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Dmodules :: WeylClosure

WeylClosure -- Weyl closure of an ideal

Synopsis

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.

See also

Ways to use WeylClosure :