The characteristic ideal of M is the annihilator of gr(M) under a good filtration with respect to the order filtration. If D is the Weyl algebra over ℂ with generators x1,…,xn and ∂1,…,∂n, then the order filtration corresponds to the weight vector (0,...,0,1,...,1). The characteristic ideal lives in the associated graded ring of D with respect to the order filtration, and this is a commutative polynomial ring ℂ[x1,…,xn,ξ1,…,ξn]. Here the ξi is the principal symbol of ∂i, that is, the image of ∂i in the associated graded ring. The zero locus of the characteristic ideal is equal to the characteristic variety of D/I which is an invariant of a D-module.
The algorithm to compute the characteristic ideal consists of computing the initial ideal of I with respect to the weight vector (0,...,0,1...,1). More details can be found in [SST, Section 1.4].
i1 : makeWA(QQ[x,y]) o1 = QQ[x, y, dx, dy] o1 : PolynomialRing, 2 differential variables |
i2 : I = ideal (x*dx+2*y*dy-3, dx^2-dy) 2 o2 = ideal (x*dx + 2y*dy - 3, dx - dy) o2 : Ideal of QQ[x, y, dx, dy] |
i3 : charIdeal I 2 o3 = ideal (dx , x*dx + 2y*dy) o3 : Ideal of QQ[x, y, dx, dy] |