Degree of a rational map of the Riemann sphere

After cancelling common factors, the rational map $p/q$ on the Riemann sphere has degree $\max (\mathrm{deg}p,\mathrm{deg}q)$. It is an analytic isomorphism exactly when this degree is one, equivalently when it is a Möbius transformation.

For a nonconstant rational function of degree $n$, its derivative has degree between $n-1$ and $2n$. A finite pole of order $m$ becomes one of order $m+1$. If infinity is a pole of order $p-q>0$, it contributes $p-q-1$ to the derivative's pole count, where $p,q$ are the degrees of coprime numerator and denominator. This proves the lower bound; the quotient rule denominator $Q^2$ proves the upper bound. The functions $z^n$ and $1/(z^n-1)$ attain the bounds.

Let a nonconstant rational function $f$ have degree $d$, let $r$ be its number of distinct finite poles, and let $m_{\infty }$ be its pole order at infinity, taken as zero when infinity is not a pole. Differentiation raises each finite pole order by one and changes a pole of order $m_{\infty }>0$ at infinity into one of order $m_{\infty }-1$. HenceIn particular, $d-1\le \mathrm{deg}{f}^{\prime }\le 2d$.