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arXiv · 2407.21779

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

Grigoriy Blekherman·Khazhgali Kozhasov·Bruce Reznick
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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.

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Grigoriy Blekherman, Khazhgali Kozhasov, Bruce Reznick. 2024-07-31. On odd powers of nonnegative polynomials that are not sums of squares. https://arxiv.org/abs/2407.21779

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