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

Synopsis

Function: bFunction
Usage:

b = bFunction(M,w,m)
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 => ..., default value IntRing, 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

The algorithm represents M as F/N where F is free and N is a submodule of F. Then it computes b-functions b_i(s) for N \cap F_i (i-th component of F) and outputs lcm{ b_i(s-m_i) }

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

o1 = R

o1 : PolynomialRing, 1 differential variable(s)

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

The Weyl algebra should not have any parameters. Similarly, it should not be a homogeneous Weyl algebra

Ways to use this method: