The Duality of SONC: Advances in Circuit-based Certificates

TL;DR

This paper introduces the DSONC cone, a new subset of the SONC cone, with properties enabling linear programming membership tests, and explores its geometric and algebraic relationships with other cones and tropical geometry.

Contribution

We construct the DSONC cone as an extension of the dual SONC cone, analyze its properties, and relate it to tropical geometry and other polynomial cones.

Findings

DSONC is a proper, full-dimensional cone.

Functions in DSONC cannot have real zeros.

DSONC's boundary points relate to equilibria and tropical geometry.

Abstract

The cone of sums of nonnegative circuits (SONCs) is a subset of the cone of nonnegative polynomials / exponential sums, which has been studied extensively in recent years. In this article, we construct a subset of the SONC cone which we call the DSONC cone. The DSONC cone can be seen as an extension of the dual SONC cone; membership can be tested via linear programming. We show that the DSONC cone is a proper, full-dimensional cone, we provide a description of its extreme rays, and collect several properties that parallel those of the SONC cone. Moreover, we show that functions in the DSONC cone cannot have real zeros, which yields that DSONC cone does not intersect the boundary of the SONC cone. Furthermore, we discuss the intersection of the DSONC cone with the SOS and SDSOS cones. Finally, we show that circuit functions in the boundary of the DSONC cone are determined by points of equilibria, which hence are the analogues to singular points in the primal SONC cone, and relate the DSONC cone to tropical geometry.