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.

中文翻译：

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计算与$μ$恒定变形相关的$ b $函数的方法-内模2的情况-

引入了用于计算参数局部$ b $函数的新方法，用于半加权均质奇异点的$ \ mu $常数变形。该方法的关键是Poincar \'e-Birkhoff-Witt代数中的综合Gr'obner系统和完整的$ {\ mathcal D} $模块。结果表明，半加权同质性的使用降低了与$ \ mu $恒定变形相关的$ b $函数在内部模态2的情况下，通过结果方法获得与$ \ mu $恒定变形相关的局部$ b $函数并给出参数列表本地$ b $功能。