bFunction(Module,List,List) -- b-function of a holonomic D-module

Synopsis

• Usage:

  b = bFunction(M,w,m)

• Function: bFunction
• Inputs:

  • M, a module, a holonomic module over a Weyl algebra A_n(K)
  • w, a list, a list of integer weights corresponding to the differential variables in the Weyl algebra
  • m, a list, a list of integers, each of which is the shift for the corresponding component
• Optional inputs:

  • Strategy => ..., -- specify strategy for computing b-function
• Outputs:

  • b, a ring element, a polynomial b(s) which is the b-function of M with respect to w and m

Description

i1 : R = QQ[x, dx, WeylAlgebra => {x=>dx}]

o1 = R

o1 : PolynomialRing

i2 : M = cokernel matrix {{x^2, 0, 0}, {0, dx^3, 0}, {0, 0, x^3}}

o2 = cokernel | x2 0    0  |
              | 0  dx^3 0  |
              | 0  0    x3 |

                            3
o2 : R-module, quotient of R

i3 : factorBFunction bFunction(M, {1}, {0,0,0})

o3 = (s)(s - 2)(s - 1)(s + 1)(s + 2)(s + 3)

o3 : Expression of class Product

i4 : factorBFunction bFunction(M, {1}, {1,2,3})

o4 = (s)(s - 4)(s - 3)(s - 2)(s - 1)(s + 1)

o4 : Expression of class Product

Caveat

See also

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