Sublinear circuits for polyhedral sets

TL;DR

This paper explores the combinatorial structure of sublinear circuits associated with polyhedral sets, providing criteria, relations, and explicit enumerations for specific cases like the non-negative orthant and the cube.

Contribution

It introduces new criteria and characterizations for sublinear circuits in polyhedral sets, extending the understanding of their combinatorial properties.

Findings

Established relations between sublinear circuits and their supports

Provided necessary and sufficient conditions for sublinear circuits

Enumerated sublinear circuits for the non-negative orthant and the cube

Abstract

Sublinear circuits are generalizations of the affine circuits in matroid theory, and they arise as the convex-combinatorial core underlying constrained non-negativity certificates of exponential sums and of polynomials based on the arithmetic-geometric inequality. Here, we study the polyhedral combinatorics of sublinear circuits for polyhedral constraint sets. We give results on the relation between the sublinear circuits and their supports and provide necessary as well as sufficient criteria for sublinear circuits. Based on these characterizations, we provide some explicit results and enumerations for two prominent polyhedral cases, namely the non-negative orthant and the cube $[-1,1]^n$.