Weak Limited Augmented Zarankiewicz Number

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Original 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, $$ \operatorname{BSR}(5,3) \ge 10. $$

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