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

## Synopsis

• Function: AnnFs
• Usage:
AnnFs f
• Inputs:
• f, , a polynomial in a Weyl algebra An (should contain no differential variables)
• Outputs:
• an ideal, an ideal of An[s]

## 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.