Let D be the Weyl algebra with generators x1,…,xn and ∂1,…,∂n over a field K of characteristic zero, and denote R = K(x1..xn)<∂1..∂n>, 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 R I with D. It consists of all operators which vanish on the common holomorphic solutions of D 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 : makeWA(QQ[x]) o1 = QQ[x, dx] o1 : PolynomialRing, 1 differential variables |
i2 : I = ideal(x*dx-2) o2 = ideal(x*dx - 2) o2 : Ideal of QQ[x, dx] |
i3 : holonomicRank I o3 = 1 |
i4 : WeylClosure I 3 2 o4 = ideal (x*dx - 2, x*dx - 2, dx , x*dx - dx) o4 : Ideal of QQ[x, dx] |
The ideal I should be of finite holonomic rank, which can be tested manually by using the function holonomicRank. The Weyl closure of non-finite rank ideals or arbitrary submodules has not been implemented.