normalForm -- computes the normal form within the rational Weyl algebra

Usage:

normalForm(P, Q)

normalForm(P, G)
Inputs:
- P, a ring element, in the Weyl algebra
- Q, a ring element, in the Weyl algebra
- G, a list, of elements in the Weyl algebra (typically a Gröbner basis of a $D_n$-ideal)
Outputs:
- normalForm, a ring element, normalForm, computed in rational Weyl algebra with respect to a list of elements in the Weyl algebra

Description

This method computes the normal form of an element $P$ in the Weyl algebra $D_n$ with respect to another element in the Weyl algebra, or a whole list of such elements. The reduction step is carried out over the rational Weyl algebra $R_n$.

i1 : D = makeWeylAlgebra(QQ[x,y], {1,1});

i2 : P = dx^2 ; Q = x*dx+1;

i4 : normalForm(P, Q)

      2
o4 = --
      2
     x

o4 : frac(QQ[x..y])[dx, dy]

We can compute the normal form with respect to the Gröbner basis of a $D$-ideal.

i5 : D = makeWeylAlgebra(QQ[x, y]);

i6 : I = ideal(x*dx^2-y*dy^2+2*dx-2*dy, x*dx+y*dy+1);

o6 : Ideal of D

i7 : G = first entries mingens gb I;

i8 : P = dx;

i9 : normalForm(P,G)

       y      1
o9 = - -*dy - -
       x      x

o9 : frac(QQ[x..y])[dx, dy]

References

See [SST, Theorem 1.1.7].

Caveat

Due to technical limitations, the output lives in the graded associative algebra of the rational Weyl algebra, which is a commutative ring over the base fraction field of $D_n$ where the partial differentials are adjoined as commuting variables.

Ways to use normalForm:

normalForm(RingElement,List)
normalForm(RingElement,RingElement)
normalForm(RingElement,GroebnerBasis) (missing documentation)

For the programmer

The object normalForm is a method function.

The source of this document is in ConnectionMatrices/docs.m2:105:0.