| version 1.13, 2004/09/13 09:23:30 |
version 1.23, 2019/09/13 09:31:00 |
|
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| @comment $OpenXM: OpenXM/src/asir-doc/parts/groebner.texi,v 1.12 2003/12/27 11:52:07 takayama Exp $ |
@comment $OpenXM: OpenXM/src/asir-doc/parts/groebner.texi,v 1.22 2019/03/29 04:54:25 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 |
|
|
| * $BM-M}<0$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:: |
| |
* $BB?9`<04D>e$N2C72(B:: |
| * $B%0%l%V%J4pDl$K4X$9$kH!?t(B:: |
* $B%0%l%V%J4pDl$K4X$9$kH!?t(B:: |
| \E |
\E |
| \BEG |
\BEG |
|
|
| * Groebner basis computation with rational function coefficients:: |
* Groebner basis computation with rational function coefficients:: |
| * Change of ordering:: |
* Change of ordering:: |
| * Weyl algebra:: |
* Weyl algebra:: |
| |
* Module over a polynomial ring:: |
| * Functions for Groebner basis computation:: |
* Functions for Groebner basis computation:: |
| \E |
\E |
| @end menu |
@end menu |
| Line 201 In an @b{Asir} session, it is displayed in the form li |
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| Line 203 In an @b{Asir} session, it is displayed in the form li |
|
| \EG and also can be input in such a form. |
\EG and also can be input in such a form. |
| |
|
| \BJP |
\BJP |
| @itemx $BF,C19`<0(B (head monomial) |
|
| @item $BF,9`(B (head term) |
@item $BF,9`(B (head term) |
| |
@itemx $BF,C19`<0(B (head monomial) |
| @itemx $BF,78?t(B (head coefficient) |
@itemx $BF,78?t(B (head coefficient) |
| $BJ,;6I=8=B?9`<0$K$*$1$k3FC19`<0$O(B, $B9`=g=x$K$h$j@0Ns$5$l$k(B. $B$3$N;~=g(B |
$BJ,;6I=8=B?9`<0$K$*$1$k3FC19`<0$O(B, $B9`=g=x$K$h$j@0Ns$5$l$k(B. $B$3$N;~=g(B |
| $B=x:GBg$NC19`<0$rF,C19`<0(B, $B$=$l$K8=$l$k9`(B, $B78?t$r$=$l$>$lF,9`(B, $BF,78?t(B |
$B=x:GBg$NC19`<0$rF,C19`<0(B, $B$=$l$K8=$l$k9`(B, $B78?t$r$=$l$>$lF,9`(B, $BF,78?t(B |
| $B$H8F$V(B. |
$B$H8F$V(B. |
| \E |
\E |
| \BEG |
\BEG |
| @itemx head monomial |
|
| @item head term |
@item head term |
| |
@itemx head monomial |
| @itemx head coefficient |
@itemx head coefficient |
| |
|
| Monomials in a distributed polynomial is sorted by a total order. |
Monomials in a distributed polynomial is sorted by a total order. |
| Line 220 the head term and the head coefficient respectively. |
|
| Line 222 the head term and the head coefficient respectively. |
|
| \E |
\E |
| @end table |
@end table |
| |
|
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@noindent |
| |
ChangeLog |
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@itemize @bullet |
| \BJP |
\BJP |
| |
@item $BJ,;6I=8=B?9`<0$OG$0U$N%*%V%8%'%/%H$r78?t$K$b$F$k$h$&$K$J$C$?(B. |
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$B$^$?2C72$N(Bk$B@.J,$NMWAG$r<!$N7A<0(B <<d0,d1,...:k>> $B$GI=8=$9$k$h$&$K$J$C$?(B (2017-08-31). |
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\E |
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\BEG |
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@item Distributed polynomials accept objects as coefficients. |
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The k-th element of a free module is expressed as <<d0,d1,...:k>> (2017-08-31). |
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\E |
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@item |
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1.15 algnum.c, |
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1.53 ctrl.c, |
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1.66 dp-supp.c, |
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1.105 dp.c, |
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1.73 gr.c, |
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1.4 reduct.c, |
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1.16 _distm.c, |
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1.17 dalg.c, |
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1.52 dist.c, |
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1.20 distm.c, |
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1.8 gmpq.c, |
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1.238 engine/nd.c, |
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1.102 ca.h, |
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1.411 version.h, |
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1.28 cpexpr.c, |
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1.42 pexpr.c, |
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1.20 pexpr_body.c, |
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1.40 spexpr.c, |
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1.27 arith.c, |
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1.77 eval.c, |
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1.56 parse.h, |
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1.37 parse.y, |
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1.8 stdio.c, |
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1.31 plotf.c |
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@end itemize |
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|
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\BJP |
| @node $B%U%!%$%k$NFI$_9~$_(B,,, $B%0%l%V%J4pDl$N7W;;(B |
@node $B%U%!%$%k$NFI$_9~$_(B,,, $B%0%l%V%J4pDl$N7W;;(B |
| @section $B%U%!%$%k$NFI$_9~$_(B |
@section $B%U%!%$%k$NFI$_9~$_(B |
| \E |
\E |
| Line 1069 beforehand, and some heuristic trial may be inevitable |
|
| Line 1109 beforehand, and some heuristic trial may be inevitable |
|
| $B$h$j0lHLE*$J$b$N$H$J$k(B. |
$B$h$j0lHLE*$J$b$N$H$J$k(B. |
| \E |
\E |
| \BEG |
\BEG |
| Term orders introduced in the previous section can be generalized |
Term orderings introduced in the previous section can be generalized |
| by setting a weight for each variable. |
by setting a weight for each variable. |
| \E |
\E |
| @example |
@example |
| Line 1097 In this example, the weights for the first, the second |
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| Line 1137 In this example, the weights for the first, the second |
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| variable are set to 1, 2 and 3 respectively. |
variable are set to 1, 2 and 3 respectively. |
| Therefore the total degree of @code{<<1,1,1>>} under this weight, |
Therefore the total degree of @code{<<1,1,1>>} under this weight, |
| which is called the weight of the monomial, is @code{1*1+1*2+1*3=6}. |
which is called the weight of the monomial, is @code{1*1+1*2+1*3=6}. |
| By setting weights, different term orders can be set under a term |
By setting weights, different term orderings can be set under a type of |
| order type. For example, a polynomial can be made weighted homogeneous |
term ordeing. In some case a polynomial can |
| by setting an appropriate weight. |
be made weighted homogeneous by setting an appropriate weight. |
| \E |
\E |
| |
|
| \BJP |
\BJP |
| Line 1131 is also considered as a refinement of comparison by we |
|
| Line 1171 is also considered as a refinement of comparison by we |
