= Order of a meromorphic function along a hypersurface
{title2=$\operatorname{ord}_Y(f)$}
At a generic smooth point of an irreducible <complex analytic hypersurface> $Y$, a nonzero <meromorphic function> has the form $f=t^m u$, where $t$ is a reduced transverse local equation and $u$ is a holomorphic unit. The integer $m$ is its order along $Y$: positive for a zero and negative for a pole. The <unique factorization domain> property of the <local ring> of a <complex manifold> makes this independent of the equation and propagates it along $Y$. These orders are the coefficients of the <principal divisor on a complex manifold> of $f$.
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