{{Short description|Polynomial related to differential operators}} In mathematics, the '''Bernstein–Sato polynomial''' is a polynomial related to differential operators, introduced independently by Joseph Bernstein (1971)<ref>{{Cite journal |last=Bernshtein |first=I. N. |date=1971 |title=Modules over a ring of differential operators. Study of the fundamental solutions of equations with constant coefficients |url=http://link.springer.com/10.1007/BF01076413 |journal=Functional Analysis and Its Applications |language=en |volume=5 |issue=2 |pages=89–101 |doi=10.1007/BF01076413 |issn=0016-2663|url-access=subscription }}</ref> and {{harvs|txt|author1-link=Mikio Sato|last1=Sato|first1=Mikio|last2=Shintani|first2=Takuro|year1=1972|year2=1974}},<ref>{{Cite journal |last1=Sato |first1=Mikio |last2=Shintani |first2=Takuro |date=June 1972 |title=On Zeta Functions Associated with Prehomogeneous Vector Spaces |journal=Proceedings of the National Academy of Sciences |language=en |volume=69 |issue=5 |pages=1081–1082 |doi=10.1073/pnas.69.5.1081 |doi-access=free |pmid=16591979 |issn=0027-8424|pmc=426633 }}</ref><ref>{{Cite journal |last1=Sato |first1=Mikio |last2=Shintani |first2=Takuro |date=July 1974 |title=On Zeta Functions Associated with Prehomogeneous Vector Spaces |url=https://www.jstor.org/stable/1970844 |journal=The Annals of Mathematics |volume=100 |issue=1 |pages=131 |doi=10.2307/1970844|jstor=1970844 |url-access=subscription }}</ref> Sato (1990).<ref>{{Cite journal |last1=Sato |first1=Mikio |last2=Shintani |first2=Takuro |date=December 1990 |title=Theory of prehomogeneous vector spaces (algebraic part)—the English translation of Sato's lecture from Shintani's note |url=https://www.cambridge.org/core/product/identifier/S0027763000003214/type/journal_article |journal=Nagoya Mathematical Journal |language=en |volume=120 |pages=1–34 |doi=10.1017/S0027763000003214 |issn=0027-7630|url-access=subscription }}</ref> 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.

{{harvs|txt|first=Severino|last=Coutinho|year=1995}}<ref>{{cite book |last=Coutinho |first=Severino C. |title=A primer of algebraic D-modules |publisher=Cambridge University Press |year=1995 |isbn=0-521-55908-1 |series=London Mathematical Society Student Texts |volume=33 |publication-place=Cambridge, UK}}</ref> gives an elementary introduction, while {{harvs|txt|first=Armand|last=Borel|authorlink=Armand Borel|year=1987}}<ref>{{cite book |last=Borel |first=Armand |author-link=Armand Borel |title=Algebraic D-Modules |publisher=Academic Press |year=1987 |isbn=0-12-117740-8 |series=Perspectives in Mathematics |volume=2 |publication-place=Boston, MA}}</ref> and {{harvs|txt|last=Kashiwara | first=Masaki | author-link=Masaki Kashiwara|year=2003}}<ref>{{cite book |last=Kashiwara |first=Masaki |author-link=Masaki Kashiwara |title=D-modules and microlocal calculus |publisher=American Mathematical Society |year=2003 |isbn=978-0-8218-2766-6 |series=Translations of Mathematical Monographs |volume=217 |location=Providence, R.I. |mr=1943036}}</ref> give more advanced accounts.

==Definition and properties==

If <math>f(x)</math> is a polynomial in several variables, then there are a non-zero polynomial <math>b(s)</math> and a differential operator <math>P(s)</math> with polynomial coefficients such that

:<math>P(s)f(x)^{s+1} = b(s)f(x)^s.</math>

The Bernstein–Sato polynomial is the monic polynomial of smallest degree amongst such polynomials <math>b(s)</math>. 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.<ref>{{Cite journal |last=Kashiwara |first=Masaki |date=February 1976 |title=B-functions and holonomic systems: Rationality of roots ofB-functions |url=http://link.springer.com/10.1007/BF01390168 |journal=Inventiones Mathematicae |language=en |volume=38 |issue=1 |pages=33–53 |doi=10.1007/BF01390168 |issn=0020-9910|url-access=subscription }}</ref>

The Bernstein–Sato polynomial can also be defined for products of powers of several polynomials.<ref>{{Cite journal |last=Sabbah |first=C. |date=1987 |title=Proximité évanescente. I. La structure polaire d'un $\mathcal {D}$-module |url=https://www.numdam.org/item/?id=CM_1987__62_3_283_0 |journal=Compositio Mathematica |language=fr |volume=62 |issue=3 |pages=283–328 |issn=1570-5846}}</ref> In this case it is a product of linear factors with rational coefficients.{{Citation needed|date=July 2014}}

{{harvs|txt| last1=Budur | first1=Nero | last2=Mustață | first2=Mircea | author2-link=Mircea Mustață| last3=Saito | first3=Morihiko |author3-link=Morihiko Saito| year=2006}} generalized the Bernstein–Sato polynomial to arbitrary varieties.<ref>{{Cite journal |last1=Budur |first1=Nero |last2=Mustata |first2=Mircea |last3=Saito |first3=Morihiko |date=May 2006 |title=Bernstein–Sato polynomials of arbitrary varieties |url=http://www.journals.cambridge.org/abstract_S0010437X06002193 |journal=Compositio Mathematica |language=en |volume=142 |issue=3 |pages=779–797 |doi=10.1112/S0010437X06002193 |issn=0010-437X}}</ref>

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.

