, @code{simp_ff()} $B$J$I$G(B, 
+$B9M$($kM-8BBN>e$K<M1F$5$l$F$$$kI,MW$,$"$k(B. 
+@item
 $B%U%i%0(B @var{homo} $B$,(B 0 $B$G$J$$;~(B, $BF~NO$r@F<!2=$7$F$+$i(B Buchberger $B%"%k%4%j%:%`(B
 $B$r<B9T$9$k(B. 
 @item
@@ -1906,8 +2186,15 @@ z^32+11405*z^31+20868*z^30+21602*z^29+...
 @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.
+are all interfaces to these functions. Functions whose names
+contain weyl are those for computation in Weyl algebra.
 @item
+@code{dp_gr_f_main()} and @code{dp_weyl_gr_f_main()}
+are functions for Groebner basis computation
+over various finite fields. Coefficients of input polynomials 
+must be converted to elements of a finite field 
+currently specified by @code{setmod_ff()}.
+@item
 If @var{homo} is not equal to 0, homogenization is applied before entering
 Buchberger algorithm
 @item
@@ -1945,19 +2232,24 @@ Actual computation is controlled by various parameters
 @fref{dp_ord},
 @fref{dp_gr_flags dp_gr_print},
 @fref{gr hgr gr_mod},
+@fref{setmod_ff},
 \JP @fref{$B7W;;$*$h$SI=<($N@)8f(B}.
 \EG @fref{Controlling Groebner basis computations}
 @end table
 
-\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}
+\JP @node dp_f4_main dp_f4_mod_main dp_weyl_f4_main dp_weyl_f4_mod_main,,, $B%0%l%V%J4pDl$K4X$9$kH!?t(B
+\EG @node dp_f4_main dp_f4_mod_main dp_weyl_f4_main dp_weyl_f4_mod_main,,, Functions for Groebner basis computation
+@subsection @code{dp_f4_main}, @code{dp_f4_mod_main}, @code{dp_weyl_f4_main}, @code{dp_weyl_f4_mod_main}
 @findex dp_f4_main
 @findex dp_f4_mod_main
+@findex dp_weyl_f4_main
+@findex dp_weyl_f4_mod_main
 
 @table @t
 @item dp_f4_main(@var{plist},@var{vlist},@var{order})
 @itemx dp_f4_mod_main(@var{plist},@var{vlist},@var{order})
+@itemx dp_weyl_f4_main(@var{plist},@var{vlist},@var{order})
+@itemx dp_weyl_f4_mod_main(@var{plist},@var{vlist},@var{order})
 \JP :: F4 $B%"%k%4%j%:%`$K$h$k%0%l%V%J4pDl$N7W;;(B ($BAH$_9~$_H!?t(B)
 \EG :: Groebner basis computation by F4 algorithm (built-in functions)
 @end table
@@ -1966,7 +2258,7 @@ Actual computation is controlled by various parameters
 @item return
 \JP $B%j%9%H(B
 \EG list
-@item plist, vlist
+@item plist  vlist
 \JP $B%j%9%H(B
 \EG list
 @item order
@@ -1983,7 +2275,9 @@ F4 $B%"%k%4%j%:%`$O(B, J.C. Faugere $B$K$h$jDs>'$5$
 $B;;K!$G$"$j(B, $BK\<BAu$O(B, $BCf9q>jM>DjM}$K$h$k@~7AJ}Dx<05a2r$rMQ$$$?(B
 $B;n83E*$J<BAu$G$"$k(B. 
 @item
-$B0z?t$*$h$SF0:n$O$=$l$>$l(B @code{dp_gr_main()}, @code{dp_gr_mod_main()}
+$B@F<!2=$N0z?t$,$J$$$3$H$r=|$1$P(B, $B0z?t$*$h$SF0:n$O$=$l$>$l(B 
+@code{dp_gr_main()}, @code{dp_gr_mod_main()},
+@code{dp_weyl_gr_main()}, @code{dp_weyl_gr_mod_main()}
 $B$HF1MM$G$"$k(B. 
 \E
 \BEG
@@ -1995,7 +2289,9 @@ invented by J.C. Faugere. The current implementation o
 uses Chinese Remainder theorem and not highly optimized.
 @item
 Arguments and actions are the same as those of 
-@code{dp_gr_main()}, @code{dp_gr_mod_main()}.
+@code{dp_gr_main()}, @code{dp_gr_mod_main()},
+@code{dp_weyl_gr_main()}, @code{dp_weyl_gr_mod_main()},
+except for lack of the argument for controlling homogenization.
 \E
 @end itemize
 
@@ -2009,6 +2305,152 @@ Arguments and actions are the same as those of 
 \EG @fref{Controlling Groebner basis computations}
 @end table
 
