Let $R = \mathbb{C}[x\_1,...,x\_n]$, $p \in R$. There exists a monic (of lowest degree) $b\_p(x) \in \mathbb{C}[s]$ and a differential operator $D(s)$ such that
$$b\_p(s) p^s = D(s)p^{s+1}.$$
The polynomial $b\_p(s)$ is called the Bernstein-Sato polynomial of $p$. The calculation of $b\_p(s)$ is very complicated.


Consider $n=1$ and $R = \mathbb{C}[x]$. We have $p = (x-a\_1)^{n\_1}...(x-a\_t)^{n\_t}$ with $n\_1 \leq n\_2 \leq \cdots \leq n\_t$. We have a formula for $b\_p(s)$ in the following cases


1. if $t=1$ we have $b\_p(s) = (s+\frac{1}{n\_1})\cdots (s+\frac{n\_1-1}{n\_1}) (s+1)$.
2. if $n\_1 = \cdots = n\_t=1$, then $b\_p(s) = (s+1)$.


**Question.** Does there exist a formula for $b\_p(s)$ in term $(n\_1,...,n\_t)$?



Bernstein-Sato polynomial (one variable)
It is known that the (global) Bernstein-Sato polynomial is the least common multiple of all local Bernstein-Sato polynomials. In the case at hand, the local polynomials are given by $b\_{p,i}(s) = (s+\frac{1}{n\_i})\cdots (s+\frac{n\_i-1}{n\_i}) (s+1)$ (localization at $a\_i$) or 1 (localization somewhere else), which gives your desired formula.


Experimenting with these polynomials online is possible with Macauley2 at <http://habanero.math.cornell.edu:3690/>. For example, 



```
needsPackage "Dmodules"

R = QQ[x]

f = x^3*(x-1)^5

b = globalBFunction f

c = factor b

```

returns 



```
                                                              1
o9 = (s + 1)(3s + 1)(3s + 2)(5s + 1)(5s + 2)(5s + 3)(5s + 4)(----)
                                                             5625

```

