View source for Bernstein–Sato polynomial

{{Short description|Polynomial related to differential operators}}
{{format footnotes |date=May 2024|reason=Parenthetical referencing has been [[WP:PARREF|deprecated]]; convert to shortened footnotes.}}
In [[mathematics]], the '''Bernstein–Sato polynomial''' is a polynomial related to [[differential operator]]s,  introduced independently by {{harvs|txt=yes|authorlink=Joseph Bernstein|first=Joseph|last=Bernstein|year=1971}} and {{harvs|txt|author1-link=Mikio Sato|last1=Sato|first1=Mikio|last2=Shintani|first2=Takuro|year1=1972|year2=1974}}, {{harvtxt|Sato|1990}}. It is also known as the '''b-function''',  the '''b-polynomial''', and the '''Bernstein polynomial''', though it is not related to the [[Bernstein polynomial]]s used in [[approximation theory]]. It has applications to [[singularity theory]], [[monodromy theory]], and [[quantum field theory]].

{{harvs|txt|first=Severino|last=Coutinho|year=1995}} gives an elementary introduction, while {{harvs|txt|first=Armand|last=Borel|authorlink=Armand Borel|year=1987}} and {{harvs|txt|last=Kashiwara | first=Masaki | author-link=Masaki Kashiwara|year=2003}} give more advanced accounts.

==Definition and properties==

If <math>f(x)</math> is a polynomial in several variables, then there is 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-module]]s.

{{harvtxt|Kashiwara|1976}} proved that all roots of the Bernstein–Sato polynomial are negative [[rational number]]s.

The Bernstein–Sato polynomial can also be defined for products of  powers of several polynomials {{harv|Sabbah|1987}}. 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.

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]].

{{harvs|txt|first1=Daniel|last1=Andres | first2=Viktor|last2= Levandovskyy | first3 = Jorge |last3=Martín-Morales| year=2009}} 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.

