# Ddim -- dimension of a D-module

## Synopsis

• Usage:
Ddim M
Ddim I
• Inputs:
• M, , over the Weyl algebra $D$
• I, an ideal, which represents the module $M=D/I$
• Outputs:

## Description

The dimension of $M$ is equal to the dimension of the associated graded module with respect to the Bernstein filtration. If $D$ is the Weyl algebra over &#x2102; with generators $x_1,\dots,x_n$ and $\partial_1,\dots,\partial_n$, then the Bernstein filtration corresponds to the weight vector $(1,...,1,1,...,1)$.

 i1 : makeWA(QQ[x,y]) o1 = QQ[x..y, dx, dy] o1 : PolynomialRing, 2 differential variables 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 : Ddim I o3 = 2