Detecting integral polyhedral functions

We study the class of real-valued functions on convex subsets of R^n which are computed by the maximum of finitely many affine functionals with integer slopes. We prove several results to the effect that this property of a function can be detected by sampling on small subsets of the domain. In so doing, we recover in a unified way some prior results of the first author (some joint with Liang Xiao). We also prove that a function on R^2 is a tropical polynomial if and only if its restriction to each translate of a generic tropical line is a tropical polynomial.