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Dmodules :: AnnFs(RingElement)

AnnFs(RingElement) -- the annihilating ideal of f^s

Synopsis

Description

The annihilator ideal is needed to compute a D-module representation of the localization of k[x1,...,xn] at f.
i1 : R = QQ[x_1..x_4, z, d_1..d_4, Dz, WeylAlgebra => toList(1..4)/(i -> x_i => d_i) | {z=>Dz}]

o1 = R

o1 : PolynomialRing, 5 differential variables
i2 : f = x_1 + x_2 * z + x_3 * z^2 + x_4 * z^3

        3      2
o2 = x z  + x z  + x z + x
      4      3      2     1

o2 : R
i3 : AnnFs f

             2                                  2                          
o3 = ideal (d  - d d , d d  - d d , z*d  - d , d  - d d , z*d  - d , x d  +
             3    2 4   2 3    1 4     3    4   2    1 3     2    3   2 2  
     ------------------------------------------------------------------------
                                                                            
     2x d  + 3x d  - z*Dz, z*d  - d , x d  + 2x d  + 3x d  - Dz, x d  - x d 
       3 3     4 4            1    2   2 1     3 2     4 3        1 1    3 3
     ------------------------------------------------------------------------
                                    2
     - 2x d  + z*Dz - s, 3x z*d  - z Dz + x d  + 2x d )
         4 4               4   4           2 3     3 4

o3 : Ideal of QQ[x ..x , z, d ..d , Dz, s]
                  1   4      1   4

Caveat

The ring of f should not have any parameters, i.e., it should be a pure Weyl algebra. Also this ring should not be a homogeneous Weyl algebra.

See also