Newton polygons and curve gonalities

We give a combinatorial upper bound for the gonality of a curve that is defined by a bivariate Laurent polynomial with given Newton polygon. We conjecture that this bound is generically attained, and provide proofs in a considerable number of special cases. One proof technique uses re-cent work of M. Baker on linear systems on graphs, by means of which we reduce our conjecture to a purely combinatorial statement. MSC2010: 14H51, 14M25, 52B20