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Diff for /OpenXM/src/asir-doc/parts/groebner.texi between version 1.2 and 1.23

-version 1.2, 1999/12/21 02:47:31
+version 1.23, 2019/09/13 09:31:00
 Line 1
 Line 1
 Line 1
- @comment $OpenXM$
+ @comment $OpenXM: OpenXM/src/asir-doc/parts/groebner.texi,v 1.22 2019/03/29 04:54:25 noro Exp $
  \BJP
  @node $B%0%l%V%J4pDl$N7W;;(B,,, Top
  @chapter $B%0%l%V%J4pDl$N7W;;(B
 Line 15
 Line 15
 Line 15
  * $B4pK\E*$JH!?t(B::
  * $B7W;;$*$h$SI=<($N@)8f(B::
  * $B9`=g=x$N@_Dj(B::
+ * Weight::
  * $BM-M}<0$r78?t$H$9$k%0%l%V%J4pDl7W;;(B::
  * $B4pDlJQ49(B::
+ * Weyl $BBe?t(B::
+ * $BB?9`<04D>e$N2C72(B::
  * $B%0%l%V%J4pDl$K4X$9$kH!?t(B::
  \E
  \BEG
-Line 25
+Line 28
 Line 25
 Line 28
  * Fundamental functions::
  * Controlling Groebner basis computations::
  * Setting term orderings::
+ * Weight::
  * Groebner basis computation with rational function coefficients::
  * Change of ordering::
+ * Weyl algebra::
+ * Module over a polynomial ring::
  * Functions for Groebner basis computation::
  \E
  @end menu
-Line 197  In an @b{Asir} session, it is displayed in the form li
+Line 203  In an @b{Asir} session, it is displayed in the form li
 Line 197  In an @b{Asir} session, it is displayed in the form li
 Line 203  In an @b{Asir} session, it is displayed in the form li
  \EG and also can be input in such a form.
  \BJP
- @itemx $BF,C19`<0(B (head monomial)
  @item $BF,9`(B (head term)
+ @itemx $BF,C19`<0(B (head monomial)
  @itemx $BF,78?t(B (head coefficient)
  $BJ,;6I=8=B?9`<0$K$*$1$k3FC19`<0$O(B, $B9`=g=x$K$h$j@0Ns$5$l$k(B. $B$3$N;~=g(B
  $B=x:GBg$NC19`<0$rF,C19`<0(B, $B$=$l$K8=$l$k9`(B, $B78?t$r$=$l$>$lF,9`(B, $BF,78?t(B
  $B$H8F$V(B.
  \E
  \BEG
- @itemx head monomial
  @item head term
+ @itemx head monomial
  @itemx head coefficient
  Monomials in a distributed polynomial is sorted by a total order.
-Line 216  the head term and the head coefficient respectively.
+Line 222  the head term and the head coefficient respectively.
 Line 216  the head term and the head coefficient respectively.
 Line 222  the head term and the head coefficient respectively.
  \E
  @end table
+ @noindent
+ ChangeLog
+ @itemize @bullet
  \BJP
+ @item $BJ,;6I=8=B?9`<0$OG$0U$N%*%V%8%'%/%H$r78?t$K$b$F$k$h$&$K$J$C$?(B.
+ $B$^$?2C72$N(Bk$B@.J,$NMWAG$r<!$N7A<0(B <<d0,d1,...:k>> $B$GI=8=$9$k$h$&$K$J$C$?(B (2017-08-31).
+ \E
+ \BEG
+ @item Distributed polynomials accept objects as coefficients.
+ The k-th element of a free module is expressed as <<d0,d1,...:k>> (2017-08-31).
+ \E
+ @item
+.15 algnum.c,
+.53 ctrl.c,
+.66 dp-supp.c,
+.105 dp.c,
+.73 gr.c,
+.4 reduct.c,
+.16 _distm.c,
+.17 dalg.c,
+.52 dist.c,
+.20 distm.c,
+.8  gmpq.c,
+.238 engine/nd.c,
+.102  ca.h,
+.411  version.h,
+.28 cpexpr.c,
+.42 pexpr.c,
+.20 pexpr_body.c,
+.40 spexpr.c,
+.27 arith.c,
+.77 eval.c,
+.56 parse.h,
+.37 parse.y,
+.8 stdio.c,
+.31 plotf.c
+ @end itemize
+ \BJP
  @node $B%U%!%$%k$NFI$_9~$_(B,,, $B%0%l%V%J4pDl$N7W;;(B
  @section $B%U%!%$%k$NFI$_9~$_(B
  \E
-Line 228  the head term and the head coefficient respectively.
+Line 272  the head term and the head coefficient respectively.
 Line 228  the head term and the head coefficient respectively.
 Line 272  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}
+ $B%i%$%V%i%j%G%#%l%/%H%j$KCV$+$l$F$$$k(B.
- $B$rFC$K0[$J$k%Q%9$K@_Dj$7$J$$8B$j(B, $B%U%!%$%kL>$N$_$GFI$_9~$`$3$H$,$G$-$k(B.
  \E
  \BEG
- Facilities for computing Groebner bases are provided not by built-in
+ Facilities for computing Groebner bases are
- functions but by a set of user functions written in @b{Asir}.
+ @code{dp_gr_main()}, @code{dp_gr_mod_main()}and @code{dp_gr_f_main()}.
- The set of functions is provided as a file (sometimes called package),
+ To call these functions,
- named @samp{gr}.
+ 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
+ directory of @b{Asir}.
- its file name, unless the environment variable @code{ASIR_LIBDIR}
- is set to a non-standard one.
  \E
  @example
-Line 350  These parameters can be set and examined by a built-in
+Line 394  These parameters can be set and examined by a built-in
 Line 350  These parameters can be set and examined by a built-in
 Line 394  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,
+ [Demand,0,NoSugar,0,NoCriB,0,NoGC,0,NoMC,0,NoRA,0,NoGCD,0,Top,0,
- Print,1,Stat,0,Reverse,0,InterReduce,0,Multiple,0]
+ ShowMag,1,Print,1,Stat,0,Reverse,0,InterReduce,0,Multiple,0]
  [101]
  @end example
-Line 447  If `on', various informations during a Groebner basis
+Line 491  If `on', various informations during a Groebner basis
 Line 447  If `on', various informations during a Groebner basis
 Line 491  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=<($5(B
-Line 469  is shown after every normal computation.  After comlet
+Line 520  is shown after every normal computation.  After comlet
 Line 469  is shown after every normal computation.  After comlet
 Line 520  is shown after every normal computation.  After comlet
  computation the maximal value among the sums is shown.
  \E
- @item Multiple
+ @item Content
+ @itemx Multiple
  \BJP
-$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
+$B$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{Multiple} $BG\$K$J$k$4$H$K78?tA4BN$N(B GCD $B$,7W;;$5$l(B, $B$=$N(B GCD $B$G(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{Multiple} $B$,(B 1 $B$J$i$P(B, $B4JLs$9$k$4$H$K(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{Multiple} $B$r(B 2 $BDxEY(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
-Line 530  membercheck
+Line 585  membercheck
 Line 530  membercheck
 Line 585  membercheck
  (0,0)(0,0)(0,0)(0,0)
  gbcheck total 8 pairs
  ........
- UP=(0,0)SP=(0,0)SPM=(0,0)NF=(0,0)NFM=(0.010002,0)ZNFM=(0.010002,0)PZ=(0,0)
+ UP=(0,0)SP=(0,0)SPM=(0,0)NF=(0,0)NFM=(0.010002,0)ZNFM=(0.010002,0)
- NP=(0,0)MP=(0,0)RA=(0,0)MC=(0,0)GC=(0,0)T=40,B=0 M=8 F=6 D=12 ZR=5 NZR=6
+ PZ=(0,0)NP=(0,0)MP=(0,0)RA=(0,0)MC=(0,0)GC=(0,0)T=40,B=0 M=8 F=6
- Max_mag=6
+ D=12 ZR=5 NZR=6 Max_mag=6
  [94]
  @end example
-Line 992  time as well as the choice of types of term orderings.
+Line 1047  time as well as the choice of types of term orderings.
 Line 992  time as well as the choice of types of term orderings.
 Line 1047  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
+ (y+t^2*z^2)*x+y^7+(20*t^2+6*t+1)*y^2+(-t^17+6*t^14-21*t^12-15*t^11
- +20*t^8-35*t^7-126*t^6-15*t^5+70*t^4+84*t^3-t^2+5*t-9)*z^2*y+(6*t^16-36*t^13
+ +84*t^9+20*t^8-35*t^7-126*t^6-15*t^5+70*t^4+84*t^3-t^2+5*t-9)*z^2*y
- +14*t^11+90*t^10-56*t^8-120*t^7-14*t^6+64*t^5+84*t^4+27*t^3-16*t^2-30*t+7)*z^4,
+ +(6*t^16-36*t^13+14*t^11+90*t^10-56*t^8-120*t^7-14*t^6+64*t^5+84*t^4
- (t^3-1)*x-y^6+(-6*t^13+24*t^10-20*t^8-36*t^7+40*t^5+24*t^4-6*t^3-20*t^2-6*t-1)*y
+ +27*t^3-16*t^2-30*t+7)*z^4,
- +(t^17-6*t^14+9*t^12+15*t^11-36*t^9-20*t^8-5*t^7+54*t^6+15*t^5+10*t^4-36*t^3
+ (t^3-1)*x-y^6+(-6*t^13+24*t^10-20*t^8-36*t^7+40*t^5+24*t^4-6*t^3-20*t^2
- -11*t^2-5*t+9)*z^2,
+ -6*t-1)*y+(t^17-6*t^14+9*t^12+15*t^11-36*t^9-20*t^8-5*t^7+54*t^6+15*t^5
+ +10*t^4-36*t^3-11*t^2-5*t+9)*z^2,
  -y^8-8*t*y^3+16*z^2*y^2+(-8*t^16+48*t^13-56*t^11-120*t^10+224*t^8+160*t^7
- -56*t^6-336*t^5-112*t^4+112*t^3+224*t^2+24*t-56)*z^4*y+(t^24-8*t^21+20*t^19
+ -56*t^6-336*t^5-112*t^4+112*t^3+224*t^2+24*t-56)*z^4*y+(t^24-8*t^21
- +28*t^18-120*t^16-56*t^15+14*t^14+300*t^13+70*t^12-56*t^11-400*t^10-84*t^9
+ +20*t^19+28*t^18-120*t^16-56*t^15+14*t^14+300*t^13+70*t^12-56*t^11
- +84*t^8+268*t^7+84*t^6-56*t^5-63*t^4-36*t^3+46*t^2-12*t+1)*z,
+ -400*t^10-84*t^9+84*t^8+268*t^7+84*t^6-56*t^5-63*t^4-36*t^3+46*t^2
-*t*y^5+z*y^2+(-2*t^11+8*t^8-20*t^6-12*t^5+40*t^3+8*t^2-10*t-20)*z^3*y+8*t^14
+ -12*t+1)*z,2*t*y^5+z*y^2+(-2*t^11+8*t^8-20*t^6-12*t^5+40*t^3+8*t^2
- -32*t^11+48*t^8-t^7-32*t^5-6*t^4+9*t^2-t,
+ -10*t-20)*z^3*y+8*t^14-32*t^11+48*t^8-t^7-32*t^5-6*t^4+9*t^2-t,
  -z*y^3+(t^7-2*t^4+3*t^2+t)*y+(-2*t^6+4*t^3+2*t-2)*z^2,
-*t^2*y^3+z^2*y^2+(-2*t^5+4*t^2-6)*z^4*y+(4*t^8-t^7-8*t^5+2*t^4-4*t^3+5*t^2-t)*z,
+*t^2*y^3+z^2*y^2+(-2*t^5+4*t^2-6)*z^4*y
+ +(4*t^8-t^7-8*t^5+2*t^4-4*t^3+5*t^2-t)*z,
  z^3*y^2+2*t^3*y+(-t^7+2*t^4+t^2-t)*z^2,
  -t*z*y^2-2*z^3*y+t^8-2*t^5-t^3+t^2,
- -t^3*y^2-2*t^2*z^2*y+(t^6-2*t^3-t+1)*z^4,
+ -t^3*y^2-2*t^2*z^2*y+(t^6-2*t^3-t+1)*z^4,z^5-t^4]
- z^5-t^4]
  [93] gr(B,[t,z,y,x],2);
  [x^10-t,x^8-z,x^31-x^6-x-y]
  @end example
-Line 1041  beforehand, and some heuristic trial may be inevitable
+Line 1097  beforehand, and some heuristic trial may be inevitable
 Line 1041  beforehand, and some heuristic trial may be inevitable
 Line 1097  beforehand, and some heuristic trial may be inevitable
  \E
  \BJP
+ @node Weight ,,, $B%0%l%V%J4pDl$N7W;;(B
+ @section Weight
+ \E
+ \BEG
+ @node Weight,,, Groebner basis computation
+ @section Weight
+ \E
+ \BJP
+ $BA0@a$G>R2p$7$?9`=g=x$O(B, $B3FJQ?t$K(B weight ($B=E$_(B) $B$r@_Dj$9$k$3$H$G(B
+ $B$h$j0lHLE*$J$b$N$H$J$k(B.
+ \E
+ \BEG
+ Term orderings introduced in the previous section can be generalized
+ by setting a weight for each variable.
+ \E
+ @example
+ [0] dp_td(<<1,1,1>>);
+ [1] dp_set_weight([1,2,3])$
+ [2] dp_td(<<1,1,1>>);
+ @end example
+ \BJP
+ $BC19`<0$NA4<!?t$r7W;;$9$k:](B, $B%G%U%)%k%H$G$O(B
+ $B3FJQ?t$N;X?t$NOB$rA4<!?t$H$9$k(B. $B$3$l$O3FJQ?t$N(B weight $B$r(B 1 $B$H(B
+ $B9M$($F$$$k$3$H$KAjEv$9$k(B. $B$3$NNc$G$O(B, $BBh0l(B, $BBhFs(B, $BBh;0JQ?t$N(B
+ weight $B$r$=$l$>$l(B 1,2,3 $B$H;XDj$7$F$$$k(B. $B$3$N$?$a(B, @code{<<1,1,1>>}
+ $B$NA4<!?t(B ($B0J2<$G$O$3$l$rC19`<0$N(B weight $B$H8F$V(B) $B$,(B @code{1*1+1*2+1*3=6} $B$H$J$k(B.
