“Bernstein–Sato polynomial”的意思、由来-开放百科全书

词条

Bernstein–Sato polynomial

释义

Definition and properties
Examples
Applications
Notes
References

In mathematics, the Bernstein–Sato polynomial is a polynomial related to differential operators, 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 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}} 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 is a polynomial in several variables, then there is a non-zero polynomial and a differential operator with polynomial coefficients such that

The Bernstein–Sato polynomial is the monic polynomial of smallest degree amongst such polynomials . Its existence can be shown using the notion of holonomic D-modules.

{{harvtxt|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 {{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=Mustaţǎ | first2=Mircea | last3=Saito | first3=Morihiko | 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 then

so the Bernstein–Sato polynomial is

If then

so

The Bernstein–Sato polynomial of x² + y³ is

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

which follows from

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

Applications

If is a non-negative polynomial then , 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

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;^[1] 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 times the inverse of

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.

{{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 , 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 and 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

References

{{Citation |first1=Daniel |last1=Andres |first2=Viktor |last2=Levandovskyy |first3=Jorge |last3=Martín-Morales |title=Principal Intersection and Bernstein-Sato Polynomial of an Affine Variety |arxiv=1002.3644 |doi=10.1145/1576702.1576735 |year=2009 |journal=Proc. ISSAC 2009 |publisher=Association for Computing Machinery |pages=231 }}

{{cite journal |first1=Christine |last1=Berkesch |first2=Anton |last2=Leykin |title=Algorithms for Bernstein-Sato polynomials and multiplier ideals |arxiv=1002.1475 |year=2010 |journal=Proc. ISSAC 2010 |bibcode=2010arXiv1002.1475B}}

{{cite journal |first=Joseph |last= Bernstein |authorlink=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}}

{{cite journal |last1=Budur |first1=Nero |last2=Mustaţǎ |first2=Mircea |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 |arxiv=math/0408408}}

{{cite book |authorlink=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 |authorlink=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 |authorlink=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}}

{{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}}

{{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 |pmc=426633}}

{{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 |origyear=1970 |url=http://projecteuclid.org/getRecord?id=euclid.nmj/1118782193 |mr=1086566 |year=1990 |journal=Nagoya Mathematical Journal |volume=120 |pages=1–34}}

{{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= Nucl. Instrum. Methods A | volume=389 | pages=309–313| bibcode=1997NIMPA.389..309T }}

2 : Polynomials|Differential operators

随便看

开放百科全书收录14589846条英语、德语、日语等多语种百科知识，基本涵盖了大多数领域的百科知识，是一部内容自由、开放的电子版国际百科全书。