Let R = K(x
1..x
n)<d
1..d
n> 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
|