We know by the Riemann Existence Theorem that any Riemann surface can arise holmorphically as the branched cover of a sphere:

Therefore, any branched cover of the sphere can be represented by a rational function ~~polynomial~~ $\frac{p (z)}{q(z)}: X \to \mathbb{C}$.

Given arbitrary polynomial $p$, do there exist ways to compute the monodromy around each critical points?

For example, $z \mapsto z^2 + \frac{1}{z^2}$ (not a polynomial) is 4-sheeted cover of the Riemann sphere $\hat{\mathbb{C}}$. It has 3 branch points, where $$p'(z) = 2z - \frac{2}{z^3} = 0 $$ It seems to have 4 branch points at $z = 1, i, -1, -i$.

**How do we calculate of which sheets were permuted given $p(z)$?**

This is motivated by trying to understand a proof of the Riemann Existence Theorem in the case of genus

**0**branched covers of the sphere $S^2$.

This question is the reverse direction of my previous question on branched covers: polynomial branched cover of the sphere with specified monodromy

Our example has a symmetry - and I am trying to avoid techniques which exploit them too much.

dessin? Maybe I can e-mail you, or post my entire train of thought in yet another question. $\endgroup$ – john mangual Nov 18 '13 at 22:31