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

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1 Answer 1

up vote 3 down vote accepted

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
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Thanks you. Based on your answer I think we can give an explicit answer for some questions of Lyubeznik in this paper ams.org/journals/proc/1997-125-07/S0002-9939-97-03774-X in the case one variable. –  Pham Hung Quy Oct 8 '13 at 12:28

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