Bernstein–Sato polynomial explained

In mathematics, the Bernstein–Sato polynomial is a polynomial related to differential operators, introduced independently by and, . 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.

gives an elementary introduction, while and give more advanced accounts.

Definition and properties

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 polynomials

b(s)

. Its existence can be shown using the notion of holonomic D-modules.

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 . In this case it is a product of linear factors with rational coefficients.

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.

presented algorithms to compute the Bernstein–Sato polynomial of an affine variety together with an implementation in the computer algebra system SINGULAR.

described some of the algorithms for computing Bernstein–Sato polynomials by computer.

Examples

2
f(x)=x
n

then
n
\sum
i=1
2
\partial
i

f(x)s+1=4(s+1)\left(s+

n
2

\right)f(x)s

so the Bernstein–Sato polynomial is

b(s)=(s+1)\left(s+n
2

\right).

n1
f(x)=x
1
n2
x
2

nr
x
r
then
nj
\prod
xj

f(x)s+1

nj
=\prod
i=1

(njs+i)f(x)s

so

nj
b(s)=\prod\left(s+
i=1
i
nj

\right).

(s+1)\left(s+5\right)\left(s+
6
7
6

\right).

(s+1)(s+2)(s+n)

which follows from

\Omega(\det(tij)s)=s(s+1)(s+n-1)\det(tij)s-1

where Ω is Cayley's omega process, which in turn follows from the Capelli identity.

Applications

f(x)

is a non-negative polynomial then

f(x)s

, initially defined for s with non-negative real part, can be analytically continued to a meromorphic distribution-valued function of s by repeatedly using the functional equation

f(x)s={1\overb(s)}P(s)f(x)s+1.

It may have poles whenever b(s + n) is zero for a non-negative integer n.

\barf(x)

times the inverse of

\barf(x)f(x).

s1
(f
1(x))
s2
(f
2(x))
, with x having 2-6 scalar components, and the pair of polynomials having orders 2 and 3. Unfortunately, a brute force determination of the corresponding differential operators

P(s1,s2)

and

b(s1,s2)

for such cases has so far proved prohibitively cumbersome. Devising ways to bypass the combinatorial explosion of the brute force algorithm would be of great value in such applications.

References

Notes and References

  1. 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.