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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 $f\colon(\mathbbC^n,0)\to\mathbb C$ such that the Bruce-Roberts number $\mu_{BR}(f,X)$ isfinite. 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 isolatedcomplete intersection singularity. Second, we show that the logarithmiccharacteristic variety $LC(X,0)$ is Cohen-Macaulay. Both theorems generalizethe results of a previous paper by some of the authors, in which thehypersurface $(X,0)$ was assumed to be weighted homogeneous.

The Bruce-Roberts number of a function on a hypersurface with isolated singularity