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

## Synopsis

• Function: paramBpoly
• Usage:
paramBpoly(f,file)
• Inputs:
• f, , 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

Along with the finite list of B.-S. polynomials this function 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.