+ weight $B$r@_Dj$9$k$3$H$G(B, $BF1$89`=g=x7?$N$b$H$G0[$J$k9`=g=x$,Dj5A$G$-$k(B.
+ $BNc$($P(B, weight $B$r$&$^$/@_Dj$9$k$3$H$G(B, $BB?9`<0$r(B weighted homogeneous
+ $B$K$9$k$3$H$,$G$-$k>l9g$,$"$k(B.
+ \E
+ \BEG
+ By default, the total degree of a monomial is equal to
+ the sum of all exponents. This means that the weight for each variable
+ is set to 1.
+ In this example, the weights for the first, the second and the third
+ variable are set to 1, 2 and 3 respectively.
+ Therefore the total degree of @code{<<1,1,1>>} under this weight,
+ which is called the weight of the monomial, is @code{1*1+1*2+1*3=6}.
+ By setting weights, different term orderings can be set under a type of
+ term ordeing. In some case a polynomial can
+ be made weighted homogeneous by setting an appropriate weight.
+ \E
+ \BJP
+ $B3FJQ?t$KBP$9$k(B weight $B$r$^$H$a$?$b$N$r(B weight vector $B$H8F$V(B.
+ $B$9$Y$F$N@.J,$,@5$G$"$j(B, $B%0%l%V%J4pDl7W;;$K$*$$$F(B, $BA4<!?t$N(B
+ $BBe$o$j$KMQ$$$i$l$k$b$N$rFC$K(B sugar weight $B$H8F$V$3$H$K$9$k(B.
+ sugar strategy $B$K$*$$$F(B, $BA4<!?t$NBe$o$j$K;H$o$l$k$+$i$G$"$k(B.
+ $B0lJ}$G(B, $B3F@.J,$,I,$:$7$b@5$H$O8B$i$J$$(B weight vector $B$O(B,
+ sugar weight $B$H$7$F@_Dj$9$k$3$H$O$G$-$J$$$,(B, $B9`=g=x$N0lHL2=$K$O(B
+ $BM-MQ$G$"$k(B. $B$3$l$i$O(B, $B9TNs$K$h$k9`=g=x$N@_Dj$K$9$G$K8=$l$F(B
+ $B$$$k(B. $B$9$J$o$A(B, $B9`=g=x$rDj5A$9$k9TNs$N3F9T$,(B, $B0l$D$N(B weight vector
+ $B$H8+$J$5$l$k(B. $B$^$?(B, $B%V%m%C%/=g=x$O(B, $B3F%V%m%C%/$N(B
+ $BJQ?t$KBP1~$9$k@.J,$N$_(B 1 $B$GB>$O(B 0 $B$N(B weight vector $B$K$h$kHf3S$r(B
+ $B:G=i$K9T$C$F$+$i(B, $B3F%V%m%C%/Kh$N(B tie breaking $B$r9T$&$3$H$KAjEv$9$k(B.
+ \E
+ \BEG
+ A list of weights for all variables is called a weight vector.
+ A weight vector is called a sugar weight vector if
+ its elements are all positive and it is used for computing
+ a weighted total degree of a monomial, because such a weight
+ is used instead of total degree in sugar strategy.
+ On the other hand, a weight vector whose elements are not necessarily
+ positive cannot be set as a sugar weight, but it is useful for
+ generalizing term order. In fact, such a weight vector already
+ appeared in a matrix order. That is, each row of a matrix defining
+ a term order is regarded as a weight vector. A block order
+ is also considered as a refinement of comparison by weight vectors.
+ It compares two terms by using a weight vector whose elements
+ corresponding to variables in a block is 1 and 0 otherwise,
+ then it applies a tie breaker.
+ \E
+ \BJP
+ weight vector $B$N@_Dj$O(B @code{dp_set_weight()} $B$G9T$&$3$H$,$G$-$k(B
+ $B$,(B, $B9`=g=x$r;XDj$9$k:]$NB>$N%Q%i%a%?(B ($B9`=g=x7?(B, $BJQ?t=g=x(B) $B$H(B
+ $B$^$H$a$F@_Dj$G$-$k$3$H$,K>$^$7$$(B. $B$3$N$?$a(B, $B<!$N$h$&$J7A$G$b(B
+ $B9`=g=x$,;XDj$G$-$k(B.
+ \E
+ \BEG
+ A weight vector can be set by using @code{dp_set_weight()}.
+ However it is more preferable if a weight vector can be set
+ together with other parapmeters such as a type of term ordering
+ and a variable order. This is realized as follows.
+ \E
+ @example
+ [64] B=[x+y+z-6,x*y+y*z+z*x-11,x*y*z-6]$
+ [65] dp_gr_main(B|v=[x,y,z],sugarweight=[3,2,1],order=0);
+ [z^3-6*z^2+11*z-6,x+y+z-6,-y^2+(-z+6)*y-z^2+6*z-11]
+ [66] dp_gr_main(B|v=[y,z,x],order=[[1,1,0],[0,1,0],[0,0,1]]);
+ [x^3-6*x^2+11*x-6,x+y+z-6,-x^2+(-y+6)*x-y^2+6*y-11]
+ [67] dp_gr_main(B|v=[y,z,x],order=[[x,1,y,2,z,3]]);
+ [x+y+z-6,x^3-6*x^2+11*x-6,-x^2+(-y+6)*x-y^2+6*y-11]
+ @end example
+ \BJP
+ $B$$$:$l$NNc$K$*$$$F$b(B, $B9`=g=x$O(B option $B$H$7$F;XDj$5$l$F$$$k(B.
+ $B:G=i$NNc$G$O(B @code{v} $B$K$h$jJQ?t=g=x$r(B, @code{sugarweight} $B$K$h$j(B
+ sugar weight vector $B$r(B, @code{order}$B$K$h$j9`=g=x7?$r;XDj$7$F$$$k(B.
+ $BFs$DL\$NNc$K$*$1$k(B @code{order} $B$N;XDj$O(B matrix order $B$HF1MM$G$"$k(B.
+ $B$9$J$o$A(B, $B;XDj$5$l$?(B weight vector $B$r:8$+$i=g$K;H$C$F(B weight $B$NHf3S(B
+ $B$r9T$&(B. $B;0$DL\$NNc$bF1MM$G$"$k$,(B, $B$3$3$G$O(B weight vector $B$NMWAG$r(B
+ $BJQ?tKh$K;XDj$7$F$$$k(B. $B;XDj$,$J$$$b$N$O(B 0 $B$H$J$k(B. $B;0$DL\$NNc$G$O(B,
+ @code{order} $B$K$h$k;XDj$G$O9`=g=x$,7hDj$7$J$$(B. $B$3$N>l9g$K$O(B,
+ tie breaker $B$H$7$FA4<!?t5U<-=q<0=g=x$,<+F0E*$K@_Dj$5$l$k(B.
+ $B$3$N;XDjJ}K!$O(B, @code{dp_gr_main}, @code{dp_gr_mod_main} $B$J$I(B
+ $B$NAH$_9~$_4X?t$G$N$_2DG=$G$"$j(B, @code{gr} $B$J$I$N%f!<%6Dj5A4X?t(B
+ $B$G$OL$BP1~$G$"$k(B.
+ \E
+ \BEG
+ In each example, a term ordering is specified as options.
+ In the first example, a variable order, a sugar weight vector
+ and a type of term ordering are specified by options @code{v},
+ @code{sugarweight} and @code{order} respectively.
+ In the second example, an option @code{order} is used
+ to set a matrix ordering. That is, the specified weight vectors
+ are used from left to right for comparing terms.
+ The third example shows a variant of specifying a weight vector,
+ where each component of a weight vector is specified variable by variable,
+ and unspecified components are set to zero. In this example,
+ a term order is not determined only by the specified weight vector.
+ In such a case a tie breaker by the graded reverse lexicographic ordering
+ is set automatically.
+ This type of a term ordering specification can be applied only to builtin
+ functions such as @code{dp_gr_main()}, @code{dp_gr_mod_main()}, not to
+ user defined functions such as @code{gr()}.
+ \E
+ \BJP
  @node $BM-M}<0$r78?t$H$9$k%0%l%V%J4pDl7W;;(B,,, $B%0%l%V%J4pDl$N7W;;(B
  @section $BM-M}<0$r78?t$H$9$k%0%l%V%J4pDl7W;;(B
  \E
-Line 1200  Refer to the sections for each functions.
+Line 1389  Refer to the sections for each functions.
 Line 1200  Refer to the sections for each functions.
 Line 1389  Refer to the sections for each functions.
  \E
  \BJP
+ @node Weyl $BBe?t(B,,, $B%0%l%V%J4pDl$N7W;;(B
+ @section Weyl $BBe?t(B
+ \E
+ \BEG
+ @node Weyl algebra,,, Groebner basis computation
+ @section Weyl algebra
+ \E
+ @noindent
+ \BJP
+ $B$3$l$^$G$O(B, $BDL>o$N2D49$JB?9`<04D$K$*$1$k%0%l%V%J4pDl7W;;$K$D$$$F(B
+ $B=R$Y$F$-$?$,(B, $B%0%l%V%J4pDl$NM}O@$O(B, $B$"$k>r7o$rK~$?$9Hs2D49$J(B
+ $B4D$K$b3HD%$G$-$k(B. $B$3$N$h$&$J4D$NCf$G(B, $B1~MQ>e$b=EMW$J(B,
+ Weyl $BBe?t(B, $B$9$J$o$AB?9`<04D>e$NHyJ,:nMQAG4D$N1i;;$*$h$S(B
+ $B%0%l%V%J4pDl7W;;$,(B Risa/Asir $B$K<BAu$5$l$F$$$k(B.
+ $BBN(B @code{K} $B>e$N(B @code{n} $B<!85(B Weyl $BBe?t(B
+ @code{D=K<x1,@dots{},xn,D1,@dots{},Dn>} $B$O(B
+ \E
+ \BEG
+ So far we have explained Groebner basis computation in
+ commutative polynomial rings. However Groebner basis can be
+ considered in more general non-commutative rings.
+ Weyl algebra is one of such rings and
+ Risa/Asir implements fundamental operations
+ in Weyl algebra and Groebner basis computation in Weyl algebra.
+ The @code{n} dimensional Weyl algebra over a field @code{K},
+ @code{D=K<x1,@dots{},xn,D1,@dots{},Dn>} is a non-commutative
+ algebra which has the following fundamental relations:
+ \E
+ @code{xi*xj-xj*xi=0}, @code{Di*Dj-Dj*Di=0}, @code{Di*xj-xj*Di=0} (@code{i!=j}),
+ @code{Di*xi-xi*Di=1}
+ \BJP
+ $B$H$$$&4pK\4X78$r;}$D4D$G$"$k(B. @code{D} $B$O(B $BB?9`<04D(B @code{K[x1,@dots{},xn]} $B$r78?t(B
+ $B$H$9$kHyJ,:nMQAG4D$G(B,  @code{Di} $B$O(B @code{xi} $B$K$h$kHyJ,$rI=$9(B. $B8r494X78$K$h$j(B,
+ @code{D} $B$N85$O(B, @code{x1^i1*@dots{}*xn^in*D1^j1*@dots{}*Dn^jn} $B$J$kC19`(B
+ $B<0$N(B @code{K} $B@~7A7k9g$H$7$F=q$-I=$9$3$H$,$G$-$k(B.
+ Risa/Asir $B$K$*$$$F$O(B, $B$3$NC19`<0$r(B, $B2D49$JB?9`<0$HF1MM$K(B
+ @code{<<i1,@dots{},in,j1,@dots{},jn>>} $B$GI=$9(B. $B$9$J$o$A(B, @code{D} $B$N85$b(B
+ $BJ,;6I=8=B?9`<0$H$7$FI=$5$l$k(B. $B2C8:;;$O(B, $B2D49$N>l9g$HF1MM$K(B, @code{+}, @code{-}
+ $B$K$h$j(B
+ $B<B9T$G$-$k$,(B, $B>h;;$O(B, $BHs2D49@-$r9MN8$7$F(B @code{dp_weyl_mul()} $B$H$$$&4X?t(B
+ $B$K$h$j<B9T$9$k(B.
+ \E
+ \BEG
+ @code{D} is the ring of differential operators whose coefficients
+ are polynomials in @code{K[x1,@dots{},xn]} and
+ @code{Di} denotes the differentiation with respect to  @code{xi}.
+ According to the commutation relation,
+ elements of @code{D} can be represented as a @code{K}-linear combination
+ of monomials @code{x1^i1*@dots{}*xn^in*D1^j1*@dots{}*Dn^jn}.
+ In Risa/Asir, this type of monomial is represented
+ by @code{<<i1,@dots{},in,j1,@dots{},jn>>} as in the case of commutative
+ polynomial.
+ That is, elements of @code{D} are represented by distributed polynomials.
+ Addition and subtraction can be done by @code{+}, @code{-},
+ but multiplication is done by calling @code{dp_weyl_mul()} because of
+ the non-commutativity of @code{D}.
+ \E
+ @example
+ [0] A=<<1,2,2,1>>;
+ (1)*<<1,2,2,1>>
+ [1] B=<<2,1,1,2>>;
+ (1)*<<2,1,1,2>>
+ [2] A*B;
+ (1)*<<3,3,3,3>>
+ [3] dp_weyl_mul(A,B);
+ (1)*<<3,3,3,3>>+(1)*<<3,2,3,2>>+(4)*<<2,3,2,3>>+(4)*<<2,2,2,2>>
+ +(2)*<<1,3,1,3>>+(2)*<<1,2,1,2>>
+ @end example
+ \BJP
+ $B%0%l%V%J4pDl7W;;$K$D$$$F$b(B, Weyl $BBe?t@lMQ$N4X?t$H$7$F(B,
+ $B<!$N4X?t$,MQ0U$7$F$"$k(B.
+ \E
+ \BEG
+ The following functions are avilable for Groebner basis computation
+ in Weyl algebra:
+ \E
+ @code{dp_weyl_gr_main()},
+ @code{dp_weyl_gr_mod_main()},
+ @code{dp_weyl_gr_f_main()},
+ @code{dp_weyl_f4_main()},
+ @code{dp_weyl_f4_mod_main()}.
+ \BJP
+ $B$^$?(B, $B1~MQ$H$7$F(B, global b $B4X?t$N7W;;$,<BAu$5$l$F$$$k(B.
