Methods for computing $b$-functions associated with $\mu$-constant deformations -- Case of inner modality 2 --

Abstract

New methods for computing parametric local $b$-functions are introduced for $\mu$-constant deformations of semi-weighted homogeneous singularities. The keys of the methods are comprehensive Gröbner systems in Poincaré-Birkhoff-Witt algebra and holonomic $\mathcal{D}$-modules. It is shown that the use of semi-weighted homogeneity reduces the computational complexity of $b$-functions associated with $\mu$-constant deformations. In the case of inner modality 2, local $b$-functions associated with $\mu$-constant deformations are obtained by the resulting method and given the list of parametric local $b$-functions.