Comparison of theoretical complexities of two methods for computing annihilating ideals of polynomials

Let f1,…,fp be polynomials in C[x1,…,xn] and let D=Dn be the n-th Weyl algebra. We provide upper bounds for the complexity of computing the annihilating ideal of fs=f1s1⋯fpsp in D[s]=D[s1,…,sp]. These bounds provide an initial explanation of the differences between the running times of the two methods known to obtain the so-called Bernstein-Sato ideals.