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Dmodules :: BMM

BMM -- the characteristic cycle of the localized $D$-module

Synopsis

Description

Provided a characteristic cycle in the form {I_1 => m_1, ..., I_k => m_k} with associated prime ideals I_1,...,I_k and the multiplicities m_1,...,m_k of M along them, the routine computes the characteristic cycle of M_f.

The method is based on a geometric formula given by V.Ginsburg in Characteristic varieties and vanishing cycles, Invent. Math. 84 (1986), 327--402. and reinterpreted by J.Briancon, P.Maisonobe and M.Merle in Localisation de systemes differentiels, stratifications de Whitney et condition de Thom, Invent. Math. 117 (1994), 531--550.

i1 : A =  QQ[x_1,x_2,a_1,a_2]

o1 = A

o1 : PolynomialRing
i2 : cc = {ideal A => 1} -- the characteristic ideal of R = CC[x_1,x_2] 

o2 = {ideal () => 1}

o2 : List
i3 : cc1 = BMM(cc,x_1)   -- cc of R_{x_1}

o3 = {ideal () => 1, ideal x  => 1}
                            1

o3 : List
i4 : cc12 = BMM(cc1,x_2) -- cc of R_{x_1x_2}

o4 = {ideal () => 1, ideal x  => 1, ideal x  => 1, ideal (x , x ) => 1}
                            2              1               2   1

o4 : List

Caveat

The module has to be a regular holonomic complex-analytic module; while the holomicity can be checked by isHolonomic there is no algorithm to check the regularity.

See also

Ways to use BMM :

For the programmer

The object BMM is a method function.