Real zeros of SONC polynomials

JOURNAL OF PURE AND APPLIED ALGEBRA

卷:225

Article; Proceedings Paper

Dressler, Mareike¹ 

[1] Univ Calif San Diego, Dept Math, 9500 Gilman Dr, La Jolla, CA 92093 USA

关键词: Nonnegative polynomial;    Sums of nonnegative circuit;    polynomials;    Sums of squares;    Real zeros;    Faces;    Dimensional differences;   

DOI  :  10.1016/j.jpaa.2020.106602

来源: Elsevier

【 摘 要 】

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,B- 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,B- 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. (C) 2020 Elsevier B.V. All rights reserved.

【 授权许可】

Free