# Computing Bernstein–Sato ideals and s-parametric annihilators¶

First, we need to load the D-module library “dmod_lib”.

[1]:

LIB "dmod.lib";

[1]:

// ** loaded /usr/local/bin/../Cellar/singular/4.2.1_1/bin/../share/singular/LIB/dmod.lib (4.1.2.0,Feb_2019)



## Example 3.7¶

The following code computes the $$s$$-parametric annihilator $$\hbox{Ann}_{D_n[s]}(f^s)$$ of the polynomial $$f = (x-1)(x-2).$$

[2]:

ring r = 0,x,dp;
poly f = (x-1)*(x-2);
def A = operatorBM(f); setring A;


The ideal LD of the ring $$A$$ is the $$s$$-parametric annihilator of $$f.$$

[3]:

LD; // s-parametric annihilator of f

[3]:

LD[1]=x^2*Dx-3*x*Dx-2*x*s+2*Dx+3*s



We also obtain the Bernstein-Sato polynomial $$b_f$$ of $$f$$ as well as its roots with multiplicities.

[4]:

bs; // the Bernstein-Sato polynomial of f
BS; // roots of the Bernstein-Sato polynomial of f and their multiplicities

[4]:

s+1
[1]:
_[1]=-1
[2]:
1



We now compute a Bernstein-Sato operator of $$f$$. Note that Bernstein-Sato operators are unique only modulo $$\hbox{Ann}_{D_n[s]}(f^{s+1}).$$

[5]:

PS; // a Bernstein-Sato operator of f

[5]:

2*x*Dx-3*Dx-4*s-4