Order of vanishing
= Order of vanishing
{title2=$\operatorname{ord}_p(f)$}
The order of vanishing of a nonzero <rational function> $f$ at a point $p$ is the exponent of a local parameter in its local factorization. A negative order is the order of a pole.
The local multiplicity in <zeros and poles> is positive for a zero and negative for a pole; it is zero at a regular nonvanishing point.