Singular

7.7.5.0. GBWeight

Usage:

GBWeight(I,u,v [,s,t,w]);
I ideal, u,v intvecs, s,t optional ints, w an optional intvec

Return:

ideal, Groebner basis of I w.r.t. the weights u and v

Assume:

The basering is the n-th Weyl algebra over a field of characteristic 0
and for all 1<=i<=n the identity var(i+n)*var(i)=var(i)*var(i+1)+1
holds, i.e. the sequence of variables is given by
x(1),...,x(n),D(1),...,D(n), where D(i) is the differential operator
belonging to x(i).

Purpose:

computes a Groebner basis with respect to given weights

Note:

The weights u and v are understood as weight vectors for x(i) and D(i),
respectively. According to (SST), one computes the homogenization of a
given ideal relative to (u,v), then one computes a Groebner basis and
returns the dehomogenization of the result.
If s<>0, std is used for Groebner basis computations,
otherwise, and by default, slimgb is used.
If t<>0, a matrix ordering is used for Groebner basis computations,
otherwise, and by default, a block ordering is used.
If w is given and consists of exactly 2*n strictly positive entries,
w is used for constructing the weighted homogenized Weyl algebra,
see Noro (2002). Otherwise, and by default, the homogenization weight
(1,...,1) is used.

Display:

If printlevel=1, progress debug messages will be printed,
if printlevel>=2, all the debug messages will be printed.

LIB "dmodapp.lib";
ring r = 0,(x,y,Dx,Dy),dp;
def D2 = Weyl();
setring D2;
ideal I = 3*x^2*Dy+2*y*Dx,2*x*Dx+3*y*Dy+6;
intvec u = -2,-3;
intvec v = -u;
GBWeight(I,u,v);
==> _[1]=2*x*Dx+3*y*Dy+6
==> _[2]=3*x^2*Dy+2*y*Dx
==> _[3]=9*x*y*Dy^2-4*y*Dx^2+15*x*Dy
==> _[4]=27*y^2*Dy^3+8*y*Dx^3+135*y*Dy^2+105*Dy
ideal J = std(I);
GBWeight(J,u,v); // same as above
==> _[1]=2*x*Dx+3*y*Dy+6
==> _[2]=3*x^2*Dy+2*y*Dx
==> _[3]=9*x*y*Dy^2-4*y*Dx^2+15*x*Dy
==> _[4]=27*y^2*Dy^3+8*y*Dx^3+135*y*Dy^2+105*Dy
u = 0,1;
GBWeight(I,u,v);
==> _[1]=2*x*Dx+3*y*Dy+6
==> _[2]=3*x^2*Dy+2*y*Dx
==> _[3]=-x^3*Dx+y^2*Dx-3*x^2