|
| It compares two terms by using a weight vector whose elements |
It compares two terms by using a weight vector whose elements |
| corresponding to variables in a block is 1 and 0 otherwise, |
corresponding to variables in a block is 1 and 0 otherwise, |
| then it applies a tie breaker. |
then it applies a tie breaker. |
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\E |
| |
|
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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$$(B. $B$3$N$?$a(B, $B<!$N$h$&$J7A$G$b(B |
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$B9`=g=x$,;XDj$G$-$k(B. |
| \E |
\E |
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\BEG |
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A weight vector can be set by using @code{dp_set_weight()}. |
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However it is more preferable if a weight vector can be set |
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together with other parapmeters such as a type of term ordering |
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and a variable order. This is realized as follows. |
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\E |
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|
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@example |
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[64] B=[x+y+z-6,x*y+y*z+z*x-11,x*y*z-6]$ |
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[65] dp_gr_main(B|v=[x,y,z],sugarweight=[3,2,1],order=0); |
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[z^3-6*z^2+11*z-6,x+y+z-6,-y^2+(-z+6)*y-z^2+6*z-11] |
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[66] dp_gr_main(B|v=[y,z,x],order=[[1,1,0],[0,1,0],[0,0,1]]); |
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[x^3-6*x^2+11*x-6,x+y+z-6,-x^2+(-y+6)*x-y^2+6*y-11] |
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[67] dp_gr_main(B|v=[y,z,x],order=[[x,1,y,2,z,3]]); |
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[x+y+z-6,x^3-6*x^2+11*x-6,-x^2+(-y+6)*x-y^2+6*y-11] |
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@end example |
| |
|
| \BJP |
\BJP |
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$B$$$:$l$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 1393 Computation of the global b function is implemented as |
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| Line 1488 Computation of the global b function is implemented as |
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| \E |
\E |
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|
| \BJP |
\BJP |
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@node $BB?9`<04D>e$N2C72(B,,, $B%0%l%V%J4pDl$N7W;;(B |
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@section $BB?9`<04D>e$N2C72(B |
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\E |
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\BEG |
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@node Module over a polynomial ring,,, Groebner basis computation |
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@section Module over a polynomial ring |
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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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$BB?9`<04D>e$N<+M32C72$N85$O(B, $B2C72C19`<0(B te_i $B$N@~7?OB$H$7$FFbItI=8=$5$l$k(B. |
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$B$3$3$G(B t $B$OB?9`<04D$NC19`<0(B, e_i $B$O<+M32C72$NI8=`4pDl$G$"$k(B. $B2C72C19`<0$O(B, $BB?9`<04D$NC19`<0(B |
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$B$K0LCV(B i $B$rDI2C$7$?(B @code{<<a,b,...,c:i>>} $B$GI=$9(B. $B2C72B?9`<0(B, $B$9$J$o$A2C72C19`<0$N@~7?OB$O(B, |
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$B@_Dj$5$l$F$$$k2C729`=g=x$K$7$?$,$C$F9_=g$K@0Ns$5$l$k(B. $B2C729`=g=x$K$O0J2<$N(B3$B<oN`$,$"$k(B. |
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|
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@table @code |
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@item TOP $B=g=x(B |
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$B$3$l$O(B, te_i > se_j $B$H$J$k$N$O(B t>s $B$^$?$O(B (t=s $B$+$D(B i<j) $B$H$J$k$h$&$J9`=g=x$G$"$k(B. $B$3$3$G(B, |
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t, s $B$NHf3S$OB?9`<04D$K@_Dj$5$l$F$$$k=g=x$G9T$&(B. |
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$B$3$N7?$N=g=x$O(B, @code{dp_ord([0,Ord])} $B$K(B |
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$B$h$j@_Dj$9$k(B. $B$3$3$G(B, @code{Ord} $B$OB?9`<04D$N=g=x7?$G$"$k(B. |
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|
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@item POT $B=g=x(B |
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$B$3$l$O(B, te_i > se_j $B$H$J$k$N$O(B i<j $B$^$?$O(B (i=j $B$+$D(B t>s) $B$H$J$k$h$&$J9`=g=x$G$"$k(B. $B$3$3$G(B, |
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t, s $B$NHf3S$OB?9`<04D$K@_Dj$5$l$F$$$k=g=x$G9T$&(B. |
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$B$3$N7?$N=g=x$O(B, @code{dp_ord([1,Ord])} $B$K(B |
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$B$h$j@_Dj$9$k(B. $B$3$3$G(B, @code{Ord} $B$OB?9`<04D$N=g=x7?$G$"$k(B. |
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|
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@item Schreyer $B7?=g=x(B |
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|
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$B3FI8=`4pDl(B e_i $B$KBP$7(B, $BJL$N<+M32C72$N2C72C19`<0(B T_i $B$,M?$($i$l$F$$$F(B, te_i > se_j $B$H$J$k$N$O(B |
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tT_i > sT_j $B$^$?$O(B (tT_i=sT_j $B$+$D(B i<j) $B$H$J$k$h$&$J9`=g=x$G$"$k(B. $B$3$3$G(B tT_i, sT_j $B$N(B |
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$BHf3S$O(B, $B$3$l$i$,=jB0$9$k<+M32C72$K@_Dj$5$l$F$$$k=g=x$G9T$&(B. |
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$B$3$N7?$N=g=x$O(B, $BDL>o:F5"E*$K@_Dj$5$l$k(B. $B$9$J$o$A(B, T_i $B$,=jB0$9$k<+M32C72$N=g=x$b(B Schreyer $B7?(B |
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$B$G$"$k$+(B, $B$^$?$O%\%H%`$H$J$k(B TOP, POT $B$J$I$N9`=g=x$H$J$k(B. |
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$B$3$N7?$N=g=x$O(B @code{dpm_set_schreyer([H_1,H_2,...])} $B$K$h$j;XDj$9$k(B. $B$3$3$G(B, |
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@code{H_i=[T_1,T_2,...]} $B$O2C72C19`<0$N%j%9%H$G(B, @code{[H_2,...]} $B$GDj5A$5$l$k(B Schreyer $B7?9`=g=x$r(B |
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@code{tT_i} $B$i$KE,MQ$9$k$H$$$&0UL#$G$"$k(B. |
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@end table |
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|
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$B2C72B?9`<0$rF~NO$9$kJ}K!$H$7$F$O(B, @code{<<a,b,...:i>>} $B$J$k7A<0$GD>@\F~NO$9$kB>$K(B, |
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$BB?9`<0%j%9%H$r:n$j(B, @code{dpm_ltod()} $B$K$h$jJQ49$9$kJ}K!$b$"$k(B. |
