= Critical point of a rational map
{title2=$e_p(R)\geq2$}
= Critical points of a rational map
{synonym}
= Rational critical point
{synonym}
For a nonconstant rational map, choose local source and target coordinates centred at $p$ and $R(p)$. If the local expression is $a u^e+O(u^{e+1})$ with $a\ne0$, its local degree is $e$, and $p$ is critical exactly when $e\geq2$. This definition covers poles and infinity; a simple pole is not critical merely because the usual finite-coordinate expression is unbounded. Coordinate changes have nonzero first derivatives and leave $e$ unchanged.
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