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.

中文翻译：

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具有孤立奇异性的超曲面上的函数的布鲁斯-罗伯茨数

令| $（X，0）$ | 是由| $ \ phi \ colon（{\ mathbb {C}} ^ n，0）\ to（{\ mathbb {C}}，0）$ |定义的孤立的超表面奇点 和| $ f \冒号（{\ mathbb {C}} ^ n，0）\至{\ mathbb {C}} $$ | 这样Bruce–Roberts数| $ \ mu _ {BR}（f，X）$ | 是有限的。我们首先证明| $ \ mu _ {BR}（f，X）= \ mu（f）+ \ mu（\ phi，f）+ \ mu（X，0）-\ tau（X，0）$ | ⁠，其中| $ \ mu $ | 和| $ \ tau $ | 是函数或孤立的完整交点奇点的Milnor和Tjurina数。其次，我们显示对数特征变量| $ LC（X，0）$ |是科恩·马考雷。这两个定理都推广了一些作者的前一篇论文的结果，其中超曲面| $（X，0）$ | 假定加权均质。