Bernstein–Sato polynomial

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In mathematics, the Bernstein-Sato polynomial is a construction of Joseph Bernstein and Mikio Sato, based on an algebraic theory of differential operators. It is also known as the Bernstein polynomial, the b-function, and the b-polynomial (it is not related to the Bernstein polynomial used in approximation theory). It has turned out to have many mathematical connections, for example to singularity theory and monodromy theory.

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[edit] Definition

If f(x) is a polynomial in several variables then there is a non-zero polynomial b(s) and a differential operator P(s) with polynomial coefficients such that

P(s)f(x)s + 1 = b(s)f(x)s.

The Bernstein-Sato polynomial is the monic polynomial of smallest degree amongst such b(s).

[edit] Examples

  • If f(x)=x_1^2+\cdots+x_n^2 then
\sum_{i=1}^n \partial_i^2 f(x)^{s+1} = 4(s+1)\left(s+\frac{n}{2}\right)f(x)^s
so the Bernstein-Sato polynomial is
b(s)=(s+1)\left(s+\frac{n}{2}\right).
  • If f(x)=x_1^{n_1}x_2^{n_2}\cdots x_r^{n_r} then
\prod_{j=1}^r\partial_{x_j}^{n_j}\quad f(x)^{s+1} =\prod_{j=1}^r\prod_{i=1}^{n_j}(n_js+i)\quad f(x)^s
so
b(s)=\prod_{j=1}^r\prod_{i=1}^{n_j}\left(s+\frac{i}{n_j}\right).
  • The Bernstein-Sato polynomial of x2 + y3 is
(s+1)\left(s+\frac{5}{6}\right)\left(s+\frac{7}{6}\right).

[edit] Properties

Masaki Kashiwara proved that all roots of the Bernstein-Sato polynomial are negative rational numbers.

[edit] Applications

f(x)^s={1\over b(s)} P(s)f(x)^{s+1}.
It may have poles whenever b(s + n) is zero for a non-negative integer n.
  • If f(x) is a polynomial, not identically zero, then it has an inverse g that is a distribution; in other words, fg = 1 as distributions. (Warning: the inverse is not unique in general, because if f has zeros then there are distributions whose product with f is zero, and adding one of these to an inverse of f is another inverse of f. The usual proof of uniqueness of inverses fails because the product of distributions is not always defined, and need not be associative even when it is defined.) If f(x) is non-negative the inverse can be constructed using the Bernstein-Sato polynomial by taking the constant term of the Laurent expansion of f(x)s at s = −1. For arbitrary f(x) just take \bar f(x) times the inverse of \bar f(x)f(x).
  • The Malgrange-Ehrenpreis theorem states that every differential operator with polynomial coefficients has a Green's function. By taking Fourier transforms this follows from the fact that every polynomial has a distributional inverse, which is proved in the paragraph above.

[edit] History

The Bernstein-Sato polynomial was introduced independently by Joseph Bernstein (Modules over a ring of differential operators, Functional Analysis and its Applications 5 (1971)) and Mikio Sato.

[edit] Further reading

Some more advanced books are:

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