If is a polynomial in several variables, then there is a non-zero polynomial and a differential operator with polynomial coefficients such that
The BernsteinâSato polynomial is the
monic polynomial of smallest degree amongst such polynomials . Its existence can be shown using the notion of holonomic
D-modules.
The BernsteinâSato 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]
Note, that the BernsteinâSato 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.
Daniel Andres, Viktor Levandovskyy, and Jorge MartĂn-Morales (
2009) presented algorithms to compute the BernsteinâSato polynomial of an affine variety together with an implementation in the computer algebra system
SINGULAR.
Christine Berkesch and Anton Leykin (
2010) described some of the algorithms for computing BernsteinâSato polynomials by computer.
Examples
If then
so the BernsteinâSato polynomial is
If then
so
The BernsteinâSato polynomial of x2 + y3 is
If tij are n2 variables, then the BernsteinâSato polynomial of det(tij) is given by
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;[a] in other words, f g = 1 as distributions. If f(x) is non-negative the inverse can be constructed using the BernsteinâSato 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 Bernstein-Sato functional equation is used in computations of some of the more complex kinds of singular integrals occurring in
quantum field theory 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 , 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
^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.
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. 99â106.
arXiv:1002.1475.
doi:
10.1145/1837934.1837958.
ISBN9781450301503.
S2CID33730581.
Bernstein, Joseph (1971). "Modules over a ring of differential operators. Study of the fundamental solutions of equations with constant coefficients". Functional Analysis and Its Applications. 5 (2): 89â101.
doi:
10.1007/BF01076413.
MR0290097.
S2CID124605141.
Coutinho, Severino C. (1995). A primer of algebraic D-modules. London Mathematical Society Student Texts. Vol. 33. Cambridge, UK:
Cambridge University Press.
ISBN0-521-55908-1.
If is a polynomial in several variables, then there is a non-zero polynomial and a differential operator with polynomial coefficients such that
The BernsteinâSato polynomial is the
monic polynomial of smallest degree amongst such polynomials . Its existence can be shown using the notion of holonomic
D-modules.
The BernsteinâSato 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]
Note, that the BernsteinâSato 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.
Daniel Andres, Viktor Levandovskyy, and Jorge MartĂn-Morales (
2009) presented algorithms to compute the BernsteinâSato polynomial of an affine variety together with an implementation in the computer algebra system
SINGULAR.
Christine Berkesch and Anton Leykin (
2010) described some of the algorithms for computing BernsteinâSato polynomials by computer.
Examples
If then
so the BernsteinâSato polynomial is
If then
so
The BernsteinâSato polynomial of x2 + y3 is
If tij are n2 variables, then the BernsteinâSato polynomial of det(tij) is given by
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;[a] in other words, f g = 1 as distributions. If f(x) is non-negative the inverse can be constructed using the BernsteinâSato 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 Bernstein-Sato functional equation is used in computations of some of the more complex kinds of singular integrals occurring in
quantum field theory 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 , 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
^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.
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. 99â106.
arXiv:1002.1475.
doi:
10.1145/1837934.1837958.
ISBN9781450301503.
S2CID33730581.
Bernstein, Joseph (1971). "Modules over a ring of differential operators. Study of the fundamental solutions of equations with constant coefficients". Functional Analysis and Its Applications. 5 (2): 89â101.
doi:
10.1007/BF01076413.
MR0290097.
S2CID124605141.
Coutinho, Severino C. (1995). A primer of algebraic D-modules. London Mathematical Society Student Texts. Vol. 33. Cambridge, UK:
Cambridge University Press.
ISBN0-521-55908-1.