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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 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 where P and Q are polynomials of degree . If then is invertible, and is called a fractional linear transformation (or, sometimes, a Mobius transformation). The critical points are the zeroes of ; note that this is a polynomial of degree (not ) and the images of these points under are the critical values. Again, generically, there will be critical values; let’s call them . Precomposing with a fractional linear transformation will not change the set of critical values.
The map cannot usually be recovered from (even up to precomposition with a fractional linear transformation); one needs to specify some extra global topological information. If we let denote the preimage of under , and let denote the subset consisting of critical points, then the restriction is a covering map of degree , and to specify the rational map we must specify both 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 and a representation (here denotes the group of permutations of the set ) which describes how the branches of are permuted by monodromy about . Such a representation is not arbitrary, of course; first of all it must be irreducible (i.e. not conjugate into for any ) so that the cover is connected. Second of all, the cover must be topologically a sphere. Let’s call the (branched) cover for the moment, before we know what it is. The Riemann-Hurwitz formula lets one compute the Euler characteristic of from the representation . A nice presentation for has generators represented by small loops around the points , and the relation . For each define to be the number of orbits of on the set . Then
If each is a transposition (i.e. in the generic case), then and we recover the fact that .
This raises the following natural question:
Basic Question: Given a set of points in the Riemann sphere, and an irreducible representation satisfying , what are the coefficients of the rational function that they determine (up to precomposition by a fractional linear transformation)?