---
tag: 7M5P2A0BZ6
kind: concept
name: Bernstein–Sato polynomial
msc: [14F10, 32S40, 32C38]
sources:
  - url: https://en.wikipedia.org/wiki/Bernstein%E2%80%93Sato_polynomial
same_as:
  wikidata: Q17003059
  wikipedia: Bernstein–Sato polynomial
---

## Summary

The Bernstein–Sato polynomial, or $b$-function, of a polynomial $f$ is a monic polynomial $b(s)$, see the [[definition|R2AV47AX0Y]].
It is the one of least degree for which a differential operator turns $f^{s+1}$ into $b(s) f^s$.
Joseph Bernstein proved its existence for polynomials.
Mikio Sato introduced it independently in his theory of prehomogeneous vector spaces.
Masaki Kashiwara proved that its roots are negative rational numbers.
It is a fine invariant of the singularities of $f^{-1}(0)$.
Its roots are linked with the eigenvalues of the [[monodromy|DQP63REKEK]] and with the log canonical threshold.
It also gives the meromorphic continuation of the distribution $\abs{f}^s$ in $s$.

## Explanations

### L1 human

For $f(x) = x$, differentiating $x^{s+1}$ gives $(s+1) x^s$.
So one derivative turns $x^{s+1}$ into $x^s$ times the polynomial $s + 1$.
Bernstein showed that for every polynomial $f$ in any number of variables there is a differential operator doing the same job, up to a factor $b(s)$.
The best possible factor is the Bernstein–Sato polynomial, and its roots record how singular $f$ is.