Low Dimensional Test Sets for Nonnegativity of Even Symmetric Forms

TL;DR

This paper develops low-dimensional test sets for deciding nonnegativity of even symmetric forms, improving bounds beyond Timofte's theorem by analyzing subspaces where nonnegativity can be checked at fewer points.

Contribution

It introduces a novel approach to verify nonnegativity of symmetric forms using subspaces with fewer test points, independent of the form's degree, and explores their structural properties.

Findings

Nonnegativity can be checked at fewer points in certain subspaces.

The dimension of these subspaces can be maximized for fixed test point counts.

The geometric and topological structure of these sets is characterized.

Abstract

An important theorem by Timofte states that nonnegativity of real $n$-variate symmetric polynomials of degree $d$ can be decided at test sets given by all points with at most $\lfloor\frac{d}{2}\rfloor$ distinct components. However, if the degree is sufficiently larger than the number of variables, then the theorem obviously does not provide nontrivial information. Our approach is to look at $(m + 1)$-dimensional subspaces of even symmetric forms of degree 4d, at which nonnegativity can be checked at $(m - 1)$-points, i.e., points with at most $m - 1 \in \N$ distinct components, where $m$ is independent of the degree of the forms and better than Timofte's bound. Furthermore, for fixed $k \in \N$, we tackle problems concerning the maximum dimension of such subspaces, at which nonnegativity can be checked at all $k$-points, as well as the geometrical and topological structure of the set of all forms whose nonnegativity can be decided at all $k$-points.