On odd powers of nonnegative polynomials that are not sums of squares
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arXiv
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| Format: | Preprint |
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2024
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| _version_ | 1866916341716353024 |
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| author | Blekherman, Grigoriy Kozhasov, Khazhgali Reznick, Bruce |
| author_facet | Blekherman, Grigoriy Kozhasov, Khazhgali Reznick, Bruce |
| contents | 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. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2407_21779 |
| institution | arXiv |
| publishDate | 2024 |
| record_format | arxiv |
| spellingShingle | On odd powers of nonnegative polynomials that are not sums of squares Blekherman, Grigoriy Kozhasov, Khazhgali Reznick, Bruce Algebraic Geometry Optimization and Control 14B05, 14H20, 14H45, 14P25, 26C10, 90C22, 90C23 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. |
| title | On odd powers of nonnegative polynomials that are not sums of squares |
| topic | Algebraic Geometry Optimization and Control 14B05, 14H20, 14H45, 14P25, 26C10, 90C22, 90C23 |
| url | https://arxiv.org/abs/2407.21779 |