Tropical Hurwitz numbers

Hurwitz numbers count genus g, degree d covers of a"(TM)(1) with fixed branch locus. This equals the degree of a natural branch map defined on the Hurwitz space. In tropical geometry, algebraic curves are replaced by certain piece-wise linear objects called tropical curves. This paper develops a tropical counterpart of the branch map and shows that its degree recovers classical Hurwitz numbers. Further, the combinatorial techniques developed are applied to recover results of Goulden et al. (in Adv. Math. 198:43-92, 2005) and Shadrin et al. (in Adv. Math. 217(1):79-96, 2008) on the piecewise polynomial structure of double Hurwitz numbers in genus 0.