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\E |
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\BEG |
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not yet |
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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 1409 Computation of the global b function is implemented as |
|
| Line 1555 Computation of the global b function is implemented as |
|
| * tolexm minipolym:: |
* tolexm minipolym:: |
| * 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_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_weyl_f4_main dp_weyl_f4_mod_main:: |
* dp_f4_main dp_f4_mod_main dp_weyl_f4_main dp_weyl_f4_mod_main:: |
| |
* nd_gr nd_gr_trace nd_f4 nd_f4_trace nd_weyl_gr nd_weyl_gr_trace:: |
| |
* nd_gr_postproc nd_weyl_gr_postproc:: |
| * dp_gr_flags dp_gr_print:: |
* dp_gr_flags dp_gr_print:: |
| * dp_ord:: |
* dp_ord:: |
| |
* dp_set_weight dp_set_top_weight dp_weyl_set_weight:: |
| * dp_ptod:: |
* dp_ptod:: |
| * dp_dtop:: |
* dp_dtop:: |
| * dp_mod dp_rat:: |
* dp_mod dp_rat:: |
| * dp_homo dp_dehomo:: |
* dp_homo dp_dehomo:: |
| * dp_ptozp dp_prim:: |
* dp_ptozp dp_prim:: |
| * dp_nf dp_nf_mod dp_true_nf dp_true_nf_mod:: |
* dp_nf dp_nf_mod dp_true_nf dp_true_nf_mod dp_weyl_nf dp_weyl_nf_mod:: |
| * dp_hm dp_ht dp_hc dp_rest:: |
* dp_hm dp_ht dp_hc dp_rest:: |
| |
* dpm_hm dpm_ht dpm_hc dpm_hp dpm_rest:: |
| * dp_td dp_sugar:: |
* dp_td dp_sugar:: |
| * dp_lcm:: |
* dp_lcm:: |
| * dp_redble:: |
* dp_redble:: |
| Line 1474 Computation of the global b function is implemented as |
|
| Line 1624 Computation of the global b function is implemented as |
|
| @item |
@item |
| $BI8=`%i%$%V%i%j$N(B @samp{gr} $B$GDj5A$5$l$F$$$k(B. |
$BI8=`%i%$%V%i%j$N(B @samp{gr} $B$GDj5A$5$l$F$$$k(B. |
| @item |
@item |
| |
gr $B$rL>A0$K4^$`4X?t$O8=:_%a%s%F$5$l$F$$$J$$(B. @code{nd_gr}$B7O$N4X?t$rBe$o$j$KMxMQ$9$Y$-$G$"$k(B(@fref{nd_gr nd_gr_trace nd_f4 nd_f4_trace nd_weyl_gr nd_weyl_gr_trace}). |
| |
@item |
| $B$$$:$l$b(B, $BB?9`<0%j%9%H(B @var{plist} $B$N(B, $BJQ?t=g=x(B @var{vlist}, $B9`=g=x7?(B |
$B$$$:$l$b(B, $BB?9`<0%j%9%H(B @var{plist} $B$N(B, $BJQ?t=g=x(B @var{vlist}, $B9`=g=x7?(B |
| @var{order} $B$K4X$9$k%0%l%V%J4pDl$r5a$a$k(B. @code{gr()}, @code{hgr()} |
@var{order} $B$K4X$9$k%0%l%V%J4pDl$r5a$a$k(B. @code{gr()}, @code{hgr()} |
| $B$O(B $BM-M}?t78?t(B, @code{gr_mod()} $B$O(B GF(@var{p}) $B78?t$H$7$F7W;;$9$k(B. |
$B$O(B $BM-M}?t78?t(B, @code{gr_mod()} $B$O(B GF(@var{p}) $B78?t$H$7$F7W;;$9$k(B. |
| Line 1485 Computation of the global b function is implemented as |
|
| Line 1637 Computation of the global b function is implemented as |
|
| 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. |
| Line 1505 CPU $B;~4V$G$"$j(B, $B$3$NH!?t$N>l9g$O$[$H$s$IDL?.$ |
|
| Line 1657 CPU $B;~4V$G$"$j(B, $B$3$NH!?t$N>l9g$O$[$H$s$IDL?.$ |
|
| @item |
@item |
| These functions are defined in @samp{gr} in the standard library |
These functions are defined in @samp{gr} in the standard library |
| directory. |
directory. |
| |
@item |
| |
Functions of which names contains gr are obsolted. |
| |
Functions of @code{nd_gr} families should be used (@fref{nd_gr nd_gr_trace nd_f4 nd_f4_trace nd_weyl_gr nd_weyl_gr_trace}). |
| @item |
@item |
| They compute a Groebner basis of a polynomial list @var{plist} with |
They compute a Groebner basis of a polynomial list @var{plist} with |
| respect to the variable order @var{vlist} and the order type @var{order}. |
respect to the variable order @var{vlist} and the order type @var{order}. |
| Line 2248 except for lack of the argument for controlling homoge |
|
| Line 2403 except for lack of the argument for controlling homoge |
|
| \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}[|@var{option=value,...}]) |
| |
@itemx nd_gr_trace(@var{plist},@var{vlist},@var{homo},@var{p},@var{order}[|@var{option=value,...}]) |
| |
@itemx nd_f4(@var{plist},@var{vlist},@var{modular},@var{order}[|@var{option=value,...}]) |
| |
@itemx nd_f4_trace(@var{plist},@var{vlist},@var{homo},@var{p},@var{order}[|@var{option=value,...}]) |
| |
@itemx nd_weyl_gr(@var{plist},@var{vlist},@var{p},@var{order}[|@var{option=value,...}]) |
| |
@itemx nd_weyl_gr_trace(@var{plist},@var{vlist},@var{homo},@var{p},@var{order}[|@var{option=value,...}]) |
| |
\JP :: $B%0%l%V%J4pDl$N7W;;(B ($BAH$_9~$_H!?t(B) |
| |
\EG :: Groebner basis computation (built-in functions) |
| |
@end table |
| |
|
| |
@table @var |
| |
@item return |
| |
\JP $B%j%9%H(B |
| |
\EG list |
| |
@item plist vlist |
| |
\JP $B%j%9%H(B |
| |
\EG list |
| |
@item order |
| |
\JP $B?t(B, $B%j%9%H$^$?$O9TNs(B |
| |
\EG number, list or matrix |
| |
@item homo |
| |
\JP $B%U%i%0(B |
| |
\EG flag |
| |
@item modular |
| |
\JP $B%U%i%0$^$?$OAG?t(B |
| |
\EG flag or prime |
| |
@end table |
| |
|
| |
\BJP |
| |
@itemize @bullet |
| |
@item |
| |
$B$3$l$i$NH!?t$O(B, $B%0%l%V%J4pDl7W;;AH$_9~$_4X?t$N?7<BAu$G$"$k(B. |
| |
@item @code{nd_gr} $B$O(B, @code{p} $B$,(B 0 $B$N$H$-M-M}?tBN>e$N(B Buchberger |
| |
$B%"%k%4%j%:%`$r<B9T$9$k(B. @code{p} $B$,(B 2 $B0J>e$N<+A3?t$N$H$-(B, GF(p) $B>e$N(B |
| |
Buchberger $B%"%k%4%j%:%`$r<B9T$9$k(B. |
| |
@item @code{nd_gr_trace} $B$*$h$S(B @code{nd_f4_trace} |
| |
$B$OM-M}?tBN>e$G(B trace $B%"%k%4%j%:%`$r<B9T$9$k(B. |
| |
@var{p} $B$,(B 0 $B$^$?$O(B 1 $B$N$H$-(B, $B<+F0E*$KA*$P$l$?AG?t$rMQ$$$F(B, $B@.8y$9$k(B |
| |
$B$^$G(B trace $B%"%k%4%j%:%`$r<B9T$9$k(B. |
| |
@var{p} $B$,(B 2 $B0J>e$N$H$-(B, trace $B$O(BGF(p) $B>e$G7W;;$5$l$k(B. trace $B%"%k%4%j%:%`(B |
| |
$B$,<:GT$7$?>l9g(B 0 $B$,JV$5$l$k(B. @var{p} $B$,Ii$N>l9g(B, $B%0%l%V%J4pDl%A%'%C%/$O(B |
| |
$B9T$o$J$$(B. $B$3$N>l9g(B, @var{p} $B$,(B -1 $B$J$i$P<+F0E*$KA*$P$l$?AG?t$,(B, |
| |
$B$=$l0J30$O;XDj$5$l$?AG?t$rMQ$$$F%0%l%V%J4pDl8uJd$N7W;;$,9T$o$l$k(B. |
| |
@code{nd_f4_trace} $B$O(B, $B3FA4<!?t$K$D$$$F(B, $B$"$kM-8BBN>e$G(B F4 $B%"%k%4%j%:%`(B |
| |
$B$G9T$C$?7k2L$r$b$H$K(B, $B$=$NM-8BBN>e$G(B 0 $B$G$J$$4pDl$rM?$($k(B S-$BB?9`<0$N$_$r(B |
| |
$BMQ$$$F9TNs@8@.$r9T$$(B, $B$=$NA4<!?t$K$*$1$k4pDl$r@8@.$9$kJ}K!$G$"$k(B. $BF@$i$l$k(B |
| |
$BB?9`<0=89g$O$d$O$j%0%l%V%J4pDl8uJd$G$"$j(B, @code{nd_gr_trace} $B$HF1MM$N(B |
| |
$B%A%'%C%/$,9T$o$l$k(B. |
| |
@item |
| |
@code{nd_f4} $B$O(B @code{modular} $B$,(B 0 $B$N$H$-M-M}?tBN>e$N(B, @code{modular} $B$,(B |
| |
$B%^%7%s%5%$%:AG?t$N$H$-M-8BBN>e$N(B F4 $B%"%k%4%j%:%`$r<B9T$9$k(B. |
| |
@item |
| |
@var{plist} $B$,B?9`<0%j%9%H$N>l9g(B, @var{plist}$B$G@8@.$5$l$k%$%G%"%k$N%0%l%V%J!<4pDl$,(B |
| |
$B7W;;$5$l$k(B. @var{plist} $B$,B?9`<0%j%9%H$N%j%9%H$N>l9g(B, $B3FMWAG$OB?9`<04D>e$N<+M32C72$N85$H8+$J$5$l(B, |
| |
$B$3$l$i$,@8@.$9$kItJ,2C72$N%0%l%V%J!<4pDl$,7W;;$5$l$k(B. $B8e<T$N>l9g(B, $B9`=g=x$O2C72$KBP$9$k9`=g=x$r(B |
| |
$B;XDj$9$kI,MW$,$"$k(B. $B$3$l$O(B @var{[s,ord]} $B$N7A$G;XDj$9$k(B. @var{s} $B$,(B 0 $B$J$i$P(B TOP (Term Over Position), |
| |
1 $B$J$i$P(B POT (Position Over Term) $B$r0UL#$7(B, @var{ord} $B$OB?9`<04D$NC19`<0$KBP$9$k9`=g=x$G$"$k(B. |
| |
@item |
| |
