| version 1.4, 2003/04/19 15:44:56 |
version 1.17, 2006/09/06 23:53:31 |
|
|
| @comment $OpenXM: OpenXM/src/asir-doc/parts/groebner.texi,v 1.3 1999/12/24 04:38:04 noro Exp $ |
@comment $OpenXM: OpenXM/src/asir-doc/parts/groebner.texi,v 1.16 2004/10/20 00:30:55 fujiwara Exp $ |
| \BJP |
\BJP |
| @node $B%0%l%V%J4pDl$N7W;;(B,,, Top |
@node $B%0%l%V%J4pDl$N7W;;(B,,, Top |
| @chapter $B%0%l%V%J4pDl$N7W;;(B |
@chapter $B%0%l%V%J4pDl$N7W;;(B |
|
|
| * $B4pK\E*$JH!?t(B:: |
* $B4pK\E*$JH!?t(B:: |
| * $B7W;;$*$h$SI=<($N@)8f(B:: |
* $B7W;;$*$h$SI=<($N@)8f(B:: |
| * $B9`=g=x$N@_Dj(B:: |
* $B9`=g=x$N@_Dj(B:: |
| |
* Weight:: |
| * $BM-M}<0$r78?t$H$9$k%0%l%V%J4pDl7W;;(B:: |
* $BM-M}<0$r78?t$H$9$k%0%l%V%J4pDl7W;;(B:: |
| * $B4pDlJQ49(B:: |
* $B4pDlJQ49(B:: |
| |
* Weyl $BBe?t(B:: |
| * $B%0%l%V%J4pDl$K4X$9$kH!?t(B:: |
* $B%0%l%V%J4pDl$K4X$9$kH!?t(B:: |
| \E |
\E |
| \BEG |
\BEG |
|
|
| * Fundamental functions:: |
* Fundamental functions:: |
| * Controlling Groebner basis computations:: |
* Controlling Groebner basis computations:: |
| * Setting term orderings:: |
* Setting term orderings:: |
| |
* Weight:: |
| * Groebner basis computation with rational function coefficients:: |
* Groebner basis computation with rational function coefficients:: |
| * Change of ordering:: |
* Change of ordering:: |
| |
* Weyl algebra:: |
| * Functions for Groebner basis computation:: |
* Functions for Groebner basis computation:: |
| \E |
\E |
| @end menu |
@end menu |
| Line 228 the head term and the head coefficient respectively. |
|
| Line 232 the head term and the head coefficient respectively. |
|
| @noindent |
@noindent |
| \BJP |
\BJP |
| $B%0%l%V%J4pDl$r7W;;$9$k$?$a$N4pK\E*$JH!?t$O(B @code{dp_gr_main()} $B$*$h$S(B |
$B%0%l%V%J4pDl$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$$$k$N$,JXMx$G$"$k(B. |
$B@_Dj$J$I$r9T$C$?$N$A$3$l$i$r8F$S=P$9%f!<%6H!?t$rMQ$$$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$3$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 |
$B$_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 |
\E |
| \BEG |
\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 |
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 |
@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 |
\E |
| |
|
| @example |
@example |
| Line 350 These parameters can be set and examined by a built-in |
|
| Line 354 These parameters can be set and examined by a built-in |
|
| |
|
| @example |
@example |
| [100] dp_gr_flags(); |
[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] |
[101] |
| @end example |
@end example |
| |
|
| Line 447 If `on', various informations during a Groebner basis |
|
| Line 451 If `on', various informations during a Groebner basis |
|
| displayed. |
displayed. |
| \E |
\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 |
@item Stat |
| \BJP |
\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 |
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 480 is shown after every normal computation. After comlet |
|
| computation the maximal value among the sums is shown. |
computation the maximal value among the sums is shown. |
| \E |
\E |
| |
|
| @item Multiple |
@item Content |
| |
@itemx Multiple |
| \BJP |
\BJP |
| 0 $B$G$J$$@0?t$N;~(B, $BM-M}?t>e$N@55,7A7W;;$K$*$$$F(B, $B78?t$N%S%C%HD9$NOB$,(B |
0 $B$G$J$$M-M}?t$N;~(B, $BM-M}?t>e$N@55,7A7W;;$K$*$$$F(B, $B78?t$N%S%C%HD9$NOB$,(B |
| @code{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. |
$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 |
\E |
| \BEG |
\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 |
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 |
larger than a registered value, which is set to the magnitude of the |
| input polynomial. After each content removal the registered value is |
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. |
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. |
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 |
\E |
| |
|
| @item Demand |
@item Demand |
|
|
| (0,0)(0,0)(0,0)(0,0) |
(0,0)(0,0)(0,0)(0,0) |
| gbcheck total 8 pairs |
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] |
[94] |
| @end example |
@end example |
| |
|
| Line 992 time as well as the choice of types of term orderings. |
|
| Line 1007 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 |
-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 |
+(-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, |
-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 |
-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 |
| 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, |
-z*y^3+(t^7-2*t^4+3*t^2+t)*y+(-2*t^6+4*t^3+2*t-2)*z^2, |
| 2*t^2*y^3+z^2*y^2+(-2*t^5+4*t^2-6)*z^4*y+(4*t^8-t^7-8*t^5+2*t^4-4*t^3+5*t^2-t)*z, |
2*t^2*y^3+z^2*y^2+(-2*t^5+4*t^2-6)*z^4*y |
| |
+(4*t^8-t^7-8*t^5+2*t^4-4*t^3+5*t^2-t)*z, |
| z^3*y^2+2*t^3*y+(-t^7+2*t^4+t^2-t)*z^2, |
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*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); |
