| version 1.5, 2003/04/20 08:01:25 |
version 1.17, 2006/09/06 23:53:31 |
|
|
| @comment $OpenXM: OpenXM/src/asir-doc/parts/groebner.texi,v 1.4 2003/04/19 15:44:56 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:: |
* Weyl $BBe?t(B:: |
|
|
| * 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:: |
* Weyl algebra:: |
| Line 449 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 471 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 |
| Line 1044 beforehand, and some heuristic trial may be inevitable |
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| Line 1057 beforehand, and some heuristic trial may be inevitable |
|
| \E |
\E |
| |
|
| \BJP |
\BJP |
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@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 |
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@section Weight |
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\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 |
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by setting a weight for each variable. |
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\E |
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@example |
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[0] dp_td(<<1,1,1>>); |
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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 |
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\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>>} |
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$B$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$,$"$k(B. |
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\E |
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\BEG |
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By default, the total degree of a monomial is equal to |
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the sum of all exponents. This means that the weight for each variable |
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is set to 1. |
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In this example, the weights for the first, the second and the third |
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variable are set to 1, 2 and 3 respectively. |
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Therefore the total degree of @code{<<1,1,1>>} under this weight, |
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which is called the weight of the monomial, is @code{1*1+1*2+1*3=6}. |
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By setting weights, different term orderings can be set under a type of |
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term ordeing. In some case a polynomial can |
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be made weighted homogeneous by setting an appropriate weight. |
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\E |
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|
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\BJP |
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$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$*$$$F(B, $BA4<!?t$N(B |
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$BBe$o$j$KMQ$$$i$l$k$b$N$rFC$K(B sugar weight $B$H8F$V$3$H$K$9$k(B. |
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sugar strategy $B$K$*$$$F(B, $BA4<!?t$NBe$o$j$K;H$o$l$k$+$i$G$"$k(B. |
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$B0lJ}$G(B, $B3F@.J,$,I,$:$7$b@5$H$O8B$i$J$$(B weight vector $B$O(B, |
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sugar weight $B$H$7$F@_Dj$9$k$3$H$O$G$-$J$$$,(B, $B9`=g=x$N0lHL2=$K$O(B |
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$BM-MQ$G$"$k(B. $B$3$l$i$O(B, $B9TNs$K$h$k9`=g=x$N@_Dj$K$9$G$K8=$l$F(B |
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$B$$$k(B. $B$9$J$o$A(B, $B9`=g=x$rDj5A$9$k9TNs$N3F9T$,(B, $B0l$D$N(B weight vector |
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$B$H8+$J$5$l$k(B. $B$^$?(B, $B%V%m%C%/=g=x$O(B, $B3F%V%m%C%/$N(B |
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$BJQ?t$KBP1~$9$k@.J,$N$_(B 1 $B$GB>$O(B 0 $B$N(B weight vector $B$K$h$kHf3S$r(B |
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$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. |
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\E |
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|
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\BEG |
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A list of weights for all variables is called a weight vector. |
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A weight vector is called a sugar weight vector if |
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its elements are all positive and it is used for computing |
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a weighted total degree of a monomial, because such a weight |
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is used instead of total degree in sugar strategy. |