+\JP @node nd_gr nd_gr_trace nd_f4 nd_f4_trace nd_weyl_gr nd_weyl_gr_trace,,, $B%0%l%V%J4pDl$K4X$9$kH!?t(B
+\EG @node nd_gr nd_gr_trace nd_f4 nd_f4_trace nd_weyl_gr nd_weyl_gr_trace,,, Functions for Groebner basis computation
+@subsection @code{nd_gr}, @code{nd_gr_trace}, @code{nd_f4}, @code{nd_f4_trace}, @code{nd_weyl_gr}, @code{nd_weyl_gr_trace}
+@findex nd_gr
+@findex nd_gr_trace
+@findex nd_f4
+@findex nd_f4_trace
+@findex nd_weyl_gr
+@findex nd_weyl_gr_trace
+
+@table @t
+@item nd_gr(@var{plist},@var{vlist},@var{p},@var{order})
+@itemx nd_gr_trace(@var{plist},@var{vlist},@var{homo},@var{p},@var{order})
+@itemx nd_f4(@var{plist},@var{vlist},@var{modular},@var{order})
+@itemx nd_f4_trace(@var{plist},@var{vlist},@var{homo},@var{p},@var{order})
+@item nd_weyl_gr(@var{plist},@var{vlist},@var{p},@var{order})
+@itemx nd_weyl_gr_trace(@var{plist},@var{vlist},@var{homo},@var{p},@var{order})
+\JP :: $B%0%l%V%J4pDl$N7W;;(B ($BAH$_9~$_H!?t(B)
+\EG :: Groebner basis computation (built-in functions)
+@end table
+
+@table @var
+@item return
+\JP $B%j%9%H(B
+\EG list
+@item plist  vlist
+\JP $B%j%9%H(B
+\EG list
+@item order
+\JP $B?t(B, $B%j%9%H$^$?$O9TNs(B
+\EG number, list or matrix
+@item homo
+\JP $B%U%i%0(B
+\EG flag
+@item modular
+\JP $B%U%i%0$^$?$OAG?t(B
+\EG flag or prime
+@end table
+
+\BJP
+@itemize @bullet
+@item
+$B$3$l$i$NH!?t$O(B, $B%0%l%V%J4pDl7W;;AH$_9~$_4X?t$N?7<BAu$G$"$k(B.
+@item @code{nd_gr} $B$O(B, @code{p} $B$,(B 0 $B$N$H$-M-M}?tBN>e$N(B Buchberger
+$B%"%k%4%j%:%`$r<B9T$9$k(B. @code{p} $B$,(B 2 $B0J>e$N<+A3?t$N$H$-(B, GF(p) $B>e$N(B
+Buchberger $B%"%k%4%j%:%`$r<B9T$9$k(B.
+@item @code{nd_gr_trace} $B$*$h$S(B @code{nd_f4_trace}
+$B$OM-M}?tBN>e$G(B trace $B%"%k%4%j%:%`$r<B9T$9$k(B.
+@var{p} $B$,(B 0 $B$^$?$O(B 1 $B$N$H$-(B, $B<+F0E*$KA*$P$l$?AG?t$rMQ$$$F(B, $B@.8y$9$k(B
+$B$^$G(B trace $B%"%k%4%j%:%`$r<B9T$9$k(B.
+@var{p} $B$,(B 2 $B0J>e$N$H$-(B, trace $B$O(BGF(p) $B>e$G7W;;$5$l$k(B. trace $B%"%k%4%j%:%`(B
+$B$,<:GT$7$?>l9g(B 0 $B$,JV$5$l$k(B. @var{p} $B$,Ii$N>l9g(B, $B%0%l%V%J4pDl%A%'%C%/$O(B
+$B9T$o$J$$(B. $B$3$N>l9g(B, @var{p} $B$,(B -1 $B$J$i$P<+F0E*$KA*$P$l$?AG?t$,(B,
+$B$=$l0J30$O;XDj$5$l$?AG?t$rMQ$$$F%0%l%V%J4pDl8uJd$N7W;;$,9T$o$l$k(B. 
+@code{nd_f4_trace} $B$O(B, $B3FA4<!?t$K$D$$$F(B, $B$"$kM-8BBN>e$G(B F4 $B%"%k%4%j%:%`(B
+$B$G9T$C$?7k2L$r$b$H$K(B, $B$=$NM-8BBN>e$G(B 0 $B$G$J$$4pDl$rM?$($k(B S-$BB?9`<0$N$_$r(B
+$BMQ$$$F9TNs@8@.$r9T$$(B, $B$=$NA4<!?t$K$*$1$k4pDl$r@8@.$9$kJ}K!$G$"$k(B. $BF@$i$l$k(B
+$BB?9`<0=89g$O$d$O$j%0%l%V%J4pDl8uJd$G$"$j(B, @code{nd_gr_trace} $B$HF1MM$N(B
+$B%A%'%C%/$,9T$o$l$k(B.
+@item
+@code{nd_f4} $B$O(B @code{modular} $B$,(B 0 $B$N$H$-M-M}?tBN>e$N(B, @code{modular} $B$,(B
+$B%^%7%s%5%$%:AG?t$N$H$-M-8BBN>e$N(B F4 $B%"%k%4%j%:%`$r<B9T$9$k(B.
+@item
+@var{plist} $B$,B?9`<0%j%9%H$N>l9g(B, @var{plist}$B$G@8@.$5$l$k%$%G%"%k$N%0%l%V%J!<4pDl$,(B
+$B7W;;$5$l$k(B. @var{plist} $B$,B?9`<0%j%9%H$N%j%9%H$N>l9g(B, $B3FMWAG$OB?9`<04D>e$N<+M32C72$N85$H8+$J$5$l(B,
+$B$3$l$i$,@8@.$9$kItJ,2C72$N%0%l%V%J!<4pDl$,7W;;$5$l$k(B. $B8e<T$N>l9g(B, $B9`=g=x$O2C72$KBP$9$k9`=g=x$r(B
+$B;XDj$9$kI,MW$,$"$k(B. $B$3$l$O(B @var{[s,ord]} $B$N7A$G;XDj$9$k(B. @var{s} $B$,(B 0 $B$J$i$P(B TOP (Term Over Position),
+1 $B$J$i$P(B POT (Position Over Term) $B$r0UL#$7(B, @var{ord} $B$OB?9`<04D$NC19`<0$KBP$9$k9`=g=x$G$"$k(B.
+@item
+@code{nd_weyl_gr}, @code{nd_weyl_gr_trace} $B$O(B Weyl $BBe?tMQ$G$"$k(B.
+@item
+@code{f4} $B7O4X?t0J30$O$9$Y$FM-M}4X?t78?t$N7W;;$,2DG=$G$"$k(B.
+@item
+$B0lHL$K(B @code{dp_gr_main}, @code{dp_gr_mod_main} $B$h$j9bB.$G$"$k$,(B,
+$BFC$KM-8BBN>e$N>l9g82Cx$G$"$k(B.
+@end itemize
+\E
+
+\BEG
+@itemize @bullet
+@item
+These functions are new implementations for computing Groebner bases.
+@item @code{nd_gr} executes Buchberger algorithm over the rationals
+if  @code{p} is 0, and that over GF(p) if @code{p} is a prime.
+@item @code{nd_gr_trace} executes the trace algorithm over the rationals.
+If @code{p} is 0 or 1, the trace algorithm is executed until it succeeds
+by using automatically chosen primes.
+If @code{p} a positive prime, 
+the trace is comuted over GF(p). 
+If the trace algorithm fails 0 is returned.
+If @code{p} is negative, 
+the Groebner basis check and ideal-membership check are omitted.
+In this case, an automatically chosen prime if @code{p} is 1,
+otherwise the specified prime is used to compute a Groebner basis
+candidate.
+Execution of @code{nd_f4_trace} is done as follows:
+For each total degree, an F4-reduction of S-polynomials over a finite field
+is done, and S-polynomials which give non-zero basis elements are gathered.
+Then F4-reduction over Q is done for the gathered S-polynomials.
+The obtained polynomial set is a Groebner basis candidate and the same
+check procedure as in the case of @code{nd_gr_trace} is done.
+@item
+@code{nd_f4} executes F4 algorithm over Q if @code{modular} is equal to 0,
+or over a finite field GF(@code{modular}) 
+if @code{modular} is a prime number of machine size (<2^29).
+If @var{plist} is a list of polynomials, then a Groebner basis of the ideal generated by @var{plist}
+is computed. If @var{plist} is a list of lists of polynomials, then each list of polynomials are regarded
+as an element of a free module over a polynomial ring and a Groebner basis of the sub-module generated by @var{plist}
+in the free module. In the latter case a term order in the free module should be specified.
+This is specified by @var{[s,ord]}. If @var{s} is 0 then it means TOP (Term Over Position).
+If @var{s} is 1 then it means POT 1 (Position Over Term). @var{ord} is a term order in the base polynomial ring.
+@item
+@code{nd_weyl_gr}, @code{nd_weyl_gr_trace} are for Weyl algebra computation.
+@item
+Functions except for F4 related ones can handle rational coeffient cases.
+@item
+In general these functions are more efficient than 
+@code{dp_gr_main}, @code{dp_gr_mod_main}, especially over finite fields.
+@end itemize
+\E
+
+@example
+[38] load("cyclic")$
+[49] C=cyclic(7)$
+[50] V=vars(C)$
+[51] cputime(1)$
+[52] dp_gr_mod_main(C,V,0,31991,0)$
+26.06sec + gc : 0.313sec(26.4sec)
+[53] nd_gr(C,V,31991,0)$
+ndv_alloc=1477188
+5.737sec + gc : 0.1837sec(5.921sec)
+[54] dp_f4_mod_main(C,V,31991,0)$  
+3.51sec + gc : 0.7109sec(4.221sec)
+[55] nd_f4(C,V,31991,0)$
+1.906sec + gc : 0.126sec(2.032sec)
+@end example
+
+@table @t
+\JP @item $B;2>H(B
+\EG @item References
+@fref{dp_ord},
+@fref{dp_gr_flags dp_gr_print},
+\JP @fref{$B7W;;$*$h$SI=<($N@)8f(B}.
+\EG @fref{Controlling Groebner basis computations}
+@end table
+
 \JP @node 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}
@@ -2017,7 +2459,7 @@ Arguments and actions are the same as those of 
 
