2019/09/30 by Mareike Dressler
Computer Science · Mathematics · #Combinatorics #Cone (formal languages) #Degree (music) #Dimension (graph theory) #Discrete mathematics #Explained sum of squares #Infimum and supremum #Mathematics #Matrix Theory and Algorithms #Numerical Methods and Algorithms #Polynomial and algebraic computation #math.AG #math.CO #math.OC #msc:05E99 #msc:12D15 #msc:14P99 #msc:26C10 #msc:52A99 #msc:90C25
paper · pdf · doi:10.1016/j.jpaa.2020.106602
Minor revision; final version; to appear in Journal of Pure and Applied Algebra; 27 pages, 3 figures, 4 tables
openalex publication_date 2020/10/23 · arxiv created 2020/11/07 · arxiv updated 2020/11/10 · openalex created_date 2025/10/10 · openalex updated_date 2026/08/05
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 Bn+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 Bn+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.