+ \E
+ \BEG
+ Computation of the global b function is implemented as an application.
+ \E
+ \BJP
+ @node $BB?9`<04D>e$N2C72(B,,, $B%0%l%V%J4pDl$N7W;;(B
+ @section $BB?9`<04D>e$N2C72(B
+ \E
+ \BEG
+ @node Module over a polynomial ring,,, Groebner basis computation
+ @section Module over a polynomial ring
+ \E
+ @noindent
+ \BJP
+ $BB?9`<04D>e$N<+M32C72$N85$O(B, $B2C72C19`<0(B te_i $B$N@~7?OB$H$7$FFbItI=8=$5$l$k(B.
+ $B$3$3$G(B t $B$OB?9`<04D$NC19`<0(B, e_i $B$O<+M32C72$NI8=`4pDl$G$"$k(B. $B2C72C19`<0$O(B, $BB?9`<04D$NC19`<0(B
+ $B$K0LCV(B i $B$rDI2C$7$?(B @code{<<a,b,...,c:i>>} $B$GI=$9(B. $B2C72B?9`<0(B, $B$9$J$o$A2C72C19`<0$N@~7?OB$O(B,
+ $B@_Dj$5$l$F$$$k2C729`=g=x$K$7$?$,$C$F9_=g$K@0Ns$5$l$k(B. $B2C729`=g=x$K$O0J2<$N(B3$B<oN`$,$"$k(B.
+ @table @code
+ @item TOP $B=g=x(B
+ $B$3$l$O(B, te_i > se_j $B$H$J$k$N$O(B t>s $B$^$?$O(B (t=s $B$+$D(B i<j) $B$H$J$k$h$&$J9`=g=x$G$"$k(B. $B$3$3$G(B,
+ t, s $B$NHf3S$OB?9`<04D$K@_Dj$5$l$F$$$k=g=x$G9T$&(B.
+ $B$3$N7?$N=g=x$O(B, @code{dp_ord([0,Ord])} $B$K(B
+ $B$h$j@_Dj$9$k(B. $B$3$3$G(B, @code{Ord} $B$OB?9`<04D$N=g=x7?$G$"$k(B.
+ @item POT $B=g=x(B
+ $B$3$l$O(B, te_i > se_j $B$H$J$k$N$O(B i<j $B$^$?$O(B (i=j $B$+$D(B t>s) $B$H$J$k$h$&$J9`=g=x$G$"$k(B. $B$3$3$G(B,
+ t, s $B$NHf3S$OB?9`<04D$K@_Dj$5$l$F$$$k=g=x$G9T$&(B.
+ $B$3$N7?$N=g=x$O(B, @code{dp_ord([1,Ord])} $B$K(B
+ $B$h$j@_Dj$9$k(B. $B$3$3$G(B, @code{Ord} $B$OB?9`<04D$N=g=x7?$G$"$k(B.
+ @item Schreyer $B7?=g=x(B
+ $B3FI8=`4pDl(B e_i $B$KBP$7(B, $BJL$N<+M32C72$N2C72C19`<0(B T_i $B$,M?$($i$l$F$$$F(B, te_i > se_j $B$H$J$k$N$O(B
+ tT_i > sT_j $B$^$?$O(B (tT_i=sT_j $B$+$D(B i<j) $B$H$J$k$h$&$J9`=g=x$G$"$k(B. $B$3$3$G(B tT_i, sT_j $B$N(B
+ $BHf3S$O(B, $B$3$l$i$,=jB0$9$k<+M32C72$K@_Dj$5$l$F$$$k=g=x$G9T$&(B.
+ $B$3$N7?$N=g=x$O(B, $BDL>o:F5"E*$K@_Dj$5$l$k(B. $B$9$J$o$A(B, T_i $B$,=jB0$9$k<+M32C72$N=g=x$b(B Schreyer $B7?(B
+ $B$G$"$k$+(B, $B$^$?$O%\%H%`$H$J$k(B TOP, POT $B$J$I$N9`=g=x$H$J$k(B.
+ $B$3$N7?$N=g=x$O(B @code{dpm_set_schreyer([H_1,H_2,...])} $B$K$h$j;XDj$9$k(B. $B$3$3$G(B,
+ @code{H_i=[T_1,T_2,...]} $B$O2C72C19`<0$N%j%9%H$G(B, @code{[H_2,...]} $B$GDj5A$5$l$k(B Schreyer $B7?9`=g=x$r(B
+ @code{tT_i} $B$i$KE,MQ$9$k$H$$$&0UL#$G$"$k(B.
+ @end table
+ $B2C72B?9`<0$rF~NO$9$kJ}K!$H$7$F$O(B, @code{<<a,b,...:i>>} $B$J$k7A<0$GD>@\F~NO$9$kB>$K(B,
+ $BB?9`<0%j%9%H$r:n$j(B, @code{dpm_ltod()} $B$K$h$jJQ49$9$kJ}K!$b$"$k(B.
+ \E
+ \BEG
+ not yet
+ \E
+ \BJP
  @node $B%0%l%V%J4pDl$K4X$9$kH!?t(B,,, $B%0%l%V%J4pDl$N7W;;(B
  @section $B%0%l%V%J4pDl$K4X$9$kH!?t(B
  \E
-Line 1214  Refer to the sections for each functions.
+Line 1553  Refer to the sections for each functions.
 Line 1214  Refer to the sections for each functions.
 Line 1553  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_gr_main dp_gr_mod_main dp_gr_f_main dp_weyl_gr_main dp_weyl_gr_mod_main dp_weyl_gr_f_main::
- * dp_f4_main dp_f4_mod_main::
+ * dp_f4_main dp_f4_mod_main dp_weyl_f4_main dp_weyl_f4_mod_main::
+ * nd_gr nd_gr_trace nd_f4 nd_f4_trace nd_weyl_gr nd_weyl_gr_trace::
+ * nd_gr_postproc nd_weyl_gr_postproc::
  * dp_gr_flags dp_gr_print::
  * dp_ord::
+ * dp_set_weight dp_set_top_weight dp_weyl_set_weight::
  * dp_ptod::
  * dp_dtop::
  * dp_mod dp_rat::
  * dp_homo dp_dehomo::
  * dp_ptozp dp_prim::
- * dp_nf dp_nf_mod dp_true_nf dp_true_nf_mod::
+ * dp_nf dp_nf_mod dp_true_nf dp_true_nf_mod dp_weyl_nf dp_weyl_nf_mod::
  * dp_hm dp_ht dp_hc dp_rest::
+ * dpm_hm dpm_ht dpm_hc dpm_hp dpm_rest::
  * dp_td dp_sugar::
  * dp_lcm::
  * dp_redble::
-Line 1239  Refer to the sections for each functions.
+Line 1582  Refer to the sections for each functions.
 Line 1239  Refer to the sections for each functions.
 Line 1582  Refer to the sections for each functions.
  * katsura hkatsura cyclic hcyclic::
  * dp_vtoe dp_etov::
  * lex_hensel_gsl tolex_gsl tolex_gsl_d::
+ * primadec primedec::
+ * primedec_mod::
+ * bfunction bfct generic_bfct ann ann0::
  @end menu
  \JP @node gr hgr gr_mod,,, $B%0%l%V%J4pDl$K4X$9$kH!?t(B
-Line 1262  Refer to the sections for each functions.
+Line 1608  Refer to the sections for each functions.
 Line 1262  Refer to the sections for each functions.
 Line 1608  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
-Line 1278  Refer to the sections for each functions.
+Line 1624  Refer to the sections for each functions.
 Line 1278  Refer to the sections for each functions.
 Line 1624  Refer to the sections for each functions.
  @item
  $BI8=`%i%$%V%i%j$N(B @samp{gr} $B$GDj5A$5$l$F$$$k(B.
  @item
+ gr $B$rL>A0$K4^$`4X?t$O8=:_%a%s%F$5$l$F$$$J$$(B. @code{nd_gr}$B7O$N4X?t$rBe$o$j$KMxMQ$9$Y$-$G$"$k(B(@fref{nd_gr nd_gr_trace nd_f4 nd_f4_trace nd_weyl_gr nd_weyl_gr_trace}).
+ @item
  $B$$$:$l$b(B, $BB?9`<0%j%9%H(B @var{plist} $B$N(B, $BJQ?t=g=x(B @var{vlist}, $B9`=g=x7?(B
  @var{order} $B$K4X$9$k%0%l%V%J4pDl$r5a$a$k(B. @code{gr()}, @code{hgr()}
  $B$O(B $BM-M}?t78?t(B, @code{gr_mod()} $B$O(B GF(@var{p}) $B78?t$H$7$F7W;;$9$k(B.
-Line 1289  Refer to the sections for each functions.
+Line 1637  Refer to the sections for each functions.
 Line 1289  Refer to the sections for each functions.
 Line 1637  Refer to the sections for each functions.
  strategy $B$K$h$k7W;;(B, @code{hgr()} $B$O(B trace-lifting $B$*$h$S(B
  $B@F<!2=$K$h$k(B $B6:@5$5$l$?(B sugar strategy $B$K$h$k7W;;$r9T$&(B.
  @item
- @code{dgr()} $B$O(B, @code{gr()}, @code{dgr()} $B$r(B
+ @code{dgr()} $B$O(B, @code{gr()}, @code{hgr()} $B$r(B
  $B;R%W%m%;%9%j%9%H(B @var{procs} $B$N(B 2 $B$D$N%W%m%;%9$K$h$jF1;~$K7W;;$5$;(B,
  $B@h$K7k2L$rJV$7$?J}$N7k2L$rJV$9(B. $B7k2L$OF10l$G$"$k$,(B, $B$I$A$i$NJ}K!$,(B
  $B9bB.$+0lHL$K$OITL@$N$?$a(B, $B<B:]$N7P2a;~4V$rC;=L$9$k$N$KM-8z$G$"$k(B.
  @item
  @code{dgr()} $B$GI=<($5$l$k;~4V$O(B, $B$3$NH!?t$,<B9T$5$l$F$$$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
  These functions are defined in @samp{gr} in the standard library
  directory.
+ @item
+ Functions of which names contains gr are obsolted.
+ Functions of @code{nd_gr} families should be used (@fref{nd_gr nd_gr_trace nd_f4 nd_f4_trace nd_weyl_gr nd_weyl_gr_trace}).
  @item
  They compute a Groebner basis of a polynomial list @var{plist} with
  respect to the variable order @var{vlist} and the order type @var{order}.
-Line 1324  Therefore this function is useful to reduce the actual
+Line 1683  Therefore this function is useful to reduce the actual
 Line 1324  Therefore this function is useful to reduce the actual
 Line 1683  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
-Line 1341  for communication.
+Line 1707  for communication.
 Line 1341  for communication.
 Line 1707  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 dp_gr_f_main dp_weyl_gr_main dp_weyl_gr_mod_main dp_weyl_gr_f_main},
- @fref{dp_gr_main dp_gr_mod_main},
  @fref{dp_ord}.
  @end table
-Line 1371  for communication.
+Line 1736  for communication.
 Line 1371  for communication.
 Line 1736  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
-Line 1559  processes.
+Line 1924  processes.
 Line 1559  processes.
 Line 1924  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
-Line 1575  processes.
+Line 1940  processes.
 Line 1575  processes.
 Line 1940  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})
+ @item tolex_gsl(@var{plist},@var{vlist1},@var{order},@var{vlist2})
- @itemx tolex_gsl_d(@var{plist},@var{vlist1},@var{order},@var{vlist2},@var{homo},@var{procs})
+ @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
-Line 1585  processes.
+Line 1950  processes.
 Line 1585  processes.
 Line 1950  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
-Line 1661  processes.
+Line 2026  processes.
 Line 1661  processes.
 Line 2026  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
-Line 1691  processes.
+Line 2057  processes.
 Line 1691  processes.
 Line 2057  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
-Line 1788  for @code{gr_minipoly()}.
+Line 2154  for @code{gr_minipoly()}.
 Line 1788  for @code{gr_minipoly()}.
 Line 2154  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
-Line 1836  z^32+11405*z^31+20868*z^30+21602*z^29+...
+Line 2202  z^32+11405*z^31+20868*z^30+21602*z^29+...
 Line 1836  z^32+11405*z^31+20868*z^30+21602*z^29+...
 Line 2202  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
+ \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,,, Functions for Groebner basis computation
+ \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}
+ @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
-Line 1853  z^32+11405*z^31+20868*z^30+21602*z^29+...
+Line 2227  z^32+11405*z^31+20868*z^30+21602*z^29+...
 Line 1853  z^32+11405*z^31+20868*z^30+21602*z^29+...
 Line 2227  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
-Line 1872  z^32+11405*z^31+20868*z^30+21602*z^29+...
+Line 2246  z^32+11405*z^31+20868*z^30+21602*z^29+...
 Line 1872  z^32+11405*z^31+20868*z^30+21602*z^29+...
 Line 2246  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
-Line 1905  z^32+11405*z^31+20868*z^30+21602*z^29+...
+Line 2284  z^32+11405*z^31+20868*z^30+21602*z^29+...
 Line 1905  z^32+11405*z^31+20868*z^30+21602*z^29+...
 Line 2284  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
-Line 1944  Actual computation is controlled by various parameters
+Line 2330  Actual computation is controlled by various parameters
 Line 1944  Actual computation is controlled by various parameters
 Line 2330  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
+ \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,,, Functions for Groebner basis computation
+ \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}
+ @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
-Line 1965  Actual computation is controlled by various parameters
+Line 2356  Actual computation is controlled by various parameters
 Line 1965  Actual computation is controlled by various parameters
 Line 2356  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
-Line 1982  F4 $B%"%k%4%j%:%`$O(B, J.C. Faugere $B$K$h$jDs>'$5$
+Line 2373  F4 $B%"%k%4%j%:%`$O(B, J.C. Faugere $B$K$h$jDs>'$5$
 Line 1982  F4 $B%"%k%4%j%:%`$O(B, J.C. Faugere $B$K$h$jDs>'$5$
 Line 2373  F4 $B%"%k%4%j%:%`$O(B, J.C. Faugere $B$K$h$jDs>'$5$
  $B;;K!$G$"$j(B, $BK\<BAu$O(B, $BCf9q>jM>DjM}$K$h$k@~7AJ}Dx<05a2r$rMQ$$$?(B
  $B;n83E*$J<BAu$G$"$k(B.
