Note on Milnor numbers of irreducible germs

Abstract

Let $(\mathbf{V},0)\subset ({\mathbb{C}}^{\mathbf{n}},0)$ be a germ of a complex hypersurface and let $f:({\mathbb{C}}^n,0)\to ({\mathbb{C}}^n,0)$ be a germ of a finite holomorphic mapping. If germs $(\mathbf{V},0)$ and $\mathbf{W}:=(F^{-1}(\mathbf{V})),0)$ are irreducible and with isolated singularities, then $$\mu (F^{-1}(\mathbf{V}))\ge \mu (\mathbf{V}),$$ where $\mu$ denotes the Milnor number.