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arXiv · 2509.04314

Macaulay representation of the prolongation matrix and the SOS conjecture

Zhiwei Wang·Chenlong Yue·Xiangyu Zhou
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Abstract

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 $n\geq 7$.

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Zhiwei Wang, Chenlong Yue, Xiangyu Zhou. 2025-09-04. Macaulay representation of the prolongation matrix and the SOS conjecture. https://arxiv.org/abs/2509.04314

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