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

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}, a...

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Bibliographic Details
Published in:arXiv.org 2024-07
Main Authors: Blekherman, Grigoriy, Kozhasov, Khazhgali, Reznick, Bruce
Format: Article
Language:English
Subjects:
Curves
Delta rays
Invariants
Polynomials
Online Access:Get full text
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Summary: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.
ISSN:2331-8422