Real Zeros of SONC Polynomials

TL;DR

This paper characterizes the real zeros of SONC polynomials, determines the maximum number of zeros for these polynomials, and explores the structure of the SONC cone's exposed faces, highlighting differences from other cones.

Contribution

It provides an explicit characterization of zeros of SONC polynomials and determines the exact maximum number of zeros, advancing understanding of the SONC cone's geometry.

Findings

Complete characterization of zeros of SONC polynomials

Exact determination of the maximum number of zeros for SONC polynomials

Analysis of the exposed faces of the SONC cone and their dimensional properties

Abstract

We provide a complete and explicit characterization of the real zeros of sums of nonnegative circuit (SONC) polynomials, a recent certificate for nonnegative polynomials independent of sums of squares. As a consequence, we derive an exact determination of the number $B''_{n+1,2d}$ for all $n$ and $d$. $B''_{n+1,2d}$ is defined to be the supremum of the number of zeros of all homogeneous $n+1$-variate polynomials of degree $2d$ in the SONC cone. The analogously defined numbers $B_{n+1,2d}$ and $B'_{n+1,2d}$ for the nonnegativity cone and the cone of sums of squares were first introduced and studied by Choi, Lam, and Reznick. In strong contrast to our case, the determination of both $B_{n+1,2d}$ and $B'_{n+1,2d}$ for general $n$ and $d$ is still an open question. Moreover, we initiate the study of the exposed faces of the SONC cone. In particular, we explicitly consider small dimensions and analyze dimension bounds on the exposed faces. When comparing the exposed faces of the SONC cone with those of the nonnegativity cone we observe dimensional differences between them.