Singular

7.7.5.0. initialIdealW

Usage:

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

Return:

ideal, GB of initial ideal of the input ideal wrt the weights u and v

Assume:

The basering is the n-th Weyl algebra in 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 the initial ideal with respect to given weights.

Note:

u and v are understood as weight vectors for x(1..n) and D(1..n)
respectively.
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 as homogenization weight.
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;
initialIdealW(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);
initialIdealW(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;
initialIdealW(I,u,v);
==> _[1]=Dx
==> _[2]=Dy