Macaulay2 » Documentation
Packages » BernsteinSato :: paramBpoly(RingElement,File)
next | previous | forward | backward | up | index | toc

# paramBpoly(RingElement,File) -- compute the list of all possible Bernstein-Sato polynomials for a parametric polynomial

- Function: paramBpoly
- Usage:
    `paramBpoly(f,file)`
- Inputs:
  - f, a ring element, a polynomial in Weyl algebra A_n(Q)
  - file, a file, a file for TeX output (or stdio)
- Optional inputs:
  - ground field
- Outputs:
  - a list, all possible Bernstein-Sato polynomials

## Description

This function computes the finite list of all possible Bernstein-Sato polynomials for a polynomial with parametric coefficients and creates latex code describing strata in the parameter space corresponding to the B.-S. polynomials --- each stratum is a constructible set. This is an implementation of the algorithmic approach in Anton Leykin. Constructibility of the Set of Polynomials with a Fixed Bernstein-Sato Polynomial: an Algorithmic Approach. Journal of Symbolic Computation, 32(6):663–675, 2001.

    i1 : A =  (QQ [a,b,c]) [x, y, Dx, Dy, WeylAlgebra => {x=>Dx, y=>Dy}]

    o1 = A

    o1 : PolynomialRing, 2 differential variable(s)

    i2 : paramBpoly(a*x^2 + b*x*y + c*y^2, stdio)
    \begin{itemize}
    \item $b(s)=(S+1)^2$ corresponds to
    $V(0)$$\setminus$$V(b^2-4*a*c)$
    \item $b(s)=(S+1/2)*(S+1)$ corresponds to
    $V(b^2-4*a*c)$
    \end{itemize}

                 2       1
    o2 = {(S + 1) , (S + -)(S + 1)}
                         2

    o2 : List

## Caveat

A finite field ZZ/p is used to speed up computations. Option "ground field" may be used to change the characteristic p. If p=0 the computation will be attempted over QQ.

## See also

- globalBFunction -- global b-function (else known as the Bernstein-Sato polynomial)

## Ways to use this method:

- paramBpoly(RingElement,File) -- compute the list of all possible Bernstein-Sato polynomials for a parametric polynomial

The source of this document is in /build/macaulay2-88fgJW/macaulay2-1.25.11+ds/M2/Macaulay2/packages/BernsteinSato/DOC/bFunctions.m2:407:0.