inw -- get the initial term or ideal with respect to a weight vector

Synopsis

Usage:

inF = inw(F, w)

inI = inw(I, w)

inM = inw(M, w)
Inputs:
- F, a ring element,
- I, an ideal,
- M, a matrix,
- w, a list, of weights
Outputs:
- inF, a ring element, the initial form of F with respect to the weight vector
- inI, an ideal, the initial ideal of I with respect to the weight vector
- inM, a matrix, with the columns generating the initial module of the image of M with respect to the weight vector

Description

This routine computes the initial ideal 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+v > 0, then the ideal lives in the associated graded ring which is a commutative ring. In the case where u+v = 0, then the ideal lives in the associated graded ring which is again the Weyl algebra. In the general case u+v >= 0 the associated graded ring is somewhere between. There are two strategies to compute the initial ideal. One is to homogenize to an ideal of the homogeneous Weyl algebra. The other is to homogenize with respect to the weight vector w. For more details, see [SST, Sections 1.1 and 1.2].

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 : inw(I, {1,3,3,-1})

                    2              2  2
o3 = ideal (x*dx, dx , 2y*dx*dy, 4y dy  + 2x*dx)

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

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

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

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

Caveat

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

Ways to use inw :

inw(Ideal,List)
inw(Matrix,List)
inw(RingElement,List)

For the programmer

The object inw is a method function.