弱受限增强扎兰凯维茨数
A Weak Condition for Limited Augmented Zarankiewicz Numbers
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中文总结 AI 辅助
研究通过放宽条件引入弱受限增强扎兰凯维茨数\(z_{WL}(m,n)\)，证明其相关双二次形式不可约，构建不等式链。通过\(5×3\)增强图应用得出\(z_{WL}(5,3) \ge 10\)，改进了\(z_L(5,3)\)的值，提升了对双二次形式SOS秩下限的认知。
中文摘要 AI 辅助
我们通过放宽先前用于建立双二次平方和（SOS）秩下界的广义无环条件，引入了弱受限增强扎兰凯维茨数\(z_{WL}(m,n)\)。关键创新在于对条件2的递归弱化：在非退化2 - 边定义一个依赖图并要求其无环，同时有一个技术条件，即若一个非退化2 - 边的两个相对单元被1 - 边占据，则相关双二次形式必须分解为独立块的直和。我们证明这些弱条件足以使相关双简单双二次形式不可约，得到不等式链\(\operatorname{BSR}(m,n) \ge z_{WL}(m,n) \ge z_L(m,n) \ge z(m,n)\)。作为具体应用，构造了一个\(5×3\)的增强图，有两个2 - 边满足弱条件但违反原定义，得出\(z_{WL}(5,3) \ge 10\)，改进了之前的受限增强值\(z_L(5,3)=9\)，进而得出\(\operatorname{BSR}(5,3) \ge 10\)。
英文摘要
This paper introduces the weak augmented Zarankiewicz number $z_{wA}(m,n)$ and the weak limited augmented Zarankiewicz number $z_{wL}(m,n)$, which are combinatorial extensions of the classical Zarankiewicz number obtained by relaxing the original admissibility conditions for augmented bipartite graphs. We show that the resulting weak framework still guarantees irreducibility of the associated doubly simple biquadratic forms, with SOS rank equal to the total number of edges. This yields the inequality chain \[ \mathrm{BSR}(m,n) \geq z_{wA}(m,n) \geq z_{wL}(m,n) \geq z_L(m,n) \geq z(m,n). \] We provide three complementary constructions demonstrating the power of the weak framework. First, a $5\times 3$ construction using degenerate 2-edges yields $z_{wL}(5,3)\ge 10>9=z_L(5,3)$, giving $\mathrm{BSR}(5,3)\ge 10$. Second, a $15\times 6$ construction on the incidence graph of $K_6$ with 14 nondegenerate 2-edges gives $z_{wL}(15,6)\ge 44>43$, improving the previously known bound. Third, a critical $6\times 3$ construction with complementary 2-cycles gives $z_{wL}(6,3)\ge 12>11=z_L(6,3)$, yielding $\mathrm{BSR}(6,3)\ge 12$ and demonstrating that complementary 2-cycles are safe.
