gbw -- compute a Gröbner basis with respect to a weight vector

Synopsis

Usage:

gbI = gbw(I, w)

gbM = gbw(M, w)
Inputs:
- I, an ideal, in the Weyl algebra
- M, a matrix, with entries in the Weyl algebra
- w, a list, of weights
Optional inputs:
- Strategy => a thing, default value null, if set to homogenize the input is homogenized in the homogeneous Weyl algebra
Outputs:
- gbI, an ideal, with the generators forming a Gröbner basis of the ideal with respect to the weight vector
- gbM, a matrix, with the columns forming a Gröbner basis of the submodule generated by the columns of the matrix with respect to the weight vector

Description

This routine computes a Gröbner basis of a left ideal I of the Weyl algebra with respect to a weight vector w = (u,v) where u+v >= 0. In the case where u_i+v_i > 0 for all i, the ordinary Buchberger algorithm works for any term order refining the weight order. If there exists i so that u_i+v_i = 0 the Buchberger algorithm needs to be adapted to guarantee termination. There are two strategies for doing this. One is to homogenize to an ideal of the homogeneous Weyl algebra.

i1 : makeWA(QQ[x,y])

o1 = QQ[x..y, dx, dy]

o1 : PolynomialRing, 2 differential variable(s)

i2 : I = ideal (x*dx+2*y*dy-3, dx^2-dy)

                                2
o2 = ideal (x*dx + 2y*dy - 3, dx  - dy)

o2 : Ideal of QQ[x..y, dx, dy]

i3 : gbw(I, {-1,-3,1,3})

                                  2
o3 = ideal (x*dx + 2y*dy - 3, - dx  + dy)

o3 : Ideal of QQ[x..y, dx, dy]

The other is to homogenize with respect to the weight vector w. More details can be found in [SST, Sections 1.1 and 1.2].

i4 : gbw(I, {1,3,3,-1})

                                2                                2  2    2
o4 = ideal (x*dx + 2y*dy - 3, dx  - dy, 2y*dx*dy + x*dy - 2dx, 4y dy  - x dy
     ------------------------------------------------------------------------
     + 2x*dx - 2y*dy)

o4 : Ideal of QQ[x..y, dx, dy]

Caveat

The weight vector w = (u,v) must have u+v>=0.

Ways to use gbw:

gbw(Ideal,List)
gbw(Matrix,List)

For the programmer

The object gbw is a method function with options.