stafford -- computes 2 generators for a given ideal in the Weyl algebra

Synopsis

Usage:

stafford I
Inputs:
- I, an ideal, of the Weyl algebra
Outputs:
- an ideal, with 2 generators (that has the same extension as I in k(x)<dx>)

Description

A theorem of Stafford says that every ideal in the Weyl algebra can be generated by 2 elements. This routine is the implementation of the effective version of this theorem following the constructive proof in A.Leykin, `Algorithmic proofs of two theorems of Stafford', Journal of Symbolic Computation, 38(6):1535-1550, 2004.
The current implementation provides a weaker result: the 2 generators produced are guaranteed to generate only the extension of the ideal I in the Weyl algebra with rational-function coefficients.

i1 : makeWA(QQ[x_1..x_4])

o1 = QQ[x ..x , dx ..dx ]
         1   4    1    4

o1 : PolynomialRing, 4 differential variables

i2 : stafford ideal (dx_1,dx_2,dx_3,dx_4)

                  4         2         3
o2 = ideal (dx , x x dx  + x x dx  + x dx  + x dx  + dx )
              1   1 4  4    1 3  3    1  4    1  3     2

o2 : Ideal of QQ[x ..x , dx ..dx ]
                  1   4    1    4

Caveat

The input should be generated by at least 2 generators. The output and input ideals are not equal necessarily.

Ways to use `stafford` :

"stafford(Ideal)"

For the programmer

The object stafford is a method function.