@code{nd_weyl_gr}, @code{nd_weyl_gr_trace} $B$O(B Weyl $BBe?tMQ$G$"$k(B. |
| |
@item |
| |
@code{f4} $B7O4X?t0J30$O$9$Y$FM-M}4X?t78?t$N7W;;$,2DG=$G$"$k(B. |
| |
@item |
| |
$B0lHL$K(B @code{dp_gr_main}, @code{dp_gr_mod_main} $B$h$j9bB.$G$"$k$,(B, |
| |
$BFC$KM-8BBN>e$N>l9g82Cx$G$"$k(B. |
| |
@item |
| |
$B0J2<$N%*%W%7%g%s$,;XDj$G$-$k(B. |
| |
@table @code |
| |
@item homo |
| |
1 $B$N$H$-(B, $B@F<!2=$r7PM3$7$F7W;;$9$k(B. (@code{nd_gr}, @code{nd_f4} $B$N$_(B) |
| |
@item dp |
| |
1 $B$N$H$-(B, $BJ,;6I=8=B?9`<0(B ($B2C72$N>l9g$K$O2C72B?9`<0(B) $B$r7k2L$H$7$FJV$9(B. |
| |
@item nora |
| |
1 $B$N$H$-(B, $B7k2L$NAj8_4JLs$r9T$o$J$$(B. |
| |
@end table |
| |
@end itemize |
| |
\E |
| |
|
| |
\BEG |
| |
@itemize @bullet |
| |
@item |
| |
These functions are new implementations for computing Groebner bases. |
| |
@item @code{nd_gr} executes Buchberger algorithm over the rationals |
| |
if @code{p} is 0, and that over GF(p) if @code{p} is a prime. |
| |
@item @code{nd_gr_trace} executes the trace algorithm over the rationals. |
| |
If @code{p} is 0 or 1, the trace algorithm is executed until it succeeds |
| |
by using automatically chosen primes. |
| |
If @code{p} a positive prime, |
| |
the trace is comuted over GF(p). |
| |
If the trace algorithm fails 0 is returned. |
| |
If @code{p} is negative, |
| |
the Groebner basis check and ideal-membership check are omitted. |
| |
In this case, an automatically chosen prime if @code{p} is 1, |
| |
otherwise the specified prime is used to compute a Groebner basis |
| |
candidate. |
| |
Execution of @code{nd_f4_trace} is done as follows: |
| |
For each total degree, an F4-reduction of S-polynomials over a finite field |
| |
is done, and S-polynomials which give non-zero basis elements are gathered. |
| |
Then F4-reduction over Q is done for the gathered S-polynomials. |
| |
The obtained polynomial set is a Groebner basis candidate and the same |
| |
check procedure as in the case of @code{nd_gr_trace} is done. |
| |
@item |
| |
@code{nd_f4} executes F4 algorithm over Q if @code{modular} is equal to 0, |
| |
or over a finite field GF(@code{modular}) |
| |
if @code{modular} is a prime number of machine size (<2^29). |
| |
If @var{plist} is a list of polynomials, then a Groebner basis of the ideal generated by @var{plist} |
| |
is computed. If @var{plist} is a list of lists of polynomials, then each list of polynomials are regarded |
| |
as an element of a free module over a polynomial ring and a Groebner basis of the sub-module generated by @var{plist} |
| |
in the free module. In the latter case a term order in the free module should be specified. |
| |
This is specified by @var{[s,ord]}. If @var{s} is 0 then it means TOP (Term Over Position). |
| |
If @var{s} is 1 then it means POT 1 (Position Over Term). @var{ord} is a term order in the base polynomial ring. |
| |
@item |
| |
@code{nd_weyl_gr}, @code{nd_weyl_gr_trace} are for Weyl algebra computation. |
| |
@item |
| |
Functions except for F4 related ones can handle rational coeffient cases. |
| |
@item |
| |
In general these functions are more efficient than |
| |
@code{dp_gr_main}, @code{dp_gr_mod_main}, especially over finite fields. |
| |
@item |
| |
The fallowing options can be specified. |
| |
@table @code |
| |
@item homo |
| |
If set to 1, the computation is done via homogenization. (only for @code{nd_gr} and @code{nd_f4}) |
| |
@item dp |
| |
If set to 1, the functions return a list of distributed polynomials (a list of |
| |
module polynomials when the input is a sub-module). |
| |
@item nora |
| |
If set to 1, the inter-reduction is not performed. |
| |
@end table |
| |
@end itemize |
| |
\E |
| |
|
| |
@example |
| |
[38] load("cyclic")$ |
| |
[49] C=cyclic(7)$ |
| |
[50] V=vars(C)$ |
| |
[51] cputime(1)$ |
| |
[52] dp_gr_mod_main(C,V,0,31991,0)$ |
| |
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 nd_gr_postproc nd_weyl_gr_postproc,,, $B%0%l%V%J4pDl$K4X$9$kH!?t(B |
| |
\EG @node nd_gr_postproc nd_weyl_gr_postproc,,, Functions for Groebner basis computation |
| |
@subsection @code{nd_gr_postproc}, @code{nd_weyl_gr_postproc} |
| |
@findex nd_gr_postproc |
| |
@findex nd_weyl_gr_postproc |
| |
|
| |
@table @t |
| |
@item nd_gr_postproc(@var{plist},@var{vlist},@var{p},@var{order},@var{check}) |
| |
@itemx nd_weyl_gr_postproc(@var{plist},@var{vlist},@var{p},@var{order},@var{check}) |
| |
\JP :: $B%0%l%V%J4pDl8uJd$N%A%'%C%/$*$h$SAj8_4JLs(B |
| |
\EG :: Check of Groebner basis candidate and inter-reduction |
| |
@end table |
| |
|
| |
@table @var |
| |
@item return |
| |
\JP $B%j%9%H(B $B$^$?$O(B 0 |
| |
\EG list or 0 |
| |
@item plist vlist |
| |
\JP $B%j%9%H(B |
| |
\EG list |
| |
@item p |
| |
\JP $BAG?t$^$?$O(B 0 |
| |
\EG prime or 0 |
| |
@item order |
| |
\JP $B?t(B, $B%j%9%H$^$?$O9TNs(B |
| |
\EG number, list or matrix |
| |
@item check |
| |
\JP 0 $B$^$?$O(B 1 |
| |
\EG 0 or 1 |
| |
@end table |
| |
|
| |
@itemize @bullet |
| |
\BJP |
| |
@item |
| |
$B%0%l%V%J4pDl(B($B8uJd(B)$B$NAj8_4JLs$r9T$&(B. |
| |
@item |
| |
@code{nd_weyl_gr_postproc} $B$O(B Weyl $BBe?tMQ$G$"$k(B. |
| |
@item |
| |
@var{check=1} $B$N>l9g(B, @var{plist} $B$,(B, @var{vlist}, @var{p}, @var{order} $B$G;XDj$5$l$kB?9`<04D(B, $B9`=g=x$G%0%l%V%J!<4pDl$K$J$C$F$$$k$+(B |
| |
$B$N%A%'%C%/$b9T$&(B. |
| |
@item |
| |
$B@F<!2=$7$F7W;;$7$?%0%l%V%J!<4pDl$rHs@F<!2=$7$?$b$N$rAj8_4JLs$r9T$&(B, CRT $B$G7W;;$7$?%0%l%V%J!<4pDl8uJd$N%A%'%C%/$r9T$&$J$I$N>l9g$KMQ$$$k(B. |
| |
\E |
| |
\BEG |
| |
@item |
| |
Perform the inter-reduction for a Groebner basis (candidate). |
| |
@item |
| |
@code{nd_weyl_gr_postproc} is for Weyl algebra. |
| |
@item |
| |
If @var{check=1} then the check whether @var{plist} is a Groebner basis with respect to a term order in a polynomial ring |
| |
or Weyl algebra specified by @var{vlist}, @var{p} and @var{order}. |
| |
@item |
| |
This function is used for inter-reduction of a non-reduced Groebner basis that is obtained by dehomogenizing a Groebner basis |
| |
computed via homogenization, or Groebner basis check of a Groebner basis candidate computed by CRT. |
| |
\E |
| |
@end itemize |
| |
|
| |
@example |
| |
afo |
| |
@end example |
| |
|
| \JP @node dp_gr_flags dp_gr_print,,, $B%0%l%V%J4pDl$K4X$9$kH!?t(B |
\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 2378 uses the value as a flag for showing intermediate info |
|
| Line 2761 uses the value as a flag for showing intermediate info |
|
| @item |
@item |
| $B%H%C%W%l%Y%kH!?t0J30$NH!?t$rD>@\8F$S=P$9>l9g$K$O(B, $B$3$NH!?t$K$h$j(B |
$B%H%C%W%l%Y%kH!?t0J30$NH!?t$rD>@\8F$S=P$9>l9g$K$O(B, $B$3$NH!?t$K$h$j(B |
| $BJQ?t=g=x7?$r@5$7$/@_Dj$7$J$1$l$P$J$i$J$$(B. |
$BJQ?t=g=x7?$r@5$7$/@_Dj$7$J$1$l$P$J$i$J$$(B. |
| |
|
| |
@item |
| |
$B0z?t$,%j%9%H$N>l9g(B, $B<+M32C72$K$*$1$k9`=g=x7?$r@_Dj$9$k(B. $B0z?t$,(B@code{[0,Ord]} $B$N>l9g(B, |
| |
$BB?9`<04D>e$G(B @code{Ord} $B$G;XDj$5$l$k9`=g=x$K4p$E$/(B TOP $B=g=x(B, $B0z?t$,(B @code{[1,Ord]} $B$N>l9g(B |
| |
OPT $B=g=x$r@_Dj$9$k(B. |
| |