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On the other hand, a weight vector whose elements are not necessarily |
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positive cannot be set as a sugar weight, but it is useful for |
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generalizing term order. In fact, such a weight vector already |
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appeared in a matrix order. That is, each row of a matrix defining |
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a term order is regarded as a weight vector. A block order |
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is also considered as a refinement of comparison by weight vectors. |
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It compares two terms by using a weight vector whose elements |
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corresponding to variables in a block is 1 and 0 otherwise, |
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then it applies a tie breaker. |
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\E |
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|
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\BJP |
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weight vector $B$N@_Dj$O(B @code{dp_set_weight()} $B$G9T$&$3$H$,$G$-$k(B |
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$B$,(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 |
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$B$^$H$a$F@_Dj$G$-$k$3$H$,K>$^$7$$(B. $B$3$N$?$a(B, $B<!$N$h$&$J7A$G$b(B |
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$B9`=g=x$,;XDj$G$-$k(B. |
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\E |
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\BEG |
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A weight vector can be set by using @code{dp_set_weight()}. |
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However it is more preferable if a weight vector can be set |
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together with other parapmeters such as a type of term ordering |
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and a variable order. This is realized as follows. |
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\E |
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|
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@example |
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[64] B=[x+y+z-6,x*y+y*z+z*x-11,x*y*z-6]$ |
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[65] dp_gr_main(B|v=[x,y,z],sugarweight=[3,2,1],order=0); |
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[z^3-6*z^2+11*z-6,x+y+z-6,-y^2+(-z+6)*y-z^2+6*z-11] |
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[66] dp_gr_main(B|v=[y,z,x],order=[[1,1,0],[0,1,0],[0,0,1]]); |
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[x^3-6*x^2+11*x-6,x+y+z-6,-x^2+(-y+6)*x-y^2+6*y-11] |
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[67] dp_gr_main(B|v=[y,z,x],order=[[x,1,y,2,z,3]]); |
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[x+y+z-6,x^3-6*x^2+11*x-6,-x^2+(-y+6)*x-y^2+6*y-11] |
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@end example |
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|
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\BJP |
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$B$$$:$l$NNc$K$*$$$F$b(B, $B9`=g=x$O(B option $B$H$7$F;XDj$5$l$F$$$k(B. |
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$B:G=i$NNc$G$O(B @code{v} $B$K$h$jJQ?t=g=x$r(B, @code{sugarweight} $B$K$h$j(B |
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sugar weight vector $B$r(B, @code{order}$B$K$h$j9`=g=x7?$r;XDj$7$F$$$k(B. |
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$BFs$DL\$NNc$K$*$1$k(B @code{order} $B$N;XDj$O(B matrix order $B$HF1MM$G$"$k(B. |
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$B$9$J$o$A(B, $B;XDj$5$l$?(B weight vector $B$r:8$+$i=g$K;H$C$F(B weight $B$NHf3S(B |
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$B$r9T$&(B. $B;0$DL\$NNc$bF1MM$G$"$k$,(B, $B$3$3$G$O(B weight vector $B$NMWAG$r(B |
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$BJQ?tKh$K;XDj$7$F$$$k(B. $B;XDj$,$J$$$b$N$O(B 0 $B$H$J$k(B. $B;0$DL\$NNc$G$O(B, |
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@code{order} $B$K$h$k;XDj$G$O9`=g=x$,7hDj$7$J$$(B. $B$3$N>l9g$K$O(B, |
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tie breaker $B$H$7$FA4<!?t5U<-=q<0=g=x$,<+F0E*$K@_Dj$5$l$k(B. |
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$B$3$N;XDjJ}K!$O(B, @code{dp_gr_main}, @code{dp_gr_mod_main} $B$J$I(B |
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$B$NAH$_9~$_4X?t$G$N$_2DG=$G$"$j(B, @code{gr} $B$J$I$N%f!<%6Dj5A4X?t(B |
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$B$G$OL$BP1~$G$"$k(B. |
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\E |
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\BEG |
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In each example, a term ordering is specified as options. |