  @item
- $B0z?t$*$h$SF0:n$O$=$l$>$l(B @code{dp_gr_main()}, @code{dp_gr_mod_main()}
+ $B@F<!2=$N0z?t$,$J$$$3$H$r=|$1$P(B, $B0z?t$*$h$SF0:n$O$=$l$>$l(B
+ @code{dp_gr_main()}, @code{dp_gr_mod_main()},
+ @code{dp_weyl_gr_main()}, @code{dp_weyl_gr_mod_main()}
  $B$HF1MM$G$"$k(B.
  \E
  \BEG
-Line 1994  invented by J.C. Faugere. The current implementation o
+Line 2387  invented by J.C. Faugere. The current implementation o
 Line 1994  invented by J.C. Faugere. The current implementation o
 Line 2387  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
-Line 2008  Arguments and actions are the same as those of
+Line 2403  Arguments and actions are the same as those of
 Line 2008  Arguments and actions are the same as those of
 Line 2403  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}[|@var{option=value,...}])
+ @itemx nd_gr_trace(@var{plist},@var{vlist},@var{homo},@var{p},@var{order}[|@var{option=value,...}])
+ @itemx nd_f4(@var{plist},@var{vlist},@var{modular},@var{order}[|@var{option=value,...}])
+ @itemx nd_f4_trace(@var{plist},@var{vlist},@var{homo},@var{p},@var{order}[|@var{option=value,...}])
+ @itemx nd_weyl_gr(@var{plist},@var{vlist},@var{p},@var{order}[|@var{option=value,...}])
+ @itemx nd_weyl_gr_trace(@var{plist},@var{vlist},@var{homo},@var{p},@var{order}[|@var{option=value,...}])
+ \JP :: $B%0%l%V%J4pDl$N7W;;(B ($BAH$_9~$_H!?t(B)
+ \EG :: Groebner basis computation (built-in functions)
+ @end table
+ @table @var
+ @item return
+ \JP $B%j%9%H(B
+ \EG list
+ @item plist  vlist
+ \JP $B%j%9%H(B
+ \EG list
+ @item order
+ \JP $B?t(B, $B%j%9%H$^$?$O9TNs(B
+ \EG number, list or matrix
+ @item homo
+ \JP $B%U%i%0(B
+ \EG flag
+ @item modular
+ \JP $B%U%i%0$^$?$OAG?t(B
+ \EG flag or prime
+ @end table
+ \BJP
+ @itemize @bullet
+ @item
+ $B$3$l$i$NH!?t$O(B, $B%0%l%V%J4pDl7W;;AH$_9~$_4X?t$N?7<BAu$G$"$k(B.
+ @item @code{nd_gr} $B$O(B, @code{p} $B$,(B 0 $B$N$H$-M-M}?tBN>e$N(B Buchberger
+ $B%"%k%4%j%:%`$r<B9T$9$k(B. @code{p} $B$,(B 2 $B0J>e$N<+A3?t$N$H$-(B, GF(p) $B>e$N(B
+ Buchberger $B%"%k%4%j%:%`$r<B9T$9$k(B.
+ @item @code{nd_gr_trace} $B$*$h$S(B @code{nd_f4_trace}
+ $B$OM-M}?tBN>e$G(B trace $B%"%k%4%j%:%`$r<B9T$9$k(B.
+ @var{p} $B$,(B 0 $B$^$?$O(B 1 $B$N$H$-(B, $B<+F0E*$KA*$P$l$?AG?t$rMQ$$$F(B, $B@.8y$9$k(B
+ $B$^$G(B trace $B%"%k%4%j%:%`$r<B9T$9$k(B.
+ @var{p} $B$,(B 2 $B0J>e$N$H$-(B, trace $B$O(BGF(p) $B>e$G7W;;$5$l$k(B. trace $B%"%k%4%j%:%`(B
+ $B$,<:GT$7$?>l9g(B 0 $B$,JV$5$l$k(B. @var{p} $B$,Ii$N>l9g(B, $B%0%l%V%J4pDl%A%'%C%/$O(B
+ $B9T$o$J$$(B. $B$3$N>l9g(B, @var{p} $B$,(B -1 $B$J$i$P<+F0E*$KA*$P$l$?AG?t$,(B,
+ $B$=$l0J30$O;XDj$5$l$?AG?t$rMQ$$$F%0%l%V%J4pDl8uJd$N7W;;$,9T$o$l$k(B.
+ @code{nd_f4_trace} $B$O(B, $B3FA4<!?t$K$D$$$F(B, $B$"$kM-8BBN>e$G(B F4 $B%"%k%4%j%:%`(B
+ $B$G9T$C$?7k2L$r$b$H$K(B, $B$=$NM-8BBN>e$G(B 0 $B$G$J$$4pDl$rM?$($k(B S-$BB?9`<0$N$_$r(B
+ $BMQ$$$F9TNs@8@.$r9T$$(B, $B$=$NA4<!?t$K$*$1$k4pDl$r@8@.$9$kJ}K!$G$"$k(B. $BF@$i$l$k(B
+ $BB?9`<0=89g$O$d$O$j%0%l%V%J4pDl8uJd$G$"$j(B, @code{nd_gr_trace} $B$HF1MM$N(B
+ $B%A%'%C%/$,9T$o$l$k(B.
+ @item
+ @code{nd_f4} $B$O(B @code{modular} $B$,(B 0 $B$N$H$-M-M}?tBN>e$N(B, @code{modular} $B$,(B
+ $B%^%7%s%5%$%:AG?t$N$H$-M-8BBN>e$N(B F4 $B%"%k%4%j%:%`$r<B9T$9$k(B.
+ @item
+ @var{plist} $B$,B?9`<0%j%9%H$N>l9g(B, @var{plist}$B$G@8@.$5$l$k%$%G%"%k$N%0%l%V%J!<4pDl$,(B
+ $B7W;;$5$l$k(B. @var{plist} $B$,B?9`<0%j%9%H$N%j%9%H$N>l9g(B, $B3FMWAG$OB?9`<04D>e$N<+M32C72$N85$H8+$J$5$l(B,
+ $B$3$l$i$,@8@.$9$kItJ,2C72$N%0%l%V%J!<4pDl$,7W;;$5$l$k(B. $B8e<T$N>l9g(B, $B9`=g=x$O2C72$KBP$9$k9`=g=x$r(B
+ $B;XDj$9$kI,MW$,$"$k(B. $B$3$l$O(B @var{[s,ord]} $B$N7A$G;XDj$9$k(B. @var{s} $B$,(B 0 $B$J$i$P(B TOP (Term Over Position),
+$B$J$i$P(B POT (Position Over Term) $B$r0UL#$7(B, @var{ord} $B$OB?9`<04D$NC19`<0$KBP$9$k9`=g=x$G$"$k(B.
+ @item
+ @code{nd_weyl_gr}, @code{nd_weyl_gr_trace} $B$O(B Weyl $BBe?tMQ$G$"$k(B.
+ @item
+ @code{f4} $B7O4X?t0J30$O$9$Y$FM-M}4X?t78?t$N7W;;$,2DG=$G$"$k(B.
+ @item
+ $B0lHL$K(B @code{dp_gr_main}, @code{dp_gr_mod_main} $B$h$j9bB.$G$"$k$,(B,
+ $BFC$KM-8BBN>e$N>l9g82Cx$G$"$k(B.
+ @item
+ $B0J2<$N%*%W%7%g%s$,;XDj$G$-$k(B.
+ @table @code
+ @item homo
+$B$N$H$-(B, $B@F<!2=$r7PM3$7$F7W;;$9$k(B. (@code{nd_gr}, @code{nd_f4} $B$N$_(B)
+ @item dp
+$B$N$H$-(B, $BJ,;6I=8=B?9`<0(B ($B2C72$N>l9g$K$O2C72B?9`<0(B) $B$r7k2L$H$7$FJV$9(B.
+ @item nora
+$B$N$H$-(B, $B7k2L$NAj8_4JLs$r9T$o$J$$(B.
+ @end table
+ @end itemize
+ \E
+ \BEG
+ @itemize @bullet
+ @item
+ These functions are new implementations for computing Groebner bases.
+ @item @code{nd_gr} executes Buchberger algorithm over the rationals
+ if  @code{p} is 0, and that over GF(p) if @code{p} is a prime.
+ @item @code{nd_gr_trace} executes the trace algorithm over the rationals.
+ If @code{p} is 0 or 1, the trace algorithm is executed until it succeeds
+ by using automatically chosen primes.
+ If @code{p} a positive prime,
+ the trace is comuted over GF(p).
+ If the trace algorithm fails 0 is returned.
+ If @code{p} is negative,
+ the Groebner basis check and ideal-membership check are omitted.
+ In this case, an automatically chosen prime if @code{p} is 1,
+ otherwise the specified prime is used to compute a Groebner basis
+ candidate.
+ Execution of @code{nd_f4_trace} is done as follows:
+ For each total degree, an F4-reduction of S-polynomials over a finite field
+ is done, and S-polynomials which give non-zero basis elements are gathered.
+ Then F4-reduction over Q is done for the gathered S-polynomials.
+ The obtained polynomial set is a Groebner basis candidate and the same
+ check procedure as in the case of @code{nd_gr_trace} is done.
+ @item
+ @code{nd_f4} executes F4 algorithm over Q if @code{modular} is equal to 0,
+ or over a finite field GF(@code{modular})
+ if @code{modular} is a prime number of machine size (<2^29).
+ If @var{plist} is a list of polynomials, then a Groebner basis of the ideal generated by @var{plist}
+ is computed. If @var{plist} is a list of lists of polynomials, then each list of polynomials are regarded
+ as an element of a free module over a polynomial ring and a Groebner basis of the sub-module generated by @var{plist}
+ in the free module. In the latter case a term order in the free module should be specified.
+ This is specified by @var{[s,ord]}. If @var{s} is 0 then it means TOP (Term Over Position).
+ If @var{s} is 1 then it means POT 1 (Position Over Term). @var{ord} is a term order in the base polynomial ring.
+ @item
+ @code{nd_weyl_gr}, @code{nd_weyl_gr_trace} are for Weyl algebra computation.
+ @item
+ Functions except for F4 related ones can handle rational coeffient cases.
+ @item
+ In general these functions are more efficient than
+ @code{dp_gr_main}, @code{dp_gr_mod_main}, especially over finite fields.
+ @item
+ The fallowing options can be specified.
+ @table @code
+ @item homo
+ If set to 1, the computation is done via homogenization. (only for @code{nd_gr} and @code{nd_f4})
+ @item dp
+ If set to 1, the functions return a list of distributed polynomials (a list of
+ module polynomials when the input is a sub-module).
+ @item nora
+ If set to 1, the inter-reduction is not performed.
+ @end table
+ @end itemize
+ \E
+ @example
+ [38] load("cyclic")$
+ [49] C=cyclic(7)$
+ [50] V=vars(C)$
+ [51] cputime(1)$
+ [52] dp_gr_mod_main(C,V,0,31991,0)$
+.06sec + gc : 0.313sec(26.4sec)
+ [53] nd_gr(C,V,31991,0)$
+ ndv_alloc=1477188
+.737sec + gc : 0.1837sec(5.921sec)
+ [54] dp_f4_mod_main(C,V,31991,0)$
+.51sec + gc : 0.7109sec(4.221sec)
+ [55] nd_f4(C,V,31991,0)$
+.906sec + gc : 0.126sec(2.032sec)
+ @end example
+ @table @t
+ \JP @item $B;2>H(B
+ \EG @item References
+ @fref{dp_ord},
+ @fref{dp_gr_flags dp_gr_print},
+ \JP @fref{$B7W;;$*$h$SI=<($N@)8f(B}.
+ \EG @fref{Controlling Groebner basis computations}
+ @end table
+ \JP @node nd_gr_postproc nd_weyl_gr_postproc,,, $B%0%l%V%J4pDl$K4X$9$kH!?t(B
+ \EG @node nd_gr_postproc nd_weyl_gr_postproc,,, Functions for Groebner basis computation
+ @subsection @code{nd_gr_postproc}, @code{nd_weyl_gr_postproc}
+ @findex nd_gr_postproc
+ @findex nd_weyl_gr_postproc
+ @table @t
+ @item nd_gr_postproc(@var{plist},@var{vlist},@var{p},@var{order},@var{check})
+ @itemx nd_weyl_gr_postproc(@var{plist},@var{vlist},@var{p},@var{order},@var{check})
+ \JP :: $B%0%l%V%J4pDl8uJd$N%A%'%C%/$*$h$SAj8_4JLs(B
+ \EG :: Check of Groebner basis candidate and inter-reduction
+ @end table
+ @table @var
+ @item return
+ \JP $B%j%9%H(B $B$^$?$O(B 0
+ \EG list or 0
+ @item plist  vlist
+ \JP $B%j%9%H(B
+ \EG list
+ @item p
+ \JP $BAG?t$^$?$O(B 0
+ \EG prime or 0
+ @item order
+ \JP $B?t(B, $B%j%9%H$^$?$O9TNs(B
+ \EG number, list or matrix
+ @item check
+ \JP 0 $B$^$?$O(B 1
+ \EG 0 or 1
+ @end table
+ @itemize @bullet
+ \BJP
+ @item
+ $B%0%l%V%J4pDl(B($B8uJd(B)$B$NAj8_4JLs$r9T$&(B.
+ @item
+ @code{nd_weyl_gr_postproc} $B$O(B Weyl $BBe?tMQ$G$"$k(B.
+ @item
+ @var{check=1} $B$N>l9g(B, @var{plist} $B$,(B, @var{vlist}, @var{p}, @var{order} $B$G;XDj$5$l$kB?9`<04D(B, $B9`=g=x$G%0%l%V%J!<4pDl$K$J$C$F$$$k$+(B
+ $B$N%A%'%C%/$b9T$&(B.
+ @item
+ $B@F<!2=$7$F7W;;$7$?%0%l%V%J!<4pDl$rHs@F<!2=$7$?$b$N$rAj8_4JLs$r9T$&(B, CRT $B$G7W;;$7$?%0%l%V%J!<4pDl8uJd$N%A%'%C%/$r9T$&$J$I$N>l9g$KMQ$$$k(B.
+ \E
+ \BEG
+ @item
+ Perform the inter-reduction for a Groebner basis (candidate).
