Sum of Squares Rank of Biquadratic Forms and The Zarankiewicz Number
Fuente:
arXiv
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| Autori principali: | , , |
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| Natura: | Preprint |
| Pubblicazione: |
2026
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| _version_ | 1866908848259858432 |
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| author | Cui, Chunfeng Qi, Liqun Xu, Yi |
| author_facet | Cui, Chunfeng Qi, Liqun Xu, Yi |
| contents | Denote the maximum sos rank of $m \times n$ sum of squares (SOS) biquadratic forms by $BSR(m, n)$. In this paper, we show that $BSR(m, n) \ge z(m, n)$ and conjecture that $BSR(m, n) = z(m, n)$, where $z(m, n)$ is the Zarankiewicz number. Our result coincides with the existing results for $m = 2$, $n = 2$, and $m = n = 3$, and is superior to other previously known lower bounds. Our result also connects graph theory and SOS polynomial theory. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2602_07844 |
| institution | arXiv |
| publishDate | 2026 |
| record_format | arxiv |
| spellingShingle | Sum of Squares Rank of Biquadratic Forms and The Zarankiewicz Number Cui, Chunfeng Qi, Liqun Xu, Yi Optimization and Control Denote the maximum sos rank of $m \times n$ sum of squares (SOS) biquadratic forms by $BSR(m, n)$. In this paper, we show that $BSR(m, n) \ge z(m, n)$ and conjecture that $BSR(m, n) = z(m, n)$, where $z(m, n)$ is the Zarankiewicz number. Our result coincides with the existing results for $m = 2$, $n = 2$, and $m = n = 3$, and is superior to other previously known lower bounds. Our result also connects graph theory and SOS polynomial theory. |
| title | Sum of Squares Rank of Biquadratic Forms and The Zarankiewicz Number |
| topic | Optimization and Control |
| url | https://arxiv.org/abs/2602.07844 |