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In the first example, a variable order, a sugar weight vector |
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and a type of term ordering are specified by options @code{v}, |
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@code{sugarweight} and @code{order} respectively. |
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In the second example, an option @code{order} is used |
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to set a matrix ordering. That is, the specified weight vectors |
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are used from left to right for comparing terms. |
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The third example shows a variant of specifying a weight vector, |
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where each component of a weight vector is specified variable by variable, |
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and unspecified components are set to zero. In this example, |
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a term order is not determined only by the specified weight vector. |
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In such a case a tie breaker by the graded reverse lexicographic ordering |
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is set automatically. |
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This type of a term ordering specification can be applied only to builtin |
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functions such as @code{dp_gr_main()}, @code{dp_gr_mod_main()}, not to |
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user defined functions such as @code{gr()}. |
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\E |
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|
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\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 1203 Refer to the sections for each functions. |
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| Line 1349 Refer to the sections for each functions. |
|
| \E |
\E |
| |
|
| \BJP |
\BJP |
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@node Weyl $BBe?t(B,,, $B%0%l%V%J4pDl$N7W;;(B |
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@section Weyl $BBe?t(B |
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\E |
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\BEG |
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@node Weyl algebra,,, Groebner basis computation |
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@section Weyl algebra |
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\E |
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|
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@noindent |
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|
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\BJP |
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$B$3$l$^$G$O(B, $BDL>o$N2D49$JB?9`<04D$K$*$1$k%0%l%V%J4pDl7W;;$K$D$$$F(B |
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$B=R$Y$F$-$?$,(B, $B%0%l%V%J4pDl$NM}O@$O(B, $B$"$k>r7o$rK~$?$9Hs2D49$J(B |
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$B4D$K$b3HD%$G$-$k(B. $B$3$N$h$&$J4D$NCf$G(B, $B1~MQ>e$b=EMW$J(B, |
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Weyl $BBe?t(B, $B$9$J$o$AB?9`<04D>e$NHyJ,:nMQAG4D$N1i;;$*$h$S(B |
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$B%0%l%V%J4pDl7W;;$,(B Risa/Asir $B$K<BAu$5$l$F$$$k(B. |
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|
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$BBN(B @code{K} $B>e$N(B @code{n} $B<!85(B Weyl $BBe?t(B |
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@code{D=K<x1,@dots{},xn,D1,@dots{},Dn>} $B$O(B |
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\E |
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|
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\BEG |
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So far we have explained Groebner basis computation in |
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commutative polynomial rings. However Groebner basis can be |
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considered in more general non-commutative rings. |
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Weyl algebra is one of such rings and |
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Risa/Asir implements fundamental operations |
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in Weyl algebra and Groebner basis computation in Weyl algebra. |
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|
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The @code{n} dimensional Weyl algebra over a field @code{K}, |
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@code{D=K<x1,@dots{},xn,D1,@dots{},Dn>} is a non-commutative |
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algebra which has the following fundamental relations: |
