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# generalB(List,RingElement) -- global generalized Bernstein-Sato polynomial

## Synopsis

• Function: generalB
• Usage:
b = generalB(F,g), b = generalB F
• Inputs:
• F, a list, a list of polynomials
• g, , a polynomial
• Optional inputs:
• Exponent => ..., default value null, specify exponent m for m-generalized Bernstein-Sato polynomial
• Strategy => ..., default value InitialIdeal, specify strategy for computing generalized Bernstein-Sato polynomial
• Outputs:
• b, , the general Bernstein-Sato polynomial b(s) in Q[s]

## Description

Bernstein-Sato polynomial for an arbitrary affine variety was introduced in Budur, Mustata, and Saito Bernstein--Sato polynomials of arbitrary varieties''. If the option Exponent is specified, then the m-generalized Bernstein-Sato polynomial is computed. See Berkesch and Leykin Algorithms for Bernstein-Sato polynomials and multiplier ideals'' for definitions.
 i1 : W = makeWA(QQ[x_1..x_3]); i2 : factorBFunction generalB ({x_2^2-x_1*x_3, x_1^3-x_3^2}, x_2) 5 9 11 11 13 29 31 35 37 o2 = (s + 2)(s + -)(s + -)(s + --)(s + --)(s + --)(s + --)(s + --)(s + --)(s + --) 2 4 4 6 6 12 12 12 12 o2 : Expression of class Product

## Caveat

The input could be either in a polynomial ring or the Weyl algebra. In the latter case the algebra should not have any central variables and should not be a homogeneous Weyl algebra.