Bernstein–Sato polynomial

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In

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

If ${\displaystyle f(x)}$ is a polynomial in several variables, then there is 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.

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.^{[citation needed]}

Nero Budur, Mircea Mustață, 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,

Daniel Andres, Viktor Levandovskyy, and Jorge Martín-Morales (

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

• 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

Capelli identity

.

• 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
  meromorphic distribution-valued function of s by repeatedly using the functional equation

• If f(x) is a polynomial, not identically zero, then it has an inverse g that is a distribution;
  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
  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, see Fyodor Tkachov (1997). Such computations are needed for precision measurements in elementary particle physics as practiced for instance at CERN (see the papers citing (Tkachov 1997)). 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.

• Andres, Daniel; Levandovskyy, Viktor; Martín-Morales, Jorge (2009). "Principal intersection and bernstein-sato polynomial of an affine variety". Proceedings of the 2009 international symposium on Symbolic and algebraic computation.
  S2CID 2747775
  .
• Berkesch, Christine; Leykin, Anton (2010). "Algorithms for Bernstein--Sato polynomials and multiplier ideals". Proceedings of the 2010 International Symposium on Symbolic and Algebraic Computation. pp. 99–106.
  S2CID 33730581
  .
• S2CID 124605141
  .
• Budur, Nero;
  S2CID 6955564.{{cite journal}}: CS1 maint: DOI inactive as of February 2025 (link
  )
• ISBN 0-12-117740-8
  .
• Coutinho, Severino C. (1995). A primer of algebraic D-modules. London Mathematical Society Student Texts. Vol. 33. Cambridge, UK:
  ISBN 0-521-55908-1
  .
• MR 1701608
  . (Princeton, NJ, 1996/1997)
• S2CID 17103403
  .
• MR 1943036
  .
• Sabbah, Claude (1987). "Proximité évanescente. I. La structure polaire d'un D-module".
  MR 0901394
  .
• PMID 16591979
  .
• MR 0344230
  .
• MR 1086566
  . the English translation of Sato's lecture from Shintani's note
• Tkachov, Fyodor V. (1997). "Algebraic algorithms for multiloop calculations. The first 15 years. What's next?".
  S2CID 37109930
  .