===================================================================
RCS file: /home/cvs/OpenXM/src/asir-doc/parts/groebner.texi,v
retrieving revision 1.6
retrieving revision 1.12
diff -u -p -r1.6 -r1.12
--- OpenXM/src/asir-doc/parts/groebner.texi	2003/04/20 09:55:18	1.6
+++ OpenXM/src/asir-doc/parts/groebner.texi	2003/12/27 11:52:07	1.12
@@ -1,4 +1,4 @@
-@comment $OpenXM: OpenXM/src/asir-doc/parts/groebner.texi,v 1.5 2003/04/20 08:01:25 noro Exp $
+@comment $OpenXM: OpenXM/src/asir-doc/parts/groebner.texi,v 1.11 2003/04/28 06:43:10 noro Exp $
 \BJP
 @node $B%0%l%V%J4pDl$N7W;;(B,,, Top
 @chapter $B%0%l%V%J4pDl$N7W;;(B
@@ -449,6 +449,13 @@ displayed.
 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
@@ -471,24 +478,28 @@ computation the maximal value among the sums is shown.
 computation the maximal value among the sums is shown.
 \E
 
-@item Multiple
+@item Content
+@itemx Multiple
 \BJP
-0 $B$G$J$$@0?t$N;~(B, $BM-M}?t>e$N@55,7A7W;;$K$*$$$F(B, $B78?t$N%S%C%HD9$NOB$,(B
-@code{Multiple} $BG\$K$J$k$4$H$K78?tA4BN$N(B GCD $B$,7W;;$5$l(B, $B$=$N(B GCD $B$G(B
-$B3d$C$?B?9`<0$r4JLs$9$k(B. @code{Multiple} $B$,(B 1 $B$J$i$P(B, $B4JLs$9$k$4$H$K(B
-GCD $B7W;;$,9T$o$l0lHL$K$O8zN($,0-$/$J$k$,(B, @code{Multiple} $B$r(B 2 $BDxEY(B
+0 $B$G$J$$M-M}?t$N;~(B, $BM-M}?t>e$N@55,7A7W;;$K$*$$$F(B, $B78?t$N%S%C%HD9$NOB$,(B
+@code{Content} $BG\$K$J$k$4$H$K78?tA4BN$N(B GCD $B$,7W;;$5$l(B, $B$=$N(B GCD $B$G(B
+$B3d$C$?B?9`<0$r4JLs$9$k(B. @code{Content} $B$,(B 1 $B$J$i$P(B, $B4JLs$9$k$4$H$K(B
+GCD $B7W;;$,9T$o$l0lHL$K$O8zN($,0-$/$J$k$,(B, @code{Content} $B$r(B 2 $BDxEY(B
 $B$H$9$k$H(B, $B5pBg$J@0?t$,78?t$K8=$l$k>l9g(B, $B8zN($,NI$/$J$k>l9g$,$"$k(B. 
+backward compatibility $B$N$?$a!"(B@code{Multiple} $B$G@0?tCM$r;XDj$G$-$k(B.
 \E
 \BEG
-If a non-zero integer, in a normal form computation
+If a non-zero rational number, in a normal form computation
 over the rationals, the integer content of the polynomial being
-reduced is removed when its magnitude becomes @code{Multiple} times
+reduced is removed when its magnitude becomes @code{Content} times
 larger than a registered value, which is set to the magnitude of the
 input polynomial. After each content removal the registered value is
-set to the magnitude of the resulting polynomial. @code{Multiple} is
+set to the magnitude of the resulting polynomial. @code{Content} is
 equal to 1, the simiplification is done after every normal form computation.
-It is empirically known that it is often efficient to set @code{Multiple} to 2
+It is empirically known that it is often efficient to set @code{Content} to 2
 for the case where large integers appear during the computation.
+An integer value can be set by the keyword @code{Multiple} for
+backward compatibility.
 \E
 
