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

## TL;DR

This paper establishes a formula relating the Bruce-Roberts number of a function on a hypersurface with isolated singularity to Milnor and Tjurina numbers, and proves the Cohen-Macaulay property of the logarithmic characteristic variety, generalizing previous weighted homogeneous cases.

## Contribution

The paper provides a new explicit formula for the Bruce-Roberts number and proves the Cohen-Macaulay property of the logarithmic characteristic variety for general isolated hypersurface singularities.

## Key findings

- ormula relating ruce-Roberts number to Milnor and Tjurina numbers.
- ohen-Macaulayness of the logarithmic characteristic variety.
- xtension of previous weighted homogeneous results.

## Abstract

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.