 @table @t
 @item dp_gr_flags([@var{list}])
-@itemx dp_gr_print([@var{0|1}])
+@itemx dp_gr_print([@var{i}])
 \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.
@@ -2031,12 +2473,15 @@ and showing informations.
 @item list
 \JP $B%j%9%H(B
 \EG list
+@item i
+\JP $B@0?t(B
+\EG integer
 @end table
 
 @itemize @bullet
 \BJP
 @item
-@code{dp_gr_main()}, @code{dp_gr_mod_main()} $B<B9T;~$K$*$1$k$5$^$6$^(B
+@code{dp_gr_main()}, @code{dp_gr_mod_main()}, @code{dp_gr_f_main()}  $B<B9T;~$K$*$1$k$5$^$6$^(B
 $B$J%Q%i%a%?$r@_Dj(B, $B;2>H$9$k(B. 
 @item
 $B0z?t$,$J$$>l9g(B, $B8=:_$N@_Dj$,JV$5$l$k(B. 
@@ -2044,9 +2489,18 @@ and showing informations.
 $B0z?t$O(B, @code{["Print",1,"NoSugar",1,...]} $B$J$k7A$N%j%9%H$G(B, $B:8$+$i=g$K(B
 $B@_Dj$5$l$k(B. $B%Q%i%a%?L>$OJ8;zNs$GM?$($kI,MW$,$"$k(B. 
 @item
-@code{dp_gr_print()} $B$O(B, $BFC$K%Q%i%a%?(B @code{Print} $B$NCM$rD>@\@_Dj(B, $B;2>H(B
-$B$G$-$k(B. $B$3$l$O(B, @code{dp_gr_main()} $B$J$I$r%5%V%k!<%A%s$H$7$FMQ$$$k%f!<%6(B
-$BH!?t$K$*$$$F(B, @code{Print} $B$NCM$r8+$F(B, $B$=$N%5%V%k!<%A%s$,Cf4V>pJs$NI=<((B
+@code{dp_gr_print()} $B$O(B, $BFC$K%Q%i%a%?(B @code{Print}, @code{PrintShort} $B$NCM$rD>@\@_Dj(B, $B;2>H(B
+$B$G$-$k(B. $B@_Dj$5$l$kCM$O<!$NDL$j$G$"$k!#(B
+@table @var
+@item i=0
+@code{Print=0}, @code{PrintShort=0}
+@item i=1
+@code{Print=1}, @code{PrintShort=0}
+@item i=2
+@code{Print=0}, @code{PrintShort=1}
+@end table
+$B$3$l$O(B, @code{dp_gr_main()} $B$J$I$r%5%V%k!<%A%s$H$7$FMQ$$$k%f!<%6(B
+$BH!?t$K$*$$$F(B, $B$=$N%5%V%k!<%A%s$,Cf4V>pJs$NI=<((B
 $B$r9T$&:]$K(B, $B?WB.$K%U%i%0$r8+$k$3$H$,$G$-$k$h$&$KMQ0U$5$l$F$$$k(B. 
 \E
 \BEG
@@ -2061,8 +2515,17 @@ Arguments must be specified as a list such as
 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.
+@code{Print} and @code{PrintShort}. 
+@table @var
+@item i=0
+@code{Print=0}, @code{PrintShort=0}
+@item i=1
+@code{Print=1}, @code{PrintShort=0}
+@item i=2
+@code{Print=0}, @code{PrintShort=1}
+@end table
+This functions is prepared to get quickly the value
+when a user defined function calling @code{dp_gr_main()} etc.
 uses the value as a flag for showing intermediate informations.
 \E
 @end itemize
@@ -2164,6 +2627,79 @@ when functions other than top level functions are call
 \EG @fref{Setting term orderings}
 @end table
 
