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Let   $(X,0)$   be an isolated hypersurface singularity defined by   $\phi \colon ({\mathbb{C}}^n,0)\to ({\mathbb{C}},0)$   and   $f\colon ({\mathbb{C}}^n,0)\to{\mathbb{C}}$   such that the Bruce–Roberts number   $\mu _{BR}(f,X)$   is finite. We first prove that   $\mu _{BR}(f,X)=\mu (f)+\mu (\phi ,f)+\mu (X,0)-\tau (X,0)$  , where   $\mu $   and   $\tau $   are the Milnor and Tjurina numbers respectively of a function or an isolated complete intersection singularity. Second, we show that the logarithmic characteristic variety   $LC(X,0)$   is Cohen–Macaulay. Both theorems generalize the results of a previous paper by some of the authors, in which the hypersurface   $(X,0)$   was assumed to be weighted homogeneous.

The Bruce–Roberts Number of A Function on A Hypersurface with Isolated Singularity