# WeylClosure -- Weyl closure of an ideal

## Synopsis

• Usage:
WeylClosure I, WeylClosure(I,f)
• Inputs:
• I, an ideal, a left ideal of the Weyl Algebra
• f, , a polynomial
• Outputs:
• an ideal, the Weyl closure (w.r.t. f) of I

## Description

Let R = K(x1..xn)<d1..dn> denote the ring of differential operators with rational function coefficients. The Weyl closure of an ideal I in D is the intersection of the extended ideal RI with D. It consists of all operators which vanish on the common holomorphic solutions of I and is thus analogous to the radical operation on a commutative ideal.

The partial Weyl closure of I with respect to a polynomial f is the intersection of the extended ideal D[f-1] I with D.

The Weyl closure is computed by localizing D/I with respect to a polynomial f vanishing on the singular locus, and computing the kernel of the map D --> D/I --> (D/I)[f^{-1}].
 ```i1 : W = QQ[x,Dx, WeylAlgebra => {x=>Dx}] o1 = W o1 : PolynomialRing``` ```i2 : I = ideal(x*Dx-2) o2 = ideal(x*Dx - 2) o2 : Ideal of W``` ```i3 : WeylClosure I 3 2 o3 = ideal (x*Dx - 2, x*Dx - 2, Dx , x*Dx - Dx) o3 : Ideal of W```

## Caveat

The ideal I should be finite holonomic rank, which can be tested manually by holonomicRank.The Weyl closure of non-finite rank ideals or arbitrary submodules has not been implemented.