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\E |
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|
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@code{xi*xj-xj*xi=0}, @code{Di*Dj-Dj*Di=0}, @code{Di*xj-xj*Di=0} (@code{i!=j}), |
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@code{Di*xi-xi*Di=1} |
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|
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\BJP |
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$B$H$$$&4pK\4X78$r;}$D4D$G$"$k(B. @code{D} $B$O(B $BB?9`<04D(B @code{K[x1,@dots{},xn]} $B$r78?t(B |
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$B$H$9$kHyJ,:nMQAG4D$G(B, @code{Di} $B$O(B @code{xi} $B$K$h$kHyJ,$rI=$9(B. $B8r494X78$K$h$j(B, |
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@code{D} $B$N85$O(B, @code{x1^i1*@dots{}*xn^in*D1^j1*@dots{}*Dn^jn} $B$J$kC19`(B |
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$B<0$N(B @code{K} $B@~7A7k9g$H$7$F=q$-I=$9$3$H$,$G$-$k(B. |
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Risa/Asir $B$K$*$$$F$O(B, $B$3$NC19`<0$r(B, $B2D49$JB?9`<0$HF1MM$K(B |
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@code{<<i1,@dots{},in,j1,@dots{},jn>>} $B$GI=$9(B. $B$9$J$o$A(B, @code{D} $B$N85$b(B |
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$BJ,;6I=8=B?9`<0$H$7$FI=$5$l$k(B. $B2C8:;;$O(B, $B2D49$N>l9g$HF1MM$K(B, @code{+}, @code{-} |
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$B$K$h$j(B |
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$B<B9T$G$-$k$,(B, $B>h;;$O(B, $BHs2D49@-$r9MN8$7$F(B @code{dp_weyl_mul()} $B$H$$$&4X?t(B |
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$B$K$h$j<B9T$9$k(B. |
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\E |
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|
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\BEG |
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@code{D} is the ring of differential operators whose coefficients |
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are polynomials in @code{K[x1,@dots{},xn]} and |
| |
@code{Di} denotes the differentiation with respect to @code{xi}. |
| |
According to the commutation relation, |
| |
elements of @code{D} can be represented as a @code{K}-linear combination |
| |
of monomials @code{x1^i1*@dots{}*xn^in*D1^j1*@dots{}*Dn^jn}. |
| |
In Risa/Asir, this type of monomial is represented |
| |
by @code{<<i1,@dots{},in,j1,@dots{},jn>>} as in the case of commutative |
| |
polynomial. |
| |
That is, elements of @code{D} are represented by distributed polynomials. |
| |
Addition and subtraction can be done by @code{+}, @code{-}, |
| |
but multiplication is done by calling @code{dp_weyl_mul()} because of |
| |
the non-commutativity of @code{D}. |
| |
\E |
| |
|
| |
@example |
| |
[0] A=<<1,2,2,1>>; |
| |
(1)*<<1,2,2,1>> |
| |
[1] B=<<2,1,1,2>>; |
| |
(1)*<<2,1,1,2>> |
| |
[2] A*B; |
| |
(1)*<<3,3,3,3>> |
| |
[3] dp_weyl_mul(A,B); |
| |
(1)*<<3,3,3,3>>+(1)*<<3,2,3,2>>+(4)*<<2,3,2,3>>+(4)*<<2,2,2,2>> |
| |
+(2)*<<1,3,1,3>>+(2)*<<1,2,1,2>> |
| |
@end example |
| |
|
| |
\BJP |
| |
$B%0%l%V%J4pDl7W;;$K$D$$$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 1217 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_f_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 1244 Refer to the sections for each functions. |
|
| Line 1490 Refer to the sections for each functions. |
|
| * lex_hensel_gsl tolex_gsl tolex_gsl_d:: |
* lex_hensel_gsl tolex_gsl tolex_gsl_d:: |
| * primadec primedec:: |
* primadec primedec:: |
| * primedec_mod:: |
* 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 1294 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 1329 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 1346 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 dp_gr_f_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 dp_gr_f_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 dp_gr_f_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 |
| Line 1842 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 dp_gr_f_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 dp_gr_f_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}, @code{dp_gr_f_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_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_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 1880 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()} $B$O(B, $B<o!9$NM-8BBN>e$N%0%l%V%J4pDl$r7W;;$9$k(B |
@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$$$k(B. $BF~NO$O(B, $B$"$i$+$8$a(B, @code{simp_ff()} $B$J$I$G(B, |
$B>l9g$KMQ$$$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$$$kI,MW$,$"$k(B. |
$B9M$($kM-8BBN>e$K<M1F$5$l$F$$$kI,MW$,$"$k(B. |
| @item |
@item |
| Line 1917 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()} is a function for Groebner basis computation |
@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 |
over various finite fields. Coefficients of input polynomials |
| must be converted to elements of a finite field |
