Strictly positive polynomials in the boundary of the SOS cone

TL;DR

This paper investigates the boundary of the cone of sums of squares (SOS) polynomials, providing new computational examples that confirm theoretical bounds and exploring properties of strictly positive polynomials on this boundary.

Contribution

It combines theoretical insights with computational methods to verify bounds on SOS decompositions and constructs novel examples of boundary polynomials with specific properties.

Findings

Confirmed optimality of bounds for SOS decompositions across degrees and variables.

Constructed examples of boundary polynomials with fewer squares and complex roots.

Identified polynomials with boundary length exceeding expected dimension.

Abstract

We study the boundary of the cone of real polynomials that can be decomposed as a sum of squares (SOS) of real polynomials. This cone is included in the cone of nonnegative polynomials and both cones share a part of their boundary, which corresponds to polynomials that vanish at at least one point. We focus on the part of the boundary which is not shared, corresponding to strictly positive polynomials. For the cases of polynomials of degree 6 in 3 variables and degree 4 in 4 variables, this boundary has been completely characterized by G. Blekherman. For the cases of more variables or higher degree, results by G. Blekherman, R. Sinn and M. Velasco and other authors based on general conjectures give bounds for the maximum number of polynomials that can appear in a SOS decomposition and the maximum rank of the matrices in the Gram spectrahedron. Combining theoretical results and computational techniques, we compute examples that allow us to prove the optimality of the bounds for all degrees and number of variables. Additionally, we give examples for the following problems: examples in the boundary of the cone that are the sum of less than $n$ squares and have common complex roots, and examples of polynomials in the boundary with length larger than the expected from the dimension.