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Dmodules :: WeylClosure

WeylClosure -- Weyl closure of an ideal

Synopsis

Description

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]

Caveat

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.

See also

Ways to use WeylClosure :