+ @item
+ @code{nd_weyl_gr_postproc} is for Weyl algebra.
+ @item
+ If @var{check=1} then the check whether @var{plist} is a Groebner basis with respect to a term order in a polynomial ring
+ or Weyl algebra specified by @var{vlist}, @var{p} and @var{order}.
+ @item
+ This function is used for inter-reduction of a non-reduced Groebner basis that is obtained by dehomogenizing a Groebner basis
+ computed via homogenization, or Groebner basis check of a Groebner basis candidate computed by CRT.
+ \E
+ @end itemize
+ @example
+ afo
+ @end example
  \JP @node dp_gr_flags dp_gr_print,,, $B%0%l%V%J4pDl$K4X$9$kH!?t(B
  \EG @node dp_gr_flags dp_gr_print,,, Functions for Groebner basis computation
  @subsection @code{dp_gr_flags}, @code{dp_gr_print}
-Line 2016  Arguments and actions are the same as those of
+Line 2639  Arguments and actions are the same as those of
 Line 2016  Arguments and actions are the same as those of
 Line 2639  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.
-Line 2030  and showing informations.
+Line 2653  and showing informations.
 Line 2030  and showing informations.
 Line 2653  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$$>l9g(B, $B8=:_$N@_Dj$,JV$5$l$k(B.
-Line 2043  and showing informations.
+Line 2669  and showing informations.
 Line 2043  and showing informations.
 Line 2669  and showing informations.
  $B0z?t$O(B, @code{["Print",1,"NoSugar",1,...]} $B$J$k7A$N%j%9%H$G(B, $B:8$+$i=g$K(B
  $B@_Dj$5$l$k(B. $B%Q%i%a%?L>$OJ8;zNs$GM?$($kI,MW$,$"$k(B.
  @item
- @code{dp_gr_print()} $B$O(B, $BFC$K%Q%i%a%?(B @code{Print} $B$NCM$rD>@\@_Dj(B, $B;2>H(B
+ @code{dp_gr_print()} $B$O(B, $BFC$K%Q%i%a%?(B @code{Print}, @code{PrintShort} $B$NCM$rD>@\@_Dj(B, $B;2>H(B
- $B$G$-$k(B. $B$3$l$O(B, @code{dp_gr_main()} $B$J$I$r%5%V%k!<%A%s$H$7$FMQ$$$k%f!<%6(B
+ $B$G$-$k(B. $B@_Dj$5$l$kCM$O<!$NDL$j$G$"$k!#(B
- $BH!?t$K$*$$$F(B, @code{Print} $B$NCM$r8+$F(B, $B$=$N%5%V%k!<%A%s$,Cf4V>pJs$NI=<((B
+ @table @var
+ @item i=0
+ @code{Print=0}, @code{PrintShort=0}
+ @item i=1
+ @code{Print=1}, @code{PrintShort=0}
+ @item i=2
+ @code{Print=0}, @code{PrintShort=1}
+ @end table
+ $B$3$l$O(B, @code{dp_gr_main()} $B$J$I$r%5%V%k!<%A%s$H$7$FMQ$$$k%f!<%6(B
+ $BH!?t$K$*$$$F(B, $B$=$N%5%V%k!<%A%s$,Cf4V>pJs$NI=<((B
  $B$r9T$&:]$K(B, $B?WB.$K%U%i%0$r8+$k$3$H$,$G$-$k$h$&$KMQ0U$5$l$F$$$k(B.
  \E
  \BEG
-Line 2060  Arguments must be specified as a list such as
+Line 2695  Arguments must be specified as a list such as
 Line 2060  Arguments must be specified as a list such as
 Line 2695  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} and @code{PrintShort}.
- @code{Print} when a user defined function calling @code{dp_gr_main()} etc.
+ @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
-Line 2117  uses the value as a flag for showing intermediate info
+Line 2761  uses the value as a flag for showing intermediate info
 Line 2117  uses the value as a flag for showing intermediate info
 Line 2761  uses the value as a flag for showing intermediate info
  @item
  $B%H%C%W%l%Y%kH!?t0J30$NH!?t$rD>@\8F$S=P$9>l9g$K$O(B, $B$3$NH!?t$K$h$j(B
  $BJQ?t=g=x7?$r@5$7$/@_Dj$7$J$1$l$P$J$i$J$$(B.
+ @item
+ $B0z?t$,%j%9%H$N>l9g(B, $B<+M32C72$K$*$1$k9`=g=x7?$r@_Dj$9$k(B. $B0z?t$,(B@code{[0,Ord]} $B$N>l9g(B,
+ $BB?9`<04D>e$G(B @code{Ord} $B$G;XDj$5$l$k9`=g=x$K4p$E$/(B TOP $B=g=x(B, $B0z?t$,(B @code{[1,Ord]} $B$N>l9g(B
+ OPT $B=g=x$r@_Dj$9$k(B.
  \E
  \BEG
  @item
-Line 2144  that such polynomials were generated under the same or
+Line 2794  that such polynomials were generated under the same or
 Line 2144  that such polynomials were generated under the same or
 Line 2794  that such polynomials were generated under the same or
  @item
  Type of term ordering must be correctly set by this function
  when functions other than top level functions are called directly.
+ @item
+ If the argument is a list, then an ordering type in a free module is set.
+ If the argument is @code{[0,Ord]} then a TOP ordering based on the ordering type specified
+ by @code{Ord} is set.
+ If the argument is @code{[1,Ord]} then a POT ordering is set.
  \E
  @end itemize
-Line 2163  when functions other than top level functions are call
+Line 2819  when functions other than top level functions are call
 Line 2163  when functions other than top level functions are call
 Line 2819  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}
-Line 2211  the coefficient field.
+Line 2940  the coefficient field.
 Line 2211  the coefficient field.
 Line 2940  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
-Line 2221  the coefficient field.
+Line 2951  the coefficient field.
 Line 2221  the coefficient field.
 Line 2951  the coefficient field.
  @fref{dp_ord}.
  @end table
+ \JP @node dpm_dptodpm,,, $B%0%l%V%J4pDl$K4X$9$kH!?t(B
+ \EG @node dpm_dptodpm,,, Functions for Groebner basis computation
+ @subsection @code{dpm_dptodpm}
+ @findex dpm_dptodpm
+ @table @t
+ @item dpm_dptodpm(@var{dpoly},@var{pos})
+ \JP :: $BJ,;6I=8=B?9`<0$r2C72B?9`<0$KJQ49$9$k(B.
+ \EG :: Converts a distributed polynomial into a module polynomial.
+ @end table
+ @table @var
+ @item return
+ \JP $B2C72B?9`<0(B
+ \EG module polynomial
+ @item dpoly
+ \JP $BJ,;6I=8=B?9`<0(B
+ \EG distributed polynomial
+ @item pos
+ \JP $B@5@0?t(B
+ \EG positive integer
+ @end table
+ @itemize @bullet
+ \BJP
+ @item
+ $BJ,;6I=8=B?9`<0$r2C72B?9`<0$KJQ49$9$k(B.
+ @item
+ $B=PNO$O2C72B?9`<0(B @code{dpoly e_pos} $B$G$"$k(B.
+ \E
+ \BEG
+ @item
+ This function converts a distributed polynomial into a module polynomial.
+ @item
+ The output is @code{dpoly e_pos}.
+ \E
+ @end itemize
+ @example
+ [50] dp_ord([0,0])$
+ [51] D=dp_ptod((x+y+z)^2,[x,y,z]);
+ (1)*<<2,0,0>>+(2)*<<1,1,0>>+(1)*<<0,2,0>>+(2)*<<1,0,1>>+(2)*<<0,1,1>>
+ +(1)*<<0,0,2>>
+ [52] dp_dptodpm(D,2);
+ (1)*<<2,0,0:2>>+(2)*<<1,1,0:2>>+(1)*<<0,2,0:2>>+(2)*<<1,0,1:2>>
+ +(2)*<<0,1,1:2>>+(1)*<<0,0,2:2>>
+ @end example
+ @table @t
+ \JP @item $B;2>H(B
+ \EG @item References
+ @fref{dp_ptod},
+ @fref{dp_ord}.
+ @end table
+ \JP @node dpm_ltod,,, $B%0%l%V%J4pDl$K4X$9$kH!?t(B
+ \EG @node dpm_ltod,,, Functions for Groebner basis computation
+ @subsection @code{dpm_ltod}
+ @findex dpm_ltod
+ @table @t
+ @item dpm_dptodpm(@var{plist},@var{vlist})
+ \JP :: $BB?9`<0%j%9%H$r2C72B?9`<0$KJQ49$9$k(B.
+ \EG :: Converts a list of polynomials into a module polynomial.
+ @end table
+ @table @var
+ @item return
+ \JP $B2C72B?9`<0(B
+ \EG module polynomial
+ @item plist
+ \JP $BB?9`<0%j%9%H(B
+ \EG list of polynomials
+ @item vlist
+ \JP $BJQ?t%j%9%H(B
+ \EG list of variables
+ @end table
+ @itemize @bullet
+ \BJP
+ @item
+ $BB?9`<0%j%9%H$r2C72B?9`<0$KJQ49$9$k(B.
+ @item
+ @code{[p1,...,pm]} $B$O(B @code{p1 e1+...+pm em} $B$KJQ49$5$l$k(B.
+ \E
+ \BEG
+ @item
+ This function converts a list of polynomials into a module polynomial.
+ @item
+ @code{[p1,...,pm]} is converted into @code{p1 e1+...+pm em}.
+ \E
+ @end itemize
+ @example
+ [2126] dp_ord([0,0])$
+ [2127] dpm_ltod([x^2+y^2,x,y-z],[x,y,z]);
+ (1)*<<2,0,0:1>>+(1)*<<0,2,0:1>>+(1)*<<1,0,0:2>>+(1)*<<0,1,0:3>>
+ +(-1)*<<0,0,1:3>>
+ @end example
+ @table @t
+ \JP @item $B;2>H(B
+ \EG @item References
+ @fref{dpm_dtol},
+ @fref{dp_ord}.
+ @end table
+ \JP @node dpm_dtol,,, $B%0%l%V%J4pDl$K4X$9$kH!?t(B
+ \EG @node dpm_dtol,,, Functions for Groebner basis computation
+ @subsection @code{dpm_dtol}
+ @findex dpm_dtol
+ @table @t
+ @item dpm_dptodpm(@var{poly},@var{vlist})
+ \JP :: $B2C72B?9`<0$rB?9`<0%j%9%H$KJQ49$9$k(B.
+ \EG :: Converts a module polynomial into a list of polynomials.
+ @end table
+ @table @var
+ @item return
+ \JP $BB?9`<0%j%9%H(B
+ \EG list of polynomials
+ @item poly
+ \JP $B2C72B?9`<0(B
+ \EG module polynomial
+ @item vlist
+ \JP $BJQ?t%j%9%H(B
+ \EG list of variables
+ @end table
+ @itemize @bullet
+ \BJP
+ @item
+ $B2C72B?9`<0$rB?9`<0%j%9%H$KJQ49$9$k(B.
+ @item
+ @code{p1 e1+...+pm em} $B$O(B @code{[p1,...,pm]} $B$KJQ49$5$l$k(B.
+ @item
+ $B=PNO%j%9%H$ND9$5$O(B, @code{poly} $B$K4^$^$l$kI8=`4pDl$N:GBg%$%s%G%C%/%9$H$J$k(B.
+ \E
+ \BEG
+ @item
+ This function converts a module polynomial into a list of polynomials.
+ @item
+ @code{p1 e1+...+pm em} is converted into @code{[p1,...,pm]}.
+ @item
+ The length of the output list is equal to the largest index among those of the standard bases
+ containd in @code{poly}.
+ \E
+ @end itemize
+ @example
+ [2126] dp_ord([0,0])$
+ [2127] D=(1)*<<2,0,0:1>>+(1)*<<0,2,0:1>>+(1)*<<1,0,0:2>>+(1)*<<0,1,0:3>>
+ +(-1)*<<0,0,1:3>>$
+ [2128] dpm_dtol(D,[x,y,z]);
+ [x^2+y^2,x,y-z]
+ @end example
+ @table @t
+ \JP @item $B;2>H(B
+ \EG @item References
+ @fref{dpm_ltod},
+ @fref{dp_ord}.
+ @end table
  \JP @node dp_dtop,,, $B%0%l%V%J4pDl$K4X$9$kH!?t(B
  \EG @node dp_dtop,,, Functions for Groebner basis computation
  @subsection @code{dp_dtop}
-Line 2263  variables of @var{dpoly}.
+Line 3158  variables of @var{dpoly}.
 Line 2263  variables of @var{dpoly}.
 Line 3158  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
-Line 2343  converting the coefficients into elements of a finite
+Line 3239  converting the coefficients into elements of a finite
 Line 2343  converting the coefficients into elements of a finite
 Line 3239  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
-Line 2434  These are used internally in @code{hgr()} etc.
+Line 3330  These are used internally in @code{hgr()} etc.
 Line 2434  These are used internally in @code{hgr()} etc.
 Line 3330  These are used internally in @code{hgr()} etc.
  into an integral distributed polynomial such that GCD of all its coefficients
  is 1.
  \E
- @itemx dp_prim(@var{dpoly})
+ @item dp_prim(@var{dpoly})
  \JP :: $BM-M}<0G\$7$F78?t$r@0?t78?tB?9`<078?t$+$D78?t$NB?9`<0(B GCD $B$r(B 1 $B$K$9$k(B.
  \BEG
  :: Converts a distributed polynomial @var{poly} with rational function
-Line 2487  polynomial contents included in the coefficients are n
+Line 3383  polynomial contents included in the coefficients are n
 Line 2487  polynomial contents included in the coefficients are n
 Line 3383  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
+ \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,,, Functions for Groebner basis computation
+ \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
-Line 2539  is returned in such a list as @code{[numerator, denomi
+Line 3439  is returned in such a list as @code{[numerator, denomi
 Line 2539  is returned in such a list as @code{[numerator, denomi
 Line 3439  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
-Line 2571  is returned in such a list as @code{[numerator, denomi
+Line 3473  is returned in such a list as @code{[numerator, denomi
 Line 2571  is returned in such a list as @code{[numerator, denomi
 Line 3473  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
-Line 2616  For single computation @code{p_nf} and @code{p_true_nf
+Line 3521  For single computation @code{p_nf} and @code{p_true_nf
 Line 2616  For single computation @code{p_nf} and @code{p_true_nf
 Line 3521  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
+ -11380879768451657780886122972730785203470970010204714556333530492210
- 0005716505560062087150928400876150217079820311439477560587583488*u4^15+...
