# bFunction(Ideal,List) -- b-function of an ideal

## Synopsis

• Function: bFunction
• Usage:
b = bFunction(I,w)
• Inputs:
• I, an ideal, a holonomic ideal in the Weyl algebra An(K).
• w, a list, a list of integer weights corresponding to the differential variables in the Weyl algebra.
• Optional inputs:
• Strategy => ..., default value IntRing, specify strategy for computing b-function
• Outputs:
• b, , a polynomial b(s) which is the b-function of I with respect to w

## Description

Use setHomSwitch(true) to force all the subroutines to use homogenized WeylAlgebra

Definition. The b-function b(s) is defined as the monic generator of the intersection of in(-w,w)(I) and K[s], where s = [w1t1 + ... + wntn] (here ti = xiDi).

 i1 : R = QQ[x_1,x_2,D_1,D_2,WeylAlgebra=>{x_1=>D_1,x_2=>D_2}] o1 = R o1 : PolynomialRing, 2 differential variables i2 : I = ideal(x_1, D_2-1) o2 = ideal (x , D - 1) 1 2 o2 : Ideal of R i3 : bFunction(I,{1, 0}) o3 = s + 1 o3 : QQ[s]

## Caveat

The ring of I should not have any parameters: it should be a pure Weyl algebra. Similarly, this ring should not be a homogeneous WeylAlgebra