Local defining function of a complex analytic hypersurface

A local defining function for a complex analytic hypersurface $Y$ near $x$ is a holomorphic function $f$ on a neighbourhood $U$ such that $Y\cap U=\{f=0\}$. It may be chosen reduced, with each local irreducible factor occurring once. The local ring ${\mathcal{O}}_{X,x}$ of a complex manifold is a regular local ring and hence a unique factorization domain; each height-one prime of the hypersurface germ is principal, and the product of generators for its finitely many local branches gives $f$.