Hurwitz existence problem and fiber products

With each holomorphic map f: R C P¹ between compact Riemann surfaces one can associate a combinatorial datum consisting of the genus g of R, the degree n of f, the number q of branching points of f, and the q partitions of n given by the local degrees of f at the preimages of the branching points. These quantities are related by the Riemann-Hurwitz formula, and the Hurwitz existence problem asks whether a combinatorial datum that fits this formula actually corresponds to some f. In this paper, using results and techniques related to fiber products of holomorphic maps between compact Riemann surfaces, we prove a number of results that enable us to uniformly explain the non-realizability of many previously known non-realizable branch data, and to construct a large amount of new such data. We also deduce from our results the theorem of Halphen, proven in 1880, concerning polynomial solutions of the equation A (z) ᵃ+B (z) ᵇ=C (z) ᶜ, where a, b, c are integers greater than one.