Detecting integral polyhedral functions

TL;DR

This paper characterizes integral polyhedral functions as maxima of finitely many affine functions with integer slopes and shows how to detect this property through sampling, also relating tropical polynomials to their restrictions on tropical lines.

Contribution

It provides new criteria for identifying integral polyhedral functions via sampling and establishes a novel characterization of tropical polynomials in two dimensions.

Findings

Detection of integral polyhedral functions through small subset sampling

Unified approach to previous results on polyhedral functions

Characterization of tropical polynomials via restrictions on tropical lines

Abstract

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.