Singular

7.7.5.0. restrictionIdeal

Usage:

restrictionIdeal(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 'resIdeal'

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 restriction ideal of a holonomic ideal to 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 an ideal 'resIdeal' being the restriction ideal of I wrt 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,(a,x,b,Da,Dx,Db),dp;
def D3 = Weyl();
setring D3;
ideal I = a*Db-Dx+2*Da,
x*Db-Da,
x*Da+a*Da+b*Db+1,
x*Dx-2*x*Da-a*Da,
b*Db^2+Dx*Da-Da^2+Db,
a*Dx*Da+2*x*Da^2+a*Da^2+b*Dx*Db+Dx+2*Da;
intvec w = 1,0,0;
def D2 = restrictionIdeal(I,w);
setring D2; D2;
==> //   characteristic : 0
==> //   number of vars : 4
==> //        block   1 : ordering C
==> //        block   2 : ordering dp
==> //                  : names    x b Dx Db
==> //   noncommutative relations:
==> //    Dxx=x*Dx+1
==> //    Dbb=b*Db+1
resIdeal;
==> resIdeal[1]=2*x*Db-Dx
==> resIdeal[2]=x*Dx+2*b*Db+2
==> resIdeal[3]=4*b*Db^2+Dx^2+6*Db