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I am spending a few months in Göttingen as a Courant Distinguished Visiting Professor, and talking a bit to Laurent Bartholdi about rational functions — i.e. holomorphic maps from the Riemann sphere $\widehat{\mathbb C}$ to itself. A rational function is determined (up to multiplication by a constant) by its zeroes and poles, and can therefore generically be put in the form $f:z \to P(z)/Q(z)$ where P and Q are polynomials of degree $d$. If $d=1$ then $f$ is invertible, and is called a fractional linear transformation (or, sometimes, a Mobius transformation). The critical points are the zeroes of $P'Q-Q'P$; note that this is a polynomial of degree $\le 2d-2$ (not $2d-1$) and the images of these points under $f$ are the critical values. Again, generically, there will be $2d-2$ critical values; let’s call them $V$. Precomposing $f$ with a fractional linear transformation will not change the set of critical values.

The map $f$ cannot usually be recovered from $V$ (even up to precomposition with a fractional linear transformation); one needs to specify some extra global topological information. If we let $\overline{C}$ denote the preimage of $V$ under $f$, and let $C$ denote the subset consisting of critical points, then the restriction $f:\widehat{\mathbb C} - \overline{C} \to \widehat{\mathbb C} - V$ is a covering map of degree $d$, and to specify the rational map we must specify both $V$ and the topological data of this covering. Let’s assume for convenience that 0 is not a critical value. To specify the rational map is to give both $V$ and a representation $\rho:\pi_1(\widehat{\mathbb C} - V,0) \to S_d$ (here $S_d$ denotes the group of permutations of the set $\lbrace 1,2,\cdots,d\rbrace$) which describes how the branches of $f^{-1}$ are permuted by monodromy about $V$. Such a representation is not arbitrary, of course; first of all it must be irreducible (i.e. not conjugate into $S_e \times S_{d-e}$ for any $1\le e \le d-1$) so that the cover is connected. Second of all, the cover must be topologically a sphere. Let’s call the (branched) cover $\Sigma$ for the moment, before we know what it is. The Riemann-Hurwitz formula lets one compute the Euler characteristic of $\Sigma$ from the representation $\rho$. A nice presentation for $\pi_1(\widehat{\mathbb C}-V,0)$ has generators $e_i$ represented by small loops around the points $v_i \in V$, and the relation $\prod_{i=1}^{|V|} e_i = 1$. For each $e_i$ define $o_i$ to be the number of orbits of $\rho(e_i)$ on the set $\lbrace 1,2,\cdots,d\rbrace$. Then

$\chi(\Sigma) = d\chi(S^2) - \sum_i (d-o_i)$

If each $\rho(e_i)$ is a transposition (i.e. in the generic case), then $o_i=d-1$ and we recover the fact that $|V|=2d-2$.

This raises the following natural question:

Basic Question: Given a set of points $V$ in the Riemann sphere, and an irreducible representation $\rho:\pi_1(\widehat{\mathbb C} - V,0) \to S_d$ satisfying $\sum_i (d-o_i) = 2d-2$, what are the coefficients of the rational function $z \to P(z)/Q(z)$ that they determine (up to precomposition by a fractional linear transformation)?