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

Synopsis

• Function: bFunction

• Usage:

  b = bFunction(I,w)

• Inputs:
  • I, a holonomic ideal in the Weyl algebra A_n(K).
  • w, 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

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 + ... + w_nt_n] (here t_i = x_iD_i).

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 variable(s)

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

See also

• globalBFunction -- global b-function (else known as the Bernstein-Sato polynomial)
• factorBFunction -- factorization of a b-function