# Singular

##### 7.7.4.0. operatorBM
Procedure from library `dmod.lib` (see dmod_lib).

Usage:
operatorBM(f [,eng]); f a poly, eng an optional int

Return:
ring

Purpose:
compute the B-operator and other relevant data for Ann F^s,
using e.g. algorithm by Briancon and Maisonobe for Ann F^s and BS.

Note:
activate the output ring with the `setring` command. In the output ring D[s]
- the polynomial F is the same as the input,
- the ideal LD is the annihilator of f^s in Dn[s],
- the ideal LD0 is the needed D-mod structure, where LD0 = LD|s=s0,
- the polynomial bs is the global Bernstein polynomial of f in the variable s,
- the list BS contains all the roots with multiplicities of the global Bernstein polynomial of f,
- the polynomial PS is an operator in Dn[s] such that PS*f^(s+1) = bs*f^s.
If eng <>0, `std` is used for Groebner basis computations,
otherwise and by default `slimgb` is used.

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

Example:
 ```LIB "dmod.lib"; ring r = 0,(x,y,z),Dp; poly F = x^3+y^3+z^3; printlevel = 0; def A = operatorBM(F); setring A; F; // the original polynomial itself ==> x^3+y^3+z^3 LD; // generic annihilator ==> LD[1]=x*Dx+y*Dy+z*Dz-3*s ==> LD[2]=z^2*Dy-y^2*Dz ==> LD[3]=z^2*Dx-x^2*Dz ==> LD[4]=y^2*Dx-x^2*Dy ==> LD[5]=x^3*Dz+y^3*Dz+z^3*Dz-3*z^2*s ==> LD[6]=x^3*Dy+y^3*Dy+y^2*z*Dz-3*y^2*s LD0; // annihilator ==> LD0[1]=x*Dx+y*Dy+z*Dz+6 ==> LD0[2]=z^2*Dy-y^2*Dz ==> LD0[3]=z^2*Dx-x^2*Dz ==> LD0[4]=y^2*Dx-x^2*Dy ==> LD0[5]=x^3*Dz+y^3*Dz+z^3*Dz+6*z^2 ==> LD0[6]=x^3*Dy+y^3*Dy+y^2*z*Dz+6*y^2 bs; // normalized Bernstein poly ==> s^5+7*s^4+173/9*s^3+233/9*s^2+154/9*s+40/9 BS; // roots and multiplicities of the Bernstein poly ==> [1]: ==> _[1]=-1 ==> _[2]=-4/3 ==> _[3]=-5/3 ==> _[4]=-2 ==> [2]: ==> 2,1,1,1 PS; // the operator, s.t. PS*F^{s+1} = bs*F^s mod LD ==> 2/81*y*z*Dx^3*Dy*Dz-2/81*y*z*Dy^4*Dz-4/81*y^2*Dy^2*Dz^3-2/81*y*z*Dy*Dz^4+\ 2/81*y*Dx^3*Dy*s-2/81*y*Dy^4*s+2/81*z*Dx^3*Dz*s+2/27*z*Dy^3*Dz*s+2/27*y*D\ y*Dz^3*s-2/81*z*Dz^4*s+2/27*y*Dx^3*Dy-2/27*y*Dy^4+2/27*z*Dx^3*Dz+2/27*z*D\ y^3*Dz-10/81*y*Dy*Dz^3-2/27*z*Dz^4+1/27*Dx^3*s^2+1/9*Dy^3*s^2+1/9*Dz^3*s^\ 2+5/27*Dx^3*s+11/27*Dy^3*s+11/27*Dz^3*s+20/81*Dx^3+8/27*Dy^3+16/81*Dz^3 reduce(PS*F-bs,LD); // check the property of PS ==> 0 ```