== 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 [[analytic continuation|analytically continued]] to a [[meromorphic]] [[Distribution (mathematics)|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 transform]]s this follows from the fact that every polynomial has a distributional inverse, which is proved in the paragraph above.
*{{harvs|txt|first=Pavel|last=Etingof|authorlink=Pavel Etingof|year=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]] {{harvs|txt|first=Fyodor |last=Tkachov|year=1997}}. Such computations are needed for precision measurements in elementary particle physics as practiced for instance at [[CERN]] (see the papers citing {{harv|Tkachov|1997}}). 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&nbsp;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 ==
*{{Cite book|first1=Daniel |last1=Andres |first2=Viktor |last2=Levandovskyy |first3=Jorge |last3=Martín-Morales |title=Proceedings of the 2009 international symposium on Symbolic and algebraic computation |chapter=Principal intersection and bernstein-sato polynomial of an affine variety |arxiv=1002.3644 |doi=10.1145/1576702.1576735 |year=2009  |publisher=[[Association for Computing Machinery]] |pages=231–238 |isbn=9781605586090 |s2cid=2747775 }}
*{{cite book |first1=Christine |last1=Berkesch |first2=Anton |last2=Leykin |title=Proceedings of the 2010 International Symposium on Symbolic and Algebraic Computation |chapter=Algorithms for Bernstein--Sato polynomials and multiplier ideals |arxiv=1002.1475 |year=2010 |pages=99–106 |doi=10.1145/1837934.1837958|isbn=9781450301503 |s2cid=33730581 }}
*{{cite journal |first=Joseph |last= Bernstein |author-link=Joseph Bernstein |title=Modules over a ring of differential operators. Study of the fundamental solutions of equations with constant coefficients |journal=Functional Analysis and Its Applications |volume=5 |issue=2 |year=1971 |doi=10.1007/BF01076413 |pages=89–101 |mr=0290097|s2cid= 124605141 }}
*{{cite journal |last1=Budur |first1=Nero |last2=Mustață |first2=Mircea | author2-link=Mircea Mustață| last3=Saito |first3=Morihiko |title=Bernstein-Sato polynomials of arbitrary varieties |doi=10.1112/S0010437X06002193 |mr=2231202 |year=2006 |journal=[[Compositio Mathematica]]|volume=142 |issue=3 |pages=779–797 |doi-broken-date=20 February 2025 |arxiv=math/0408408 |bibcode=2004math......8408B|s2cid=6955564 }}
*{{cite book |author-link=Armand Borel |first=Armand |last=Borel |title=Algebraic D-Modules |series=Perspectives in Mathematics |volume=2 |publisher=[[Academic Press]] |publication-place=Boston, MA |year=1987 |isbn=0-12-117740-8}}
*{{cite book |first=Severino C. |last=Coutinho |title=A primer of algebraic D-modules |series=London Mathematical Society Student Texts |volume=33 |publisher=[[Cambridge University Press]] |publication-place=Cambridge, UK |year=1995 |isbn=0-521-55908-1}}
*{{cite book |last=Etingof |first=Pavel |author-link=Pavel Etingof |title=Quantum fields and strings: A course for mathematicians |volume=1 |url=http://www.math.ias.edu/QFT/fall/ |publisher=American Mathematical Society |location=Providence, R.I. |isbn=978-0-8218-2012-4 |mr=1701608 |year=1999 |chapter=Note on dimensional regularization |pages=597–607}} (Princeton, NJ, 1996/1997)
*{{cite journal |last=Kashiwara |first=Masaki |author-link=Masaki Kashiwara |title=B-functions and holonomic systems. Rationality of roots of B-functions |doi=10.1007/BF01390168 |mr=0430304 |year=1976 |journal=[[Inventiones Mathematicae]] |volume=38 |issue=1 |pages=33–53 |bibcode=1976InMat..38...33K|s2cid=17103403 }}
*{{cite book |last=Kashiwara |first=Masaki |author-link=Masaki Kashiwara |title=D-modules and microlocal calculus |publisher=[[American Mathematical Society]] |location=Providence, R.I. |series=Translations of Mathematical Monographs |isbn=978-0-8218-2766-6 |mr=1943036 |year=2003 |volume=217}}
*{{cite journal |last1=Sabbah |first1=Claude |title=Proximité évanescente. I. La structure polaire d'un D-module |url=http://www.numdam.org/item?id=CM_1987__62_3_283_0 |mr=901394 |year=1987 |journal=[[Compositio Mathematica]] |volume=62 |issue=3 |pages=283–328}}
*{{cite journal |doi=10.1073/pnas.69.5.1081 |last1=Sato |first1=Mikio |authorlink1=Mikio Sato |last2=Shintani |first2=Takuro |title=On zeta functions associated with prehomogeneous vector spaces |jstor=61638 |mr=0296079 |year=1972 |journal=[[Proceedings of the National Academy of Sciences of the United States of America]] |volume=69 |pages=1081–1082 |issue=5 |pmc=426633 |bibcode=1972PNAS...69.1081S |pmid=16591979|doi-access=free }}
*{{cite journal |doi=10.2307/1970844 |last1=Sato |first1=Mikio |authorlink1=Mikio Sato |last2=Shintani |first2=Takuro |title=On zeta functions associated with prehomogeneous vector spaces |jstor=1970844 |mr=0344230 |year=1974 |journal=[[Annals of Mathematics]] |series=Second Series |volume=100 |issue=1 |pages=131–170}}
*{{cite journal |last1=Sato |first1=Mikio |authorlink1=Mikio Sato |title=Theory of prehomogeneous vector spaces (algebraic part) |quote=the English translation of Sato's lecture from Shintani's note |orig-year=1970 |url=http://projecteuclid.org/getRecord?id=euclid.nmj/1118782193 |mr=1086566 |year=1990 |journal=Nagoya Mathematical Journal |volume=120 |pages=1–34 |doi=10.1017/s0027763000003214|doi-access=free }}
*{{cite journal | last1=Tkachov | first1=Fyodor V. | title=Algebraic algorithms for multiloop calculations. The first 15 years. What's next? | arxiv=hep-ph/9609429 | doi=10.1016/S0168-9002(97)00110-1 | year=1997 | journal= [[Nuclear Instruments and Methods in Physics Research|Nucl. Instrum. Methods A]] | volume=389 | issue=1–2 | pages=309–313| bibcode=1997NIMPA.389..309T | s2cid=37109930 }}

[[Category:Polynomials]]
[[Category:Differential operators]]