+\JP @node dp_set_weight dp_set_top_weight dp_weyl_set_weight,,, $B%0%l%V%J4pDl$K4X$9$kH!?t(B
+\EG @node dp_set_weight dp_set_top_weight dp_weyl_set_weight,,, Functions for Groebner basis computation
+@subsection @code{dp_set_weight}, @code{dp_set_top_weight}, @code{dp_weyl_set_weight}
+@findex dp_set_weight
+@findex dp_set_top_weight
+@findex dp_weyl_set_weight
+
+@table @t
+@item dp_set_weight([@var{weight}])
+\JP :: sugar weight $B$N@_Dj(B, $B;2>H(B
+\EG :: Set and show the sugar weight.
+@item dp_set_top_weight([@var{weight}])
+\JP :: top weight $B$N@_Dj(B, $B;2>H(B
+\EG :: Set and show the top weight.
+@item dp_weyl_set_weight([@var{weight}])
+\JP :: weyl weight $B$N@_Dj(B, $B;2>H(B
+\EG :: Set and show the weyl weight.
+@end table
+
+@table @var
+@item return
+\JP $B%Y%/%H%k(B
+\EG a vector
+@item weight
+\JP $B@0?t$N%j%9%H$^$?$O%Y%/%H%k(B
+\EG a list or vector of integers
+@end table
+
+@itemize @bullet
+\BJP
+@item
+@code{dp_set_weight} $B$O(B sugar weight $B$r(B @var{weight} $B$K@_Dj$9$k(B. $B0z?t$,$J$$;~(B, 
+$B8=:_@_Dj$5$l$F$$$k(B sugar weight $B$rJV$9(B. sugar weight $B$O@5@0?t$r@.J,$H$9$k%Y%/%H%k$G(B,
+$B3FJQ?t$N=E$_$rI=$9(B. $B<!?t$D$-=g=x$K$*$$$F(B, $BC19`<0$N<!?t$r7W;;$9$k:]$KMQ$$$i$l$k(B.
+$B@F<!2=JQ?tMQ$K(B, $BKvHx$K(B 1 $B$rIU$12C$($F$*$/$H0BA4$G$"$k(B.
+@item
+@code{dp_set_top_weight} $B$O(B top weight $B$r(B @var{weight} $B$K@_Dj$9$k(B. $B0z?t$,$J$$;~(B, 
+$B8=:_@_Dj$5$l$F$$$k(B top weight $B$rJV$9(B. top weight $B$,@_Dj$5$l$F$$$k$H$-(B, 
+$B$^$:(B top weight $B$K$h$kC19`<0Hf3S$r@h$K9T$&(B. tie breaker $B$H$7$F8=:_@_Dj$5$l$F$$$k(B
+$B9`=g=x$,MQ$$$i$l$k$,(B, $B$3$NHf3S$K$O(B top weight $B$OMQ$$$i$l$J$$(B.
+
+@item
+@code{dp_weyl_set_weight} $B$O(B weyl weight $B$r(B @var{weight} $B$K@_Dj$9$k(B. $B0z?t$,$J$$;~(B, 
+$B8=:_@_Dj$5$l$F$$$k(B weyl weight $B$rJV$9(B. weyl weight w $B$r@_Dj$9$k$H(B,
+$B9`=g=x7?(B 11 $B$G$N7W;;$K$*$$$F(B, (-w,w) $B$r(B top weight, tie breaker $B$r(B graded reverse lex
+$B$H$7$?9`=g=x$,@_Dj$5$l$k(B.
+\E
+\BEG
+@item
+@code{dp_set_weight} sets the sugar weight=@var{weight}. It returns the current sugar weight.
+A sugar weight is a vector with positive integer components and it represents the weights of variables.
+It is used for computing the weight of a monomial in a graded ordering.
+It is recommended to append a component 1 at the end of the weight vector for a homogenizing variable.
+@item
+@code{dp_set_top_weight} sets the top weight=@var{weight}. It returns the current top weight.
+It a top weight is set, the weights of monomials under the top weight are firstly compared.
+If the the weights are equal then the current term ordering is applied as a tie breaker, but
+the top weight is not used in the tie breaker.
+
+@item
+@code{dp_weyl_set_weight} sets the weyl weigh=@var{weight}. It returns the current weyl weight.
+If a weyl weight w is set, in the comparsion by the term order type 11, a term order with 
+the top weight=(-w,w) and the tie breaker=graded reverse lex is applied.
+\E
+@end itemize
+
+@table @t
+\JP @item $B;2>H(B
+\EG @item References
+@fref{Weight}
+@end table
+
+
 \JP @node dp_ptod,,, $B%0%l%V%J4pDl$K4X$9$kH!?t(B
 \EG @node dp_ptod,,, Functions for Groebner basis computation
 @subsection @code{dp_ptod}
@@ -2212,7 +2748,8 @@ the coefficient field.
 (1)*<<2,0,0>>+(2)*<<1,1,0>>+(1)*<<0,2,0>>+(2)*<<1,0,1>>+(2)*<<0,1,1>>
 +(1)*<<0,0,2>>
 [52] dp_ptod((x+y+z)^2,[x,y]);  
-(1)*<<2,0>>+(2)*<<1,1>>+(1)*<<0,2>>+(2*z)*<<1,0>>+(2*z)*<<0,1>>+(z^2)*<<0,0>>
+(1)*<<2,0>>+(2)*<<1,1>>+(1)*<<0,2>>+(2*z)*<<1,0>>+(2*z)*<<0,1>>
++(z^2)*<<0,0>>
 @end example
 
 @table @t
@@ -2264,7 +2801,8 @@ variables of @var{dpoly}.
 
