On Zariski's multiplicity problem (Q2782635)

The paper presents a small but nevertheless very interesting progress in answering the celebrated question of Zariski concerning the topological invariance of the multiplicity of germs of complex hypersurfaces. Let \(f,g: ({\mathbb C}^n,0) \rightarrow({\mathbb C},0)\) be two germs of holomorphic functions and \(\varphi:({\mathbb C}^n,0) \rightarrow({\mathbb C}^n,0)\) be a germ of homeomorphism sending \(f^{-1}(0)\) onto \(g^{-1}(0)\). The main result of the paper is the proof of the following criterion:NEWLINENEWLINENEWLINEIf there are positive constants \(A,B,C,D\) such that for all \(z\) close to origin NEWLINE\[NEWLINEA|z|\leq |\varphi(z)|\leq B|z|\quad \text{and} \quad C|f(z)|\leq |g(\varphi(z))|\leq D|f(z)|NEWLINE\]NEWLINE then the germs of hypersurfaces \(f^{-1}(0)\) and \(g^{-1}(0)\) have the same multiplicity.