| version 1.5, 2003/04/20 08:01:25 |
version 1.9, 2003/04/24 08:13:24 |
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| @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 1.8 2003/04/21 08:30:01 noro 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 |
| Line 449 If `on', various informations during a Groebner basis |
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| Line 449 If `on', various informations during a Groebner basis |
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| displayed. |
displayed. |
| \E |
\E |
| |
|
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@item PrintShort |
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\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. |
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\BEG |
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If `on' and Print is `off', short information during a Groebner basis computation is |
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displayed. |
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\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 |
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| Line 478 is shown after every normal computation. After comlet |
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| computation the maximal value among the sums is shown. |
computation the maximal value among the sums is shown. |
| \E |
\E |
| |
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| @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. |
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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. |
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An integer value can be set by the keyword @code{Multiple} for |
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backward compatibility. |
| \E |
\E |
| |
|
| @item Demand |
@item Demand |
| Line 1203 Refer to the sections for each functions. |
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| Line 1214 Refer to the sections for each functions. |
|
| \E |
\E |
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| \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 |
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@code{Di} denotes the differentiation with respect to @code{xi}. |
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According to the commutation relation, |
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elements of @code{D} can be represented as a @code{K}-linear combination |
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of monomials @code{x1^i1*@dots{}*xn^in*D1^j1*@dots{}*Dn^jn}. |
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In Risa/Asir, this type of monomial is represented |
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by @code{<<i1,@dots{},in,j1,@dots{},jn>>} as in the case of commutative |
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polynomial. |
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That is, elements of @code{D} are represented by distributed polynomials. |
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Addition and subtraction can be done by @code{+}, @code{-}, |
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but multiplication is done by calling @code{dp_weyl_mul()} because of |
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the non-commutativity of @code{D}. |
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\E |
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|
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@example |
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[0] A=<<1,2,2,1>>; |
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(1)*<<1,2,2,1>> |
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[1] B=<<2,1,1,2>>; |
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(1)*<<2,1,1,2>> |
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[2] A*B; |
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(1)*<<3,3,3,3>> |
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[3] dp_weyl_mul(A,B); |
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(1)*<<3,3,3,3>>+(1)*<<3,2,3,2>>+(4)*<<2,3,2,3>>+(4)*<<2,2,2,2>> |
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+(2)*<<1,3,1,3>>+(2)*<<1,2,1,2>> |
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@end example |
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|
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\BJP |
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$B%0%l%V%J4pDl7W;;$K$D$$$F$b(B, Weyl $BBe?t@lMQ$N4X?t$H$7$F(B, |
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$B<!$N4X?t$,MQ0U$7$F$"$k(B. |
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\E |
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\BEG |
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The following functions are avilable for Groebner basis computation |
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in Weyl algebra: |
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\E |
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@code{dp_weyl_gr_main()}, |
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@code{dp_weyl_gr_mod_main()}, |
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@code{dp_weyl_gr_f_main()}, |
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@code{dp_weyl_f4_main()}, |
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@code{dp_weyl_f4_mod_main()}. |
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\BJP |
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$B$^$?(B, $B1~MQ$H$7$F(B, global b $B4X?t$N7W;;$,<BAu$5$l$F$$$k(B. |
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\E |
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\BEG |
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Computation of the global b function is implemented as an application. |
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\E |
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|
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\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. |
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| Line 1327 Refer to the sections for each functions. |
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| * 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:: |
| * 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. |
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| Line 1354 Refer to the sections for each functions. |
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| * lex_hensel_gsl tolex_gsl tolex_gsl_d:: |
* lex_hensel_gsl tolex_gsl tolex_gsl_d:: |
| * primadec primedec:: |
* primadec primedec:: |
| * primedec_mod:: |
* primedec_mod:: |
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* bfunction bfct generic_bfct:: |
| @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 1346 for communication. |
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| Line 1457 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 1952 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 |
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@findex dp_weyl_gr_mod_main |
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@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}) |
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@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 1996 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 2034 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 2085 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 2123 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 2137 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 2034 Arguments and actions are the same as those of |
