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Dmodules :: globalBFunction(RingElement)

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

Synopsis

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