On the SOS Rank of Simple and Diagonal Biquadratic Forms
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arXiv
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| Format: | Preprint |
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2026
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| _version_ | 1866918321911234560 |
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| author | Xu, Yi Cui, Chufeng Qi, Liqun |
| author_facet | Xu, Yi Cui, Chufeng Qi, Liqun |
| contents | We study the sum-of-squares (SOS) rank of simple and diagonal biquadratic forms. For simple biquadratic forms in $3 \times 3$ variables, we show that the maximum SOS rank is exactly $6$, attained by a specific six-term form. We further prove that for any $m \ge 3$, there exists an $m \times m$ simple biquadratic form whose SOS rank is exactly $2m$. Moreover, we show that for all $m, n \ge 3$, the maximum SOS rank over $m \times n$ simple biquadratic forms is at least $m+n$, which implies $\mathrm{BSR}(m,n) \ge m+n$. For diagonal biquadratic forms with nonnegative coefficients, we prove an SOS rank upper bound of $7$, improving the general bound of $8$ for $3 \times 3$ forms. These results provide new lower and upper bounds on the worst-case SOS rank of biquadratic forms and highlight the role of structure in reducing the required number of squares. |
| format | Preprint |
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arxiv_https___arxiv_org_abs_2601_19195 |
| institution | arXiv |
| publishDate | 2026 |
| record_format | arxiv |
| spellingShingle | On the SOS Rank of Simple and Diagonal Biquadratic Forms Xu, Yi Cui, Chufeng Qi, Liqun Optimization and Control We study the sum-of-squares (SOS) rank of simple and diagonal biquadratic forms. For simple biquadratic forms in $3 \times 3$ variables, we show that the maximum SOS rank is exactly $6$, attained by a specific six-term form. We further prove that for any $m \ge 3$, there exists an $m \times m$ simple biquadratic form whose SOS rank is exactly $2m$. Moreover, we show that for all $m, n \ge 3$, the maximum SOS rank over $m \times n$ simple biquadratic forms is at least $m+n$, which implies $\mathrm{BSR}(m,n) \ge m+n$. For diagonal biquadratic forms with nonnegative coefficients, we prove an SOS rank upper bound of $7$, improving the general bound of $8$ for $3 \times 3$ forms. These results provide new lower and upper bounds on the worst-case SOS rank of biquadratic forms and highlight the role of structure in reducing the required number of squares. |
| title | On the SOS Rank of Simple and Diagonal Biquadratic Forms |
| topic | Optimization and Control |
| url | https://arxiv.org/abs/2601.19195 |