Singular

7.7.5.0. integralIdeal

Usage:

integralIdeal(I,w,[,eng,m,G]);
I ideal, w intvec, eng and m optional ints, G optional ideal

Return:

ring (a Weyl algebra) containing an ideal 'intIdeal'

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).
Further, assume that I is holonomic and that w is n-dimensional with
non-negative entries.

Purpose:

computes the integral ideal of a holonomic ideal w.r.t. the subspace
defined by the variables corresponding to the non-zero entries of the
given intvec.

Note:

The output ring is the Weyl algebra defined by the zero entries of w.
It contains ideal 'intIdeal' being the integral ideal of I w.r.t. w.
If there are no zero entries, the input ring is returned.
If eng<>0, std is used for Groebner basis computations,
otherwise, and by default, slimgb is used.
The minimal integer root of the b-function of I wrt the weight (-w,w)
can be specified via the optional argument m.
The optional argument G is used for specifying a Groebner basis of I
wrt the weight (-w,w), that is, the initial form of G generates the
initial ideal of I wrt the weight (-w,w).
Further note, that the assumptions on m and G (if given) are not
checked.

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,b,Dx,Db),dp;
def D2 = Weyl();
setring D2;
ideal I = x*Dx+2*b*Db+2, x^2*Dx+b*Dx+2*x;
intvec w = 1,0;
def D1 = integralIdeal(I,w);
setring D1; D1;
==> //   characteristic : 0
==> //   number of vars : 2
==> //        block   1 : ordering C
==> //        block   2 : ordering dp
==> //                  : names    b Db
==> //   noncommutative relations:
==> //    Dbb=b*Db+1
intIdeal;
==> intIdeal[1]=2*b*Db+1