Dintegration -- integration modules of a D-module

Synopsis

Usage:

N = Dintegration(M,w), NI = Dintegration(I,w), Ni = Dintegration(i,M,w),

NIi = Dintegration(i,I,w),
Inputs:
- M, a module, over the Weyl algebra D
- I, an ideal, which represents the module M = D/I
- w, a list, a weight vector
- i, an integer, nonnegative
Optional inputs:
- Strategy => ..., default value Schreyer,
Outputs:
- Ni, a module, the i-th derived integration module of M with respect to the weight vector w
- N, a hash table, contains entries of the form i=>Ni
- NIi, a module, the i-th derived integration module of D/I with respect to the weight vector w
- NI, a hash table, contains entries of the form i=>NIi

Description

The derived integration modules of a D-module M are the derived direct images in the category of D-modules. This routine computes integration for projection to coordinate subspaces, where the subspace is determined by the strictly positive entries of the weight vector w, e.g., {x_i = 0 : w_i > 0} if D = C<x_1,...,x_n,d_1,...,d_n>. The input weight vector should be a list of n numbers to induce the weight (-w,w) on D.

The algorithm used appears in the paper 'Algorithims for D-modules' by Oaku-Takayama(1999). The method is to take the Fourier transform of M, then compute the derived restriction, then inverse Fourier transform back.

i1 : R = QQ[x_1,x_2,D_1,D_2,WeylAlgebra=>{x_1=>D_1,x_2=>D_2}]

o1 = R

o1 : PolynomialRing, 2 differential variables

i2 : I = ideal(x_1, D_2-1) 

o2 = ideal (x , D  - 1)
             1   2

o2 : Ideal of R

i3 : Dintegration(I,{1,0})

o3 = HashTable{0 => cokernel | D_2-1 |}
               1 => 0

o3 : HashTable

Caveat

The module M should be specializable to the subspace. This is true for holonomic modules.The weight vector w should be a list of n numbers if M is a module over the nth Weyl algebra.

Ways to use `Dintegration` :

"Dintegration(Ideal,List)"
"Dintegration(Module,List)"
"Dintegration(ZZ,Ideal,List)"
"Dintegration(ZZ,Module,List)"

For the programmer

The object Dintegration is a method function with options.