 @item Demand
@@ -1343,7 +1354,7 @@ Computation of the global b function is implemented as
 * lex_hensel_gsl tolex_gsl tolex_gsl_d::
 * primadec primedec::
 * primedec_mod::
-* bfunction generic_bfct::
+* bfunction bfct generic_bfct ann ann0::
 @end menu
 
 \JP @node gr hgr gr_mod,,, $B%0%l%V%J4pDl$K4X$9$kH!?t(B
@@ -1401,6 +1412,14 @@ CPU $B;~4V$G$"$j(B, $B$3$NH!?t$N>l9g$O$[$H$s$IDL?.$
 @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 
@@ -1429,6 +1448,13 @@ for communication.
 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
 
@@ -1679,8 +1705,8 @@ processes.
 @item lex_hensel_gsl(@var{plist},@var{vlist1},@var{order},@var{vlist2},@var{homo})
 \JP :: GSL $B7A<0$N%$%G%"%k4pDl$N7W;;(B
 \EG ::Computation of an GSL form ideal basis
-@item tolex_gsl(@var{plist},@var{vlist1},@var{order},@var{vlist2},@var{homo})
-@itemx tolex_gsl_d(@var{plist},@var{vlist1},@var{order},@var{vlist2},@var{homo},@var{procs})
+@item tolex_gsl(@var{plist},@var{vlist1},@var{order},@var{vlist2})
+@itemx tolex_gsl_d(@var{plist},@var{vlist1},@var{order},@var{vlist2},@var{procs})
 \JP :: $B%0%l%V%J4pDl$rF~NO$H$9$k(B, GSL $B7A<0$N%$%G%"%k4pDl$N7W;;(B
 \EG :: Computation of an GSL form ideal basis stating from a Groebner basis
 @end table
@@ -2150,7 +2176,7 @@ except for lack of the argument for controlling homoge
 
 @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.
@@ -2164,6 +2190,9 @@ 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
@@ -2177,9 +2206,18 @@ and showing informations.
 $B0z?t$O(B, @code{["Print",1,"NoSugar",1,...]} $B$J$k7A$N%j%9%H$G(B, $B:8$+$i=g$K(B
 $B@_Dj$5$l$k(B. $B%Q%i%a%?L>$OJ8;zNs$GM?$($kI,MW$,$"$k(B. 
 @item
-@code{dp_gr_print()} $B$O(B, $BFC$K%Q%i%a%?(B @code{Print} $B$NCM$rD>@\@_Dj(B, $B;2>H(B
-$B$G$-$k(B. $B$3$l$O(B, @code{dp_gr_main()} $B$J$I$r%5%V%k!<%A%s$H$7$FMQ$$$k%f!<%6(B
-$BH!?t$K$*$$$F(B, @code{Print} $B$NCM$r8+$F(B, $B$=$N%5%V%k!<%A%s$,Cf4V>pJs$NI=<((B
+@code{dp_gr_print()} $B$O(B, $BFC$K%Q%i%a%?(B @code{Print}, @code{PrintShort} $B$NCM$rD>@\@_Dj(B, $B;2>H(B
+$B$G$-$k(B. $B@_Dj$5$l$kCM$O<!$NDL$j$G$"$k!#(B
+@table @var
+@item i=0
+@code{Print=0}, @code{PrintShort=0}
+@item i=1
+@code{Print=1}, @code{PrintShort=0}
+@item i=2
+@code{Print=0}, @code{PrintShort=1}
+@end table
+$B$3$l$O(B, @code{dp_gr_main()} $B$J$I$r%5%V%k!<%A%s$H$7$FMQ$$$k%f!<%6(B
+$BH!?t$K$*$$$F(B, $B$=$N%5%V%k!<%A%s$,Cf4V>pJs$NI=<((B
 $B$r9T$&:]$K(B, $B?WB.$K%U%i%0$r8+$k$3$H$,$G$-$k$h$&$KMQ0U$5$l$F$$$k(B. 
 \E
 \BEG
@@ -2194,8 +2232,17 @@ strings.
 strings.
 @item
 @code{dp_gr_print()} is used to set and show the value of a parameter
-@code{Print}. This functions is prepared to get quickly the value of
-@code{Print} when a user defined function calling @code{dp_gr_main()} etc.
+@code{Print} and @code{PrintShort}. 
+@table @var
+@item i=0
+@code{Print=0}, @code{PrintShort=0}
+@item i=1
+@code{Print=1}, @code{PrintShort=0}
+@item i=2
+@code{Print=0}, @code{PrintShort=1}
+@end table
+This functions is prepared to get quickly the value
+when a user defined function calling @code{dp_gr_main()} etc.
 uses the value as a flag for showing intermediate informations.
 \E
 @end itemize
@@ -3831,6 +3878,9 @@ -q^3*y^4+2*q^3*y^3+(-q^3+p*q^2)*y^2],[p,q,x,y]);
 $BItJ,$r7W;;$9$k$3$H$K$h$k(B early termination $B$r9T$&(B. $B0lHL$K(B, 
 $B%$%G%"%k$N<!85$,9b$$>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
@@ -3855,6 +3905,9 @@ if the dimension is small.
 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
 
