Narrowing the Gap: SOS Ranks of $4 \times 3$ Biquadratic Forms and a Lower Bound of $8$
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arXiv
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| Natura: | Preprint |
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2026
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| _version_ | 1866910032689364992 |
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| author | Xu, Yi Cui, Chunfeng Qi, Liqun |
| author_facet | Xu, Yi Cui, Chunfeng Qi, Liqun |
| contents | We investigate the maximum sum-of-squares (SOS) rank of biquadratic forms in the critical case of $4 \times 3$ variables, where the general bounds are currently $7 \leq \mathrm{BSR}(4,3) \leq 11$. By analyzing two important structured subclasses, we obtain exact determinations and improved upper bounds that significantly narrow this gap.
For simple biquadratic forms those containing only distinct terms of the type $x_i^2 y_j^2$ we prove that the maximum achievable SOS rank is exactly 7, a value attained by a form corresponding to a $C_4$-free bipartite graph with the maximum number of edges. This settles the question for simple forms.
For $y$-deficient biquadratic forms a class introduced here that permits cross terms among two of the three $y$-variables while the third appears only in pure square terms we prove an upper bound of $9$ by combining Calderön's theorem on $m\times 2$ forms with the known value $\mathrm{BSR}(4,2) = 5$.
Our main result is a constructive proof that $\mathrm{BSR}(4,3) \geq 8$. We present an explicit non-simple, non-deficient $4\times 3$ biquadratic form and prove it requires exactly eight squares, thereby improving the general lower bound. This shows that any form achieving a rank higher than $8$ must possess a more complex algebraic structure, and it reduces the search space for determining the true value of $\mathrm{BSR}(4,3)$. Connections to Zarankiewicz numbers, extremal graph theory, and classical results on sums of squares are highlighted throughout. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2602_21570 |
| institution | arXiv |
| publishDate | 2026 |
| record_format | arxiv |
| spellingShingle | Narrowing the Gap: SOS Ranks of $4 \times 3$ Biquadratic Forms and a Lower Bound of $8$ Xu, Yi Cui, Chunfeng Qi, Liqun Optimization and Control We investigate the maximum sum-of-squares (SOS) rank of biquadratic forms in the critical case of $4 \times 3$ variables, where the general bounds are currently $7 \leq \mathrm{BSR}(4,3) \leq 11$. By analyzing two important structured subclasses, we obtain exact determinations and improved upper bounds that significantly narrow this gap. For simple biquadratic forms those containing only distinct terms of the type $x_i^2 y_j^2$ we prove that the maximum achievable SOS rank is exactly 7, a value attained by a form corresponding to a $C_4$-free bipartite graph with the maximum number of edges. This settles the question for simple forms. For $y$-deficient biquadratic forms a class introduced here that permits cross terms among two of the three $y$-variables while the third appears only in pure square terms we prove an upper bound of $9$ by combining Calderön's theorem on $m\times 2$ forms with the known value $\mathrm{BSR}(4,2) = 5$. Our main result is a constructive proof that $\mathrm{BSR}(4,3) \geq 8$. We present an explicit non-simple, non-deficient $4\times 3$ biquadratic form and prove it requires exactly eight squares, thereby improving the general lower bound. This shows that any form achieving a rank higher than $8$ must possess a more complex algebraic structure, and it reduces the search space for determining the true value of $\mathrm{BSR}(4,3)$. Connections to Zarankiewicz numbers, extremal graph theory, and classical results on sums of squares are highlighted throughout. |
| title | Narrowing the Gap: SOS Ranks of $4 \times 3$ Biquadratic Forms and a Lower Bound of $8$ |
| topic | Optimization and Control |
| url | https://arxiv.org/abs/2602.21570 |