The lattice size of a lattice polygon

We give upper bounds on the minimal degree of a model in P2P2 and the minimal bidegree of a model in P1×P1P1×P1 of the curve defined by a given Laurent polynomial, in terms of the combinatorics of the Newton polygon of the latter. We prove in various cases that this bound is sharp as soon as the polynomial is sufficiently generic with respect to its Newton polygon.