The Bruce-Roberts Numbers of a Function on an ICIS

We give formulas for the Bruce-Roberts number $\mu_{BR}(f,X)$ and its relative version $\mu_{BR}^{-}(f,X)$ of a function $f$ with respect to an ICIS $(X,0)$. We show that $\mu_{BR}^{-}(f,X)=\mu(f^{-1}(0)\cap X,0)+\mu(X,0)-\tau(X,0)$, where $\mu$ and $\tau$ are the Milnor and Tjurina numbers, respectively, of the ICIS. The formula for $\mu_{BR}(f,X)$ is more complicated and also involves $\mu(f)$ and some lengths in terms of the ideals $I_X$ and $Jf$. We also consider the logarithmic characteristic variety, $LC(X)$, and its relative version, $LC(X)^{-}$. We show that $LC(X)^{-}$ is Cohen-Macaulay and that $LC(X)$ is Cohen-Macaulay at any point not in $X\times\{0\}$. We generalize previous results presented by the authors when $(X,0)$ has codimension one and by Bruce and Roberts when it is weighted homogeneous of any codimension.