Bernstein–Sato polynomial

In mathematics, the Bernstein–Sato polynomial is a polynomial related to differential operators, introduced independently by Bernstein (1971) and Sato and Shintani (1972, 1974), Sato (1990). It is also known as the b-function, the b-polynomial, and the Bernstein polynomial, though it is not related to the Bernstein polynomials used in approximation theory. It has applications to singularity theory, monodromy theory and quantum field theory.

Coutinho (1995) gives an elementary introduction, and Borel (1987) and Kashiwara (2003) give more advanced accounts.

Definition and properties

If ƒ(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). Its existence can be shown using the notion of holonomic D-modules.

Kashiwara (1976) proved that all roots of the Bernstein–Sato polynomial are negative rational numbers.

The Bernstein–Sato polynomial can also be defined for products of powers of several polynomials (Sabbah 1987). In this case it is a product of linear factors with rational coefficients.

Nero Budur, Mircea Mustaţǎ, and Morihiko Saito (2006) generalized the Bernstein–Sato polynomial to arbitrary varieties.

Note, that the Bernstein–Sato polynomial can be computed algorithmically. However, such computations are hard in general. There are implementations of related algorithms in computer algebra systems RISA/Asir, Macaulay2 and SINGULAR.

Daniel Andres, Viktor Levandovskyy, and Jorge Martín-Morales (2009) presented algorithms to compute the Bernstein–Sato polynomial of an affine variety together with an implementation in the computer algebra system SINGULAR.

Berkesch & Leykin (2010) described some of the algorithms for computing Bernstein–Sato polynomials by computer.

Examples

\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).
\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).
(s+1)\left(s+\frac{5}{6}\right)\left(s+\frac{7}{6}\right).
s(s+1)\cdots(s+n-1)
which follows from
\Omega(\det(t_{ij})^s) = s(s+1)\cdots(s+n-1)\det(t_{ij})^{s-1}
where Ω is Cayley's omega process, which in turn follows from the Capelli identity.

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.

References