[93] gr(B,[t,z,y,x],2); |
| [x^10-t,x^8-z,x^31-x^6-x-y] |
[x^10-t,x^8-z,x^31-x^6-x-y] |
| @end example |
@end example |
| Line 1041 beforehand, and some heuristic trial may be inevitable |
|
| Line 1057 beforehand, and some heuristic trial may be inevitable |
|
| \E |
\E |
| |
|
| \BJP |
\BJP |
| |
@node Weight ,,, $B%0%l%V%J4pDl$N7W;;(B |
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@section Weight |
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\E |
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\BEG |
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@node Weight,,, Groebner basis computation |
| |
@section Weight |
| |
\E |
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\BJP |
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$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 |
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$B$h$j0lHLE*$J$b$N$H$J$k(B. |
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\E |
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\BEG |
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Term orderings introduced in the previous section can be generalized |
| |
by setting a weight for each variable. |
| |
\E |
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@example |
| |
[0] dp_td(<<1,1,1>>); |
| |
3 |
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[1] dp_set_weight([1,2,3])$ |
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[2] dp_td(<<1,1,1>>); |
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6 |
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@end example |
| |
\BJP |
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$BC19`<0$NA4<!?t$r7W;;$9$k:](B, $B%G%U%)%k%H$G$O(B |
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$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 |
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$B9M$($F$$$k$3$H$KAjEv$9$k(B. $B$3$NNc$G$O(B, $BBh0l(B, $BBhFs(B, $BBh;0JQ?t$N(B |
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weight $B$r$=$l$>$l(B 1,2,3 $B$H;XDj$7$F$$$k(B. $B$3$N$?$a(B, @code{<<1,1,1>>} |
| |
$B$NA4<!?t(B ($B0J2<$G$O$3$l$rC19`<0$N(B weight $B$H8F$V(B) $B$,(B @code{1*1+1*2+1*3=6} $B$H$J$k(B. |
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weight $B$r@_Dj$9$k$3$H$G(B, $BF1$89`=g=x7?$N$b$H$G0[$J$k9`=g=x$,Dj5A$G$-$k(B. |
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$BNc$($P(B, weight $B$r$&$^$/@_Dj$9$k$3$H$G(B, $BB?9`<0$r(B weighted homogeneous |
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$B$K$9$k$3$H$,$G$-$k>l9g$,$"$k(B. |
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\E |
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\BEG |
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By default, the total degree of a monomial is equal to |
| |
the sum of all exponents. This means that the weight for each variable |
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is set to 1. |
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In this example, the weights for the first, the second and the third |
| |
variable are set to 1, 2 and 3 respectively. |
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Therefore the total degree of @code{<<1,1,1>>} under this weight, |
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which is called the weight of the monomial, is @code{1*1+1*2+1*3=6}. |
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By setting weights, different term orderings can be set under a type of |
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term ordeing. In some case a polynomial can |
| |
be made weighted homogeneous by setting an appropriate weight. |
| |
\E |
| |
|
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\BJP |
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$B3FJQ?t$KBP$9$k(B weight $B$r$^$H$a$?$b$N$r(B weight vector $B$H8F$V(B. |
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$B$9$Y$F$N@.J,$,@5$G$"$j(B, $B%0%l%V%J4pDl7W;;$K$*$$$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 |
@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 |
@section $BM-M}<0$r78?t$H$9$k%0%l%V%J4pDl7W;;(B |
| \E |
\E |
| Line 1200 Refer to the sections for each functions. |
|
| Line 1349 Refer to the sections for each functions. |
|
| \E |
\E |
| |
|
| \BJP |
\BJP |
| |
@node Weyl $BBe?t(B,,, $B%0%l%V%J4pDl$N7W;;(B |
| |
@section Weyl $BBe?t(B |
| |
\E |
| |
\BEG |
| |
@node Weyl algebra,,, Groebner basis computation |
| |
@section Weyl algebra |
| |
\E |
| |
|
| |
@noindent |
| |
|
| |
\BJP |
| |
$B$3$l$^$G$O(B, $BDL>o$N2D49$JB?9`<04D$K$*$1$k%0%l%V%J4pDl7W;;$K$D$$$F(B |
| |
$B=R$Y$F$-$?$,(B, $B%0%l%V%J4pDl$NM}O@$O(B, $B$"$k>r7o$rK~$?$9Hs2D49$J(B |
| |
$B4D$K$b3HD%$G$-$k(B. $B$3$N$h$&$J4D$NCf$G(B, $B1~MQ>e$b=EMW$J(B, |
| |
Weyl $BBe?t(B, $B$9$J$o$AB?9`<04D>e$NHyJ,:nMQAG4D$N1i;;$*$h$S(B |
| |
$B%0%l%V%J4pDl7W;;$,(B Risa/Asir $B$K<BAu$5$l$F$$$k(B. |
| |
|
| |