@@ -3876,24 +3929,36 @@ if the dimension is small.
 @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}.
+\EG @fref{Setting term orderings},
+@fref{dp_gr_flags dp_gr_print}.
 @end table
 
-\JP @node bfunction generic_bfct,,, $B%0%l%V%J4pDl$K4X$9$kH!?t(B
-\EG @node bfunction generic_bfct,,, Functions for Groebner basis computation
-@subsection @code{bfunction}, @code{generic_bfct}
+\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})
-@item generic_bfct(@var{plist},@var{vlist},@var{dvlist},@var{weight})
-\JP :: b $B4X?t$N7W;;(B
-\EG :: Computes the global b function of a polynomial or an ideal
+@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
-@itemx f
+\JP $BB?9`<0$^$?$O%j%9%H(B
+\EG polynomial or list
+@item f
 \JP $BB?9`<0(B
 \EG polynomial
 @item plist
@@ -3907,30 +3972,45 @@ if the dimension is small.
 @itemize @bullet
 \BJP
 @item @samp{bfct} $B$GDj5A$5$l$F$$$k(B. 
-@item @code{bfunction(@var{f})} $B$OB?9`<0(B @var{f} $B$N(B global b $B4X?t(B @code{b(s)} $B$r(B
+@item @code{bfunction(@var{f})}, @code{bfct(@var{f})} $B$OB?9`<0(B @var{f} $B$N(B global @var{b} $B4X?t(B @code{b(s)} $B$r(B
 $B7W;;$9$k(B. @code{b(s)} $B$O(B, Weyl $BBe?t(B @code{D} $B>e$N0lJQ?tB?9`<04D(B @code{D[s]}
 $B$N85(B @code{P(x,s)} $B$,B8:_$7$F(B, @code{P(x,s)f^(s+1)=b(s)f^s} $B$rK~$?$9$h$&$J(B
 $BB?9`<0(B @code{b(s)} $B$NCf$G(B, $B<!?t$,:G$bDc$$$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 b $B4X?t$r7W;;$9$k(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 $B>\:Y$K$D$$$F$O(B, [SST] $B$r8+$h(B. 
+@item @code{bfunction} $B$H(B @code{bfct} $B$G$OMQ$$$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})} computes the global b-function @code{b(s)} of
+@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 b-function of a left ideal @code{I} in @code{D}
+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 See [SST] for the details.
+@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
 
@@ -3945,6 +4025,11 @@ +1278*s^4-72*s^3
 [219] generic_bfct(F,[t,z,y,x],[dt,dz,dy,dx],[1,0,0,0]);
 20000*s^10-70000*s^9+101750*s^8-79375*s^7+35768*s^6-9277*s^5
 +1278*s^4-72*s^3
+[220] P=x^3-y^2$
+[221] ann(P);
+[2*dy*x+3*dx*y^2,-3*dx*x-2*dy*y+6*s]
+[222] ann0(P);
+[-1,[2*dy*x+3*dx*y^2,-3*dx*x-2*dy*y-6]]
 @end example
 
 @table @t