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 $$b_p(s) p^s = D(x)p^{s+1}.$$ 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 exists a formula of $b_p(s)$ in term $(n_1,...,n_t)$?

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 $(n_1,...,n_t)$?

Let $R = \mathbb{C}[x_1,...,x_n]$, $p \in R$.There exist a monic (of lowest degree) $b_p(x) \in \mathbb{C}[s]$ and a differential operator $D(s)$ such that $$b_p(s) p^s = D(x)p^{s+1}.$$ The polynomial $b_p(s)$ is called the brentein-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 exists a formula of $b_p(s)$ in term $(n_1,...,n_t)$?

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 $$b_p(s) p^s = D(x)p^{s+1}.$$ 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 exists a formula of $b_p(s)$ in term $(n_1,...,n_t)$?