$BBN(B @code{K} $B>e$N(B @code{n} $B<!85(B Weyl $BBe?t(B |
| |
@code{D=K<x1,@dots{},xn,D1,@dots{},Dn>} $B$O(B |
| |
\E |
| |
|
| |
\BEG |
| |
So far we have explained Groebner basis computation in |
| |
commutative polynomial rings. However Groebner basis can be |
| |
considered in more general non-commutative rings. |
| |
Weyl algebra is one of such rings and |
| |
Risa/Asir implements fundamental operations |
| |
in Weyl algebra and Groebner basis computation in Weyl algebra. |
| |
|
| |
The @code{n} dimensional Weyl algebra over a field @code{K}, |
| |
@code{D=K<x1,@dots{},xn,D1,@dots{},Dn>} is a non-commutative |
| |
algebra which has the following fundamental relations: |
| |
\E |
| |
|
| |
@code{xi*xj-xj*xi=0}, @code{Di*Dj-Dj*Di=0}, @code{Di*xj-xj*Di=0} (@code{i!=j}), |
| |
@code{Di*xi-xi*Di=1} |
| |
|
| |
\BJP |
| |
$B$H$$$&4pK\4X78$r;}$D4D$G$"$k(B. @code{D} $B$O(B $BB?9`<04D(B @code{K[x1,@dots{},xn]} $B$r78?t(B |
| |
$B$H$9$kHyJ,:nMQAG4D$G(B, @code{Di} $B$O(B @code{xi} $B$K$h$kHyJ,$rI=$9(B. $B8r494X78$K$h$j(B, |
| |
@code{D} $B$N85$O(B, @code{x1^i1*@dots{}*xn^in*D1^j1*@dots{}*Dn^jn} $B$J$kC19`(B |
| |
$B<0$N(B @code{K} $B@~7A7k9g$H$7$F=q$-I=$9$3$H$,$G$-$k(B. |
| |
Risa/Asir $B$K$*$$$F$O(B, $B$3$NC19`<0$r(B, $B2D49$JB?9`<0$HF1MM$K(B |
| |
@code{<<i1,@dots{},in,j1,@dots{},jn>>} $B$GI=$9(B. $B$9$J$o$A(B, @code{D} $B$N85$b(B |
| |
$BJ,;6I=8=B?9`<0$H$7$FI=$5$l$k(B. $B2C8:;;$O(B, $B2D49$N>l9g$HF1MM$K(B, @code{+}, @code{-} |
| |
$B$K$h$j(B |
| |
$B<B9T$G$-$k$,(B, $B>h;;$O(B, $BHs2D49@-$r9MN8$7$F(B @code{dp_weyl_mul()} $B$H$$$&4X?t(B |
| |
$B$K$h$j<B9T$9$k(B. |
| |
\E |
| |
|
| |
\BEG |
| |
@code{D} is the ring of differential operators whose coefficients |
| |
are polynomials in @code{K[x1,@dots{},xn]} and |
| |
@code{Di} denotes the differentiation with respect to @code{xi}. |
| |
According to the commutation relation, |
| |
elements of @code{D} can be represented as a @code{K}-linear combination |
| |
of monomials @code{x1^i1*@dots{}*xn^in*D1^j1*@dots{}*Dn^jn}. |
| |
In Risa/Asir, this type of monomial is represented |
| |
by @code{<<i1,@dots{},in,j1,@dots{},jn>>} as in the case of commutative |
| |
polynomial. |
| |
That is, elements of @code{D} are represented by distributed polynomials. |
| |
Addition and subtraction can be done by @code{+}, @code{-}, |
| |
but multiplication is done by calling @code{dp_weyl_mul()} because of |
| |
the non-commutativity of @code{D}. |
| |
\E |
| |
|
| |
@example |
| |
[0] A=<<1,2,2,1>>; |
| |
(1)*<<1,2,2,1>> |
| |
[1] B=<<2,1,1,2>>; |
| |
(1)*<<2,1,1,2>> |
| |
[2] A*B; |
| |
(1)*<<3,3,3,3>> |
| |
[3] dp_weyl_mul(A,B); |
| |
(1)*<<3,3,3,3>>+(1)*<<3,2,3,2>>+(4)*<<2,3,2,3>>+(4)*<<2,2,2,2>> |
| |
+(2)*<<1,3,1,3>>+(2)*<<1,2,1,2>> |
| |
@end example |
| |
|
| |
\BJP |
| |
$B%0%l%V%J4pDl7W;;$K$D$$$F$b(B, Weyl $BBe?t@lMQ$N4X?t$H$7$F(B, |
| |
$B<!$N4X?t$,MQ0U$7$F$"$k(B. |
| |
\E |
| |
\BEG |
| |
The following functions are avilable for Groebner basis computation |
| |
in Weyl algebra: |
| |
\E |
| |
@code{dp_weyl_gr_main()}, |
| |
@code{dp_weyl_gr_mod_main()}, |
| |
@code{dp_weyl_gr_f_main()}, |
| |
@code{dp_weyl_f4_main()}, |
| |
@code{dp_weyl_f4_mod_main()}. |
| |
\BJP |
| |
$B$^$?(B, $B1~MQ$H$7$F(B, global b $B4X?t$N7W;;$,<BAu$5$l$F$$$k(B. |
| |
\E |
| |
\BEG |
| |
Computation of the global b function is implemented as an application. |
| |
\E |
| |
|
| |
\BJP |
| @node $B%0%l%V%J4pDl$K4X$9$kH!?t(B,,, $B%0%l%V%J4pDl$N7W;;(B |
@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 |
@section $B%0%l%V%J4pDl$K4X$9$kH!?t(B |
| \E |
\E |
| Line 1214 Refer to the sections for each functions. |
|
| Line 1462 Refer to the sections for each functions. |
|
| * lex_hensel_gsl tolex_gsl tolex_gsl_d:: |
* lex_hensel_gsl tolex_gsl tolex_gsl_d:: |
| * gr_minipoly minipoly:: |
* gr_minipoly minipoly:: |
| * tolexm minipolym:: |
* 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:: |
| * dp_gr_flags dp_gr_print:: |
* dp_gr_flags dp_gr_print:: |
| * dp_ord:: |
* dp_ord:: |
| * dp_ptod:: |
* dp_ptod:: |
| Line 1240 Refer to the sections for each functions. |
|
| Line 1489 Refer to the sections for each functions. |
|
| * dp_vtoe dp_etov:: |
* dp_vtoe dp_etov:: |
| * lex_hensel_gsl tolex_gsl tolex_gsl_d:: |
* lex_hensel_gsl tolex_gsl tolex_gsl_d:: |
| * primadec primedec:: |
* primadec primedec:: |
| |
* primedec_mod:: |
| |
* bfunction bfct generic_bfct ann ann0:: |
| @end menu |
@end menu |
| |
|
| \JP @node gr hgr gr_mod,,, $B%0%l%V%J4pDl$K4X$9$kH!?t(B |
\JP @node gr hgr gr_mod,,, $B%0%l%V%J4pDl$K4X$9$kH!?t(B |
| Line 1290 Refer to the sections for each functions. |
|
| Line 1541 Refer to the sections for each functions. |
|
| strategy $B$K$h$k7W;;(B, @code{hgr()} $B$O(B trace-lifting $B$*$h$S(B |
strategy $B$K$h$k7W;;(B, @code{hgr()} $B$O(B trace-lifting $B$*$h$S(B |
| $B@F<!2=$K$h$k(B $B6:@5$5$l$?(B sugar strategy $B$K$h$k7W;;$r9T$&(B. |
$B@F<!2=$K$h$k(B $B6:@5$5$l$?(B sugar strategy $B$K$h$k7W;;$r9T$&(B. |
