Separating inequalities for nonnegative polynomials that are not sums of squares

TL;DR

This paper develops a new computational method to find separating inequalities for nonnegative polynomials that are not sums of squares, especially for certain boundary cases like nonnegative ternary sextics with multiple zeros.

Contribution

The authors introduce a simplified approach for constructing separating extreme rays for specific classes of nonnegative polynomials outside the SOS cone, including a rational certificate for the Motzkin polynomial.

Findings

Method yields separating extreme rays for nonnegative ternary sextics with at least seven zeros.

Successfully computes a rational certificate proving the Motzkin polynomial is not SOS.

Simplifies the computational process for certain boundary cases.

Abstract

Ternary sextics and quaternary quartics are the smallest cases where there exist nonnegative polynomials that are not sums of squares (SOS). A complete classification of the difference between these cones was given by G. Blekherman via analyzing the extreme rays of the corresponding dual cones. However, an exact computational approach in order to build separating extreme rays for nonnegative polynomials that are not sums of squares is a widely open problem. We provide a method substantially simplifying this computation for certain classes of polynomials on the boundary of the PSD cones. In particular, our method yields separating extreme rays for every nonnegative ternary sextic with at least seven zeros. As an application to further instances, we compute a rational certificate proving that the Motzkin polynomial is not SOS.