# 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 moduleM_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.
 i1 : W = QQ[x,y,Dx,Dy, WeylAlgebra => {x=>Dx,y=>Dy}] o1 = W o1 : PolynomialRing, 2 differential variables 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 : 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

## See also

• Dlocalize -- localization of a D-module
• AnnFs -- the annihilating ideal of f^s
• Dintegration -- integration modules of a D-module

## Ways to use DlocalizeAll :

• "DlocalizeAll(Ideal,RingElement)"
• "DlocalizeAll(Module,RingElement)"

## For the programmer

The object DlocalizeAll is .