Suppose I have a function $Q(z)$ of a complex variable $z\in\mathbb P^1$, possessing square root type branch points at the positions $\left\{z\_i\right\}\_{i=1}^{2M}$. I know that the Riemann surface $\mathcal M$ on which $Q$ lives has genus $0$.


I have the following questions:


* Aside from the knowledge of the poisitions $\left\{z\_i\right\}\_{i=1}^{2M}$, which additional informations on $Q$ do I need in order to uniquely define the uniformizing variable $\omega(z)$?
* Does an explicit general formula for $\omega(z)$ exists?
* If so, can the limit $M\rightarrow\infty$ be obtained?


I remember seeing an expression of the type $\omega(z)=\frac{1}{z}+\sum\_{k=1} a\_k z^k$, but I am not sure the context corresponded to mine. Moreover there was no mention of how to fix the coefficients $a\_k$ in terms of the positions of the branch points or informations on $Q$.


Thanks a lot!


Stefano



Uniformizing variable for branched covering of the Riemann sphere
1. Besides the position of ramification points $z\_j$ you need monodromy of $Q$ to determine $M$. Once $M$ is defined, you need a normalization of your uniformizing function: it is defined up to a conformal automorphism of the sphere $L^1$.
2. No general formula exists. To find the uniformization in the case of genus $0$, you have to find a rational function with simple critical points and prescribed critical values. The degree of your function is $M+1$ by Riemann-Hurwitz formula. You write it with undetermined coefficients, add normalization, for example $f(0)=a,f(1)=b,f(\infty)=c$, where $a,b,c$ are some points distinct from your $z\_j$, and write the condition that critical values are $z\_j$. You obtain $2M+3$ algebraic equations with $2M+3$ unknowns,
this system has finitely many solutions. Different solutions correspond to different possible monodromies. The system is difficult to solve when the degree is high.


EDIT: You may look at this paper, where a special case of the problem is studied in great detail: Rational functions with real critical points and the B. and M. Shapiro conjecture in real enumerative geometry, Ann of Math., 155 (2002), 105-129.