|
| \E |
\E |
| \BEG |
\BEG |
| @item |
@item |
| Line 2405 that such polynomials were generated under the same or |
|
| Line 2794 that such polynomials were generated under the same or |
|
| @item |
@item |
| Type of term ordering must be correctly set by this function |
Type of term ordering must be correctly set by this function |
| when functions other than top level functions are called directly. |
when functions other than top level functions are called directly. |
| |
|
| |
@item |
| |
If the argument is a list, then an ordering type in a free module is set. |
| |
If the argument is @code{[0,Ord]} then a TOP ordering based on the ordering type specified |
| |
by @code{Ord} is set. |
| |
If the argument is @code{[1,Ord]} then a POT ordering is set. |
| \E |
\E |
| @end itemize |
@end itemize |
| |
|
| Line 2424 when functions other than top level functions are call |
|
| Line 2819 when functions other than top level functions are call |
|
| \EG @fref{Setting term orderings} |
\EG @fref{Setting term orderings} |
| @end table |
@end table |
| |
|
| |
\JP @node dp_set_weight dp_set_top_weight dp_weyl_set_weight,,, $B%0%l%V%J4pDl$K4X$9$kH!?t(B |
| |
\EG @node dp_set_weight dp_set_top_weight dp_weyl_set_weight,,, Functions for Groebner basis computation |
| |
@subsection @code{dp_set_weight}, @code{dp_set_top_weight}, @code{dp_weyl_set_weight} |
| |
@findex dp_set_weight |
| |
@findex dp_set_top_weight |
| |
@findex dp_weyl_set_weight |
| |
|
| |
@table @t |
| |
@item dp_set_weight([@var{weight}]) |
| |
\JP :: sugar weight $B$N@_Dj(B, $B;2>H(B |
| |
\EG :: Set and show the sugar weight. |
| |
@item dp_set_top_weight([@var{weight}]) |
| |
\JP :: top weight $B$N@_Dj(B, $B;2>H(B |
| |
\EG :: Set and show the top weight. |
| |
@item dp_weyl_set_weight([@var{weight}]) |
| |
\JP :: weyl weight $B$N@_Dj(B, $B;2>H(B |
| |
\EG :: Set and show the weyl weight. |
| |
@end table |
| |
|
| |
@table @var |
| |
@item return |
| |
\JP $B%Y%/%H%k(B |
| |
\EG a vector |
| |
@item weight |
| |
\JP $B@0?t$N%j%9%H$^$?$O%Y%/%H%k(B |
| |
\EG a list or vector of integers |
| |
@end table |
| |
|
| |
@itemize @bullet |
| |
\BJP |
| |
@item |
| |
@code{dp_set_weight} $B$O(B sugar weight $B$r(B @var{weight} $B$K@_Dj$9$k(B. $B0z?t$,$J$$;~(B, |
| |
$B8=:_@_Dj$5$l$F$$$k(B sugar weight $B$rJV$9(B. sugar weight $B$O@5@0?t$r@.J,$H$9$k%Y%/%H%k$G(B, |
| |
$B3FJQ?t$N=E$_$rI=$9(B. $B<!?t$D$-=g=x$K$*$$$F(B, $BC19`<0$N<!?t$r7W;;$9$k:]$KMQ$$$i$l$k(B. |
| |
$B@F<!2=JQ?tMQ$K(B, $BKvHx$K(B 1 $B$rIU$12C$($F$*$/$H0BA4$G$"$k(B. |
| |
@item |
| |
@code{dp_set_top_weight} $B$O(B top weight $B$r(B @var{weight} $B$K@_Dj$9$k(B. $B0z?t$,$J$$;~(B, |
| |
$B8=:_@_Dj$5$l$F$$$k(B top weight $B$rJV$9(B. top weight $B$,@_Dj$5$l$F$$$k$H$-(B, |
| |
$B$^$:(B top weight $B$K$h$kC19`<0Hf3S$r@h$K9T$&(B. tie breaker $B$H$7$F8=:_@_Dj$5$l$F$$$k(B |
| |
$B9`=g=x$,MQ$$$i$l$k$,(B, $B$3$NHf3S$K$O(B top weight $B$OMQ$$$i$l$J$$(B. |
| |
|
| |
@item |
| |
@code{dp_weyl_set_weight} $B$O(B weyl weight $B$r(B @var{weight} $B$K@_Dj$9$k(B. $B0z?t$,$J$$;~(B, |
| |
$B8=:_@_Dj$5$l$F$$$k(B weyl weight $B$rJV$9(B. weyl weight w $B$r@_Dj$9$k$H(B, |
| |
$B9`=g=x7?(B 11 $B$G$N7W;;$K$*$$$F(B, (-w,w) $B$r(B top weight, tie breaker $B$r(B graded reverse lex |
| |
$B$H$7$?9`=g=x$,@_Dj$5$l$k(B. |
| |
\E |
| |
\BEG |
| |
@item |
| |
@code{dp_set_weight} sets the sugar weight=@var{weight}. It returns the current sugar weight. |
| |
A sugar weight is a vector with positive integer components and it represents the weights of variables. |
| |
It is used for computing the weight of a monomial in a graded ordering. |
| |
It is recommended to append a component 1 at the end of the weight vector for a homogenizing variable. |
| |
@item |
| |
@code{dp_set_top_weight} sets the top weight=@var{weight}. It returns the current top weight. |
| |
It a top weight is set, the weights of monomials under the top weight are firstly compared. |
| |
If the the weights are equal then the current term ordering is applied as a tie breaker, but |
| |
the top weight is not used in the tie breaker. |
| |
|
| |
@item |
| |
@code{dp_weyl_set_weight} sets the weyl weigh=@var{weight}. It returns the current weyl weight. |
| |
If a weyl weight w is set, in the comparsion by the term order type 11, a term order with |
| |
the top weight=(-w,w) and the tie breaker=graded reverse lex is applied. |
| |
\E |
| |
@end itemize |
| |
|
| |
@table @t |
| |
\JP @item $B;2>H(B |
| |
\EG @item References |
| |
@fref{Weight} |
| |
@end table |
| |
|
| |
|
| \JP @node dp_ptod,,, $B%0%l%V%J4pDl$K4X$9$kH!?t(B |
\JP @node dp_ptod,,, $B%0%l%V%J4pDl$K4X$9$kH!?t(B |
| \EG @node dp_ptod,,, Functions for Groebner basis computation |
\EG @node dp_ptod,,, Functions for Groebner basis computation |
| @subsection @code{dp_ptod} |
@subsection @code{dp_ptod} |
| Line 2483 the coefficient field. |
|
| Line 2951 the coefficient field. |
|
| @fref{dp_ord}. |
@fref{dp_ord}. |
| @end table |
@end table |
| |
|
| |
\JP @node dpm_dptodpm,,, $B%0%l%V%J4pDl$K4X$9$kH!?t(B |
| |
\EG @node dpm_dptodpm,,, Functions for Groebner basis computation |
| |
@subsection @code{dpm_dptodpm} |
| |
@findex dpm_dptodpm |
| |
|
| |
@table @t |
| |
@item dpm_dptodpm(@var{dpoly},@var{pos}) |
| |
\JP :: $BJ,;6I=8=B?9`<0$r2C72B?9`<0$KJQ49$9$k(B. |
| |
\EG :: Converts a distributed polynomial into a module polynomial. |
| |
@end table |
| |
|
| |
@table @var |
| |
@item return |
| |
\JP $B2C72B?9`<0(B |
| |
\EG module polynomial |
| |
@item dpoly |
| |
\JP $BJ,;6I=8=B?9`<0(B |
| |
\EG distributed polynomial |
| |
@item pos |
| |
\JP $B@5@0?t(B |
| |
\EG positive integer |
| |
@end table |
| |
|
| |
@itemize @bullet |
| |
\BJP |
| |
@item |
| |
$BJ,;6I=8=B?9`<0$r2C72B?9`<0$KJQ49$9$k(B. |
| |
@item |
| |
$B=PNO$O2C72B?9`<0(B @code{dpoly e_pos} $B$G$"$k(B. |
| |
\E |
| |
\BEG |
| |
@item |
| |
This function converts a distributed polynomial into a module polynomial. |
| |
@item |
| |
The output is @code{dpoly e_pos}. |
| |
\E |
| |
@end itemize |
| |
|
| |
@example |
| |
[50] dp_ord([0,0])$ |
| |
[51] D=dp_ptod((x+y+z)^2,[x,y,z]); |
| |
(1)*<<2,0,0>>+(2)*<<1,1,0>>+(1)*<<0,2,0>>+(2)*<<1,0,1>>+(2)*<<0,1,1>> |
| |
+(1)*<<0,0,2>> |
| |
[52] dp_dptodpm(D,2); |
| |
(1)*<<2,0,0:2>>+(2)*<<1,1,0:2>>+(1)*<<0,2,0:2>>+(2)*<<1,0,1:2>> |
| |
+(2)*<<0,1,1:2>>+(1)*<<0,0,2:2>> |
| |
@end example |
| |
|
| |
@table @t |
| |
\JP @item $B;2>H(B |
| |
\EG @item References |
| |
@fref{dp_ptod}, |
| |
@fref{dp_ord}. |
| |
@end table |
| |
|
| |
\JP @node dpm_ltod,,, $B%0%l%V%J4pDl$K4X$9$kH!?t(B |
| |
\EG @node dpm_ltod,,, Functions for Groebner basis computation |
| |
@subsection @code{dpm_ltod} |
| |
@findex dpm_ltod |
| |
|
| |
@table @t |
| |
@item dpm_dptodpm(@var{plist},@var{vlist}) |
| |
\JP :: $BB?9`<0%j%9%H$r2C72B?9`<0$KJQ49$9$k(B. |
| |
\EG :: Converts a list of polynomials into a module polynomial. |
| |
@end table |
| |
|
| |
@table @var |
| |
@item return |
| |
\JP $B2C72B?9`<0(B |
| |
\EG module polynomial |
