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| Format: | Preprint |
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2025
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| Online-Zugang: | https://arxiv.org/abs/2512.07133 |
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| _version_ | 1866913063659110400 |
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| author | Wang, Zhiwei Yue, Chenlong Zhou, Xiangyu |
| author_facet | Wang, Zhiwei Yue, Chenlong Zhou, Xiangyu |
| contents | Let $z\in \mathbb C^n$ be the complex coordinates on $\mathbb C^n$, and $A(z,\bar z)$ be a real-valued Hermitian polynomial. The famous Ebenfelt's SOS conjecture asks for the minimum rank of $A(z,\bar z)\|z\|^2$ under the restriction that $A(z,\bar z)\|z\|^2$ is an SOS. Assume that $A(z,\bar z)$ is bihomogeneous. In the present note, we establish a connection between Ebenfelt's (Weak) SOS Conjecture and the theory of Newton-Okounkov bodies. By reformulating the conjecture in terms of lattice semigroups and their associated Newton-Okounkov convex bodies, we transform the problem of finding the minimal rank of a prolonged sum-of-squares polynomial into an extremal problem in convex geometry. In particular, we prove that this minimal rank is attained at the extreme points of a specific Newton-Okounkov body. Furthermore, if $A(z,\bar z)$ is moreover diagonal, we demonstrate that the relevant extreme points are finitely many rational points, thereby reducing the verification of the conjecture to a computationally tractable problem. This work provides a new tool for attacking the SOS Conjecture. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2512_07133 |
| institution | arXiv |
| publishDate | 2025 |
| record_format | arxiv |
| spellingShingle | A Newton-Okounkov Body Viewpoint on the SOS Conjecture Wang, Zhiwei Yue, Chenlong Zhou, Xiangyu Complex Variables Let $z\in \mathbb C^n$ be the complex coordinates on $\mathbb C^n$, and $A(z,\bar z)$ be a real-valued Hermitian polynomial. The famous Ebenfelt's SOS conjecture asks for the minimum rank of $A(z,\bar z)\|z\|^2$ under the restriction that $A(z,\bar z)\|z\|^2$ is an SOS. Assume that $A(z,\bar z)$ is bihomogeneous. In the present note, we establish a connection between Ebenfelt's (Weak) SOS Conjecture and the theory of Newton-Okounkov bodies. By reformulating the conjecture in terms of lattice semigroups and their associated Newton-Okounkov convex bodies, we transform the problem of finding the minimal rank of a prolonged sum-of-squares polynomial into an extremal problem in convex geometry. In particular, we prove that this minimal rank is attained at the extreme points of a specific Newton-Okounkov body. Furthermore, if $A(z,\bar z)$ is moreover diagonal, we demonstrate that the relevant extreme points are finitely many rational points, thereby reducing the verification of the conjecture to a computationally tractable problem. This work provides a new tool for attacking the SOS Conjecture. |
| title | A Newton-Okounkov Body Viewpoint on the SOS Conjecture |
| topic | Complex Variables |
| url | https://arxiv.org/abs/2512.07133 |