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Dmodules :: Dintegration

Dintegration -- integration modules of a D-module

Synopsis

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.

See also

Ways to use Dintegration :

For the programmer

The object Dintegration is a method function with options.