| |
@item plist |
| |
\JP $BB?9`<0%j%9%H(B |
| |
\EG list of polynomials |
| |
@item vlist |
| |
\JP $BJQ?t%j%9%H(B |
| |
\EG list of variables |
| |
@end table |
| |
|
| |
@itemize @bullet |
| |
\BJP |
| |
@item |
| |
$BB?9`<0%j%9%H$r2C72B?9`<0$KJQ49$9$k(B. |
| |
@item |
| |
@code{[p1,...,pm]} $B$O(B @code{p1 e1+...+pm em} $B$KJQ49$5$l$k(B. |
| |
\E |
| |
\BEG |
| |
@item |
| |
This function converts a list of polynomials into a module polynomial. |
| |
@item |
| |
@code{[p1,...,pm]} is converted into @code{p1 e1+...+pm em}. |
| |
\E |
| |
@end itemize |
| |
|
| |
@example |
| |
[2126] dp_ord([0,0])$ |
| |
[2127] dpm_ltod([x^2+y^2,x,y-z],[x,y,z]); |
| |
(1)*<<2,0,0:1>>+(1)*<<0,2,0:1>>+(1)*<<1,0,0:2>>+(1)*<<0,1,0:3>> |
| |
+(-1)*<<0,0,1:3>> |
| |
@end example |
| |
|
| |
@table @t |
| |
\JP @item $B;2>H(B |
| |
\EG @item References |
| |
@fref{dpm_dtol}, |
| |
@fref{dp_ord}. |
| |
@end table |
| |
|
| |
\JP @node dpm_dtol,,, $B%0%l%V%J4pDl$K4X$9$kH!?t(B |
| |
\EG @node dpm_dtol,,, Functions for Groebner basis computation |
| |
@subsection @code{dpm_dtol} |
| |
@findex dpm_dtol |
| |
|
| |
@table @t |
| |
@item dpm_dptodpm(@var{poly},@var{vlist}) |
| |
\JP :: $B2C72B?9`<0$rB?9`<0%j%9%H$KJQ49$9$k(B. |
| |
\EG :: Converts a module polynomial into a list of polynomials. |
| |
@end table |
| |
|
| |
@table @var |
| |
@item return |
| |
\JP $BB?9`<0%j%9%H(B |
| |
\EG list of polynomials |
| |
@item poly |
| |
\JP $B2C72B?9`<0(B |
| |
\EG module polynomial |
| |
@item vlist |
| |
\JP $BJQ?t%j%9%H(B |
| |
\EG list of variables |
| |
@end table |
| |
|
| |
@itemize @bullet |
| |
\BJP |
| |
@item |
| |
$B2C72B?9`<0$rB?9`<0%j%9%H$KJQ49$9$k(B. |
| |
@item |
| |
@code{p1 e1+...+pm em} $B$O(B @code{[p1,...,pm]} $B$KJQ49$5$l$k(B. |
| |
@item |
| |
$B=PNO%j%9%H$ND9$5$O(B, @code{poly} $B$K4^$^$l$kI8=`4pDl$N:GBg%$%s%G%C%/%9$H$J$k(B. |
| |
\E |
| |
\BEG |
| |
@item |
| |
This function converts a module polynomial into a list of polynomials. |
| |
@item |
| |
@code{p1 e1+...+pm em} is converted into @code{[p1,...,pm]}. |
| |
@item |
| |
The length of the output list is equal to the largest index among those of the standard bases |
| |
containd in @code{poly}. |
| |
\E |
| |
@end itemize |
| |
|
| |
@example |
| |
[2126] dp_ord([0,0])$ |
| |
[2127] D=(1)*<<2,0,0:1>>+(1)*<<0,2,0:1>>+(1)*<<1,0,0:2>>+(1)*<<0,1,0:3>> |
| |
+(-1)*<<0,0,1:3>>$ |
| |
[2128] dpm_dtol(D,[x,y,z]); |
| |
[x^2+y^2,x,y-z] |
| |
@end example |
| |
|
| |
@table @t |
| |
\JP @item $B;2>H(B |
| |
\EG @item References |
| |
@fref{dpm_ltod}, |
| |
@fref{dp_ord}. |
| |
@end table |
| |
|
| \JP @node dp_dtop,,, $B%0%l%V%J4pDl$K4X$9$kH!?t(B |
\JP @node dp_dtop,,, $B%0%l%V%J4pDl$K4X$9$kH!?t(B |
| \EG @node dp_dtop,,, Functions for Groebner basis computation |
\EG @node dp_dtop,,, Functions for Groebner basis computation |
| @subsection @code{dp_dtop} |
@subsection @code{dp_dtop} |
| Line 2606 converting the coefficients into elements of a finite |
|
| Line 3239 converting the coefficients into elements of a finite |
|
| @table @t |
@table @t |
| \JP @item $B;2>H(B |
\JP @item $B;2>H(B |
| \EG @item References |
\EG @item References |
| @fref{dp_nf dp_nf_mod dp_true_nf dp_true_nf_mod}, |
@fref{dp_nf dp_nf_mod dp_true_nf dp_true_nf_mod dp_weyl_nf dp_weyl_nf_mod}, |
| @fref{subst psubst}, |
@fref{subst psubst}, |
| @fref{setmod}. |
@fref{setmod}. |
| @end table |
@end table |
| Line 2697 These are used internally in @code{hgr()} etc. |
|
| Line 3330 These are used internally in @code{hgr()} etc. |
|
| into an integral distributed polynomial such that GCD of all its coefficients |
into an integral distributed polynomial such that GCD of all its coefficients |
| is 1. |
is 1. |
| \E |
\E |
| @itemx dp_prim(@var{dpoly}) |
@item dp_prim(@var{dpoly}) |
| \JP :: $BM-M}<0G\$7$F78?t$r@0?t78?tB?9`<078?t$+$D78?t$NB?9`<0(B GCD $B$r(B 1 $B$K$9$k(B. |
\JP :: $BM-M}<0G\$7$F78?t$r@0?t78?tB?9`<078?t$+$D78?t$NB?9`<0(B GCD $B$r(B 1 $B$K$9$k(B. |
| \BEG |
\BEG |
| :: Converts a distributed polynomial @var{poly} with rational function |
:: Converts a distributed polynomial @var{poly} with rational function |
| Line 2750 polynomial contents included in the coefficients are n |
|
| Line 3383 polynomial contents included in the coefficients are n |
|
| @fref{ptozp}. |
@fref{ptozp}. |
| @end table |
@end table |
| |
|
| \JP @node dp_nf dp_nf_mod dp_true_nf dp_true_nf_mod,,, $B%0%l%V%J4pDl$K4X$9$kH!?t(B |
\JP @node dp_nf dp_nf_mod dp_true_nf dp_true_nf_mod dp_weyl_nf dp_weyl_nf_mod,,, $B%0%l%V%J4pDl$K4X$9$kH!?t(B |
| \EG @node dp_nf dp_nf_mod dp_true_nf dp_true_nf_mod,,, Functions for Groebner basis computation |
\EG @node dp_nf dp_nf_mod dp_true_nf dp_true_nf_mod dp_weyl_nf dp_weyl_nf_mod,,, Functions for Groebner basis computation |
| @subsection @code{dp_nf}, @code{dp_nf_mod}, @code{dp_true_nf}, @code{dp_true_nf_mod} |
@subsection @code{dp_nf}, @code{dp_nf_mod}, @code{dp_true_nf}, @code{dp_true_nf_mod} |
| @findex dp_nf |
@findex dp_nf |
| @findex dp_true_nf |
@findex dp_true_nf |
| @findex dp_nf_mod |
@findex dp_nf_mod |
| @findex dp_true_nf_mod |
@findex dp_true_nf_mod |
| |
@findex dp_weyl_nf |
| |
@findex dp_weyl_nf_mod |
| |
|
| @table @t |
@table @t |
| @item dp_nf(@var{indexlist},@var{dpoly},@var{dpolyarray},@var{fullreduce}) |
@item dp_nf(@var{indexlist},@var{dpoly},@var{dpolyarray},@var{fullreduce}) |
| |
@item dp_weyl_nf(@var{indexlist},@var{dpoly},@var{dpolyarray},@var{fullreduce}) |
| @item dp_nf_mod(@var{indexlist},@var{dpoly},@var{dpolyarray},@var{fullreduce},@var{mod}) |
@item dp_nf_mod(@var{indexlist},@var{dpoly},@var{dpolyarray},@var{fullreduce},@var{mod}) |
| |
@item dp_weyl_nf_mod(@var{indexlist},@var{dpoly},@var{dpolyarray},@var{fullreduce},@var{mod}) |
| \JP :: $BJ,;6I=8=B?9`<0$N@55,7A$r5a$a$k(B. ($B7k2L$ODj?tG\$5$l$F$$$k2DG=@-$"$j(B) |
\JP :: $BJ,;6I=8=B?9`<0$N@55,7A$r5a$a$k(B. ($B7k2L$ODj?tG\$5$l$F$$$k2DG=@-$"$j(B) |
| |
|
| \BEG |
\BEG |
| Line 2802 is returned in such a list as @code{[numerator, denomi |
|
| Line 3439 is returned in such a list as @code{[numerator, denomi |
|
| @item |
@item |
| $BJ,;6I=8=B?9`<0(B @var{dpoly} $B$N@55,7A$r5a$a$k(B. |
$BJ,;6I=8=B?9`<0(B @var{dpoly} $B$N@55,7A$r5a$a$k(B. |
| @item |
@item |
| |
$BL>A0$K(B weyl $B$r4^$`4X?t$O%o%$%kBe?t$K$*$1$k@55,7A7W;;$r9T$&(B. $B0J2<$N@bL@$O(B weyl $B$r4^$`$b$N$KBP$7$F$bF1MM$K@.N)$9$k(B. |
| |
@item |
| @code{dp_nf_mod()}, @code{dp_true_nf_mod()} $B$NF~NO$O(B, @code{dp_mod()} $B$J$I(B |
@code{dp_nf_mod()}, @code{dp_true_nf_mod()} $B$NF~NO$O(B, @code{dp_mod()} $B$J$I(B |
| $B$K$h$j(B, $BM-8BBN>e$NJ,;6I=8=B?9`<0$K$J$C$F$$$J$1$l$P$J$i$J$$(B. |
$B$K$h$j(B, $BM-8BBN>e$NJ,;6I=8=B?9`<0$K$J$C$F$$$J$1$l$P$J$i$J$$(B. |
| @item |
@item |
| Line 2834 is returned in such a list as @code{[numerator, denomi |
|
| Line 3473 is returned in such a list as @code{[numerator, denomi |
|
| @item |
@item |
| Computes the normal form of a distributed polynomial. |
Computes the normal form of a distributed polynomial. |
| @item |
