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

## Synopsis

• Usage:
DlocalizeAll(M,f)
DlocalizeAll(I,f)
• Inputs:
• M, , over the Weyl algebra $D$
• I, an ideal, which represents the module $M=D/I$
• f, , a polynomial
• Optional inputs:
• Strategy => ..., default value OTW, strategy for computing a localization of a D-module
• Outputs:
• , which contains the localized module $M_f = M[f^{-1}]$ and some additional information

## 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