must be converted to elements of a finite field |
| currently specified by @code{setmod_ff()}. |
currently specified by @code{setmod_ff()}. |
| Line 1966 Actual computation is controlled by various parameters |
|
| Line 2236 Actual computation is controlled by various parameters |
|
| \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 2000 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 2012 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 2026 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 2034 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 2048 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 |
| Line 2061 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$$$k(B. |
| \E |
\E |
| \BEG |
\BEG |
| Line 2078 Arguments must be specified as a list such as |
|
| Line 2502 Arguments must be specified as a list such as |
|
| strings. |
strings. |
| @item |
@item |
| @code{dp_gr_print()} is used to set and show the value of a parameter |
@code{dp_gr_print()} is used to set and show the value of a parameter |
| @code{Print}. This functions is prepared to get quickly the value of |
@code{Print} and @code{PrintShort}. |
| @code{Print} when a user defined function calling @code{dp_gr_main()} etc. |
@table @var |
| |
@item i=0 |
| |
@code{Print=0}, @code{PrintShort=0} |
| |
@item i=1 |
| |
@code{Print=1}, @code{PrintShort=0} |
| |
@item i=2 |
| |
@code{Print=0}, @code{PrintShort=1} |
| |
@end table |
| |
This functions is prepared to get quickly the value |
| |
when a user defined function calling @code{dp_gr_main()} etc. |
| uses the value as a flag for showing intermediate informations. |
uses the value as a flag for showing intermediate informations. |
| \E |
\E |
| @end itemize |
@end itemize |
| Line 3665 if an input ideal is not radical. |
|
| Line 4098 if an input ideal is not radical. |
|
| \EG @fref{Setting term orderings}. |
\EG @fref{Setting term orderings}. |
| @end table |
@end table |
| |
|
| \BJP |
|
| @node Weyl $BBe?t(B,,, $B%0%l%V%J4pDl$N7W;;(B |
|
| @section Weyl $BBe?t(B |
|
| \E |
|
| \BEG |
|
| @node Weyl algebra,,, Groebner basis computation |
|
| @section Weyl algebra |
|
| \E |
|
| |
|
| @noindent |
|
| |
|
| \BJP |
|
| $B$3$l$^$G$O(B, $BDL>o$N2D49$JB?9`<04D$K$*$1$k%0%l%V%J4pDl7W;;$K$D$$$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 |
|
| |
|
| \JP @node primedec_mod,,, $B%0%l%V%J4pDl$K4X$9$kH!?t(B |
\JP @node primedec_mod,,, $B%0%l%V%J4pDl$K4X$9$kH!?t(B |
| \EG @node primedec_mod,,, Functions for Groebner basis computation |
\EG @node primedec_mod,,, Functions for Groebner basis computation |
| @subsection @code{primedec_mod} |
@subsection @code{primedec_mod} |
| Line 3814 Computation of the global b function is implemented as |
|
| Line 4148 Computation of the global b function is implemented as |
|
| $BItJ,$r7W;;$9$k$3$H$K$h$k(B early termination $B$r9T$&(B. $B0lHL$K(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 |
$B%$%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$-$$>l9g$,$"$k(B. |
$B>l9g$K$O(B overhead $B$,Bg$-$$>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 |
\E |
| \BEG |
\BEG |
| @item |
@item |
| Line 3838 is tried by computing the intersection of obtained com |
|
| Line 4175 is tried by computing the intersection of obtained com |
|
| incrementally. In general, this strategy is useful when the krull |
incrementally. In general, this strategy is useful when the krull |
| dimension of the ideal is high, but it may add some overhead |
dimension of the ideal is high, but it may add some overhead |
| if the dimension is small. |
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 |
\E |
| @end itemize |
@end itemize |
| |
|
| Line 3857 if the dimension is small. |
|
| Line 4197 if the dimension is small. |
|
| \JP @item $B;2>H(B |
\JP @item $B;2>H(B |
| \EG @item References |
\EG @item References |
| @fref{modfctr}, |
@fref{modfctr}, |
| @fref{dp_gr_main dp_gr_mod_main dp_gr_f_main}, |
@fref{dp_gr_main dp_gr_mod_main dp_gr_f_main dp_weyl_gr_main dp_weyl_gr_mod_main dp_weyl_gr_f_main}, |
| \JP @fref{$B9`=g=x$N@_Dj(B}. |
\JP @fref{$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 |
@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} |
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@findex bfunction |
| |
@findex bfct |
| |
@findex generic_bfct |
| |
@findex ann |
| |
@findex ann0 |
| |
|
| |
@table @t |
| |
@item bfunction(@var{f}) |
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@itemx bfct(@var{f}) |
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@itemx generic_bfct(@var{plist},@var{vlist},@var{dvlist},@var{weight}) |
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\JP :: @var{b} $B4X?t$N7W;;(B |
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\EG :: Computes the global @var{b} function of a polynomial or an ideal |
| |
@item ann(@var{f}) |
| |
@itemx ann0(@var{f}) |
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\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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