= 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 $f$.
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