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

TL;DR

This paper introduces new computational methods for local $b$-functions in $mbda$-constant deformations of semi-weighted homogeneous singularities, utilizing Grf6bner systems and holonomic ${\\mathcal D}$-modules to reduce complexity.

Contribution

It presents novel algorithms leveraging semi-weighted homogeneity and advanced algebraic structures to efficiently compute $b$-functions for specific singularity deformations.

Findings

Methods successfully compute local $b$-functions for $mbda$-constant deformations.

Reduction in computational complexity due to semi-weighted homogeneity.

Explicit list of parametric local $b$-functions for 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 Gr\"obner systems in Poincar\'e-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.