+ 456775930005716505560062087150928400876150217079820311439477560587583
+*u4^15+...
  [79] dp_dtop(dp_nf([4,3,2,1,0],T,DP2,1),V);
- -1138087976845165778088612297273078520347097001020471455633353049221045677593
+ -11380879768451657780886122972730785203470970010204714556333530492210
- 0005716505560062087150928400876150217079820311439477560587583488*u4^15+...
+ 456775930005716505560062087150928400876150217079820311439477560587583
+*u4^15+...
  [80] @@78==@@79;
  @end example
-Line 2638  u4^2+(6*u3+2*u2+6*u1-2)*u4+9*u3^2+(6*u2+18*u1-6)*u3+u2
+Line 3546  u4^2+(6*u3+2*u2+6*u1-2)*u4+9*u3^2+(6*u2+18*u1-6)*u3+u2
 Line 2638  u4^2+(6*u3+2*u2+6*u1-2)*u4+9*u3^2+(6*u2+18*u1-6)*u3+u2
 Line 3546  u4^2+(6*u3+2*u2+6*u1-2)*u4+9*u3^2+(6*u2+18*u1-6)*u3+u2
  @fref{p_nf p_nf_mod p_true_nf p_true_nf_mod}.
  @end table
+ \JP @node dpm_nf dpm_nf_and_quotient,,, $B%0%l%V%J4pDl$K4X$9$kH!?t(B
+ \EG @node dpm_nf dpm_nf_and_quotient,,, Functions for Groebner basis computation
+ @subsection @code{dpm_nf}, @code{dpm_nf_and_quotient}
+ @findex dpm_nf
+ @findex dpm_nf_and_quotient
+ @table @t
+ @item dpm_nf([@var{indexlist},]@var{dpoly},@var{dpolyarray},@var{fullreduce})
+ \JP :: $B2C72B?9`<0$N@55,7A$r5a$a$k(B. ($B7k2L$ODj?tG\$5$l$F$$$k2DG=@-$"$j(B)
+ \BEG
+ :: Computes the normal form of a module polynomial.
+ (The result may be multiplied by a constant in the ground field.)
+ \E
+ @item dpm_nf_and_quotient([@var{indexlist},]@var{dpoly},@var{dpolyarray})
+ \JP :: $B2C72B?9`<0$N@55,7A$H>&$r5a$a$k(B.
+ \BEG
+ :: Computes the normal form of a module polynomial and the quotient.
+ \E
+ @end table
+ @table @var
+ @item return
+ \JP @code{dpm_nf()} : $B2C72B?9`<0(B, @code{dpm_nf_and_quotient()} : $B%j%9%H(B
+ \EG @code{dpm_nf()} : module polynomial, @code{dpm_nf_and_quotient()} : list
+ @item indexlist
+ \JP $B%j%9%H(B
+ \EG list
+ @item dpoly
+ \JP $B2C72B?9`<0(B
+ \EG module polynomial
+ @item dpolyarray
+ \JP $BG[Ns(B
+ \EG array of module polynomial
+ @end table
+ @itemize @bullet
+ \BJP
+ @item
+ $B2C72B?9`<0(B @var{dpoly} $B$N@55,7A$r5a$a$k(B.
+ @item
+ $B7k2L$KM-M}?t(B, $BM-M}<0$,4^$^$l$k$N$rHr$1$k$?$a(B, @code{dpm_nf()} $B$O(B
+ $B??$NCM$NDj?tG\$NCM$rJV$9(B.
+ @item
+ @var{dpolyarray} $B$O2C72B?9`<0$rMWAG$H$9$k%Y%/%H%k(B,
+ @var{indexlist} $B$O@55,2=7W;;$KMQ$$$k(B @var{dpolyarray} $B$NMWAG$N%$%s%G%C%/%9(B
+ @item
+ @var{indexlist} $B$,M?$($i$l$F$$$k>l9g(B, @var{dpolyarray} $B$NCf$G(B, @var{indexlist} $B$G;XDj$5$l$?$b$N$N$_$,(B, $BA0$NJ}$+$iM%@hE*$K;H$o$l$k(B.
+ @var{indexlist} $B$,M?$($i$l$F$$$J$$>l9g$K$O(B, @var{dpolyarray} $B$NCf$NA4$F$NB?9`<0$,A0$NJ}$+$iM%@hE*$K;H$o$l$k(B.
+ @item
+ @code{dpm_nf_and_quotient()} $B$O(B,
+ @code{[@var{nm},@var{dn},@var{quo}]} $B$J$k7A$N%j%9%H$rJV$9(B.
+ $B$?$@$7(B, @var{nm} $B$O78?t$KJ,?t$r4^$^$J$$2C72B?9`<0(B, @var{dn} $B$O(B
+ $B?t$^$?$OB?9`<0$G(B @var{nm}/@var{dn} $B$,??$NCM$H$J$k(B.
+ @var{quo} $B$O=|;;$N>&$rI=$9G[Ns$G(B, @var{dn}@var{dpoly}=@var{nm}+@var{quo[0]dpolyarray[0]+...} $B$,@.$jN)$D(B.
+ $B$N%j%9%H(B.
+ @item
+ @var{fullreduce} $B$,(B 0 $B$G$J$$$H$-A4$F$N9`$KBP$7$F4JLs$r9T$&(B. @var{fullreduce}
+ $B$,(B 0 $B$N$H$-F,9`$N$_$KBP$7$F4JLs$r9T$&(B.
+ \E
+ \BEG
+ @item
+ Computes the normal form of a module polynomial.
+ @item
+ The result of @code{dpm_nf()} may be multiplied by a constant in the
+ ground field in order to make the result integral.
+ @item
+ @var{dpolyarray} is a vector whose components are module polynomials
+ and @var{indexlist} is a list of indices which is used for the normal form
+ computation.
+ @item
+ If @var{indexlist} is given, only the polynomials in @var{dpolyarray} specified in @var{indexlist}
+ is used in the division. An index placed at the preceding position has priority to be selected.
+ If @var{indexlist} is not given, all the polynomials in @var{dpolyarray} are used.
+ @item
+ @code{dpm_nf_and_quotient()} returns
+ such a list as @code{[@var{nm},@var{dn},@var{quo}]}.
+ Here @var{nm} is a module polynomial whose coefficients are integral
+ in the ground field, @var{dn} is an integral element in the ground
+ field and @var{nm}/@var{dn} is the true normal form.
+ @var{quo} is an array containing the quotients of the division satisfying
+ @var{dn}@var{dpoly}=@var{nm}+@var{quo[0]dpolyarray[0]+...}.
+ @item
+ When argument @var{fullreduce} has non-zero value,
+ all terms are reduced. When it has value 0,
+ only the head term is reduced.
+ \E
+ @end itemize
+ @example
+ [2126] dp_ord([1,0])$
+ [2127] S=ltov([(1)*<<0,0,2,0:1>>+(1)*<<0,0,1,1:1>>+(1)*<<0,0,0,2:1>>
+ +(-1)*<<3,0,0,0:2>>+(-1)*<<0,0,2,1:2>>+(-1)*<<0,0,1,2:2>>
+ +(1)*<<3,0,1,0:3>>+(1)*<<3,0,0,1:3>>+(1)*<<0,0,2,2:3>>,
+ (-1)*<<0,1,0,0:1>>+(-1)*<<0,0,1,0:1>>+(-1)*<<0,0,0,1:1>>
+ +(-1)*<<3,0,0,0:3>>+(1)*<<0,1,1,1:3>>,(1)*<<0,1,0,0:2>>
+ +(1)*<<0,0,1,0:2>>+(1)*<<0,0,0,1:2>>+(-1)*<<0,1,1,0:3>>
+ +(-1)*<<0,1,0,1:3>>+(-1)*<<0,0,1,1:3>>])$
+ [2128] U=dpm_sp(S[0],S[1]);
+ (1)*<<0,0,3,0:1>>+(-1)*<<0,1,1,1:1>>+(1)*<<0,0,2,1:1>>
+ +(-1)*<<0,1,0,2:1>>+(1)*<<3,1,0,0:2>>+(1)*<<0,1,2,1:2>>
+ +(1)*<<0,1,1,2:2>>+(-1)*<<3,1,1,0:3>>+(1)*<<3,0,2,0:3>>
+ +(-1)*<<3,1,0,1:3>>+(-1)*<<0,1,3,1:3>>+(-1)*<<0,1,2,2:3>>
+ [2129] dpm_nf(U,S,1);
+ [2130] L=dpm_nf_and_quotient(U,S)$
+ [2131] Q=L[2]$
+ [2132] D=L[1]$
+ [2133] D*U-(Q[1]*S[1]+Q[2]*S[2]);
+ @end example
+ @table @t
+ \JP @item $B;2>H(B
+ \EG @item References
+ @fref{dpm_sp},
+ @fref{dp_ord}.
+ @end table
  \JP @node dp_hm dp_ht dp_hc dp_rest,,, $B%0%l%V%J4pDl$K4X$9$kH!?t(B
  \EG @node dp_hm dp_ht dp_hc dp_rest,,, Functions for Groebner basis computation
  @subsection @code{dp_hm}, @code{dp_ht}, @code{dp_hc}, @code{dp_rest}
-Line 2712  The next equations hold for a distributed polynomial @
+Line 3740  The next equations hold for a distributed polynomial @
 Line 2712  The next equations hold for a distributed polynomial @
 Line 3740  The next equations hold for a distributed polynomial @
  +(-490)*<<0,0,0>>
  @end example
+ \JP @node dpm_hm dpm_ht dpm_hc dpm_hp dpm_rest,,, $B%0%l%V%J4pDl$K4X$9$kH!?t(B
+ \EG @node dpm_hm dpm_ht dpm_hc dpm_hp dpm_rest,,, Functions for Groebner basis computation
+ @subsection @code{dpm_hm}, @code{dpm_ht}, @code{dpm_hc}, @code{dpm_hp}, @code{dpm_rest}
+ @findex dpm_hm
+ @findex dpm_ht
+ @findex dpm_hc
+ @findex dpm_hp
+ @findex dpm_rest
+ @table @t
+ @item dpm_hm(@var{dpoly})
+ \JP :: $B2C72B?9`<0$NF,C19`<0$r<h$j=P$9(B.
+ \EG :: Gets the head monomial of a module polynomial.
+ @item dpm_ht(@var{dpoly})
+ \JP :: $B2C72B?9`<0$NF,9`$r<h$j=P$9(B.
+ \EG :: Gets the head term of a module polynomial.
+ @item dpm_hc(@var{dpoly})
+ \JP :: $B2C72B?9`<0$NF,78?t$r<h$j=P$9(B.
+ \EG :: Gets the head coefficient of a module polynomial.
+ @item dpm_hp(@var{dpoly})
+ \JP :: $B2C72B?9`<0$NF,0LCV$r<h$j=P$9(B.
+ \EG :: Gets the head position of a module polynomial.
+ @item dpm_rest(@var{dpoly})
+ \JP :: $B2C72B?9`<0$NF,C19`<0$r<h$j=|$$$?;D$j$rJV$9(B.
+ \EG :: Gets the remainder of a module polynomial where the head monomial is removed.
+ @end table
+ @table @var
+ \BJP
+ @item return
+ @code{dp_hm()}, @code{dp_ht()}, @code{dp_rest()} : $B2C72B?9`<0(B,
+ @code{dp_hc()} : $B?t$^$?$OB?9`<0(B
+ @item dpoly
+ $B2C72B?9`<0(B
+ \E
+ \BEG
+ @item return
+ @code{dpm_hm()}, @code{dpm_ht()}, @code{dpm_rest()} : module polynomial
+ @code{dpm_hc()} : monomial
+ @item dpoly
+ distributed polynomial
+ \E
+ @end table
+ @itemize @bullet
+ \BJP
+ @item
+ $B$3$l$i$O(B, $B2C72B?9`<0$N3FItJ,$r<h$j=P$9$?$a$NH!?t$G$"$k(B.
+ @item
+ @code{dpm_hc()} $B$O(B, @code{dpm_hm()} $B$N(B, $BI8=`4pDl$K4X$9$k78?t$G$"$kC19`<0$rJV$9(B.
+ $B%9%+%i!<78?t$r<h$j=P$9$K$O(B, $B$5$i$K(B @code{dp_hc()} $B$r<B9T$9$k(B.
+ @item
+ @code{dpm_hp()} $B$O(B, $BF,2C72C19`<0$K4^$^$l$kI8=`4pDl$N%$%s%G%C%/%9$rJV$9(B.
+ \E
+ \BEG
+ @item
+ These are used to get various parts of a module polynomial.
+ @item
+ @code{dpm_hc()} returns the monomial that is the coefficient of @code{dpm_hm()} with respect to the
+ standard base.
+ For getting its scalar coefficient apply @code{dp_hc()}.
+ @item
+ @code{dpm_hp()} returns the index of the standard base conteind in the head module monomial.
+ \E
+ @end itemize
+ @example
+ [2126] dp_ord([1,0]);
+ [1,0]
+ [2127] F=2*<<1,2,0:2>>-3*<<1,0,2:3>>+<<2,1,0:2>>;
+ (1)*<<2,1,0:2>>+(2)*<<1,2,0:2>>+(-3)*<<1,0,2:3>>
+ [2128] M=dpm_hm(F);
+ (1)*<<2,1,0:2>>
+ [2129] C=dpm_hc(F);
+ (1)*<<2,1,0>>
+ [2130] R=dpm_rest(F);
+ (2)*<<1,2,0:2>>+(-3)*<<1,0,2:3>>
+ [2131] dpm_hp(F);
+ @end example
  \JP @node dp_td dp_sugar,,, $B%0%l%V%J4pDl$K4X$9$kH!?t(B
  \EG @node dp_td dp_sugar,,, Functions for Groebner basis computation
  @subsection @code{dp_td}, @code{dp_sugar}
-Line 2790  selection strategy of critical pairs in Groebner basis
+Line 3900  selection strategy of critical pairs in Groebner basis
 Line 2790  selection strategy of critical pairs in Groebner basis
 Line 3900  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
-Line 2833  two polynomials, where coefficient is always set to 1.
