Singular

7.7.5.0. DLoc0

Usage:

DLoc0(I, f); I an ideal, f a poly

Return:

ring (a Weyl algebra) containing an ideal 'LD0' and a list 'BS'

Purpose:

compute the presentation of the localization of D/I w.r.t. f^s,
where D is a Weyl Algebra, based on the output of procedure SDLoc

Assume:

the basering is similar to the output ring of SDLoc procedure

Note:

activate the output ring with the setring command. In this ring,
- the ideal LD0 (given as Groebner basis) is the presentation of the
localization,
- the list BS contains roots and multiplicities of Bernstein
polynomial of (D/I)_f.

Display:

If printlevel =1, progress debug messages will be printed,
if printlevel>=2, all the debug messages will be printed.

LIB "dmodapp.lib";
ring r = 0,(x,y,Dx,Dy),dp;
def R = Weyl();    setring R; // Weyl algebra in variables x,y,Dx,Dy
poly F = x2-y3;
ideal I = (y^3 - x^2)*Dx - 2*x, (y^3 - x^2)*Dy + 3*y^2; // I = Dx*F, Dy*F;
// moreover I is not holonomic, since its dimension is not 2 = 4/2
gkdim(I); // 3
==> 3
def W = SDLoc(I,F);  setring W; // creates ideal LD in W = R[s]
def U = DLoc0(LD, x2-y3);  setring U; // compute in R
LD0; // Groebner basis of the presentation of localization
==> LD0[1]=3*x*Dx+2*y*Dy+12
==> LD0[2]=3*y^2*Dx+2*x*Dy
==> LD0[3]=y^3*Dy-x^2*Dy+6*y^2
BS; // description of b-function for localization
==> [1]:
==>    _[1]=0
==>    _[2]=1/6
==>    _[3]=-1/6
==> [2]:
==>    1,1,1