Proves μ_BR(f,X) = μ(f) + μ(φ,f) + μ(X,0) − τ(X,0) and that LC(X,0) is Cohen-Macaulay for isolated hypersurface singularities without assuming weighted homogeneity.
Tajima, On Polar Varieties, Logarithmic Vector Fields and Holonomi c D-modules , RIMS Kˆ okyˆ uroku Bessatsu
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The Bruce-Roberts number of a function on a hypersurface with isolated singularity
Proves μ_BR(f,X) = μ(f) + μ(φ,f) + μ(X,0) − τ(X,0) and that LC(X,0) is Cohen-Macaulay for isolated hypersurface singularities without assuming weighted homogeneity.