Bernstein–Sato polynomial

Examples

• If ${\displaystyle f(x)=x_1^2+\cdots +x_n^2}$ then

${\displaystyle \sum _{i=1}^n{\mathrm{\partial }}_i^{2}f(x{)}^{s+1}=4(s+1)\left(s+\frac{n}{2}\right)f(x{)}^s}$

so the Bernstein–Sato polynomial is

${\displaystyle b(s)=(s+1)\left(s+\frac{n}{2}\right).}$

• If ${\displaystyle f(x)=x_1^{n_1}{x}_2^{n_2}\cdots x_r^{n_r}}$ then

${\displaystyle \prod _{j=1}^r{\mathrm{\partial }}_{x_j}^{n_j}f(x{)}^{s+1}=\prod _{j=1}^r\prod _{i=1}^{n_j}(n_{j}s+i)f(x{)}^s}$

so

${\displaystyle b(s)=\prod _{j=1}^r\prod _{i=1}^{n_j}\left(s+\frac{i}{n_j}\right).}$

• The Bernstein–Sato polynomial of x² + y³ is

${\displaystyle (s+1)\left(s+\frac{5}{6}\right)\left(s+\frac{7}{6}\right).}$

• If t_ij are n² variables, then the Bernstein–Sato polynomial of det(t_ij) is given by

${\displaystyle (s+1)(s+2)\cdots (s+n)}$

which follows from

${\displaystyle \mathrm{\Omega }(det(t_{ij}{)}^s)=s(s+1)\cdots (s+n-1)det(t_{ij}{)}^{s-1}}$

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

Applications

• If ${\displaystyle f(x)}$ is a non-negative polynomial then ${\displaystyle f(x{)}^s}$, 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

${\displaystyle f(x{)}^s=\frac{1}{b(s)}P(s)f(x{)}^{s+1}.}$

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, f g = 1 as distributions. 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 ${\displaystyle \overline{f}(x)}$ times the inverse of ${\displaystyle \overline{f}(x)f(x).}$
• The Malgrange–Ehrenpreis theorem states that every differential operator with constant 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.
• Pavel Etingof (1999) showed how to use the Bernstein polynomial to define dimensional regularization rigorously, in the massive Euclidean case.
• The Bernstein-Sato functional equation is used in computations of some of the more complex kinds of singular integrals occurring in quantum field theory. Such computations are needed for precision measurements in elementary particle physics as practiced for instance at CERN (see the papers citing). However, the most interesting cases require a simple generalization of the Bernstein-Sato functional equation to the product of two polynomials ${\displaystyle (f_1(x){)}^{s_1}(f_2(x){)}^{s_2}}$, 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 ${\displaystyle P(s_1,s_2)}$ and ${\displaystyle b(s_1,s_2)}$ 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.