| @item |
@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;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 |
$B@h$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. |
$B9bB.$+0lHL$K$OITL@$N$?$a(B, $B<B:]$N7P2a;~4V$rC;=L$9$k$N$KM-8z$G$"$k(B. |
| @item |
@item |
| @code{dgr()} $B$GI=<($5$l$k;~4V$O(B, $B$3$NH!?t$,<B9T$5$l$F$$$k%W%m%;%9$G$N(B |
@code{dgr()} $B$GI=<($5$l$k;~4V$O(B, $B$3$NH!?t$,<B9T$5$l$F$$$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. |
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 |
\E |
| \BEG |
\BEG |
| @item |
@item |
| Line 1325 Therefore this function is useful to reduce the actual |
|
| Line 1584 Therefore this function is useful to reduce the actual |
|
| The CPU time shown after an exection of @code{dgr()} indicates |
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 |
that of the master process, and most of the time corresponds to the time |
| for communication. |
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 |
\E |
| @end itemize |
@end itemize |
| |
|
| Line 1342 for communication. |
|
| Line 1608 for communication. |
|
| @table @t |
@table @t |
| \JP @item $B;2>H(B |
\JP @item $B;2>H(B |
| \EG @item References |
\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}. |
@fref{dp_ord}. |
| @end table |
@end table |
| |
|
|
|
| @table @t |
@table @t |
| \JP @item $B;2>H(B |
\JP @item $B;2>H(B |
| \EG @item References |
\EG @item References |
| @fref{dp_gr_main dp_gr_mod_main}, |
@fref{dp_gr_main dp_gr_mod_main dp_gr_f_main dp_weyl_gr_main dp_weyl_gr_mod_main dp_weyl_gr_f_main}, |
| \JP @fref{dp_ord}, @fref{$BJ,;67W;;(B} |
\JP @fref{dp_ord}, @fref{$BJ,;67W;;(B} |
| \EG @fref{dp_ord}, @fref{Distributed computation} |
\EG @fref{dp_ord}, @fref{Distributed computation} |
| @end table |
@end table |
|
|
| @item lex_hensel_gsl(@var{plist},@var{vlist1},@var{order},@var{vlist2},@var{homo}) |
@item lex_hensel_gsl(@var{plist},@var{vlist1},@var{order},@var{vlist2},@var{homo}) |
| \JP :: GSL $B7A<0$N%$%G%"%k4pDl$N7W;;(B |
\JP :: GSL $B7A<0$N%$%G%"%k4pDl$N7W;;(B |
| \EG ::Computation of an GSL form ideal basis |
\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 |
\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 |
\EG :: Computation of an GSL form ideal basis stating from a Groebner basis |
| @end table |
@end table |
|
|
| [108] GSL[1]; |
[108] GSL[1]; |
| [u2,10352277157007342793600000000*u0^31-...] |
[u2,10352277157007342793600000000*u0^31-...] |
| [109] GSL[5]; |
[109] GSL[5]; |
| [u0,11771021876193064124640000000*u0^32-...,376672700038178051988480000000*u0^31-...] |
[u0,11771021876193064124640000000*u0^32-..., |
| |
376672700038178051988480000000*u0^31-...] |
| @end example |
@end example |
| |
|
| @table @t |
@table @t |
| Line 1837 z^32+11405*z^31+20868*z^30+21602*z^29+... |
|
| Line 2103 z^32+11405*z^31+20868*z^30+21602*z^29+... |
|
| @fref{gr_minipoly minipoly}. |
@fref{gr_minipoly minipoly}. |
| @end table |
@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_main |
| @findex dp_gr_mod_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 |
@table @t |
| @item dp_gr_main(@var{plist},@var{vlist},@var{homo},@var{modular},@var{order}) |
@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_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) |
\JP :: $B%0%l%V%J4pDl$N7W;;(B ($BAH$_9~$_H!?t(B) |
| \EG :: Groebner basis computation (built-in functions) |
\EG :: Groebner basis computation (built-in functions) |
| @end table |
@end table |
| Line 1873 z^32+11405*z^31+20868*z^30+21602*z^29+... |
|
| Line 2147 z^32+11405*z^31+20868*z^30+21602*z^29+... |
|
| @item |
@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()}, |
$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 |
@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$$$k$b$N$O(B, Weyl $BBe?t>e$N7W;;(B |
| |
$B$N$?$a$N4X?t$G$"$k(B. |
| @item |
@item |
| |
@code{dp_gr_f_main()}, @code{dp_weyl_f_main()} $B$O(B, $B<o!9$NM-8BBN>e$N%0%l%V%J4pDl$r7W;;$9$k(B |
| |
$B>l9g$KMQ$$$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%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. |
$B$r<B9T$9$k(B. |
| @item |
@item |
| Line 1906 z^32+11405*z^31+20868*z^30+21602*z^29+... |
|
| Line 2185 z^32+11405*z^31+20868*z^30+21602*z^29+... |
|
| @item |
@item |
| These functions are fundamental built-in functions for Groebner basis |
These functions are fundamental built-in functions for Groebner basis |
| computation and @code{gr()},@code{hgr()} and @code{gr_mod()} |
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 |
@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 |
If @var{homo} is not equal to 0, homogenization is applied before entering |
| Buchberger algorithm |
Buchberger algorithm |
| @item |
@item |
| Line 1945 Actual computation is controlled by various parameters |
|
| Line 2231 Actual computation is controlled by various parameters |
|
| @fref{dp_ord}, |
@fref{dp_ord}, |
| @fref{dp_gr_flags dp_gr_print}, |