@item |
| |
Functions whose name contain @code{weyl} compute normal forms in Weyl algebra. The description below also applies to |
| |
the functions for Weyl algebra. |
| |
@item |
| @code{dp_nf_mod()} and @code{dp_true_nf_mod()} require |
@code{dp_nf_mod()} and @code{dp_true_nf_mod()} require |
| distributed polynomials with coefficients in a finite field as arguments. |
distributed polynomials with coefficients in a finite field as arguments. |
| @item |
@item |
| Line 2904 u4^2+(6*u3+2*u2+6*u1-2)*u4+9*u3^2+(6*u2+18*u1-6)*u3+u2 |
|
| Line 3546 u4^2+(6*u3+2*u2+6*u1-2)*u4+9*u3^2+(6*u2+18*u1-6)*u3+u2 |
|
| @fref{p_nf p_nf_mod p_true_nf p_true_nf_mod}. |
@fref{p_nf p_nf_mod p_true_nf p_true_nf_mod}. |
| @end table |
@end table |
| |
|
| |
\JP @node dpm_nf dpm_nf_and_quotient,,, $B%0%l%V%J4pDl$K4X$9$kH!?t(B |
| |
\EG @node dpm_nf dpm_nf_and_quotient,,, Functions for Groebner basis computation |
| |
@subsection @code{dpm_nf}, @code{dpm_nf_and_quotient} |
| |
@findex dpm_nf |
| |
@findex dpm_nf_and_quotient |
| |
|
| |
@table @t |
| |
@item dpm_nf([@var{indexlist},]@var{dpoly},@var{dpolyarray},@var{fullreduce}) |
| |
\JP :: $B2C72B?9`<0$N@55,7A$r5a$a$k(B. ($B7k2L$ODj?tG\$5$l$F$$$k2DG=@-$"$j(B) |
| |
|
| |
\BEG |
| |
:: Computes the normal form of a module polynomial. |
| |
(The result may be multiplied by a constant in the ground field.) |
| |
\E |
| |
@item dpm_nf_and_quotient([@var{indexlist},]@var{dpoly},@var{dpolyarray}) |
| |
\JP :: $B2C72B?9`<0$N@55,7A$H>&$r5a$a$k(B. |
| |
\BEG |
| |
:: Computes the normal form of a module polynomial and the quotient. |
| |
\E |
| |
@end table |
| |
|
| |
@table @var |
| |
@item return |
| |
\JP @code{dpm_nf()} : $B2C72B?9`<0(B, @code{dpm_nf_and_quotient()} : $B%j%9%H(B |
| |
\EG @code{dpm_nf()} : module polynomial, @code{dpm_nf_and_quotient()} : list |
| |
@item indexlist |
| |
\JP $B%j%9%H(B |
| |
\EG list |
| |
@item dpoly |
| |
\JP $B2C72B?9`<0(B |
| |
\EG module polynomial |
| |
@item dpolyarray |
| |
\JP $BG[Ns(B |
| |
\EG array of module polynomial |
| |
@end table |
| |
|
| |
@itemize @bullet |
| |
\BJP |
| |
@item |
| |
$B2C72B?9`<0(B @var{dpoly} $B$N@55,7A$r5a$a$k(B. |
| |
@item |
| |
$B7k2L$KM-M}?t(B, $BM-M}<0$,4^$^$l$k$N$rHr$1$k$?$a(B, @code{dpm_nf()} $B$O(B |
| |
$B??$NCM$NDj?tG\$NCM$rJV$9(B. |
| |
@item |
| |
@var{dpolyarray} $B$O2C72B?9`<0$rMWAG$H$9$k%Y%/%H%k(B, |
| |
@var{indexlist} $B$O@55,2=7W;;$KMQ$$$k(B @var{dpolyarray} $B$NMWAG$N%$%s%G%C%/%9(B |
| |
@item |
| |
@var{indexlist} $B$,M?$($i$l$F$$$k>l9g(B, @var{dpolyarray} $B$NCf$G(B, @var{indexlist} $B$G;XDj$5$l$?$b$N$N$_$,(B, $BA0$NJ}$+$iM%@hE*$K;H$o$l$k(B. |
| |
@var{indexlist} $B$,M?$($i$l$F$$$J$$>l9g$K$O(B, @var{dpolyarray} $B$NCf$NA4$F$NB?9`<0$,A0$NJ}$+$iM%@hE*$K;H$o$l$k(B. |
| |
@item |
| |
@code{dpm_nf_and_quotient()} $B$O(B, |
| |
@code{[@var{nm},@var{dn},@var{quo}]} $B$J$k7A$N%j%9%H$rJV$9(B. |
| |
$B$?$@$7(B, @var{nm} $B$O78?t$KJ,?t$r4^$^$J$$2C72B?9`<0(B, @var{dn} $B$O(B |
| |
$B?t$^$?$OB?9`<0$G(B @var{nm}/@var{dn} $B$,??$NCM$H$J$k(B. |
| |
@var{quo} $B$O=|;;$N>&$rI=$9G[Ns$G(B, @var{dn}@var{dpoly}=@var{nm}+@var{quo[0]dpolyarray[0]+...} $B$,@.$jN)$D(B. |
| |
$B$N%j%9%H(B. |
| |
@item |
| |
@var{fullreduce} $B$,(B 0 $B$G$J$$$H$-A4$F$N9`$KBP$7$F4JLs$r9T$&(B. @var{fullreduce} |
| |
$B$,(B 0 $B$N$H$-F,9`$N$_$KBP$7$F4JLs$r9T$&(B. |
| |
\E |
| |
\BEG |
| |
@item |
| |
Computes the normal form of a module polynomial. |
| |
@item |
| |
The result of @code{dpm_nf()} may be multiplied by a constant in the |
| |
ground field in order to make the result integral. |
| |
@item |
| |
@var{dpolyarray} is a vector whose components are module polynomials |
| |
and @var{indexlist} is a list of indices which is used for the normal form |
| |
computation. |
| |
@item |
| |
If @var{indexlist} is given, only the polynomials in @var{dpolyarray} specified in @var{indexlist} |
| |
is used in the division. An index placed at the preceding position has priority to be selected. |
| |
If @var{indexlist} is not given, all the polynomials in @var{dpolyarray} are used. |
| |
@item |
| |
@code{dpm_nf_and_quotient()} returns |
| |
such a list as @code{[@var{nm},@var{dn},@var{quo}]}. |
| |
Here @var{nm} is a module polynomial whose coefficients are integral |
| |
in the ground field, @var{dn} is an integral element in the ground |
| |
field and @var{nm}/@var{dn} is the true normal form. |
| |
@var{quo} is an array containing the quotients of the division satisfying |
| |
@var{dn}@var{dpoly}=@var{nm}+@var{quo[0]dpolyarray[0]+...}. |
| |
@item |
| |
When argument @var{fullreduce} has non-zero value, |
| |
all terms are reduced. When it has value 0, |
| |
only the head term is reduced. |
| |
\E |
| |
@end itemize |
| |
|
| |
@example |
| |
[2126] dp_ord([1,0])$ |
| |
[2127] S=ltov([(1)*<<0,0,2,0:1>>+(1)*<<0,0,1,1:1>>+(1)*<<0,0,0,2:1>> |
| |
+(-1)*<<3,0,0,0:2>>+(-1)*<<0,0,2,1:2>>+(-1)*<<0,0,1,2:2>> |
| |
+(1)*<<3,0,1,0:3>>+(1)*<<3,0,0,1:3>>+(1)*<<0,0,2,2:3>>, |
| |
(-1)*<<0,1,0,0:1>>+(-1)*<<0,0,1,0:1>>+(-1)*<<0,0,0,1:1>> |
| |
+(-1)*<<3,0,0,0:3>>+(1)*<<0,1,1,1:3>>,(1)*<<0,1,0,0:2>> |
| |
+(1)*<<0,0,1,0:2>>+(1)*<<0,0,0,1:2>>+(-1)*<<0,1,1,0:3>> |
| |
+(-1)*<<0,1,0,1:3>>+(-1)*<<0,0,1,1:3>>])$ |
| |
[2128] U=dpm_sp(S[0],S[1]); |
| |
(1)*<<0,0,3,0:1>>+(-1)*<<0,1,1,1:1>>+(1)*<<0,0,2,1:1>> |
| |
+(-1)*<<0,1,0,2:1>>+(1)*<<3,1,0,0:2>>+(1)*<<0,1,2,1:2>> |
| |
+(1)*<<0,1,1,2:2>>+(-1)*<<3,1,1,0:3>>+(1)*<<3,0,2,0:3>> |
| |
+(-1)*<<3,1,0,1:3>>+(-1)*<<0,1,3,1:3>>+(-1)*<<0,1,2,2:3>> |
| |
[2129] dpm_nf(U,S,1); |
| |
0 |
| |
[2130] L=dpm_nf_and_quotient(U,S)$ |
| |
[2131] Q=L[2]$ |
| |
[2132] D=L[1]$ |
| |
[2133] D*U-(Q[1]*S[1]+Q[2]*S[2]); |
| |
0 |
| |
@end example |
| |
|
| |
@table @t |
| |
\JP @item $B;2>H(B |
| |
\EG @item References |
| |
@fref{dpm_sp}, |
| |
@fref{dp_ord}. |
| |
@end table |
| |
|
| |
|
| \JP @node dp_hm dp_ht dp_hc dp_rest,,, $B%0%l%V%J4pDl$K4X$9$kH!?t(B |
\JP @node dp_hm dp_ht dp_hc dp_rest,,, $B%0%l%V%J4pDl$K4X$9$kH!?t(B |
| \EG @node dp_hm dp_ht dp_hc dp_rest,,, Functions for Groebner basis computation |
\EG @node dp_hm dp_ht dp_hc dp_rest,,, Functions for Groebner basis computation |
| @subsection @code{dp_hm}, @code{dp_ht}, @code{dp_hc}, @code{dp_rest} |
@subsection @code{dp_hm}, @code{dp_ht}, @code{dp_hc}, @code{dp_rest} |
| Line 2978 The next equations hold for a distributed polynomial @ |
|
| Line 3740 The next equations hold for a distributed polynomial @ |
|
| +(-490)*<<0,0,0>> |
+(-490)*<<0,0,0>> |
| @end example |
@end example |
| |
|
| |
\JP @node dpm_hm dpm_ht dpm_hc dpm_hp dpm_rest,,, $B%0%l%V%J4pDl$K4X$9$kH!?t(B |
| |
\EG @node dpm_hm dpm_ht dpm_hc dpm_hp dpm_rest,,, Functions for Groebner basis computation |
| |
@subsection @code{dpm_hm}, @code{dpm_ht}, @code{dpm_hc}, @code{dpm_hp}, @code{dpm_rest} |
| |
@findex dpm_hm |
| |
@findex dpm_ht |
| |
@findex dpm_hc |
| |
@findex dpm_hp |
| |
@findex dpm_rest |
| |
|
| |
@table @t |
| |
@item dpm_hm(@var{dpoly}) |
| |
\JP :: $B2C72B?9`<0$NF,C19`<0$r<h$j=P$9(B. |
| |
