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| Main Authors: | , , |
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| Format: | Preprint |
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2025
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| Subjects: | |
| Online Access: | https://arxiv.org/abs/2509.04314 |
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| _version_ | 1866908994617999360 |
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| author | Wang, Zhiwei Yue, Chenlong Zhou, Xiangyu |
| author_facet | Wang, Zhiwei Yue, Chenlong Zhou, Xiangyu |
| contents | Let $z \in \mathbb{C}^n$, and let $A(z,\bar{z})$ be a real valued diagonal bihomogeneous Hermitian polynomial such that $A(z,\bar{z})\|z\|^2$ is a sum of squares, where $\|z\|$ denotes the Euclidean norm of $z$. In this paper, we provide an estimate for the rank of the sum of squares $A(z,\bar{z})\|z\|^2$ when $A(z,\bar{z})$ is not semipositive definite. As a consequence, we confirm the SOS conjecture proposed by Ebenfelt for $4 \leq n \leq 6$ when $A(z,\bar{z})$ is a real valued diagonal (not necessarily bihomogeneous) Hermitian polynomial, and we also give partial answers to the SOS conjecture for $n\geq 7$. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2509_04314 |
| institution | arXiv |
| publishDate | 2025 |
| record_format | arxiv |
| spellingShingle | Macaulay representation of the prolongation matrix and the SOS conjecture Wang, Zhiwei Yue, Chenlong Zhou, Xiangyu Complex Variables Let $z \in \mathbb{C}^n$, and let $A(z,\bar{z})$ be a real valued diagonal bihomogeneous Hermitian polynomial such that $A(z,\bar{z})\|z\|^2$ is a sum of squares, where $\|z\|$ denotes the Euclidean norm of $z$. In this paper, we provide an estimate for the rank of the sum of squares $A(z,\bar{z})\|z\|^2$ when $A(z,\bar{z})$ is not semipositive definite. As a consequence, we confirm the SOS conjecture proposed by Ebenfelt for $4 \leq n \leq 6$ when $A(z,\bar{z})$ is a real valued diagonal (not necessarily bihomogeneous) Hermitian polynomial, and we also give partial answers to the SOS conjecture for $n\geq 7$. |
| title | Macaulay representation of the prolongation matrix and the SOS conjecture |
| topic | Complex Variables |
| url | https://arxiv.org/abs/2509.04314 |