 @example
 [53] T=dp_ptod((x+y+z)^2,[x,y]);
-(1)*<<2,0>>+(2)*<<1,1>>+(1)*<<0,2>>+(2*z)*<<1,0>>+(2*z)*<<0,1>>+(z^2)*<<0,0>>
+(1)*<<2,0>>+(2)*<<1,1>>+(1)*<<0,2>>+(2*z)*<<1,0>>+(2*z)*<<0,1>>
++(z^2)*<<0,0>>
 [54] P=dp_dtop(T,[a,b]);
 z^2+(2*a+2*b)*z+a^2+2*b*a+b^2
 @end example
@@ -2344,7 +2882,7 @@ converting the coefficients into elements of a finite 
 @table @t
 \JP @item $B;2>H(B
 \EG @item References
-@fref{dp_nf dp_nf_mod dp_true_nf dp_true_nf_mod},
+@fref{dp_nf dp_nf_mod dp_true_nf dp_true_nf_mod dp_weyl_nf dp_weyl_nf_mod},
 @fref{subst psubst},
 @fref{setmod}.
 @end table
@@ -2488,17 +3026,21 @@ polynomial contents included in the coefficients are n
 @fref{ptozp}.
 @end table
 
-\JP @node dp_nf dp_nf_mod dp_true_nf dp_true_nf_mod,,, $B%0%l%V%J4pDl$K4X$9$kH!?t(B
-\EG @node dp_nf dp_nf_mod dp_true_nf dp_true_nf_mod,,, Functions for Groebner basis computation
+\JP @node dp_nf dp_nf_mod dp_true_nf dp_true_nf_mod dp_weyl_nf dp_weyl_nf_mod,,, $B%0%l%V%J4pDl$K4X$9$kH!?t(B
+\EG @node dp_nf dp_nf_mod dp_true_nf dp_true_nf_mod dp_weyl_nf dp_weyl_nf_mod,,, Functions for Groebner basis computation
 @subsection @code{dp_nf}, @code{dp_nf_mod}, @code{dp_true_nf}, @code{dp_true_nf_mod}
 @findex dp_nf
 @findex  dp_true_nf
 @findex dp_nf_mod
 @findex  dp_true_nf_mod
+@findex dp_weyl_nf
+@findex dp_weyl_nf_mod
 
 @table @t
 @item dp_nf(@var{indexlist},@var{dpoly},@var{dpolyarray},@var{fullreduce})
+@item dp_weyl_nf(@var{indexlist},@var{dpoly},@var{dpolyarray},@var{fullreduce})
 @item dp_nf_mod(@var{indexlist},@var{dpoly},@var{dpolyarray},@var{fullreduce},@var{mod})
+@item dp_weyl_nf_mod(@var{indexlist},@var{dpoly},@var{dpolyarray},@var{fullreduce},@var{mod})
 \JP :: $BJ,;6I=8=B?9`<0$N@55,7A$r5a$a$k(B. ($B7k2L$ODj?tG\$5$l$F$$$k2DG=@-$"$j(B)
 
 \BEG
@@ -2540,6 +3082,8 @@ is returned in such a list as @code{[numerator, denomi
 @item
 $BJ,;6I=8=B?9`<0(B @var{dpoly} $B$N@55,7A$r5a$a$k(B. 
 @item
+$BL>A0$K(B weyl $B$r4^$`4X?t$O%o%$%kBe?t$K$*$1$k@55,7A7W;;$r9T$&(B. $B0J2<$N@bL@$O(B weyl $B$r4^$`$b$N$KBP$7$F$bF1MM$K@.N)$9$k(B.
+@item
 @code{dp_nf_mod()}, @code{dp_true_nf_mod()} $B$NF~NO$O(B, @code{dp_mod()} $B$J$I(B
 $B$K$h$j(B, $BM-8BBN>e$NJ,;6I=8=B?9`<0$K$J$C$F$$$J$1$l$P$J$i$J$$(B. 
 @item
@@ -2572,6 +3116,9 @@ is returned in such a list as @code{[numerator, denomi
 @item
 Computes the normal form of a distributed polynomial.
 @item
+Functions whose name contain @code{weyl} compute normal forms in Weyl algebra. The description below also applies to
+the functions for Weyl algebra.
+@item
 @code{dp_nf_mod()} and @code{dp_true_nf_mod()} require
 distributed polynomials with coefficients in a finite field as arguments.
 @item
@@ -2617,15 +3164,18 @@ For single computation @code{p_nf} and @code{p_true_nf
 [74] DP2=newvect(length(G),map(dp_ptod,G,V))$
 [75] T=dp_ptod((u0-u1+u2-u3+u4)^2,V)$
 [76] dp_dtop(dp_nf([0,1,2,3,4],T,DP1,1),V);
-u4^2+(6*u3+2*u2+6*u1-2)*u4+9*u3^2+(6*u2+18*u1-6)*u3+u2^2+(6*u1-2)*u2+9*u1^2-6*u1+1
+u4^2+(6*u3+2*u2+6*u1-2)*u4+9*u3^2+(6*u2+18*u1-6)*u3+u2^2
++(6*u1-2)*u2+9*u1^2-6*u1+1
 [77] dp_dtop(dp_nf([4,3,2,1,0],T,DP1,1),V);
 -5*u4^2+(-4*u3-4*u2-4*u1)*u4-u3^2-3*u3-u2^2+(2*u1-1)*u2-2*u1^2-3*u1+1
 [78] dp_dtop(dp_nf([0,1,2,3,4],T,DP2,1),V);
--1138087976845165778088612297273078520347097001020471455633353049221045677593
-0005716505560062087150928400876150217079820311439477560587583488*u4^15+...
+-11380879768451657780886122972730785203470970010204714556333530492210
+456775930005716505560062087150928400876150217079820311439477560587583
+488*u4^15+...
 [79] dp_dtop(dp_nf([4,3,2,1,0],T,DP2,1),V);
--1138087976845165778088612297273078520347097001020471455633353049221045677593
-0005716505560062087150928400876150217079820311439477560587583488*u4^15+...
+-11380879768451657780886122972730785203470970010204714556333530492210
+456775930005716505560062087150928400876150217079820311439477560587583
+488*u4^15+...
 [80] @@78==@@79;
 1
 @end example
@@ -2791,7 +3341,7 @@ selection strategy of critical pairs in Groebner basis
 @item return
 \JP $BJ,;6I=8=B?9`<0(B
 \EG distributed polynomial
-@item dpoly1, dpoly2
+@item dpoly1  dpoly2
 \JP $BJ,;6I=8=B?9`<0(B
 \EG distributed polynomial
 @end table
@@ -2834,7 +3384,7 @@ two polynomials, where coefficient is always set to 1.
 @item return
 \JP $B@0?t(B
 \EG integer
-@item dpoly1, dpoly2
+@item dpoly1  dpoly2
 \JP $BJ,;6I=8=B?9`<0(B
 \EG distributed polynomial
 @end table
@@ -2889,7 +3439,7 @@ Used for finding candidate terms at reduction of polyn
 @item return
 \JP $BJ,;6I=8=B?9`<0(B
 \EG distributed polynomial
-@item dpoly1, dpoly2
+@item dpoly1  dpoly2
 \JP $BJ,;6I=8=B?9`<0(B
 \EG distributed polynomial
 @end table
@@ -3113,7 +3663,7 @@ values of @code{dp_mag()} for intermediate basis eleme
 @item return
 \JP $B%j%9%H(B
 \EG list
-@item dpoly1, dpoly2, dpoly3
+@item dpoly1  dpoly2  dpoly3
 \JP $BJ,;6I=8=B?9`<0(B
 \EG distributed polynomial
 @item vlist
@@ -3137,7 +3687,7 @@ values of @code{dp_mag()} for intermediate basis eleme
 $B$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
@@ -3156,7 +3706,7 @@ the divisibility of the head term of @var{dpoly2} by t
 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(dpoly1 + dpoly2)-bt dpoly3}.
+@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
@@ -3170,8 +3720,8 @@ The result is a list @code{[@var{a dpoly1},@var{a dpol
 [159] C=12*<<1,1,1,0,0>>+(1)*<<0,1,1,1,0>>+(1)*<<1,1,0,0,1>>; 
 (12)*<<1,1,1,0,0>>+(1)*<<0,1,1,1,0>>+(1)*<<1,1,0,0,1>>
 [160] dp_red(D,R,C);                                         
-[(6)*<<2,1,0,0,0>>+(6)*<<1,2,0,0,0>>+(2)*<<0,3,0,0,0>>,(-1)*<<0,1,1,1,0>>
-+(-1)*<<1,1,0,0,1>>]
+[(6)*<<2,1,0,0,0>>+(6)*<<1,2,0,0,0>>+(2)*<<0,3,0,0,0>>,
+(-1)*<<0,1,1,1,0>>+(-1)*<<1,1,0,0,1>>]
 @end example
 
