On odd powers of nonnegative polynomials that are not sums of squares

TL;DR

This paper introduces the concept of stubborn polynomials, nonnegative polynomials whose odd powers are never sums of squares, and develops a new invariant to analyze their properties and extremality.

Contribution

It defines the SOS-invariant, relates it to singularity invariants, and uses it to characterize stubborn polynomials and their extremal properties in the cone of nonnegative forms.

Findings

Stubborn polynomials are characterized by the SOS-invariant.

Extreme rays of the cone of nonnegative ternary sextic forms are generated by stubborn polynomials.

Non-stubborn nonnegative polynomials form a convex cone with strictly positive polynomials in its interior.

Abstract

We initiate a systematic study of nonnegative polynomials $P$ such that $P^k$ is not a sum of squares for any odd $k\geq 1$, calling such $P$ \emph{stubborn}. We develop a new invariant of a real isolated zero of a nonnegative polynomial in the plane, that we call \emph{the SOS-invariant}, and relate it to the well-known delta invariant of a plane curve singularity. Using the SOS-invariant we show that any polynomial that spans an extreme ray of the convex cone of nonnegative ternary forms of degree 6 is stubborn. We also show how to use the SOS-invariant to prove stubbornness of ternary forms in higher degree. Furthermore, we prove that in a given degree and number of variables, nonnegative polynomials that are not stubborn form a convex cone, whose interior consists of all strictly positive polynomials.