@fref{dp_gr_flags dp_gr_print}, |
| @fref{gr hgr gr_mod}, |
@fref{gr hgr gr_mod}, |
| |
@fref{setmod_ff}, |
| \JP @fref{$B7W;;$*$h$SI=<($N@)8f(B}. |
\JP @fref{$B7W;;$*$h$SI=<($N@)8f(B}. |
| \EG @fref{Controlling Groebner basis computations} |
\EG @fref{Controlling Groebner basis computations} |
| @end table |
@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_main |
| @findex dp_f4_mod_main |
@findex dp_f4_mod_main |
| |
@findex dp_weyl_f4_main |
| |
@findex dp_weyl_f4_mod_main |
| |
|
| @table @t |
@table @t |
| @item dp_f4_main(@var{plist},@var{vlist},@var{order}) |
@item dp_f4_main(@var{plist},@var{vlist},@var{order}) |
| @itemx dp_f4_mod_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) |
\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) |
\EG :: Groebner basis computation by F4 algorithm (built-in functions) |
| @end table |
@end table |
| Line 1983 F4 $B%"%k%4%j%:%`$O(B, J.C. Faugere $B$K$h$jDs>'$5$ |
|
| Line 2274 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 |
$B;;K!$G$"$j(B, $BK\<BAu$O(B, $BCf9q>jM>DjM}$K$h$k@~7AJ}Dx<05a2r$rMQ$$$?(B |
| $B;n83E*$J<BAu$G$"$k(B. |
$B;n83E*$J<BAu$G$"$k(B. |
| @item |
@item |
| $B0z?t$*$h$SF0:n$O$=$l$>$l(B @code{dp_gr_main()}, @code{dp_gr_mod_main()} |
$B@F<!2=$N0z?t$,$J$$$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. |
$B$HF1MM$G$"$k(B. |
| \E |
\E |
| \BEG |
\BEG |
| Line 1995 invented by J.C. Faugere. The current implementation o |
|
| Line 2288 invented by J.C. Faugere. The current implementation o |
|
| uses Chinese Remainder theorem and not highly optimized. |
uses Chinese Remainder theorem and not highly optimized. |
| @item |
@item |
| Arguments and actions are the same as those of |
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 |
\E |
| @end itemize |
@end itemize |
| |
|
| Line 2009 Arguments and actions are the same as those of |
|
| Line 2304 Arguments and actions are the same as those of |
|
| \EG @fref{Controlling Groebner basis computations} |
\EG @fref{Controlling Groebner basis computations} |
| @end table |
@end table |
| |
|
| |
\JP @node nd_gr nd_gr_trace nd_f4 nd_f4_trace nd_weyl_gr nd_weyl_gr_trace,,, $B%0%l%V%J4pDl$K4X$9$kH!?t(B |
| |
\EG @node nd_gr nd_gr_trace nd_f4 nd_f4_trace nd_weyl_gr nd_weyl_gr_trace,,, Functions for Groebner basis computation |
| |
@subsection @code{nd_gr}, @code{nd_gr_trace}, @code{nd_f4}, @code{nd_f4_trace}, @code{nd_weyl_gr}, @code{nd_weyl_gr_trace} |
| |
@findex nd_gr |
| |
@findex nd_gr_trace |
| |
@findex nd_f4 |
| |
@findex nd_f4_trace |
| |
@findex nd_weyl_gr |
| |
@findex nd_weyl_gr_trace |
| |
|
| |
@table @t |
| |
@item nd_gr(@var{plist},@var{vlist},@var{p},@var{order}) |
| |
@itemx nd_gr_trace(@var{plist},@var{vlist},@var{homo},@var{p},@var{order}) |
| |
@itemx nd_f4(@var{plist},@var{vlist},@var{modular},@var{order}) |
| |
@itemx nd_f4_trace(@var{plist},@var{vlist},@var{homo},@var{p},@var{order}) |
| |
@item nd_weyl_gr(@var{plist},@var{vlist},@var{p},@var{order}) |
| |
@itemx nd_weyl_gr_trace(@var{plist},@var{vlist},@var{homo},@var{p},@var{order}) |
| |
\JP :: $B%0%l%V%J4pDl$N7W;;(B ($BAH$_9~$_H!?t(B) |
| |
\EG :: Groebner basis computation (built-in functions) |
| |
@end table |
| |
|
| |
@table @var |
| |
@item return |
| |
\JP $B%j%9%H(B |
| |
\EG list |
| |
@item plist vlist |
| |
\JP $B%j%9%H(B |
| |
\EG list |
| |
@item order |
| |
\JP $B?t(B, $B%j%9%H$^$?$O9TNs(B |
| |
\EG number, list or matrix |
| |
@item homo |
| |
\JP $B%U%i%0(B |
| |
\EG flag |
| |
@item modular |
| |
\JP $B%U%i%0$^$?$OAG?t(B |
| |
\EG flag or prime |
| |
@end table |
| |
|
| |
\BJP |
| |
@itemize @bullet |
| |
@item |
| |
$B$3$l$i$NH!?t$O(B, $B%0%l%V%J4pDl7W;;AH$_9~$_4X?t$N?7<BAu$G$"$k(B. |
| |
@item @code{nd_gr} $B$O(B, @code{p} $B$,(B 0 $B$N$H$-M-M}?tBN>e$N(B Buchberger |
| |
$B%"%k%4%j%:%`$r<B9T$9$k(B. @code{p} $B$,(B 2 $B0J>e$N<+A3?t$N$H$-(B, GF(p) $B>e$N(B |
| |
Buchberger $B%"%k%4%j%:%`$r<B9T$9$k(B. |
| |
@item @code{nd_gr_trace} $B$*$h$S(B @code{nd_f4_trace} |
| |
$B$OM-M}?tBN>e$G(B trace $B%"%k%4%j%:%`$r<B9T$9$k(B. |
| |
@code{p} $B$,(B 0 $B$^$?$O(B 1 $B$N$H$-(B, $B<+F0E*$KA*$P$l$?AG?t$rMQ$$$F(B, $B@.8y$9$k(B |
| |
$B$^$G(B trace $B%"%k%4%j%:%`$r<B9T$9$k(B. |
| |
@code{p} $B$,(B 2 $B0J>e$N$H$-(B, trace $B$O(BGF(p) $B>e$G7W;;$5$l$k(B. trace $B%"%k%4%j%:%`(B |
| |
$B$,<:GT$7$?>l9g(B 0 $B$,JV$5$l$k(B. @code{p} $B$,Ii$N>l9g(B, $B%0%l%V%J4pDl%A%'%C%/$O(B |
| |
$B9T$o$J$$(B. $B$3$N>l9g(B, @code{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$$(B, $B$=$NA4<!?t$K$*$1$k4pDl$r@8@.$9$kJ}K!$G$"$k(B. $BF@$i$l$k(B |
| |
$BB?9`<0=89g$O$d$O$j%0%l%V%J4pDl8uJd$G$"$j(B, @code{nd_gr_trace} $B$HF1MM$N(B |
| |
$B%A%'%C%/$,9T$o$l$k(B. |
| |
@item |
| |
@code{nd_f4} $B$O(B @code{modular} $B$,(B 0 $B$N$H$-M-M}?tBN>e$N(B, @code{modular} $B$,(B |
| |
$B%^%7%s%5%$%:AG?t$N$H$-M-8BBN>e$N(B F4 $B%"%k%4%j%:%`$r<B9T$9$k(B. |
| |
@item |
| |
@code{nd_weyl_gr}, @code{nd_weyl_gr_trace} $B$O(B Weyl $BBe?tMQ$G$"$k(B. |
| |