 @table @t
@@ -3197,7 +3747,7 @@ The result is a list @code{[@var{a dpoly1},@var{a dpol
 @item return
 \JP $BJ,;6I=8=B?9`<0(B
 \EG distributed polynomial
-@item dpoly1, dpoly2
+@item dpoly1  dpoly2
 \JP $BJ,;6I=8=B?9`<0(B
 \EG distributed polynomial
 @item mod
@@ -3273,7 +3823,7 @@ as a form of @code{[numerator, denominator]})
 @item poly
 \JP $BB?9`<0(B
 \EG polynomial
-@item plist,vlist
+@item plist vlist
 \JP $B%j%9%H(B
 \EG list
 @item order
@@ -3409,9 +3959,9 @@ exists.
 @example
 [233] G=gr(katsura(5),[u5,u4,u3,u2,u1,u0],2)$
 [234] p_terms(G[0],[u5,u4,u3,u2,u1,u0],2);
-[u5,u0^31,u0^30,u0^29,u0^28,u0^27,u0^26,u0^25,u0^24,u0^23,u0^22,u0^21,u0^20,
-u0^19,u0^18,u0^17,u0^16,u0^15,u0^14,u0^13,u0^12,u0^11,u0^10,u0^9,u0^8,u0^7,
-u0^6,u0^5,u0^4,u0^3,u0^2,u0,1]
+[u5,u0^31,u0^30,u0^29,u0^28,u0^27,u0^26,u0^25,u0^24,u0^23,u0^22,
+u0^21,u0^20,u0^19,u0^18,u0^17,u0^16,u0^15,u0^14,u0^13,u0^12,u0^11,
+u0^10,u0^9,u0^8,u0^7,u0^6,u0^5,u0^4,u0^3,u0^2,u0,1]
 @end example
 
 \JP @node gb_comp,,, $B%0%l%V%J4pDl$K4X$9$kH!?t(B
@@ -3428,7 +3978,7 @@ u0^6,u0^5,u0^4,u0^3,u0^2,u0,1]
 @table @var
 \JP @item return 0 $B$^$?$O(B 1
 \EG @item return 0 or 1
-@item plist1, plist2
+@item plist1  plist2
 @end table
 