+Line 3943  two polynomials, where coefficient is always set to 1.
 Line 2833  two polynomials, where coefficient is always set to 1.
 Line 3943  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
-Line 2873  Used for finding candidate terms at reduction of polyn
+Line 3983  Used for finding candidate terms at reduction of polyn
 Line 2873  Used for finding candidate terms at reduction of polyn
 Line 3983  Used for finding candidate terms at reduction of polyn
  @fref{dp_red dp_red_mod}.
  @end table
+ \JP @node dpm_redble,,, $B%0%l%V%J4pDl$K4X$9$kH!?t(B
+ \EG @node dpm_redble,,, Functions for Groebner basis computation
+ @subsection @code{dpm_redble}
+ @findex dpm_redble
+ @table @t
+ @item dpm_redble(@var{dpoly1},@var{dpoly2})
+ \JP :: $BF,9`$I$&$7$,@0=|2DG=$+$I$&$+D4$Y$k(B.
+ \EG :: Checks whether one head term is divisible by the other head term.
+ @end table
+ @table @var
+ @item return
+ \JP $B@0?t(B
+ \EG integer
+ @item dpoly1  dpoly2
+ \JP $B2C72B?9`<0(B
+ \EG module polynomial
+ @end table
+ @itemize @bullet
+ \BJP
+ @item
+ @var{dpoly1} $B$NF,9`$,(B @var{dpoly2} $B$NF,9`$G3d$j@Z$l$l$P(B 1, $B3d$j@Z$l$J$1$l$P(B
+$B$rJV$9(B.
+ @item
+ $BB?9`<0$N4JLs$r9T$&:](B, $B$I$N9`$r4JLs$G$-$k$+$rC5$9$N$KMQ$$$k(B.
+ \E
+ \BEG
+ @item
+ Returns 1 if the head term of @var{dpoly2} divides the head term of
+ @var{dpoly1}; otherwise 0.
+ @item
+ Used for finding candidate terms at reduction of polynomials.
+ \E
+ @end itemize
  \JP @node dp_subd,,, $B%0%l%V%J4pDl$K4X$9$kH!?t(B
  \EG @node dp_subd,,, Functions for Groebner basis computation
  @subsection @code{dp_subd}
-Line 2888  Used for finding candidate terms at reduction of polyn
+Line 4035  Used for finding candidate terms at reduction of polyn
 Line 2888  Used for finding candidate terms at reduction of polyn
 Line 4035  Used for finding candidate terms at reduction of polyn
  @item return
  \JP $BJ,;6I=8=B?9`<0(B
  \EG distributed polynomial
- @item dpoly1, dpoly2
+ @item dpoly1  dpoly2
  \JP $BJ,;6I=8=B?9`<0(B
  \EG distributed polynomial
  @end table
-Line 3112  values of @code{dp_mag()} for intermediate basis eleme
+Line 4259  values of @code{dp_mag()} for intermediate basis eleme
 Line 3112  values of @code{dp_mag()} for intermediate basis eleme
 Line 4259  values of @code{dp_mag()} for intermediate basis eleme
  @item return
  \JP $B%j%9%H(B
  \EG list
- @item dpoly1, dpoly2, dpoly3
+ @item dpoly1  dpoly2  dpoly3
  \JP $BJ,;6I=8=B?9`<0(B
  \EG distributed polynomial
  @item vlist
-Line 3136  values of @code{dp_mag()} for intermediate basis eleme
+Line 4283  values of @code{dp_mag()} for intermediate basis eleme
 Line 3136  values of @code{dp_mag()} for intermediate basis eleme
 Line 4283  values of @code{dp_mag()} for intermediate basis eleme
  $B$J$i$J$$(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
-Line 3155  the divisibility of the head term of @var{dpoly2} by t
+Line 4302  the divisibility of the head term of @var{dpoly2} by t
 Line 3155  the divisibility of the head term of @var{dpoly2} by t
 Line 4302  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
-Line 3169  The result is a list @code{[@var{a dpoly1},@var{a dpol
+Line 4316  The result is a list @code{[@var{a dpoly1},@var{a dpol
 Line 3169  The result is a list @code{[@var{a dpoly1},@var{a dpol
 Line 4316  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>>
+ [(6)*<<2,1,0,0,0>>+(6)*<<1,2,0,0,0>>+(2)*<<0,3,0,0,0>>,
- +(-1)*<<1,1,0,0,1>>]
+ (-1)*<<0,1,1,1,0>>+(-1)*<<1,1,0,0,1>>]
  @end example
  @table @t
-Line 3196  The result is a list @code{[@var{a dpoly1},@var{a dpol
+Line 4343  The result is a list @code{[@var{a dpoly1},@var{a dpol
 Line 3196  The result is a list @code{[@var{a dpoly1},@var{a dpol
 Line 4343  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
-Line 3240  make the result integral.
+Line 4387  make the result integral.
 Line 3240  make the result integral.
 Line 4387  make the result integral.
  \EG @item References
  @fref{dp_mod dp_rat}.
  @end table
+ \JP @node dpm_sp,,, $B%0%l%V%J4pDl$K4X$9$kH!?t(B
+ \EG @node dmp_sp,,, Functions for Groebner basis computation
+ @subsection @code{dpm_sp}
+ @findex dpm_sp
+ @table @t
+ @item dpm_sp(@var{dpoly1},@var{dpoly2}[|coef=1])
+ \JP :: S-$BB?9`<0$N7W;;(B
+ \EG :: Computation of an S-polynomial
+ @end table
+ @table @var
+ @item return
+ \JP $B2C72B?9`<0$^$?$O%j%9%H(B
+ \EG module polynomial or list
+ @item dpoly1  dpoly2
+ \JP $B2C72B?9`<0(B
+ \EG module polynomial
+ \JP $BJ,;6I=8=B?9`<0(B
+ @end table
+ @itemize @bullet
+ \BJP
+ @item
+ @var{dpoly1}, @var{dpoly2} $B$N(B S-$BB?9`<0$r7W;;$9$k(B.
+ @item
+ $B%*%W%7%g%s(B @var{coef=1} $B$,;XDj$5$l$F$$$k>l9g(B, @code{[S,t1,t2]} $B$J$k%j%9%H$rJV$9(B.
+ $B$3$3$G(B, @code{t1}, @code{t2} $B$O(BS-$BB?9`<0$r:n$k:]$N78?tC19`<0$G(B @code{S=t1 dpoly1-t2 dpoly2}
+ $B$rK~$?$9(B.
+ \E
+ \BEG
+ @item
+ This function computes the S-polynomial of @var{dpoly1} and @var{dpoly2}.
+ @item
+ If an option @var{coef=1} is specified, it returns a list @code{[S,t1,t2]},
+ where @code{S} is the S-polynmial and @code{t1}, @code{t2} are monomials satisfying @code{S=t1 dpoly1-t2 dpoly2}.
+ \E
+ @end itemize
  \JP @node p_nf p_nf_mod p_true_nf p_true_nf_mod,,, $B%0%l%V%J4pDl$K4X$9$kH!?t(B
  \EG @node p_nf p_nf_mod p_true_nf p_true_nf_mod,,, Functions for Groebner basis computation
  @subsection @code{p_nf}, @code{p_nf_mod}, @code{p_true_nf}, @code{p_true_nf_mod}
-Line 3272  as a form of @code{[numerator, denominator]})
+Line 4459  as a form of @code{[numerator, denominator]})
 Line 3272  as a form of @code{[numerator, denominator]})
 Line 4459  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
-Line 3349  refer to @code{dp_true_nf()} and @code{dp_true_nf_mod(
+Line 4536  refer to @code{dp_true_nf()} and @code{dp_true_nf_mod(
 Line 3349  refer to @code{dp_true_nf()} and @code{dp_true_nf_mod(
 Line 4536  refer to @code{dp_true_nf()} and @code{dp_true_nf_mod(
  @fref{dp_ptod},
  @fref{dp_dtop},
  @fref{dp_ord},
- @fref{dp_nf dp_nf_mod dp_true_nf dp_true_nf_mod}.
+ @fref{dp_nf dp_nf_mod dp_true_nf dp_true_nf_mod dp_weyl_nf dp_weyl_nf_mod}.
  @end table
  \JP @node p_terms,,, $B%0%l%V%J4pDl$K4X$9$kH!?t(B
-Line 3408  exists.
+Line 4595  exists.
 Line 3408  exists.
 Line 4595  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,
+ [u5,u0^31,u0^30,u0^29,u0^28,u0^27,u0^26,u0^25,u0^24,u0^23,u0^22,
- u0^19,u0^18,u0^17,u0^16,u0^15,u0^14,u0^13,u0^12,u0^11,u0^10,u0^9,u0^8,u0^7,
+ u0^21,u0^20,u0^19,u0^18,u0^17,u0^16,u0^15,u0^14,u0^13,u0^12,u0^11,
- u0^6,u0^5,u0^4,u0^3,u0^2,u0,1]
+ u0^10,u0^9,u0^8,u0^7,u0^6,u0^5,u0^4,u0^3,u0^2,u0,1]
  @end example
  \JP @node gb_comp,,, $B%0%l%V%J4pDl$K4X$9$kH!?t(B
-Line 3427  u0^6,u0^5,u0^4,u0^3,u0^2,u0,1]
+Line 4614  u0^6,u0^5,u0^4,u0^3,u0^2,u0,1]
 Line 3427  u0^6,u0^5,u0^4,u0^3,u0^2,u0,1]
 Line 4614  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
-Line 3518  Polynomial set @code{cyclic} is sometimes called by ot
+Line 4705  Polynomial set @code{cyclic} is sometimes called by ot
 Line 3518  Polynomial set @code{cyclic} is sometimes called by ot
 Line 4705  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,
-*u3*u0+2*u1*u4-u3+(2*u1+2*u5)*u2,2*u2*u0+2*u2*u4+(2*u1+2*u5)*u3-u2+u1^2,
+*u3*u0+2*u1*u4-u3+(2*u1+2*u5)*u2,2*u2*u0+2*u2*u4+(2*u1+2*u5)*u3
-*u1*u0+(2*u3+2*u5)*u4+2*u2*u3+2*u1*u2-u1,
+ -u2+u1^2,2*u1*u0+(2*u3+2*u5)*u4+2*u2*u3+2*u1*u2-u1,
  u0^2-u0+2*u4^2+2*u3^2+2*u2^2+2*u1^2+2*u5^2]
  [90] hkatsura(5);
  [-t+u0+2*u4+2*u3+2*u2+2*u1+2*u5,
-Line 3545  u0^2-u0+2*u4^2+2*u3^2+2*u2^2+2*u1^2+2*u5^2]
+Line 4732  u0^2-u0+2*u4^2+2*u3^2+2*u2^2+2*u1^2+2*u5^2]
 Line 3545  u0^2-u0+2*u4^2+2*u3^2+2*u2^2+2*u1^2+2*u5^2]
 Line 4732  u0^2-u0+2*u4^2+2*u3^2+2*u2^2+2*u1^2+2*u5^2]
  \JP @item $B;2>H(B
  \EG @item References
  @fref{dp_dtop}.
+ @end table
+ \JP @node primadec primedec,,, $B%0%l%V%J4pDl$K4X$9$kH!?t(B
+ \EG @node primadec primedec,,, Functions for Groebner basis computation
+ @subsection @code{primadec}, @code{primedec}
+ @findex primadec
+ @findex primedec
+ @table @t
+ @item primadec(@var{plist},@var{vlist})
+ @item primedec(@var{plist},@var{vlist})
+ \JP :: $B%$%G%"%k$NJ,2r(B
+ \EG :: Computes decompositions of ideals.
+ @end table
+ @table @var
+ @item return
+ @itemx plist
+ \JP $BB?9`<0%j%9%H(B
+ \EG list of polynomials
+ @item vlist
+ \JP $BJQ?t%j%9%H(B
+ \EG list of variables
+ @end table
+ @itemize @bullet
+ \BJP
+ @item
+ @code{primadec()}, @code{primedec} $B$O(B @samp{primdec} $B$GDj5A$5$l$F$$$k(B.
+ @item
+ @code{primadec()}, @code{primedec()} $B$O$=$l$>$lM-M}?tBN>e$G$N%$%G%"%k$N(B
+ $B=`AGJ,2r(B, $B:,4p$NAG%$%G%"%kJ,2r$r9T$&(B.
+ @item
+ $B0z?t$OB?9`<0%j%9%H$*$h$SJQ?t%j%9%H$G$"$k(B. $BB?9`<0$OM-M}?t78?t$N$_$,5v$5$l$k(B.
+ @item
+ @code{primadec} $B$O(B @code{[$B=`AG@.J,(B, $BIUB0AG%$%G%"%k(B]} $B$N%j%9%H$rJV$9(B.
+ @item
+ @code{primadec} $B$O(B $BAG0x;R$N%j%9%H$rJV$9(B.
+ @item
+ $B7k2L$K$*$$$F(B, $BB?9`<0%j%9%H$H$7$FI=<($5$l$F$$$k3F%$%G%"%k$OA4$F(B
+ $B%0%l%V%J4pDl$G$"$k(B. $BBP1~$9$k9`=g=x$O(B, $B$=$l$>$l(B
+ $BJQ?t(B @code{PRIMAORD}, @code{PRIMEORD} $B$K3JG<$5$l$F$$$k(B.
+ @item
+ @code{primadec} $B$O(B @code{[Shimoyama,Yokoyama]} $B$N=`AGJ,2r%"%k%4%j%:%`(B
+ $B$r<BAu$7$F$$$k(B.
+ @item
+ $B$b$7AG0x;R$N$_$r5a$a$?$$$J$i(B, @code{primedec} $B$r;H$&J}$,$h$$(B.
+ $B$3$l$O(B, $BF~NO%$%G%"%k$,:,4p%$%G%"%k$G$J$$>l9g$K(B, @code{primadec}
+ $B$N7W;;$KM>J,$J%3%9%H$,I,MW$H$J$k>l9g$,$"$k$+$i$G$"$k(B.
+ \E
+ \BEG
+ @item
+ Function @code{primadec()} and @code{primedec} are defined in @samp{primdec}.
+ @item
+ @code{primadec()}, @code{primedec()} are the function for primary
+ ideal decomposition and prime decomposition of the radical over the
+ rationals respectively.
+ @item
+ The arguments are a list of polynomials and a list of variables.