@item |
| |
$B$$$:$l$N4X?t$b(B, $BM-M}4X?tBN>e$N7W;;$OL$BP1~$G$"$k(B. |
| |
@item |
| |
$B0lHL$K(B @code{dp_gr_main}, @code{dp_gr_mod_main} $B$h$j9bB.$G$"$k$,(B, |
| |
$BFC$KM-8BBN>e$N>l9g82Cx$G$"$k(B. |
| |
@end itemize |
| |
\E |
| |
|
| |
\BEG |
| |
@itemize @bullet |
| |
@item |
| |
These functions are new implementations for computing Groebner bases. |
| |
@item @code{nd_gr} executes Buchberger algorithm over the rationals |
| |
if @code{p} is 0, and that over GF(p) if @code{p} is a prime. |
| |
@item @code{nd_gr_trace} executes the trace algorithm over the rationals. |
| |
If @code{p} is 0 or 1, the trace algorithm is executed until it succeeds |
| |
by using automatically chosen primes. |
| |
If @code{p} a positive prime, |
| |
the trace is comuted over GF(p). |
| |
If the trace algorithm fails 0 is returned. |
| |
If @code{p} is negative, |
| |
the Groebner basis check and ideal-membership check are omitted. |
| |
In this case, an automatically chosen prime if @code{p} is 1, |
| |
otherwise the specified prime is used to compute a Groebner basis |
| |
candidate. |
| |
Execution of @code{nd_f4_trace} is done as follows: |
| |
For each total degree, an F4-reduction of S-polynomials over a finite field |
| |
is done, and S-polynomials which give non-zero basis elements are gathered. |
| |
Then F4-reduction over Q is done for the gathered S-polynomials. |
| |
The obtained polynomial set is a Groebner basis candidate and the same |
| |
check procedure as in the case of @code{nd_gr_trace} is done. |
| |
@item |
| |
@code{nd_f4} executes F4 algorithm over Q if @code{modular} is equal to 0, |
| |
or over a finite field GF(@code{modular}) |
| |
if @code{modular} is a prime number of machine size (<2^29). |
| |
@item |
| |
@code{nd_weyl_gr}, @code{nd_weyl_gr_trace} are for Weyl algebra computation. |
| |
@item |
| |
Each function cannot handle rational function coefficient cases. |
| |
@item |
| |
In general these functions are more efficient than |
| |
@code{dp_gr_main}, @code{dp_gr_mod_main}, especially over finite fields. |
| |
@end itemize |
| |
\E |
| |
|
| |
@example |
| |
[38] load("cyclic")$ |
| |
[49] C=cyclic(7)$ |
| |
[50] V=vars(C)$ |
| |
[51] cputime(1)$ |
| |
[52] dp_gr_mod_main(C,V,0,31991,0)$ |
| |
26.06sec + gc : 0.313sec(26.4sec) |
| |
[53] nd_gr(C,V,31991,0)$ |
| |
ndv_alloc=1477188 |
| |
5.737sec + gc : 0.1837sec(5.921sec) |
| |
[54] dp_f4_mod_main(C,V,31991,0)$ |
| |
3.51sec + gc : 0.7109sec(4.221sec) |
| |
[55] nd_f4(C,V,31991,0)$ |
| |
1.906sec + gc : 0.126sec(2.032sec) |
| |
@end example |
| |
|
| |
@table @t |
| |
\JP @item $B;2>H(B |
| |
\EG @item References |
| |
@fref{dp_ord}, |
| |
@fref{dp_gr_flags dp_gr_print}, |
| |
\JP @fref{$B7W;;$*$h$SI=<($N@)8f(B}. |
| |
\EG @fref{Controlling Groebner basis computations} |
| |
@end table |
| |
|
| \JP @node dp_gr_flags dp_gr_print,,, $B%0%l%V%J4pDl$K4X$9$kH!?t(B |
\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 |
\EG @node dp_gr_flags dp_gr_print,,, Functions for Groebner basis computation |
| @subsection @code{dp_gr_flags}, @code{dp_gr_print} |
@subsection @code{dp_gr_flags}, @code{dp_gr_print} |
| Line 2017 Arguments and actions are the same as those of |
|
| Line 2446 Arguments and actions are the same as those of |
|
| |
|
| @table @t |
@table @t |
| @item dp_gr_flags([@var{list}]) |
@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 |
\JP :: $B7W;;$*$h$SI=<(MQ%Q%i%a%?$N@_Dj(B, $B;2>H(B |
| \BEG :: Set and show various parameters for cotrolling computations |
\BEG :: Set and show various parameters for cotrolling computations |
| and showing informations. |
and showing informations. |
| Line 2031 and showing informations. |
|
| Line 2460 and showing informations. |
|
| @item list |
@item list |
| \JP $B%j%9%H(B |
\JP $B%j%9%H(B |
| \EG list |
\EG list |
| |
@item i |
| |
\JP $B@0?t(B |
| |
\EG integer |
| @end table |
@end table |
| |
|
| @itemize @bullet |
@itemize @bullet |
| \BJP |
\BJP |
| @item |
@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. |
$B$J%Q%i%a%?$r@_Dj(B, $B;2>H$9$k(B. |
| @item |
@item |
| $B0z?t$,$J$$>l9g(B, $B8=:_$N@_Dj$,JV$5$l$k(B. |
$B0z?t$,$J$$>l9g(B, $B8=:_$N@_Dj$,JV$5$l$k(B. |
| Line 2044 and showing informations. |
|
| Line 2476 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 |
$B0z?t$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. |
$B@_Dj$5$l$k(B. $B%Q%i%a%?L>$OJ8;zNs$GM?$($kI,MW$,$"$k(B. |
| @item |
@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$$$k(B. |
$B$r9T$&:]$K(B, $B?WB.$K%U%i%0$r8+$k$3$H$,$G$-$k$h$&$KMQ0U$5$l$F$$$k(B. |
| \E |
\E |
| \BEG |
\BEG |
| Line 2061 Arguments must be specified as a list such as |
|
| Line 2502 Arguments must be specified as a list such as |
|
| strings. |
strings. |
| @item |
@item |
| @code{dp_gr_print()} is used to set and show the value of a parameter |
@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. |
uses the value as a flag for showing intermediate informations. |
