next | previous | forward | backward | up | index | toc

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

## Synopsis

• Function: globalBFunction
• Usage:
b = globalBFunction f
• Inputs:
• f, , a polynomial
• Optional inputs:
• Strategy => ..., default value GeneralBernsteinSato, specify strategy for computing global b-function
• Outputs:
• b, , the b-function b(s) in Q[s]

## 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] o1 = R o1 : PolynomialRing i2 : f = x^10 10 o2 = x o2 : R i3 : b = globalBFunction f 10 11 9 66 8 363 7 157773 6 180411 5 341693 4 16819 3 o3 = s + --s + --s + ---s + ------s + ------s + ------s + -----s + 2 5 20 10000 20000 100000 20000 ------------------------------------------------------------------------ 1594197 2 66429 567 --------s + -------s + ------- 12500000 6250000 1562500 o3 : QQ[s] i4 : factorBFunction b 1 1 2 3 4 1 3 7 9 o4 = (s + 1)(s + -)(s + -)(s + -)(s + -)(s + -)(s + --)(s + --)(s + --)(s + --) 2 5 5 5 5 10 10 10 10 o4 : Expression of class Product

## See also

• bFunction -- b-function
• factorBFunction -- factorization of a b-function
• generalB -- global generalized Bernstein-Sato polynomial
• globalB -- compute global b-function and b-operator for a D-module and a polynomial