\EG :: Gets the head monomial of a module polynomial. |
| |
@item dpm_ht(@var{dpoly}) |
| |
\JP :: $B2C72B?9`<0$NF,9`$r<h$j=P$9(B. |
| |
\EG :: Gets the head term of a module polynomial. |
| |
@item dpm_hc(@var{dpoly}) |
| |
\JP :: $B2C72B?9`<0$NF,78?t$r<h$j=P$9(B. |
| |
\EG :: Gets the head coefficient of a module polynomial. |
| |
@item dpm_hp(@var{dpoly}) |
| |
\JP :: $B2C72B?9`<0$NF,0LCV$r<h$j=P$9(B. |
| |
\EG :: Gets the head position of a module polynomial. |
| |
@item dpm_rest(@var{dpoly}) |
| |
\JP :: $B2C72B?9`<0$NF,C19`<0$r<h$j=|$$$?;D$j$rJV$9(B. |
| |
\EG :: Gets the remainder of a module polynomial where the head monomial is removed. |
| |
@end table |
| |
|
| |
@table @var |
| |
\BJP |
| |
@item return |
| |
@code{dp_hm()}, @code{dp_ht()}, @code{dp_rest()} : $B2C72B?9`<0(B, |
| |
@code{dp_hc()} : $B?t$^$?$OB?9`<0(B |
| |
@item dpoly |
| |
$B2C72B?9`<0(B |
| |
\E |
| |
\BEG |
| |
@item return |
| |
@code{dpm_hm()}, @code{dpm_ht()}, @code{dpm_rest()} : module polynomial |
| |
@code{dpm_hc()} : monomial |
| |
@item dpoly |
| |
distributed polynomial |
| |
\E |
| |
@end table |
| |
|
| |
@itemize @bullet |
| |
\BJP |
| |
@item |
| |
$B$3$l$i$O(B, $B2C72B?9`<0$N3FItJ,$r<h$j=P$9$?$a$NH!?t$G$"$k(B. |
| |
@item |
| |
@code{dpm_hc()} $B$O(B, @code{dpm_hm()} $B$N(B, $BI8=`4pDl$K4X$9$k78?t$G$"$kC19`<0$rJV$9(B. |
| |
$B%9%+%i!<78?t$r<h$j=P$9$K$O(B, $B$5$i$K(B @code{dp_hc()} $B$r<B9T$9$k(B. |
| |
@item |
| |
@code{dpm_hp()} $B$O(B, $BF,2C72C19`<0$K4^$^$l$kI8=`4pDl$N%$%s%G%C%/%9$rJV$9(B. |
| |
\E |
| |
\BEG |
| |
@item |
| |
These are used to get various parts of a module polynomial. |
| |
@item |
| |
@code{dpm_hc()} returns the monomial that is the coefficient of @code{dpm_hm()} with respect to the |
| |
standard base. |
| |
For getting its scalar coefficient apply @code{dp_hc()}. |
| |
@item |
| |
@code{dpm_hp()} returns the index of the standard base conteind in the head module monomial. |
| |
\E |
| |
@end itemize |
| |
|
| |
@example |
| |
[2126] dp_ord([1,0]); |
| |
[1,0] |
| |
[2127] F=2*<<1,2,0:2>>-3*<<1,0,2:3>>+<<2,1,0:2>>; |
| |
(1)*<<2,1,0:2>>+(2)*<<1,2,0:2>>+(-3)*<<1,0,2:3>> |
| |
[2128] M=dpm_hm(F); |
| |
(1)*<<2,1,0:2>> |
| |
[2129] C=dpm_hc(F); |
| |
(1)*<<2,1,0>> |
| |
[2130] R=dpm_rest(F); |
| |
(2)*<<1,2,0:2>>+(-3)*<<1,0,2:3>> |
| |
[2131] dpm_hp(F); |
| |
2 |
| |
@end example |
| |
|
| |
|
| \JP @node dp_td dp_sugar,,, $B%0%l%V%J4pDl$K4X$9$kH!?t(B |
\JP @node dp_td dp_sugar,,, $B%0%l%V%J4pDl$K4X$9$kH!?t(B |
| \EG @node dp_td dp_sugar,,, Functions for Groebner basis computation |
\EG @node dp_td dp_sugar,,, Functions for Groebner basis computation |
| @subsection @code{dp_td}, @code{dp_sugar} |
@subsection @code{dp_td}, @code{dp_sugar} |
| Line 3139 Used for finding candidate terms at reduction of polyn |
|
| Line 3983 Used for finding candidate terms at reduction of polyn |
|
| @fref{dp_red dp_red_mod}. |
@fref{dp_red dp_red_mod}. |
| @end table |
@end table |
| |
|
| |
\JP @node dpm_redble,,, $B%0%l%V%J4pDl$K4X$9$kH!?t(B |
| |
\EG @node dpm_redble,,, Functions for Groebner basis computation |
| |
@subsection @code{dpm_redble} |
| |
@findex dpm_redble |
| |
|
| |
@table @t |
| |
@item dpm_redble(@var{dpoly1},@var{dpoly2}) |
| |
\JP :: $BF,9`$I$&$7$,@0=|2DG=$+$I$&$+D4$Y$k(B. |
| |
\EG :: Checks whether one head term is divisible by the other head term. |
| |
@end table |
| |
|
| |
@table @var |
| |
@item return |
| |
\JP $B@0?t(B |
| |
\EG integer |
| |
@item dpoly1 dpoly2 |
| |
\JP $B2C72B?9`<0(B |
| |
\EG module polynomial |
| |
@end table |
| |
|
| |
@itemize @bullet |
| |
\BJP |
| |
@item |
| |
@var{dpoly1} $B$NF,9`$,(B @var{dpoly2} $B$NF,9`$G3d$j@Z$l$l$P(B 1, $B3d$j@Z$l$J$1$l$P(B |
| |
0 $B$rJV$9(B. |
| |
@item |
| |
$BB?9`<0$N4JLs$r9T$&:](B, $B$I$N9`$r4JLs$G$-$k$+$rC5$9$N$KMQ$$$k(B. |
| |
\E |
| |
\BEG |
| |
@item |
| |
Returns 1 if the head term of @var{dpoly2} divides the head term of |
| |
@var{dpoly1}; otherwise 0. |
| |
@item |
| |
Used for finding candidate terms at reduction of polynomials. |
| |
\E |
| |
@end itemize |
| |
|
| \JP @node dp_subd,,, $B%0%l%V%J4pDl$K4X$9$kH!?t(B |
\JP @node dp_subd,,, $B%0%l%V%J4pDl$K4X$9$kH!?t(B |
| \EG @node dp_subd,,, Functions for Groebner basis computation |
\EG @node dp_subd,,, Functions for Groebner basis computation |
| @subsection @code{dp_subd} |
@subsection @code{dp_subd} |
| Line 3506 make the result integral. |
|
| Line 4387 make the result integral. |
|
| \EG @item References |
\EG @item References |
| @fref{dp_mod dp_rat}. |
@fref{dp_mod dp_rat}. |
| @end table |
@end table |
| |
|
| |
\JP @node dpm_sp,,, $B%0%l%V%J4pDl$K4X$9$kH!?t(B |
| |
\EG @node dmp_sp,,, Functions for Groebner basis computation |
| |
@subsection @code{dpm_sp} |
| |
@findex dpm_sp |
| |
|
| |
@table @t |
| |
@item dpm_sp(@var{dpoly1},@var{dpoly2}[|coef=1]) |
| |
\JP :: S-$BB?9`<0$N7W;;(B |
| |
\EG :: Computation of an S-polynomial |
| |
@end table |
| |
|
| |
@table @var |
| |
@item return |
| |
\JP $B2C72B?9`<0$^$?$O%j%9%H(B |
| |
\EG module polynomial or list |
| |
@item dpoly1 dpoly2 |
| |
\JP $B2C72B?9`<0(B |
| |
\EG module polynomial |
| |
\JP $BJ,;6I=8=B?9`<0(B |
| |
@end table |
| |
|
| |
@itemize @bullet |
| |
\BJP |
| |
@item |
| |
@var{dpoly1}, @var{dpoly2} $B$N(B S-$BB?9`<0$r7W;;$9$k(B. |
| |
@item |
| |
$B%*%W%7%g%s(B @var{coef=1} $B$,;XDj$5$l$F$$$k>l9g(B, @code{[S,t1,t2]} $B$J$k%j%9%H$rJV$9(B. |
| |
$B$3$3$G(B, @code{t1}, @code{t2} $B$O(BS-$BB?9`<0$r:n$k:]$N78?tC19`<0$G(B @code{S=t1 dpoly1-t2 dpoly2} |
| |
$B$rK~$?$9(B. |
| |
\E |
| |
\BEG |
| |
@item |
| |
This function computes the S-polynomial of @var{dpoly1} and @var{dpoly2}. |
| |
@item |
| |
If an option @var{coef=1} is specified, it returns a list @code{[S,t1,t2]}, |
| |
where @code{S} is the S-polynmial and @code{t1}, @code{t2} are monomials satisfying @code{S=t1 dpoly1-t2 dpoly2}. |
| |
\E |
| |
@end itemize |
| |
|
| \JP @node p_nf p_nf_mod p_true_nf p_true_nf_mod,,, $B%0%l%V%J4pDl$K4X$9$kH!?t(B |
\JP @node p_nf p_nf_mod p_true_nf p_true_nf_mod,,, $B%0%l%V%J4pDl$K4X$9$kH!?t(B |
| \EG @node p_nf p_nf_mod p_true_nf p_true_nf_mod,,, Functions for Groebner basis computation |
\EG @node p_nf p_nf_mod p_true_nf p_true_nf_mod,,, Functions for Groebner basis computation |
| @subsection @code{p_nf}, @code{p_nf_mod}, @code{p_true_nf}, @code{p_true_nf_mod} |
@subsection @code{p_nf}, @code{p_nf_mod}, @code{p_true_nf}, @code{p_true_nf_mod} |
| Line 3615 refer to @code{dp_true_nf()} and @code{dp_true_nf_mod( |
|
| Line 4536 refer to @code{dp_true_nf()} and @code{dp_true_nf_mod( |
|
| @fref{dp_ptod}, |
@fref{dp_ptod}, |
| @fref{dp_dtop}, |
@fref{dp_dtop}, |
| @fref{dp_ord}, |
@fref{dp_ord}, |
| @fref{dp_nf dp_nf_mod dp_true_nf dp_true_nf_mod}. |
@fref{dp_nf dp_nf_mod dp_true_nf dp_true_nf_mod dp_weyl_nf dp_weyl_nf_mod}. |
| @end table |
@end table |
| |
|
| \JP @node p_terms,,, $B%0%l%V%J4pDl$K4X$9$kH!?t(B |
\JP @node p_terms,,, $B%0%l%V%J4pDl$K4X$9$kH!?t(B |
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