| \E |
\E |
| @end itemize |
@end itemize |
| Line 2212 the coefficient field. |
|
| Line 2662 the coefficient field. |
|
| (1)*<<2,0,0>>+(2)*<<1,1,0>>+(1)*<<0,2,0>>+(2)*<<1,0,1>>+(2)*<<0,1,1>> |
(1)*<<2,0,0>>+(2)*<<1,1,0>>+(1)*<<0,2,0>>+(2)*<<1,0,1>>+(2)*<<0,1,1>> |
| +(1)*<<0,0,2>> |
+(1)*<<0,0,2>> |
| [52] dp_ptod((x+y+z)^2,[x,y]); |
[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 |
@end example |
| |
|
| @table @t |
@table @t |
| Line 2264 variables of @var{dpoly}. |
|
| Line 2715 variables of @var{dpoly}. |
|
| |
|
| @example |
@example |
| [53] T=dp_ptod((x+y+z)^2,[x,y]); |
[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]); |
[54] P=dp_dtop(T,[a,b]); |
| z^2+(2*a+2*b)*z+a^2+2*b*a+b^2 |
z^2+(2*a+2*b)*z+a^2+2*b*a+b^2 |
| @end example |
@end example |
| Line 2617 For single computation @code{p_nf} and @code{p_true_nf |
|
| Line 3069 For single computation @code{p_nf} and @code{p_true_nf |
|
| [74] DP2=newvect(length(G),map(dp_ptod,G,V))$ |
[74] DP2=newvect(length(G),map(dp_ptod,G,V))$ |
| [75] T=dp_ptod((u0-u1+u2-u3+u4)^2,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); |
[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); |
[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 |
-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); |
[78] dp_dtop(dp_nf([0,1,2,3,4],T,DP2,1),V); |
| -1138087976845165778088612297273078520347097001020471455633353049221045677593 |
-11380879768451657780886122972730785203470970010204714556333530492210 |
| 0005716505560062087150928400876150217079820311439477560587583488*u4^15+... |
456775930005716505560062087150928400876150217079820311439477560587583 |
| |
488*u4^15+... |
| [79] dp_dtop(dp_nf([4,3,2,1,0],T,DP2,1),V); |
[79] dp_dtop(dp_nf([4,3,2,1,0],T,DP2,1),V); |
| -1138087976845165778088612297273078520347097001020471455633353049221045677593 |
-11380879768451657780886122972730785203470970010204714556333530492210 |
| 0005716505560062087150928400876150217079820311439477560587583488*u4^15+... |
456775930005716505560062087150928400876150217079820311439477560587583 |
| |
488*u4^15+... |
| [80] @@78==@@79; |
[80] @@78==@@79; |
| 1 |
1 |
| @end example |
@end example |
| Line 3170 The result is a list @code{[@var{a dpoly1},@var{a dpol |
|
| Line 3625 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>>; |
[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>> |
(12)*<<1,1,1,0,0>>+(1)*<<0,1,1,1,0>>+(1)*<<1,1,0,0,1>> |
| [160] dp_red(D,R,C); |
[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 |
@end example |
| |
|
| @table @t |
@table @t |
|
|
| @example |
@example |
| [233] G=gr(katsura(5),[u5,u4,u3,u2,u1,u0],2)$ |
[233] G=gr(katsura(5),[u5,u4,u3,u2,u1,u0],2)$ |
| [234] p_terms(G[0],[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 |
@end example |
| |
|
| \JP @node gb_comp,,, $B%0%l%V%J4pDl$K4X$9$kH!?t(B |
\JP @node gb_comp,,, $B%0%l%V%J4pDl$K4X$9$kH!?t(B |
| Line 3519 Polynomial set @code{cyclic} is sometimes called by ot |
|
| Line 3974 Polynomial set @code{cyclic} is sometimes called by ot |
|
| [79] load("cyclic")$ |
[79] load("cyclic")$ |
| [89] katsura(5); |
[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, |
[u0+2*u4+2*u3+2*u2+2*u1+2*u5-1,2*u4*u0-u4+2*u1*u3+u2^2+2*u5*u1, |
| 2*u3*u0+2*u1*u4-u3+(2*u1+2*u5)*u2,2*u2*u0+2*u2*u4+(2*u1+2*u5)*u3-u2+u1^2, |
2*u3*u0+2*u1*u4-u3+(2*u1+2*u5)*u2,2*u2*u0+2*u2*u4+(2*u1+2*u5)*u3 |
| 2*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] |
u0^2-u0+2*u4^2+2*u3^2+2*u2^2+2*u1^2+2*u5^2] |
| [90] hkatsura(5); |
[90] hkatsura(5); |
| [-t+u0+2*u4+2*u3+2*u2+2*u1+2*u5, |
[-t+u0+2*u4+2*u3+2*u2+2*u1+2*u5, |
| Line 3642 if an input ideal is not radical. |
|
| Line 4097 if an input ideal is not radical. |
|
| \JP @fref{$B9`=g=x$N@_Dj(B}. |
\JP @fref{$B9`=g=x$N@_Dj(B}. |
| \EG @fref{Setting term orderings}. |
\EG @fref{Setting term orderings}. |
| @end table |
@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()} |
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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 |
| |
|
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@example |
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[0] load("primdec_mod")$ |
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[246] PP444=[x^8+x^2+t,y^8+y^2+t,z^8+z^2+t]$ |
| |
[247] primedec_mod(PP444,[x,y,z,t],0,2,1); |
| |
[[y+z,x+z,z^8+z^2+t],[x+y,y^2+y+z^2+z+1,z^8+z^2+t], |
| |
[y+z+1,x+z+1,z^8+z^2+t],[x+z,y^2+y+z^2+z+1,z^8+z^2+t], |
| |
[y+z,x^2+x+z^2+z+1,z^8+z^2+t],[y+z+1,x^2+x+z^2+z+1,z^8+z^2+t], |
| |
[x+z+1,y^2+y+z^2+z+1,z^8+z^2+t],[y+z+1,x+z,z^8+z^2+t], |
| |
[x+y+1,y^2+y+z^2+z+1,z^8+z^2+t],[y+z,x+z+1,z^8+z^2+t]] |
| |
[248] |
| |
@end example |
| |
|
| |
@table @t |
| |
\JP @item $B;2>H(B |
| |
\EG @item References |
| |
@fref{modfctr}, |
| |
@fref{dp_gr_main dp_gr_mod_main dp_gr_f_main dp_weyl_gr_main dp_weyl_gr_mod_main dp_weyl_gr_f_main}, |
| |
\JP @fref{$B9`=g=x$N@_Dj(B}. |
| |
