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 |