{{harvs|txt|first1=Daniel|last1=Andres | first2=Viktor|last2= Levandovskyy | first3 = Jorge |last3=Martín-Morales| year=2009}}<ref>{{Cite book |last1=Andres |first1=Daniel |last2=Levandovskyy |first2=Viktor |last3=Morales |first3=Jorge Martín |chapter=Principal intersection and bernstein-sato polynomial of an affine variety |date=June 28, 2009 |title=Proceedings of the 2009 international symposium on Symbolic and algebraic computation |chapter-url=https://dl.acm.org/doi/10.1145/1576702.1576735 |language=en |publisher=ACM |pages=231–238 |doi=10.1145/1576702.1576735 |isbn=978-1-60558-609-0|chapter-url-access=subscription }}</ref> presented algorithms to compute the Bernstein–Sato polynomial of an affine variety together with an implementation in the computer algebra system SINGULAR. {{harvs|txt|first1=Christine|last1=Berkesch|first2=Anton|last2=Leykin|year=2010}} described some of the algorithms for computing Bernstein–Sato polynomials by computer.<ref>{{Cite book |last1=Berkesch |first1=Christine |last2=Leykin |first2=Anton |chapter=Algorithms for Bernstein--Sato polynomials and multiplier ideals |date=2010-07-25 |title=Proceedings of the 2010 International Symposium on Symbolic and Algebraic Computation |chapter-url=https://dl.acm.org/doi/10.1145/1837934.1837958 |language=en |publisher=ACM |pages=99–106 |doi=10.1145/1837934.1837958 |isbn=978-1-4503-0150-3|chapter-url-access=subscription }}</ref>

== Examples == * If <math>f(x)=x_1^2+\cdots+x_n^2 \, </math> then

::<math>\sum_{i=1}^n \partial_i^2 f(x)^{s+1} = 4(s+1)\left(s+\frac{n}{2}\right)f(x)^s</math>

:so the Bernstein–Sato polynomial is

::<math>b(s)=(s+1)\left(s+\frac{n}{2}\right).</math>

* If <math> f(x)=x_1^{n_1}x_2^{n_2}\cdots x_r^{n_r}</math> then

::<math>\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</math>

:so

::<math>b(s)=\prod_{j=1}^r\prod_{i=1}^{n_j}\left(s+\frac{i}{n_j}\right).</math>

* The Bernstein–Sato polynomial of ''x''<sup>2</sup>&nbsp;+&nbsp;''y''<sup>3</sup> is ::<math>(s+1)\left(s+\frac{5}{6}\right)\left(s+\frac{7}{6}\right).</math>

*If ''t''<sub>''ij''</sub> are ''n''<sup>2</sup> variables, then the Bernstein–Sato polynomial of det(''t''<sub>''ij''</sub>) is given by ::<math>(s+1)(s+2)\cdots(s+n)</math> :which follows from ::<math>\Omega(\det(t_{ij})^s) = s(s+1)\cdots(s+n-1)\det(t_{ij})^{s-1}</math> :where &Omega; is Cayley's omega process, which in turn follows from the Capelli identity.

== Applications == * If <math>f(x)</math> is a non-negative polynomial then <math>f(x)^s</math>, 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

::<math>f(x)^s={1\over b(s)} P(s)f(x)^{s+1}.</math>

:It may have poles whenever ''b''(''s''&nbsp;+&nbsp;''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;<ref group=lower-alpha>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''.</ref> in other words, ''f&nbsp;g''&nbsp;=&nbsp;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'')<sup>''s''</sup> at ''s''&nbsp;=&nbsp;&minus;1. For arbitrary ''f''(''x'') just take <math>\bar f(x)</math> times the inverse of <math>\bar f(x)f(x).</math> * 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)<ref>{{Citation |title=Quantum fields and strings: a course for mathematicians. Vol. 1 |date=2000 |editor-last=Deligne |editor-first=Pierre |editor-link=Pierre Deligne |access-date= |others=Institute for Advanced Study |edition=2. Nachdr. |place=Providence, RI |publisher=American Mathematical Society [u.a.] |isbn=978-0-8218-2012-4}}</ref> 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.<ref name=":0">{{Cite journal |last=Tkachov |first=Fyodor V |date=April 1997 |title=Algebraic algorithms for multiloop calculations The first 15 years. What's next? |url=https://linkinghub.elsevier.com/retrieve/pii/S0168900297001101 |journal=Nuclear Instruments and Methods in Physics Research Section A: Accelerators, Spectrometers, Detectors and Associated Equipment |language=en |volume=389 |issue=1–2 |pages=309–313 |doi=10.1016/S0168-9002(97)00110-1|arxiv=hep-ph/9609429 }}</ref> Such computations are needed for precision measurements in elementary particle physics as practiced for instance at CERN (see the papers citing<ref name=":0" />). However, the most interesting cases require a simple generalization of the Bernstein-Sato functional equation to the product of two polynomials <math>(f_1(x))^{s_1}(f_2(x))^{s_2}</math>, with ''x'' having 2-6&nbsp;scalar components, and the pair of polynomials having orders 2 and 3. Unfortunately, a brute force determination of the corresponding differential operators <math>P(s_1,s_2)</math> and <math>b(s_1,s_2)</math> 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 == {{notelist}}

== References == {{Reflist}}{{DEFAULTSORT:Bernstein-Sato polynomial}} Category:Polynomials Category:Differential operators