On algebraic bi-Lipschitz homeomorphisms

Abstract

Let $X\subset {\mathbb{C}}^n;Y\subset {\mathbb{C}}^m$ be closed affine varieties and let $\varphi :X\to Y$ be an algebraic bi-Lipschitz homeomorphism. Then $\mathrm{deg}X=\mathrm{deg}Y.$ Similarly, let $(X,0)\subset ({\mathbb{C}}^n,0),(Y,0)\subset ({\mathbb{C}}^m,0)$ be germs of analytic sets and let $f:(X,0)\to (Y,0)$ be a c-holomorphic and bi-Lipschitz mapping. Then ${\mathrm{mult}}_{0}X={\mathrm{mult}}_{0}Y.$ Finally we show that the normality is not a bi-Lipschitz invariant.