Biquadratic SOS Rank and Augmented Zarankiewicz Number
Fuente:
arXiv
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| Autori principali: | , , |
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| Natura: | Preprint |
| Pubblicazione: |
2026
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| _version_ | 1866913000386985984 |
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| author | Qi, Liqun Cui, Chunfeng Xu, Yi |
| author_facet | Qi, Liqun Cui, Chunfeng Xu, Yi |
| contents | This paper introduces the concepts of the augmented Zarankiewicz number $z_A(m,n)$ and the limited augmented Zarankiewicz number $z_L(m,n)$, which are natural combinatorial extensions of the classical Zarankiewicz number. These numbers arise from augmented bipartite graphs that may contain both standard edges (1-edges) and pairs of edges representing squares of binomials (2-edges). The main theoretical result establishes the inequality chain $\operatorname{BSR}(m, n) \geq z_A(m, n) \geq z_L(m, n) \geq z(m, n)$, linking the maximum biquadratic sum-of-squares (SOS) rank to these extremal graph parameters. We determine the exact values of $z_L(m, n)$ for the cases $(m,2)$, $(3,3)$, $(4, 3)$ and $(4,4)$, and provide new lower bounds for the cases $(5,3)$, $(5,4)$, and $(5,5)$. These results yield improved lower bounds for the maximum SOS rank of biquadratic forms, demonstrating that $z_L(m,n)$ can exceed the classical Zarankiewicz number, thereby offering a refined combinatorial perspective on the SOS rank problem. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2603_04912 |
| institution | arXiv |
| publishDate | 2026 |
| record_format | arxiv |
| spellingShingle | Biquadratic SOS Rank and Augmented Zarankiewicz Number Qi, Liqun Cui, Chunfeng Xu, Yi Optimization and Control This paper introduces the concepts of the augmented Zarankiewicz number $z_A(m,n)$ and the limited augmented Zarankiewicz number $z_L(m,n)$, which are natural combinatorial extensions of the classical Zarankiewicz number. These numbers arise from augmented bipartite graphs that may contain both standard edges (1-edges) and pairs of edges representing squares of binomials (2-edges). The main theoretical result establishes the inequality chain $\operatorname{BSR}(m, n) \geq z_A(m, n) \geq z_L(m, n) \geq z(m, n)$, linking the maximum biquadratic sum-of-squares (SOS) rank to these extremal graph parameters. We determine the exact values of $z_L(m, n)$ for the cases $(m,2)$, $(3,3)$, $(4, 3)$ and $(4,4)$, and provide new lower bounds for the cases $(5,3)$, $(5,4)$, and $(5,5)$. These results yield improved lower bounds for the maximum SOS rank of biquadratic forms, demonstrating that $z_L(m,n)$ can exceed the classical Zarankiewicz number, thereby offering a refined combinatorial perspective on the SOS rank problem. |
| title | Biquadratic SOS Rank and Augmented Zarankiewicz Number |
| topic | Optimization and Control |
| url | https://arxiv.org/abs/2603.04912 |