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We consider the relative Bruce–Roberts number $\mu _{\textrm {BR}}^{-}(f,\,X)$
of a function on an isolated hypersurface singularity $(X,\,0)$
. We show that $\mu _{\textrm {BR}}^{-}(f,\,X)$
is equal to the sum of the Milnor number of the fibre $\mu (f^{-1}(0)\cap X,\,0)$
plus the difference $\mu (X,\,0)-\tau (X,\,0)$
between the Milnor and the Tjurina numbers of $(X,\,0)$
. As an application, we show that the usual Bruce–Roberts number $\mu _{\textrm {BR}}(f,\,X)$
is equal to $\mu (f)+\mu _{\textrm {BR}}^{-}(f,\,X)$
. We also deduce that the relative logarithmic characteristic variety $LC(X)^{-}$
, obtained from the logarithmic characteristic variety $LC(X)$
by eliminating the component corresponding to the complement of $X$
in the ambient space, is Cohen–Macaulay.