+ These functions accept ideals with rational function coefficients only.
+ @item
+ @code{primadec} returns the list of pair lists consisting a primary component
+ and its associated prime.
+ @item
+ @code{primedec} returns the list of prime components.
+ @item
+ Each component is a Groebner basis and the corresponding term order
+ is indicated by the global variables @code{PRIMAORD}, @code{PRIMEORD}
+ respectively.
+ @item
+ @code{primadec} implements the primary decompostion algorithm
+ in @code{[Shimoyama,Yokoyama]}.
+ @item
+ If one only wants to know the prime components of an ideal, then
+ use @code{primedec} because @code{primadec} may need additional costs
+ if an input ideal is not radical.
+ \E
+ @end itemize
+ @example
+ [84] load("primdec")$
+ [102] primedec([p*q*x-q^2*y^2+q^2*y,-p^2*x^2+p^2*x+p*q*y,
+ (q^3*y^4-2*q^3*y^3+q^3*y^2)*x-q^3*y^4+q^3*y^3,
+ -q^3*y^4+2*q^3*y^3+(-q^3+p*q^2)*y^2],[p,q,x,y]);
+ [[y,x],[y,p],[x,q],[q,p],[x-1,q],[y-1,p],[(y-1)*x-y,q*y^2-2*q*y-p+q]]
+ [103] primadec([x,z*y,w*y^2,w^2*y-z^3,y^3],[x,y,z,w]);
+ [[[x,z*y,y^2,w^2*y-z^3],[z,y,x]],[[w,x,z*y,z^3,y^3],[w,z,y,x]]]
+ @end example
+ @table @t
+ \JP @item $B;2>H(B
+ \EG @item References
+ @fref{fctr sqfr},
+ \JP @fref{$B9`=g=x$N@_Dj(B}.
+ \EG @fref{Setting term orderings}.
+ @end table
+ \JP @node primedec_mod,,, $B%0%l%V%J4pDl$K4X$9$kH!?t(B
+ \EG @node primedec_mod,,, Functions for Groebner basis computation
+ @subsection @code{primedec_mod}
+ @findex primedec_mod
+ @table @t
+ @item primedec_mod(@var{plist},@var{vlist},@var{ord},@var{mod},@var{strategy})
+ \JP :: $B%$%G%"%k$NJ,2r(B
+ \EG :: Computes decompositions of ideals over small finite fields.
+ @end table
+ @table @var
+ @item return
+ @itemx plist
+ \JP $BB?9`<0%j%9%H(B
+ \EG list of polynomials
+ @item vlist
+ \JP $BJQ?t%j%9%H(B
+ \EG list of variables
+ @item ord
+ \JP $B?t(B, $B%j%9%H$^$?$O9TNs(B
+ \EG number, list or matrix
+ @item mod
+ \JP $B@5@0?t(B
+ \EG positive integer
+ @item strategy
+ \JP $B@0?t(B
+ \EG integer
+ @end table
+ @itemize @bullet
+ \BJP
+ @item
+ @code{primedec_mod()} $B$O(B @samp{primdec_mod}
+ $B$GDj5A$5$l$F$$$k(B. @code{[Yokoyama]} $B$NAG%$%G%"%kJ,2r%"%k%4%j%:%`(B
+ $B$r<BAu$7$F$$$k(B.
+ @item
+ @code{primedec_mod()} $B$OM-8BBN>e$G$N%$%G%"%k$N(B
+ $B:,4p$NAG%$%G%"%kJ,2r$r9T$$(B, $BAG%$%G%"%k$N%j%9%H$rJV$9(B.
+ @item
+ @code{primedec_mod()} $B$O(B, GF(@var{mod}) $B>e$G$NJ,2r$rM?$($k(B.
+ $B7k2L$N3F@.J,$N@8@.85$O(B, $B@0?t78?tB?9`<0$G$"$k(B.
+ @item
+ $B7k2L$K$*$$$F(B, $BB?9`<0%j%9%H$H$7$FI=<($5$l$F$$$k3F%$%G%"%k$OA4$F(B
+ [@var{vlist},@var{ord}] $B$G;XDj$5$l$k9`=g=x$K4X$9$k%0%l%V%J4pDl$G$"$k(B.
+ @item
+ @var{strategy} $B$,(B 0 $B$G$J$$$H$-(B, incremental $B$K(B component $B$N6&DL(B
+ $BItJ,$r7W;;$9$k$3$H$K$h$k(B early termination $B$r9T$&(B. $B0lHL$K(B,
+ $B%$%G%"%k$N<!85$,9b$$>l9g$KM-8z$@$,(B, 0 $B<!85$N>l9g$J$I(B, $B<!85$,>.$5$$(B
+ $B>l9g$K$O(B overhead $B$,Bg$-$$>l9g$,$"$k(B.
+ @item
+ $B7W;;ESCf$GFbIt>pJs$r8+$?$$>l9g$K$O!"(B
+ $BA0$b$C$F(B @code{dp_gr_print(2)} $B$r<B9T$7$F$*$1$P$h$$(B.
+ \E
+ \BEG
+ @item
+ Function @code{primedec_mod()}
+ is defined in @samp{primdec_mod} and implements the prime decomposition
+ algorithm in @code{[Yokoyama]}.
+ @item
+ @code{primedec_mod()}
+ is the function for prime ideal decomposition
+ of the radical of a polynomial ideal over small finite field,
+ and they return a list of prime ideals, which are associated primes
+ of the input ideal.
+ @item
+ @code{primedec_mod()} gives the decomposition over GF(@var{mod}).
+ The generators of each resulting component consists of integral polynomials.
+ @item
+ Each resulting component is a Groebner basis with respect to
+ a term order specified by [@var{vlist},@var{ord}].
+ @item
+ If @var{strategy} is non zero, then the early termination strategy
+ is tried by computing the intersection of obtained components
+ incrementally. In general, this strategy is useful when the krull
+ dimension of the ideal is high, but it may add some overhead
+ if the dimension is small.
+ @item
+ If you want to see internal information during the computation,
+ execute @code{dp_gr_print(2)} in advance.
+ \E
+ @end itemize
+ @example
+ [0] load("primdec_mod")$
+ [246] PP444=[x^8+x^2+t,y^8+y^2+t,z^8+z^2+t]$
+ [247] primedec_mod(PP444,[x,y,z,t],0,2,1);
+ [[y+z,x+z,z^8+z^2+t],[x+y,y^2+y+z^2+z+1,z^8+z^2+t],
+ [y+z+1,x+z+1,z^8+z^2+t],[x+z,y^2+y+z^2+z+1,z^8+z^2+t],
+ [y+z,x^2+x+z^2+z+1,z^8+z^2+t],[y+z+1,x^2+x+z^2+z+1,z^8+z^2+t],
+ [x+z+1,y^2+y+z^2+z+1,z^8+z^2+t],[y+z+1,x+z,z^8+z^2+t],
+ [x+y+1,y^2+y+z^2+z+1,z^8+z^2+t],[y+z,x+z+1,z^8+z^2+t]]
+ [248]
+ @end example
+ @table @t
+ \JP @item $B;2>H(B
+ \EG @item References
+ @fref{modfctr},
+ @fref{dp_gr_main dp_gr_mod_main dp_gr_f_main dp_weyl_gr_main dp_weyl_gr_mod_main dp_weyl_gr_f_main},
+ \JP @fref{$B9`=g=x$N@_Dj(B}.
+ \EG @fref{Setting term orderings},
+ @fref{dp_gr_flags dp_gr_print}.
+ @end table
+ \JP @node bfunction bfct generic_bfct ann ann0,,, $B%0%l%V%J4pDl$K4X$9$kH!?t(B
+ \EG @node bfunction bfct generic_bfct ann ann0,,, Functions for Groebner basis computation
+ @subsection @code{bfunction}, @code{bfct}, @code{generic_bfct}, @code{ann}, @code{ann0}
+ @findex bfunction
+ @findex bfct
+ @findex generic_bfct
+ @findex ann
+ @findex ann0
+ @table @t
+ @item bfunction(@var{f})
+ @itemx bfct(@var{f})
+ @itemx generic_bfct(@var{plist},@var{vlist},@var{dvlist},@var{weight})
+ \JP :: @var{b} $B4X?t$N7W;;(B
+ \EG :: Computes the global @var{b} function of a polynomial or an ideal
+ @item ann(@var{f})
+ @itemx ann0(@var{f})
+ \JP :: $BB?9`<0$N%Y%-$N(B annihilator $B$N7W;;(B
+ \EG :: Computes the annihilator of a power of polynomial
+ @end table
+ @table @var
+ @item return
+ \JP $BB?9`<0$^$?$O%j%9%H(B
+ \EG polynomial or list
+ @item f
+ \JP $BB?9`<0(B
+ \EG polynomial
+ @item plist
+ \JP $BB?9`<0%j%9%H(B
+ \EG list of polynomials
+ @item vlist dvlist
+ \JP $BJQ?t%j%9%H(B
+ \EG list of variables
+ @end table
+ @itemize @bullet
+ \BJP
+ @item @samp{bfct} $B$GDj5A$5$l$F$$$k(B.
+ @item @code{bfunction(@var{f})}, @code{bfct(@var{f})} $B$OB?9`<0(B @var{f} $B$N(B global @var{b} $B4X?t(B @code{b(s)} $B$r(B
+ $B7W;;$9$k(B. @code{b(s)} $B$O(B, Weyl $BBe?t(B @code{D} $B>e$N0lJQ?tB?9`<04D(B @code{D[s]}
+ $B$N85(B @code{P(x,s)} $B$,B8:_$7$F(B, @code{P(x,s)f^(s+1)=b(s)f^s} $B$rK~$?$9$h$&$J(B
+ $BB?9`<0(B @code{b(s)} $B$NCf$G(B, $B<!?t$,:G$bDc$$$b$N$G$"$k(B.
+ @item @code{generic_bfct(@var{f},@var{vlist},@var{dvlist},@var{weight})}
+ $B$O(B, @var{plist} $B$G@8@.$5$l$k(B @code{D} $B$N:8%$%G%"%k(B @code{I} $B$N(B,
+ $B%&%'%$%H(B @var{weight} $B$K4X$9$k(B global @var{b} $B4X?t$r7W;;$9$k(B.
+ @var{vlist} $B$O(B @code{x}-$BJQ?t(B, @var{vlist} $B$OBP1~$9$k(B @code{D}-$BJQ?t(B
+ $B$r=g$KJB$Y$k(B.
+ @item @code{bfunction} $B$H(B @code{bfct} $B$G$OMQ$$$F$$$k%"%k%4%j%:%`$,(B
+ $B0[$J$k(B. $B$I$A$i$,9bB.$+$OF~NO$K$h$k(B.
+ @item @code{ann(@var{f})} $B$O(B, @code{@var{f}^s} $B$N(B annihilator ideal
+ $B$N@8@.7O$rJV$9(B. @code{ann(@var{f})} $B$O(B, @code{[@var{a},@var{list}]}
+ $B$J$k%j%9%H$rJV$9(B. $B$3$3$G(B, @var{a} $B$O(B @var{f} $B$N(B @var{b} $B4X?t$N:G>.@0?t:,(B,
+ @var{list} $B$O(B @code{ann(@var{f})} $B$N7k2L$N(B @code{s}$ $B$K(B, @var{a} $B$r(B
+ $BBeF~$7$?$b$N$G$"$k(B.
+ @item $B>\:Y$K$D$$$F$O(B, [Saito,Sturmfels,Takayama] $B$r8+$h(B.
+ \E
+ \BEG
+ @item These functions are defined in @samp{bfct}.
+ @item @code{bfunction(@var{f})} and @code{bfct(@var{f})} compute the global @var{b}-function @code{b(s)} of
+ a polynomial @var{f}.
+ @code{b(s)} is a polynomial of the minimal degree
+ such that there exists @code{P(x,s)} in D[s], which is a polynomial
+ ring over Weyl algebra @code{D}, and @code{P(x,s)f^(s+1)=b(s)f^s} holds.
+ @item @code{generic_bfct(@var{f},@var{vlist},@var{dvlist},@var{weight})}
+ computes the global @var{b}-function of a left ideal @code{I} in @code{D}
+ generated by @var{plist}, with respect to @var{weight}.
+ @var{vlist} is the list of @code{x}-variables,
+ @var{vlist} is the list of corresponding @code{D}-variables.
+ @item @code{bfunction(@var{f})} and @code{bfct(@var{f})} implement
+ different algorithms and the efficiency depends on inputs.
+ @item @code{ann(@var{f})} returns the generator set of the annihilator
+ ideal of @code{@var{f}^s}.
+ @code{ann(@var{f})} returns a list @code{[@var{a},@var{list}]},
+ where @var{a} is the minimal integral root of the global @var{b}-function
+ of @var{f}, and @var{list} is a list of polynomials obtained by
+ substituting @code{s} in @code{ann(@var{f})} with @var{a}.
+ @item See [Saito,Sturmfels,Takayama] for the details.
+ \E
+ @end itemize
+ @example
+ [0] load("bfct")$
+ [216] bfunction(x^3+y^3+z^3+x^2*y^2*z^2+x*y*z);
+ -9*s^5-63*s^4-173*s^3-233*s^2-154*s-40
+ [217] fctr(@@);
+ [[-1,1],[s+2,1],[3*s+4,1],[3*s+5,1],[s+1,2]]
+ [218] F = [4*x^3*dt+y*z*dt+dx,x*z*dt+4*y^3*dt+dy,
+ x*y*dt+5*z^4*dt+dz,-x^4-z*y*x-y^4-z^5+t]$
+ [219] generic_bfct(F,[t,z,y,x],[dt,dz,dy,dx],[1,0,0,0]);
+*s^10-70000*s^9+101750*s^8-79375*s^7+35768*s^6-9277*s^5
+ +1278*s^4-72*s^3
+ [220] P=x^3-y^2$
+ [221] ann(P);
+ [2*dy*x+3*dx*y^2,-3*dx*x-2*dy*y+6*s]
+ [222] ann0(P);
+ [-1,[2*dy*x+3*dx*y^2,-3*dx*x-2*dy*y-6]]
+ @end example
+ @table @t
+ \JP @item $B;2>H(B
+ \EG @item References
+ \JP @fref{Weyl $BBe?t(B}.
+ \EG @fref{Weyl algebra}.
  @end table

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