|
| Line 2161 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 2175 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 2191 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$$$k(B. |
| \E |
\E |
| \BEG |
\BEG |
| Line 2078 Arguments must be specified as a list such as |
|
| Line 2217 Arguments must be specified as a list such as |
|
| strings. |
strings. |
| @item |
@item |
| @code{dp_gr_print()} is used to set and show the value of a parameter |
@code{dp_gr_print()} is used to set and show the value of a parameter |
| @code{Print}. This functions is prepared to get quickly the value of |
@code{Print} and @code{PrintShort}. |
| @code{Print} when a user defined function calling @code{dp_gr_main()} etc. |
@table @var |
| |
@item i=0 |
| |
@code{Print=0}, @code{PrintShort=0} |
| |
@item i=1 |
| |
@code{Print=1}, @code{PrintShort=0} |
| |
@item i=2 |
| |
@code{Print=0}, @code{PrintShort=1} |
| |
@end table |
| |
This functions is prepared to get quickly the value |
| |
when a user defined function calling @code{dp_gr_main()} etc. |
| uses the value as a flag for showing intermediate informations. |
uses the value as a flag for showing intermediate informations. |
| \E |
\E |
| @end itemize |
@end itemize |
| Line 3665 if an input ideal is not radical. |
|
| Line 3813 if an input ideal is not radical. |
|
| \EG @fref{Setting term orderings}. |
\EG @fref{Setting term orderings}. |
| @end table |
@end table |
| |
|
| \BJP |
|
| @node Weyl $BBe?t(B,,, $B%0%l%V%J4pDl$N7W;;(B |
|
| @section Weyl $BBe?t(B |
|
| \E |
|
| \BEG |
|
| @node Weyl algebra,,, Groebner basis computation |
|
| @section Weyl algebra |
|
| \E |
|
| |
|
| @noindent |
|
| |
|
| \BJP |
|
| $B$3$l$^$G$O(B, $BDL>o$N2D49$JB?9`<04D$K$*$1$k%0%l%V%J4pDl7W;;$K$D$$$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 3863 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 3890 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 3912 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,,, $B%0%l%V%J4pDl$K4X$9$kH!?t(B |
| |
\EG @node bfunction bfct generic_bfct,,, Functions for Groebner basis computation |
| |
@subsection @code{bfunction}, @code{bfct}, @code{generic_bfct} |
| |
@findex bfunction |
| |
@findex bfct |
| |
@findex generic_bfct |
| |
|
| |
@table @t |
| |
@item bfunction(@var{f}) |
| |
@item bfct(@var{f}) |
| |
@item generic_bfct(@var{plist},@var{vlist},@var{dvlist},@var{weight}) |
| |
\JP :: b $B4X?t$N7W;;(B |
| |
\EG :: Computes the global b function of a polynomial or an ideal |
| |
@end table |
| |
@table @var |
| |
@item return |
| |
@itemx 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$$$k(B. |
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@item @code{bfunction(@var{f})}, @code{bfct(@var{f})} $B$OB?9`<0(B @var{f} $B$N(B global b $B4X?t(B @code{b(s)} $B$r(B |
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$B7W;;$9$k(B. @code{b(s)} $B$O(B, Weyl $BBe?t(B @code{D} $B>e$N0lJQ?tB?9`<04D(B @code{D[s]} |
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$B$N85(B @code{P(x,s)} $B$,B8:_$7$F(B, @code{P(x,s)f^(s+1)=b(s)f^s} $B$rK~$?$9$h$&$J(B |
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$BB?9`<0(B @code{b(s)} $B$NCf$G(B, $B<!?t$,:G$bDc$$$b$N$G$"$k(B. |
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@item @code{generic_bfct(@var{f},@var{vlist},@var{dvlist},@var{weight})} |
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$B$O(B, @var{plist} $B$G@8@.$5$l$k(B @code{D} $B$N:8%$%G%"%k(B @code{I} $B$N(B, |
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$B%&%'%$%H(B @var{weight} $B$K4X$9$k(B global b $B4X?t$r7W;;$9$k(B. |
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@var{vlist} $B$O(B @code{x}-$BJQ?t(B, @var{vlist} $B$OBP1~$9$k(B @code{D}-$BJQ?t(B |
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$B$r=g$KJB$Y$k(B. |
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@item @code{bfunction} $B$H(B @code{bfct} $B$G$OMQ$$$F$$$k%"%k%4%j%:%`$,(B |
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$B0[$J$k(B. $B$I$A$i$,9bB.2=$OF~NO$K$h$k(B. |
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@item $B>\:Y$K$D$$$F$O(B, [Saito,Sturmfels,Takayama] $B$r8+$h(B. |
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\E |
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\BEG |
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@item These functions are defined in @samp{bfct}. |
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@item @code{bfunction(@var{f})} and @code{bfct(@var{f})} compute the global b-function @code{b(s)} of |
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a polynomial @var{f}. |
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@code{b(s)} is a polynomial of the minimal degree |
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such that there exists @code{P(x,s)} in D[s], which is a polynomial |
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ring over Weyl algebra @code{D}, and @code{P(x,s)f^(s+1)=b(s)f^s} holds. |
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@item @code{generic_bfct(@var{f},@var{vlist},@var{dvlist},@var{weight})} |
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computes the global b-function of a left ideal @code{I} in @code{D} |
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generated by @var{plist}, with respect to @var{weight}. |
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@var{vlist} is the list of @code{x}-variables, |
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@var{vlist} is the list of corresponding @code{D}-variables. |
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@item @code{bfunction(@var{f})} and @code{bfct(@var{f})} implement |
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different algorithms and the efficiency depends on inputs. |
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@item See [Saito,Sturmfels,Takayama] for the details. |
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\E |
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@end itemize |
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|
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@example |
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[0] load("bfct")$ |
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[216] bfunction(x^3+y^3+z^3+x^2*y^2*z^2+x*y*z); |
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-9*s^5-63*s^4-173*s^3-233*s^2-154*s-40 |
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[217] fctr(@@); |
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[[-1,1],[s+2,1],[3*s+4,1],[3*s+5,1],[s+1,2]] |
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[218] F = [4*x^3*dt+y*z*dt+dx,x*z*dt+4*y^3*dt+dy, |
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x*y*dt+5*z^4*dt+dz,-x^4-z*y*x-y^4-z^5+t]$ |
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[219] generic_bfct(F,[t,z,y,x],[dt,dz,dy,dx],[1,0,0,0]); |
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20000*s^10-70000*s^9+101750*s^8-79375*s^7+35768*s^6-9277*s^5 |
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+1278*s^4-72*s^3 |
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@end example |
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|
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@table @t |
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\JP @item $B;2>H(B |
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\EG @item References |
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\JP @fref{Weyl $BBe?t(B}. |
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\EG @fref{Weyl algebra}. |
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@end table |
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