Singular

7.7.4.0. annfs

Usage:

annfs(f [,S,eng]); f a poly, S a string, eng an optional int

Return:

ring

Purpose:

compute the D-module structure of basering[1/f]*f^s with the algorithm
given in S and with the Groebner basis engine given in "eng"

Note:

activate the output ring with the setring command.
String S; S can be one of the following:
'bm' (default) - for the algorithm of Briancon and Maisonobe,
'ot' - for the algorithm of Oaku and Takayama,
'lot' - for the Levandovskyy's modification of the algorithm of OT.
If eng <>0, std is used for Groebner basis computations,
otherwise and by default slimgb is used.
In the output ring:
- the ideal LD (which is a Groebner basis) is the needed D-module structure,
- the list BS contains roots and multiplicities of a BS-polynomial of f.

Display:

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

LIB "dmod.lib";
ring r = 0,(x,y,z),Dp;
poly F = z*x^2+y^3;
def A  = annfs(F); // here, the default BM algorithm will be used
setring A; // the Weyl algebra in (x,y,z,Dx,Dy,Dz)
LD; //the annihilator of F^{-1} over A
==> LD[1]=y*Dy+3*z*Dz+3
==> LD[2]=x*Dx-2*z*Dz
==> LD[3]=x^2*Dy-3*y^2*Dz
==> LD[4]=3*y^2*Dx-2*x*z*Dy
==> LD[5]=y^3*Dz+x^2*z*Dz+x^2
==> LD[6]=2*x*z*Dy^2+9*y*z*Dx*Dz+3*y*Dx
==> LD[7]=9*y*z*Dx^2*Dz+4*z^2*Dy^2*Dz+3*y*Dx^2+2*z*Dy^2
==> LD[8]=4*z^2*Dy^3*Dz-27*z^2*Dx^2*Dz^2+2*z*Dy^3-54*z*Dx^2*Dz-6*Dx^2
BS; // roots with multiplicities of BS polynomial
==> [1]:
==>    _[1]=-7/6
==>    _[2]=-5/3
==>    _[3]=-4/3
==>    _[4]=-5/6
==>    _[5]=-1
==> [2]:
==>    1,1,1,1,1