Enumeration of almost polynomial rational functions with given critical values

Enumerating ramified coverings of the sphere with fixed ramification types is a well-known problem first considered by A. Hurwitz [5]. Up to now, explicit solutions have been obtained only for some families of ramified coverings, for instant, those realized by polynomials in one complex variable. In this paper we obtain an explicit answer for a large new family of coverings, namely, the coverings realized by simple almost polynomials, defined below. Unlike most other results in the field, our formula is obtained by elementary methods. 1 Rational functions and minimal factorizations of permutations Let f : C → C be a rational function of degree n in one complex variable. A critical point of f is a point z ∈ C such that f (z) = 0. Its degree is the number a ≥ 2 such that f looks like f(z) = z in the neighborhood of the critical point. A critical value of f is its value at a critical point. (Note that we do not count poles as critical points.) IPDE, IHES, Le Bois-Marie, 35, route de Chartres F-91440, Bures-sur-Yvette, France. E-mail: [email protected] . Institut Mathématique de Jussieu, Université Paris VI, 175, rue du Chevaleret, 75013 Paris, France. E-mail: [email protected] . The second author was partially suported by EAGER European Algebraic Geometry Research Training Network, contract No. HPRN-CT-2000-00099 (BBW) and by the Russian Foundation of Basic Research grant 02-01-22004.