Bernstein–Sato polynomial

In mathematics, the Bernstein–Sato polynomial is a polynomial related to differential operators, introduced independently by Joseph Bernstein (1971)^[1] and Mikio Sato and Takuro Shintani (1972, 1974),^[2][3] Sato (1990).^[4] 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.

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

Definition and properties

If ${\displaystyle f(x)}$ is a polynomial in several variables, then there are a non-zero polynomial ${\displaystyle b(s)}$ and a differential operator ${\displaystyle P(s)}$ with polynomial coefficients such that

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

The Bernstein–Sato polynomial is the monic polynomial of smallest degree amongst such polynomials ${\displaystyle b(s)}$. 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.^[5]

The Bernstein–Sato polynomial can also be defined for products of powers of several polynomials.^[6] In this case it is a product of linear factors with rational coefficients.^{[citation needed]}

Nero Budur, Mircea Mustață, and Morihiko Saito (2006) generalized the Bernstein–Sato polynomial to arbitrary varieties.^[7]

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)^[8] presented algorithms to compute the Bernstein–Sato polynomial of an affine variety together with an implementation in the computer algebra system SINGULAR. Christine Berkesch and Anton Leykin (2010) described some of the algorithms for computing Bernstein–Sato polynomials by computer.^[9]

Examples

• If ${\displaystyle f(x)=x_{1}^{2}+\cdots +x_{n}^{2}\,}$ then

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

so the Bernstein–Sato polynomial is

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

• If ${\displaystyle f(x)=x_{1}^{n_{1}}x_{2}^{n_{2}}\cdots x_{r}^{n_{r}}}$ then

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

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

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

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

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

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;^[a] 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 ${\displaystyle {\bar {f}}(x)}$ times the inverse of ${\displaystyle {\bar {f}}(x)f(x).}$
• 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)^[10] 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.^[11] Such computations are needed for precision measurements in elementary particle physics as practiced for instance at CERN (see the papers citing^[11]). 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.

Notes

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.

References

1. Bernshtein, I. N. (1971). "Modules over a ring of differential operators. Study of the fundamental solutions of equations with constant coefficients". Functional Analysis and Its Applications. 5 (2): 89–101. doi:10.1007/BF01076413. ISSN 0016-2663.
2. Sato, Mikio; Shintani, Takuro (June 1972). "On Zeta Functions Associated with Prehomogeneous Vector Spaces". Proceedings of the National Academy of Sciences. 69 (5): 1081–1082. doi:10.1073/pnas.69.5.1081. ISSN 0027-8424.
3. Sato, Mikio; Shintani, Takuro (July 1974). "On Zeta Functions Associated with Prehomogeneous Vector Spaces". The Annals of Mathematics. 100 (1): 131. doi:10.2307/1970844.
4. Sato, Mikio; Shintani, Takuro (December 1990). "Theory of prehomogeneous vector spaces (algebraic part)—the English translation of Sato's lecture from Shintani's note". Nagoya Mathematical Journal. 120: 1–34. doi:10.1017/S0027763000003214. ISSN 0027-7630.
5. Kashiwara, Masaki (February 1976). "B-functions and holonomic systems: Rationality of roots ofB-functions". Inventiones Mathematicae. 38 (1): 33–53. doi:10.1007/BF01390168. ISSN 0020-9910.
6. Sabbah, C. (1987). "Proximité évanescente. I. La structure polaire d'un $\mathcal {D}$-module". Compositio Mathematica (in French). 62 (3): 283–328. ISSN 1570-5846.
7. Budur, Nero; Mustata, Mircea; Saito, Morihiko (May 2006). "Bernstein–Sato polynomials of arbitrary varieties". Compositio Mathematica. 142 (03): 779–797. doi:10.1112/S0010437X06002193. ISSN 0010-437X.
8. Andres, Daniel; Levandovskyy, Viktor; Morales, Jorge Martín (June 28, 2009). "Principal intersection and bernstein-sato polynomial of an affine variety". Proceedings of the 2009 international symposium on Symbolic and algebraic computation. ACM: 231–238. doi:10.1145/1576702.1576735. ISBN 978-1-60558-609-0.
9. Berkesch, Christine; Leykin, Anton (2010-07-25). "Algorithms for Bernstein--Sato polynomials and multiplier ideals". Proceedings of the 2010 International Symposium on Symbolic and Algebraic Computation. ACM: 99–106. doi:10.1145/1837934.1837958. ISBN 978-1-4503-0150-3.
10. Deligne, Pierre, ed. (2000), Quantum fields and strings: a course for mathematicians. Vol. 1, Institute for Advanced Study (2. Nachdr. ed.), Providence, RI: American Mathematical Society [u.a.], ISBN 978-0-8218-2012-4
11. Tkachov, Fyodor V (April 1997). "Algebraic algorithms for multiloop calculations The first 15 years. What's next?". Nuclear Instruments and Methods in Physics Research Section A: Accelerators, Spectrometers, Detectors and Associated Equipment. 389 (1–2): 309–313. arXiv:hep-ph/9609429. doi:10.1016/S0168-9002(97)00110-1.

• Borel, Armand (1987). Algebraic D-Modules. Perspectives in Mathematics. Vol. 2. Boston, MA: Academic Press. ISBN 0-12-117740-8.
• Coutinho, Severino C. (1995). A primer of algebraic D-modules. London Mathematical Society Student Texts. Vol. 33. Cambridge, UK: Cambridge University Press. ISBN 0-521-55908-1.
• Kashiwara, Masaki (2003). D-modules and microlocal calculus. Translations of Mathematical Monographs. Vol. 217. Providence, R.I.: American Mathematical Society. ISBN 978-0-8218-2766-6. MR 1943036.