 @itemize @bullet
@@ -3519,8 +4069,8 @@ Polynomial set @code{cyclic} is sometimes called by ot
 [79] load("cyclic")$ 
 [89] katsura(5);
 [u0+2*u4+2*u3+2*u2+2*u1+2*u5-1,2*u4*u0-u4+2*u1*u3+u2^2+2*u5*u1,
-2*u3*u0+2*u1*u4-u3+(2*u1+2*u5)*u2,2*u2*u0+2*u2*u4+(2*u1+2*u5)*u3-u2+u1^2,
-2*u1*u0+(2*u3+2*u5)*u4+2*u2*u3+2*u1*u2-u1,
+2*u3*u0+2*u1*u4-u3+(2*u1+2*u5)*u2,2*u2*u0+2*u2*u4+(2*u1+2*u5)*u3
+-u2+u1^2,2*u1*u0+(2*u3+2*u5)*u4+2*u2*u3+2*u1*u2-u1,
 u0^2-u0+2*u4^2+2*u3^2+2*u2^2+2*u1^2+2*u5^2]
 [90] hkatsura(5);
 [-t+u0+2*u4+2*u3+2*u2+2*u1+2*u5,
@@ -3642,3 +4192,215 @@ if an input ideal is not radical.
 \JP @fref{$B9`=g=x$N@_Dj(B}.
 \EG @fref{Setting term orderings}.
 @end table
+
+\JP @node primedec_mod,,, $B%0%l%V%J4pDl$K4X$9$kH!?t(B
+\EG @node primedec_mod,,, Functions for Groebner basis computation
+@subsection @code{primedec_mod}
+@findex primedec_mod
+
+@table @t
+@item primedec_mod(@var{plist},@var{vlist},@var{ord},@var{mod},@var{strategy})
+\JP :: $B%$%G%"%k$NJ,2r(B
+\EG :: Computes decompositions of ideals over small finite fields.
+@end table
+
+@table @var
+@item return
+@itemx plist
+\JP $BB?9`<0%j%9%H(B
+\EG list of polynomials
+@item vlist
+\JP $BJQ?t%j%9%H(B
+\EG list of variables
+@item ord
+\JP $B?t(B, $B%j%9%H$^$?$O9TNs(B
+\EG number, list or matrix
+@item mod
+\JP $B@5@0?t(B
+\EG positive integer
+@item strategy
+\JP $B@0?t(B
+\EG integer
+@end table
+
+@itemize @bullet
+\BJP
+@item
+@code{primedec_mod()} $B$O(B @samp{primdec_mod}
+$B$GDj5A$5$l$F$$$k(B. @code{[Yokoyama]} $B$NAG%$%G%"%kJ,2r%"%k%4%j%:%`(B
+$B$r<BAu$7$F$$$k(B. 
+@item
+@code{primedec_mod()} $B$OM-8BBN>e$G$N%$%G%"%k$N(B
+$B:,4p$NAG%$%G%"%kJ,2r$r9T$$(B, $BAG%$%G%"%k$N%j%9%H$rJV$9(B. 
+@item
+@code{primedec_mod()} $B$O(B, GF(@var{mod}) $B>e$G$NJ,2r$rM?$($k(B. 
+$B7k2L$N3F@.J,$N@8@.85$O(B, $B@0?t78?tB?9`<0$G$"$k(B. 
+@item
+$B7k2L$K$*$$$F(B, $BB?9`<0%j%9%H$H$7$FI=<($5$l$F$$$k3F%$%G%"%k$OA4$F(B
+[@var{vlist},@var{ord}] $B$G;XDj$5$l$k9`=g=x$K4X$9$k%0%l%V%J4pDl$G$"$k(B.
+@item
+@var{strategy} $B$,(B 0 $B$G$J$$$H$-(B, incremental $B$K(B component $B$N6&DL(B
+$BItJ,$r7W;;$9$k$3$H$K$h$k(B early termination $B$r9T$&(B. $B0lHL$K(B, 
+$B%$%G%"%k$N<!85$,9b$$>l9g$KM-8z$@$,(B, 0 $B<!85$N>l9g$J$I(B, $B<!85$,>.$5$$(B
+$B>l9g$K$O(B overhead $B$,Bg$-$$>l9g$,$"$k(B. 
+@item
+$B7W;;ESCf$GFbIt>pJs$r8+$?$$>l9g$K$O!"(B
+$BA0$b$C$F(B @code{dp_gr_print(2)} $B$r<B9T$7$F$*$1$P$h$$(B.
+\E
+\BEG
+@item
+Function @code{primedec_mod()}
+is defined in @samp{primdec_mod} and implements the prime decomposition
+algorithm in @code{[Yokoyama]}.
+@item
+@code{primedec_mod()}
+is the function for prime ideal decomposition 
+of the radical of a polynomial ideal over small finite field,
+and they return a list of prime ideals, which are associated primes
+of the input ideal.
+@item
+@code{primedec_mod()} gives the decomposition over GF(@var{mod}).
+The generators of each resulting component consists of integral polynomials.
+@item
+Each resulting component is a Groebner basis with respect to
+a term order specified by [@var{vlist},@var{ord}].
+@item
+If @var{strategy} is non zero, then the early termination strategy
+is tried by computing the intersection of obtained components
+incrementally. In general, this strategy is useful when the krull
+dimension of the ideal is high, but it may add some overhead
+if the dimension is small.
+@item
+If you want to see internal information during the computation,
+execute @code{dp_gr_print(2)} in advance.
+\E
+@end itemize
+
+@example
+[0] load("primdec_mod")$
+[246] PP444=[x^8+x^2+t,y^8+y^2+t,z^8+z^2+t]$
+[247] primedec_mod(PP444,[x,y,z,t],0,2,1);
+[[y+z,x+z,z^8+z^2+t],[x+y,y^2+y+z^2+z+1,z^8+z^2+t],
+[y+z+1,x+z+1,z^8+z^2+t],[x+z,y^2+y+z^2+z+1,z^8+z^2+t],
+[y+z,x^2+x+z^2+z+1,z^8+z^2+t],[y+z+1,x^2+x+z^2+z+1,z^8+z^2+t],
+[x+z+1,y^2+y+z^2+z+1,z^8+z^2+t],[y+z+1,x+z,z^8+z^2+t],
+[x+y+1,y^2+y+z^2+z+1,z^8+z^2+t],[y+z,x+z+1,z^8+z^2+t]]
+[248] 
+@end example
+
+@table @t
+\JP @item $B;2>H(B
+\EG @item References
+@fref{modfctr},
+@fref{dp_gr_main dp_gr_mod_main dp_gr_f_main dp_weyl_gr_main dp_weyl_gr_mod_main dp_weyl_gr_f_main},
+\JP @fref{$B9`=g=x$N@_Dj(B}.
+\EG @fref{Setting term orderings},
+@fref{dp_gr_flags dp_gr_print}.
+@end table
+
+\JP @node bfunction bfct generic_bfct ann ann0,,, $B%0%l%V%J4pDl$K4X$9$kH!?t(B