\EG @fref{Setting term orderings}, |
| |
@fref{dp_gr_flags dp_gr_print}. |
| |
@end table |
| |
|
| |
\JP @node bfunction bfct generic_bfct ann ann0,,, $B%0%l%V%J4pDl$K4X$9$kH!?t(B |
| |
\EG @node bfunction bfct generic_bfct ann ann0,,, Functions for Groebner basis computation |
| |
@subsection @code{bfunction}, @code{bfct}, @code{generic_bfct}, @code{ann}, @code{ann0} |
| |
@findex bfunction |
| |
@findex bfct |
| |
@findex generic_bfct |
| |
@findex ann |
| |
@findex ann0 |
| |
|
| |
@table @t |
| |
@item bfunction(@var{f}) |
| |
@itemx bfct(@var{f}) |
| |
@itemx generic_bfct(@var{plist},@var{vlist},@var{dvlist},@var{weight}) |
| |
\JP :: @var{b} $B4X?t$N7W;;(B |
| |
\EG :: Computes the global @var{b} function of a polynomial or an ideal |
| |
@item ann(@var{f}) |
| |
@itemx ann0(@var{f}) |
| |
\JP :: $BB?9`<0$N%Y%-$N(B annihilator $B$N7W;;(B |
| |
\EG :: Computes the annihilator of a power of polynomial |
| |
@end table |
| |
|
| |
@table @var |
| |
@item return |
| |
\JP $BB?9`<0$^$?$O%j%9%H(B |
| |
\EG polynomial or list |
| |
@item f |
| |
\JP $BB?9`<0(B |
| |
\EG polynomial |
| |
@item plist |
| |
\JP $BB?9`<0%j%9%H(B |
| |
\EG list of polynomials |
| |
@item vlist dvlist |
| |
\JP $BJQ?t%j%9%H(B |
| |
\EG list of variables |
| |
@end table |
| |
|
| |
@itemize @bullet |
| |
\BJP |
| |
@item @samp{bfct} $B$GDj5A$5$l$F$$$k(B. |
| |
@item @code{bfunction(@var{f})}, @code{bfct(@var{f})} $B$OB?9`<0(B @var{f} $B$N(B global @var{b} $B4X?t(B @code{b(s)} $B$r(B |
| |
$B7W;;$9$k(B. @code{b(s)} $B$O(B, Weyl $BBe?t(B @code{D} $B>e$N0lJQ?tB?9`<04D(B @code{D[s]} |
| |
$B$N85(B @code{P(x,s)} $B$,B8:_$7$F(B, @code{P(x,s)f^(s+1)=b(s)f^s} $B$rK~$?$9$h$&$J(B |
| |
$BB?9`<0(B @code{b(s)} $B$NCf$G(B, $B<!?t$,:G$bDc$$$b$N$G$"$k(B. |
| |
@item @code{generic_bfct(@var{f},@var{vlist},@var{dvlist},@var{weight})} |
| |
$B$O(B, @var{plist} $B$G@8@.$5$l$k(B @code{D} $B$N:8%$%G%"%k(B @code{I} $B$N(B, |
| |
$B%&%'%$%H(B @var{weight} $B$K4X$9$k(B global @var{b} $B4X?t$r7W;;$9$k(B. |
| |
@var{vlist} $B$O(B @code{x}-$BJQ?t(B, @var{vlist} $B$OBP1~$9$k(B @code{D}-$BJQ?t(B |
| |
$B$r=g$KJB$Y$k(B. |
| |
@item @code{bfunction} $B$H(B @code{bfct} $B$G$OMQ$$$F$$$k%"%k%4%j%:%`$,(B |
| |
$B0[$J$k(B. $B$I$A$i$,9bB.$+$OF~NO$K$h$k(B. |
| |
@item @code{ann(@var{f})} $B$O(B, @code{@var{f}^s} $B$N(B annihilator ideal |
| |
$B$N@8@.7O$rJV$9(B. @code{ann(@var{f})} $B$O(B, @code{[@var{a},@var{list}]} |
| |
$B$J$k%j%9%H$rJV$9(B. $B$3$3$G(B, @var{a} $B$O(B @var{f} $B$N(B @var{b} $B4X?t$N:G>.@0?t:,(B, |
| |
@var{list} $B$O(B @code{ann(@var{f})} $B$N7k2L$N(B @code{s}$ $B$K(B, @var{a} $B$r(B |
| |
$BBeF~$7$?$b$N$G$"$k(B. |
| |
@item $B>\:Y$K$D$$$F$O(B, [Saito,Sturmfels,Takayama] $B$r8+$h(B. |
| |
\E |
| |
\BEG |
| |
@item These functions are defined in @samp{bfct}. |
| |
@item @code{bfunction(@var{f})} and @code{bfct(@var{f})} compute the global @var{b}-function @code{b(s)} of |
| |
a polynomial @var{f}. |
| |
@code{b(s)} is a polynomial of the minimal degree |
| |
such that there exists @code{P(x,s)} in D[s], which is a polynomial |
| |
ring over Weyl algebra @code{D}, and @code{P(x,s)f^(s+1)=b(s)f^s} holds. |
| |
@item @code{generic_bfct(@var{f},@var{vlist},@var{dvlist},@var{weight})} |
| |
computes the global @var{b}-function of a left ideal @code{I} in @code{D} |
| |
generated by @var{plist}, with respect to @var{weight}. |
| |
@var{vlist} is the list of @code{x}-variables, |
| |
@var{vlist} is the list of corresponding @code{D}-variables. |
| |
@item @code{bfunction(@var{f})} and @code{bfct(@var{f})} implement |
| |
different algorithms and the efficiency depends on inputs. |
| |
@item @code{ann(@var{f})} returns the generator set of the annihilator |
| |
ideal of @code{@var{f}^s}. |
| |
@code{ann(@var{f})} returns a list @code{[@var{a},@var{list}]}, |
| |
where @var{a} is the minimal integral root of the global @var{b}-function |
| |
of @var{f}, and @var{list} is a list of polynomials obtained by |
| |
substituting @code{s} in @code{ann(@var{f})} with @var{a}. |
| |
@item See [Saito,Sturmfels,Takayama] for the details. |
| |
\E |
| |
@end itemize |
| |
|
| |
@example |
| |
[0] load("bfct")$ |
| |
[216] bfunction(x^3+y^3+z^3+x^2*y^2*z^2+x*y*z); |
| |
-9*s^5-63*s^4-173*s^3-233*s^2-154*s-40 |
| |
[217] fctr(@@); |
| |
[[-1,1],[s+2,1],[3*s+4,1],[3*s+5,1],[s+1,2]] |
| |
[218] F = [4*x^3*dt+y*z*dt+dx,x*z*dt+4*y^3*dt+dy, |
| |
x*y*dt+5*z^4*dt+dz,-x^4-z*y*x-y^4-z^5+t]$ |
| |
[219] generic_bfct(F,[t,z,y,x],[dt,dz,dy,dx],[1,0,0,0]); |
| |
20000*s^10-70000*s^9+101750*s^8-79375*s^7+35768*s^6-9277*s^5 |
| |
+1278*s^4-72*s^3 |
| |
[220] P=x^3-y^2$ |
| |
[221] ann(P); |
| |
[2*dy*x+3*dx*y^2,-3*dx*x-2*dy*y+6*s] |
| |
[222] ann0(P); |
| |
[-1,[2*dy*x+3*dx*y^2,-3*dx*x-2*dy*y-6]] |
| |
@end example |
| |
|
| |
@table @t |
| |
\JP @item $B;2>H(B |
| |
\EG @item References |
| |
\JP @fref{Weyl $BBe?t(B}. |
| |
\EG @fref{Weyl algebra}. |
| |
@end table |
| |
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