In mathematics

Mathematics

Mathematics is the study of quantity, space, structure, and change. Mathematicians seek out patterns and formulate new conjectures. Mathematicians resolve the truth or falsity of conjectures by mathematical proofs, which are arguments sufficient to convince other mathematicians of their validity...

, the Bernstein–Sato polynomial is a polynomial related to differential operator

Differential operator

In mathematics, a differential operator is an operator defined as a function of the differentiation operator. It is helpful, as a matter of notation first, to consider differentiation as an abstract operation, accepting a function and returning another .This article considers only linear operators,...

s, introduced independently by and , . It is also known as the b-function, the b-polynomial, and the Bernstein polynomial, though it is not related to the Bernstein polynomial

Bernstein polynomial

In the mathematical field of numerical analysis, a Bernstein polynomial, named after Sergei Natanovich Bernstein, is a polynomial in the Bernstein form, that is a linear combination of Bernstein basis polynomials....

s used in approximation theory

Approximation theory

In mathematics, approximation theory is concerned with how functions can best be approximated with simpler functions, and with quantitatively characterizing the errors introduced thereby...

. It has applications to singularity theory

Singularity theory

-The notion of singularity:In mathematics, singularity theory is the study of the failure of manifold structure. A loop of string can serve as an example of a one-dimensional manifold, if one neglects its width. What is meant by a singularity can be seen by dropping it on the floor...

 and monodromy theory and quantum field theory

Quantum field theory

Quantum field theory provides a theoretical framework for constructing quantum mechanical models of systems classically parametrized by an infinite number of dynamical degrees of freedom, that is, fields and many-body systems. It is the natural and quantitative language of particle physics and...

.

gives an elementary introduction, and and give more advanced accounts.

Definition and properties

If ƒ(x) is a polynomial in several variables then there is a non-zero polynomial b(s) and a differential operator P(s) with polynomial coefficients such that


The Bernstein–Sato polynomial is the monic polynomial of smallest degree amongst such b(s). Its existence can be shown using the notion of holonomic D-module

D-module

In mathematics, a D-module is a module over a ring D of differential operators. The major interest of such D-modules is as an approach to the theory of linear partial differential equations...

s.

proved that all roots of the Bernstein–Sato polynomial are negative rational number

Rational number

In mathematics, a rational number is any number that can be expressed as the quotient or fraction a/b of two integers, with the denominator b not equal to zero. Since b may be equal to 1, every integer is a rational number...

s.

The Bernstein–Sato polynomial can also be defined for products of powers of several polynomials . In this case it is a product of linear factors with rational coefficients.

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

Macaulay2

Macaulay2 is a free computer algebra system developed by Daniel Grayson and Michael Stillman for computation in commutative algebra and algebraic geometry. Stillman, along with Dave Bayer had authored the predecessor, Macaulay. Macaulay2 uses its own high level programming language, intended to...

 and SINGULAR

SINGULAR

SINGULAR is a computer algebra system for polynomial computations with special emphasis on the needs of commutative algebra, algebraic geometry, and singularity theory. SINGULAR is free software released under the GNU General Public License. Problems in non-commutative algebra can be tackled with...

.

presented algorithms to compute the Bernstein–Sato polynomial of an affine variety together with an implementation in the computer algebra system SINGULAR

SINGULAR

SINGULAR is a computer algebra system for polynomial computations with special emphasis on the needs of commutative algebra, algebraic geometry, and singularity theory. SINGULAR is free software released under the GNU General Public License. Problems in non-commutative algebra can be tackled with...

.

described some of the algorithms for computing Bernstein–Sato polynomials by computer.

Examples

• If then

so the Bernstein–Sato polynomial is

• If then

• 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 f(x) is a non-negative polynomial then f(x)^s, initially defined for s with non-negative real part, can be analytically continued
  Analytic continuation
  In complex analysis, a branch of mathematics, analytic continuation is a technique to extend the domain of a given analytic function. Analytic continuation often succeeds in defining further values of a function, for example in a new region where an infinite series representation in terms of which...
  
   to a meromorphic distribution
  Distribution (mathematics)
  In mathematical analysis, distributions are objects that generalize functions. Distributions make it possible to differentiate functions whose derivatives do not exist in the classical sense. In particular, any locally integrable function has a distributional derivative...
  
  -valued function of s by repeatedly using the functional equation
  Functional equation
  In mathematics, a functional equation is any equation that specifies a function in implicit form.Often, the equation relates the value of a function at some point with its values at other points. For instance, properties of functions can be determined by considering the types of functional...
  
  

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; in other words, fg = 1 as distributions. (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. The usual proof of uniqueness of inverses fails because the product of distributions is not always defined, and need not be associative even when it is defined.) 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
  Malgrange–Ehrenpreis theorem
  In mathematics, the Malgrange–Ehrenpreis theorem states that every non-zero linear differential operator with constant coefficients has a Green's function...
  
   states that every differential operator
  Differential operator
  In mathematics, a differential operator is an operator defined as a function of the differentiation operator. It is helpful, as a matter of notation first, to consider differentiation as an abstract operation, accepting a function and returning another .This article considers only linear operators,...
  
   with constant coefficients
  Constant coefficients
  In mathematics, constant coefficients is a term applied to differential operators, and also some difference operators, to signify that they contain no functions of the independent variables, other than constant functions. In other words, it singles out special operators, within the larger class of...
  
   has a Green's function
  Green's function
  In mathematics, a Green's function is a type of function used to solve inhomogeneous differential equations subject to specific initial conditions or boundary conditions...
  
  . By taking Fourier transform
  Fourier transform
  In mathematics, Fourier analysis is a subject area which grew from the study of Fourier series. The subject began with the study of the way general functions may be represented by sums of simpler trigonometric functions...
  
  s this follows from the fact that every polynomial has a distributional inverse, which is proved in the paragraph above.

showed how to use the Bernstein polynomial to define dimensional regularization

Dimensional regularization

In theoretical physics, dimensional regularization is a method introduced by Giambiagi and Bollini for regularizing integrals in the evaluation of Feynman diagrams; in other words, assigning values to them that are meromorphic functions of an auxiliary complex parameter d, called the...

 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
  Quantum field theory
  Quantum field theory provides a theoretical framework for constructing quantum mechanical models of systems classically parametrized by an infinite number of dynamical degrees of freedom, that is, fields and many-body systems. It is the natural and quantitative language of particle physics and...
  
   . Such computations are needed for precision measurements in elementary particle physics as practiced e.g. at CERN
  CERN
  The European Organization for Nuclear Research , known as CERN , is an international organization whose purpose is to operate the world's largest particle physics laboratory, which is situated in the northwest suburbs of Geneva on the Franco–Swiss border...
  
  (see the papers citing ). 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.