+\EG @node bfunction bfct generic_bfct ann ann0,,, Functions for Groebner basis computation
+@subsection @code{bfunction}, @code{bfct}, @code{generic_bfct}, @code{ann}, @code{ann0}
+@findex bfunction
+@findex bfct
+@findex generic_bfct
+@findex ann
+@findex ann0
+
+@table @t
+@item bfunction(@var{f})
+@itemx bfct(@var{f})
+@itemx generic_bfct(@var{plist},@var{vlist},@var{dvlist},@var{weight})
+\JP :: @var{b} $B4X?t$N7W;;(B
+\EG :: Computes the global @var{b} function of a polynomial or an ideal
+@item ann(@var{f})
+@itemx ann0(@var{f})
+\JP :: $BB?9`<0$N%Y%-$N(B annihilator $B$N7W;;(B
+\EG :: Computes the annihilator of a power of polynomial
+@end table
+
+@table @var
+@item return
+\JP $BB?9`<0$^$?$O%j%9%H(B
+\EG polynomial or list
+@item f
+\JP $BB?9`<0(B
+\EG polynomial
+@item plist
+\JP $BB?9`<0%j%9%H(B
+\EG list of polynomials
+@item vlist dvlist
+\JP $BJQ?t%j%9%H(B
+\EG list of variables
+@end table
+
+@itemize @bullet
+\BJP
+@item @samp{bfct} $B$GDj5A$5$l$F$$$k(B. 
+@item @code{bfunction(@var{f})}, @code{bfct(@var{f})} $B$OB?9`<0(B @var{f} $B$N(B global @var{b} $B4X?t(B @code{b(s)} $B$r(B
+$B7W;;$9$k(B. @code{b(s)} $B$O(B, Weyl $BBe?t(B @code{D} $B>e$N0lJQ?tB?9`<04D(B @code{D[s]}
+$B$N85(B @code{P(x,s)} $B$,B8:_$7$F(B, @code{P(x,s)f^(s+1)=b(s)f^s} $B$rK~$?$9$h$&$J(B
+$BB?9`<0(B @code{b(s)} $B$NCf$G(B, $B<!?t$,:G$bDc$$$b$N$G$"$k(B. 
+@item @code{generic_bfct(@var{f},@var{vlist},@var{dvlist},@var{weight})}
+$B$O(B, @var{plist} $B$G@8@.$5$l$k(B @code{D} $B$N:8%$%G%"%k(B @code{I} $B$N(B, 
+$B%&%'%$%H(B @var{weight} $B$K4X$9$k(B global @var{b} $B4X?t$r7W;;$9$k(B. 
+@var{vlist} $B$O(B @code{x}-$BJQ?t(B, @var{vlist} $B$OBP1~$9$k(B @code{D}-$BJQ?t(B
+$B$r=g$KJB$Y$k(B. 
+@item @code{bfunction} $B$H(B @code{bfct} $B$G$OMQ$$$F$$$k%"%k%4%j%:%`$,(B
+$B0[$J$k(B. $B$I$A$i$,9bB.$+$OF~NO$K$h$k(B.
+@item @code{ann(@var{f})} $B$O(B, @code{@var{f}^s} $B$N(B annihilator ideal
+$B$N@8@.7O$rJV$9(B. @code{ann(@var{f})} $B$O(B, @code{[@var{a},@var{list}]}
+$B$J$k%j%9%H$rJV$9(B. $B$3$3$G(B, @var{a} $B$O(B @var{f} $B$N(B @var{b} $B4X?t$N:G>.@0?t:,(B,
+@var{list} $B$O(B @code{ann(@var{f})} $B$N7k2L$N(B @code{s}$ $B$K(B, @var{a} $B$r(B
+$BBeF~$7$?$b$N$G$"$k(B.
+@item $B>\:Y$K$D$$$F$O(B, [Saito,Sturmfels,Takayama] $B$r8+$h(B. 
+\E
+\BEG
+@item These functions are defined in @samp{bfct}.
+@item @code{bfunction(@var{f})} and @code{bfct(@var{f})} compute the global @var{b}-function @code{b(s)} of
+a polynomial @var{f}.
+@code{b(s)} is a polynomial of the minimal degree 
+such that there exists @code{P(x,s)} in D[s], which is a polynomial
+ring over Weyl algebra @code{D}, and @code{P(x,s)f^(s+1)=b(s)f^s} holds.
+@item @code{generic_bfct(@var{f},@var{vlist},@var{dvlist},@var{weight})}
+computes the global @var{b}-function of a left ideal @code{I} in @code{D}
+generated by @var{plist}, with respect to @var{weight}.
+@var{vlist} is the list of @code{x}-variables, 
+@var{vlist} is the list of corresponding @code{D}-variables.
+@item @code{bfunction(@var{f})} and @code{bfct(@var{f})} implement
+different algorithms and the efficiency depends on inputs.
+@item @code{ann(@var{f})} returns the generator set of the annihilator
+ideal of @code{@var{f}^s}.
+@code{ann(@var{f})} returns a list @code{[@var{a},@var{list}]},
+where @var{a} is the minimal integral root of the global @var{b}-function
+of @var{f}, and @var{list} is a list of polynomials obtained by
+substituting @code{s} in @code{ann(@var{f})} with @var{a}.
+@item See [Saito,Sturmfels,Takayama] for the details.
+\E
+@end itemize
+
+@example
+[0] load("bfct")$
+[216] bfunction(x^3+y^3+z^3+x^2*y^2*z^2+x*y*z);
+-9*s^5-63*s^4-173*s^3-233*s^2-154*s-40
+[217] fctr(@@);
+[[-1,1],[s+2,1],[3*s+4,1],[3*s+5,1],[s+1,2]]
+[218] F = [4*x^3*dt+y*z*dt+dx,x*z*dt+4*y^3*dt+dy,
+x*y*dt+5*z^4*dt+dz,-x^4-z*y*x-y^4-z^5+t]$
+[219] generic_bfct(F,[t,z,y,x],[dt,dz,dy,dx],[1,0,0,0]);
+20000*s^10-70000*s^9+101750*s^8-79375*s^7+35768*s^6-9277*s^5
++1278*s^4-72*s^3
+[220] P=x^3-y^2$
+[221] ann(P);
+[2*dy*x+3*dx*y^2,-3*dx*x-2*dy*y+6*s]
+[222] ann0(P);
+[-1,[2*dy*x+3*dx*y^2,-3*dx*x-2*dy*y-6]]
+@end example
+
+@table @t
+\JP @item $B;2>H(B
+\EG @item References
+\JP @fref{Weyl $BBe?t(B}.
+\EG @fref{Weyl algebra}.
+@end table
+