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DlocalizeAll -- localization of a D-module (extended version)

Synopsis

Description

An extension of Dlocalize that in addition computes the localization map the b-function, and the power $s$ of the generator $f^s$.

The keys of the output HashTable depend on which strategy is used. Common to each strategy are the keys LocMap and LocModule, which have the localization map and the localized module, respectively; and GeneratorPower, which is an integer $s$ such that (the images of) the generators of $M$ are $f^{-s}$ times the generators of $M_f$.

i1 : W = makeWeylAlgebra(QQ[x,y])

o1 = W

o1 : PolynomialRing, 2 differential variable(s)
 
i2 : M = W^1/ideal(x*dx + 1, dy)

o2 = cokernel | xdx+1 dy |

                            1
o2 : W-module, quotient of W
 
i3 : f = x^2 - y^3

        3    2
o3 = - y  + x

o3 : W
 
i4 : Mfall = DlocalizeAll(M, f)

o4 = HashTable{GeneratorPower => -2                                        }
                                                    4      5      5      7
               IntegrateBfunction => (s)(s + 1)(s + -)(s + -)(s + -)(s + -)
                                                    3      3      6      6
               LocMap => | y6-2x2y3+x4 |
               LocModule => cokernel | -3xdx-2ydy-15 -y3dy+x2dy-6y2 |

o4 : HashTable
 
i5 : gens image Mfall.LocMap == f^(-Mfall.GeneratorPower) * gens Mfall.LocModule

o5 = true

See also

Ways to use DlocalizeAll:

For the programmer

The object DlocalizeAll is a method function with options.