{"ID":6138115,"CreatedAt":"2026-07-09T01:07:32.349475501Z","UpdatedAt":"2026-07-11T06:53:25.465901322Z","DeletedAt":null,"paper_url":"https://arxiv.org/abs/2607.07071","arxiv_id":"2607.07071","title":"Weak Limited Augmented Zarankiewicz Number","abstract":"We introduce the weak limited augmented Zarankiewicz number $z_{WL}(m,n)$ by relaxing the generalized cycle-free conditions previously used to establish lower bounds for the biquadratic sum-of-squares (SOS) rank. The key innovation is a recursive weakening of Condition~2: we define a dependency graph on nondegenerate 2-edges and require that it be acyclic, together with a technical condition that if a nondegenerate 2-edge has both opposite cells occupied by 1-edges, then the associated biquadratic form must decompose as a direct sum of independent blocks. We prove that these weak conditions suffice for irreducibility of the associated doubly simple biquadratic form, yielding the inequality chain $$ \\operatorname{BSR}(m,n) \\ge z_{WL}(m,n) \\ge z_L(m,n) \\ge z(m,n), $$ where $\\operatorname{BSR}(m,n)$ is the maximum SOS rank among all $m\\times n$ biquadratic forms, $z_L(m,n)$ is the limited augmented Zarankiewicz number, and $z(m,n)$ is the classical Zarankiewicz number. As a concrete application, we construct a $5 \\times 3$ augmented graph with two 2-edges that satisfies the weak conditions but violates the original definition. This establishes $$ z_{WL}(5,3) \\ge 10, $$ improving the previous limited augmented value \\(z_L(5,3)=9\\). Consequently, $$ \\operatorname{BSR}(5,3) \\ge 10. $$","short_abstract":"We introduce the weak limited augmented Zarankiewicz number $z_{WL}(m,n)$ by relaxing the generalized cycle-free conditions previously used to establish lower bounds for the biquadratic sum-of-squares (SOS) rank. The key innovation is a recursive weakening of Condition~2: we define a dependency graph on nondegenerate 2...","url_abs":"https://arxiv.org/abs/2607.07071","url_pdf":"https://arxiv.org/pdf/2607.07071v1","authors":"[\"Liqun Qi\",\"Chunfeng Cui\",\"Yi Xu\"]","published":"2026-07-08T07:01:07Z","proceeding":"math.CO","tasks":"[\"math.CO\",\"math.OC\"]","methods":"[]","has_code":false}