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globalBFunction(RingElement) -- global b-function (else known as the Bernstein-Sato polynomial)

Synopsis

Usage:
b = globalBFunction f
Function: globalBFunction
Inputs:
- f, a ring element, a polynomial in a Weyl algebra, that does not contain differential variables
Outputs:
- b, a ring element, the b-function b(s) in Q[s]
Optional inputs:
- Strategy => ..., -- specify strategy for computing global b-function

Description

Definition. Let D = A_{2n}(K) = K<x_1,...,x_n,d_1,...,d_n> be a Weyl algebra. The Bernstein-Sato polynomial of a polynomial f is defined to be the monic generator of the ideal of all polynomials b(s) in K[s] such that b(s) f^s = Q(s,x,d) f^{s+1} where Q lives in D[s].

Algorithm. Let I_f = D<t,dt>*<t-f, d_1+df/dx_1*dt, ..., d_n+df/dx_n*dt> Let B(s) = bFunction(I, {1, 0, ..., 0}) where 1 in the weight that corresponds to dt. Then the global b-function is b_f = B(-s-1)

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

o1 = R

o1 : PolynomialRing

i2 : f = x^10

      10
o2 = x

o2 : R

i3 : b = globalBFunction f

              10            9             8             7             6  
o3 = 12500000s   + 68750000s  + 165000000s  + 226875000s  + 197216250s  +
     ------------------------------------------------------------------------
               5            4            3           2
     112756875s  + 42711625s  + 10511875s  + 1594197s  + 132858s + 4536

o3 : QQ [s]

i4 : factor b

o4 = (s + 1)(2s + 1)(5s + 1)(5s + 2)(5s + 3)(5s + 4)(10s + 1)(10s + 3)(10s + 7)(10s + 9)

o4 : Expression of class Product

Caveat

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

See also

bFunction -- b-function