
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 for all and . is defined to be the supremum of the number of zeros of all homogeneous -variate polynomials of degree in the SONC cone. The analogously defined numbers and 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 and for general and 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…
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