Stratification associated with local b-functions

The local b-function bf,p(s) of an n-variate polynomial f∈C[x] (x=(x1,…,xn)) at a point p∈Cn is constant on each stratum of a stratification of Cn. We propose a new method for computing such a stratification and bf,p(s) on each stratum. In the existing method proposed in Oaku (1997b), a primary ideal decomposition of an ideal in C[x,s] is needed and our experiment shows that the primary decomposition can be a bottleneck for computing the stratification. In our new method, the computation can be done by just computing ideal quotients and examining inclusions of algebraic sets. The precise form of a stratum can be obtained by computing the decomposition of the radicals of the ideals in C[x] defining the stratum. We also introduce various techniques for improving the practical efficiency of the implementation and we show results of computations for some examples.