Valency theorem

For a nonconstant analytic map $f:R\to S$ between compact connected Riemann surfaces, there is an integer $\mathrm{deg}f$ such that for every $w\in S$,

If a nonconstant holomorphic map has local-coordinate expressionthen $m=m_f(p)$ is its local degree, multiplicity, or ramification index at $p$.

For a nonconstant holomorphic map $f:X\to Y$ between compact connected Riemann surfaces, its degree isThe valency theorem says that this integer is independent of $y\in Y$.

For $k\ge 1$, the power map has $k$ distinct preimages of every nonzero finite regular value, all with positive local degree of a map between oriented manifolds. Integrating the pullback of a differential form of $\Omega =id\zeta \wedge d\bar{\zeta }/(1+|\zeta {|}^2{)}^2$ gives the same answer, since $\int \Omega =2\pi$ and $\int f^{\ast }\Omega =2\pi k$. For $k=0$ use the constant extension $1$ at infinity; its degree of a map between oriented manifolds is zero.

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.

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, $d-1\le \mathrm{deg}{f}^{\prime }\le 2d$.

The degree of a nonconstant elliptic function is the sum of the orders of its poles in a fundamental parallelogram. Equivalently, it is the degree of the induced holomorphic map from its complex torus to the Riemann sphere.

If an elliptic function $g$ has degree $d$ and $r$ distinct poles in a fundamental parallelogram, then each pole order increases by one under differentiation, soBecause $1\le r\le d$, this gives $d+1\le \mathrm{deg}{g}^{\prime }\le 2d$.

The rotation group of the octahedron has order $24$. Its vertex, face-centre, and edge-centre stabilizers have orders $4,3,2$, giving exceptional orbit sizes $6,8,12$; every other orbit has size $24$.

Let a finite group $G$ of order $d$ act analytically on a compact Riemann surface, and let an invariant analytic map $F$ have degree $d$. At a point with stabilizer of order $e$, invariance forces the local multiplicity of $F$ to be at least $e$. Its orbit has $d/e$ points, so it already contributes at least $d$ to the fibre multiplicity. The valency theorem forces equality and shows that every fibre is exactly one orbit.