# isHolonomic -- determines whether a D-module (or ideal in Weyl algebra) is holonomic

## Synopsis

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

## Description

Let $D$ be the Weyl algebra with generators $x_1,\dots,x_n$ and $\partial_1,\dots,\partial_n$. over a field. A $D$-module is holonomic if it has dimension $n$. For more details see [SST, Section 1.4].

 i1 : D = makeWA(QQ[x_1..x_3]) o1 = D o1 : PolynomialRing, 3 differential variables i2 : A = matrix{{1,1,1},{0,1,2}} o2 = | 1 1 1 | | 0 1 2 | 2 3 o2 : Matrix ZZ <--- ZZ i3 : b = {3,4} o3 = {3, 4} o3 : List i4 : I = gkz(A,b,D) 2 o4 = ideal (x dx + x dx + x dx - 3, x dx + 2x dx - 4, - dx + dx dx ) 1 1 2 2 3 3 2 2 3 3 2 1 3 o4 : Ideal of D i5 : isHolonomic I o5 = true

## Ways to use isHolonomic :

• "isHolonomic(Ideal)"
• "isHolonomic(Module)"

## For the programmer

The object isHolonomic is .