发表机构
Jiangsu Provincial Scientific Research Center of Applied Mathematics; Department of Applied Mathematics, The Hong Kong Polytechnic University; School of Mathematical Sciences, Beihang University（江苏省应用数学省部共建协同创新中心; 香港理工大学应用数学系; 北京航空航天大学数学科学学院）
机构由 AI 辅助整理，请以论文原文为准。AI 中文总结
本文引入二阶增广 Zarankiewicz 数 z_{2A}(m,n)，证明 z_{2A}(4,4)=11>z_2(4,4)=10，首次分离二阶数与其增广变体，并改进 BSR(4,4) 的下界。
AI 中文摘要
二阶 Zarankiewicz 数 $z_2(m,n)$ 与双四次平方和秩 $\mathrm{BSR}(m,n)$ 之间由无条件层级关系 \\[ \mathrm{BSR}(m,n)\\ \ge\\ z_2(m,n)\\ \ge\\ z_{SL}(m,n)\\ \ge\\ z_{RL}(m,n) \\ \ge\\ z_{wL}(m,n)\\ \ge\\ z(m,n). \\] 我们引入 \\(\emph{二阶增广 Zarankiewicz 数}\\) $z_{2A}(m,n)$，它由 $z_2(m,n)$ 删除配置必须为 \\(\emph{有限的}\\) 这一要求而得到，因此 \\[ \mathrm{BSR}(m,n)\\ \ge\\ z_{2A}(m,n)\\ \ge\\ z_2(m,n). \\] 尽管定义类被扩大，$z_{2A}$ 仍然满足 Löfberg 和 Qi 的普适单元界，因为该界仅使用单边图的 $C_4$-自由性。我们证明 \\[ \mathrm{BSR}(4,4)\\ \ge\\ z_{2A}(4,4)\\ =\\ 11\\ >\\ 10\\ =\\ z_2(4,4) \\ =\\ z_{RL}(4,4), \\] 这是二阶数与其增广变体之间首次记录的分离。此结果也为 $\mathrm{BSR}(4,4)$ 提供了更好的下界。该下界由一个显式的非有限 $4\times4$ 配置所见证，其显示长度为 $11$，其递归线闭包满足 $(\mathrm{RW}3^+)$；匹配的上界通过普适单元界以及对 $161$ 个十二方格配置的精确有限分类排除了 $12$，而 $z_2(4,4)=z_{RL}(4,4)=10$ 是 Xu 和 Yan 的精确值。
英文摘要
The second-order Zarankiewicz number $z_2(m,n)$ and the biquadratic sum-of-squares rank $\mathrm{BSR}(m,n)$ are related by the unconditional hierarchy \[ \mathrm{BSR}(m,n)\ \ge\ z_2(m,n)\ \ge\ z_{SL}(m,n)\ \ge\ z_{RL}(m,n) \ \ge\ z_{wL}(m,n)\ \ge\ z(m,n). \] We introduce the \emph{second-order augmented Zarankiewicz number} $z_{2A}(m,n)$, obtained from $z_2(m,n)$ by deleting the requirement that the configuration be \emph{limited}, so that \[ \mathrm{BSR}(m,n)\ \ge\ z_{2A}(m,n)\ \ge\ z_2(m,n). \] Although the defining class is enlarged, $z_{2A}$ still obeys the universal cell bound of Löfberg and Qi, because that bound uses only the \(C_4\)-freeness of the one-edge graph. We prove \[ \mathrm{BSR}(4,4)\ \ge\ z_{2A}(4,4)\ =\ 11\ >\ 10\ =\ z_2(4,4) \ =\ z_{RL}(4,4), \] the first recorded separation between the second-order number and its augmented variant. This result also gives a better lower bound for $\mathrm{BSR}(4,4)$. The lower bound is witnessed by an explicit non-limited $4\times4$ configuration of displayed length $11$ whose recursive-line closure satisfies $(\mathrm{RW}3^+)$; the matching upper bound excludes $12$ by the universal cell bound together with an exact finite classification of the $161$ twelve-square configurations, and \(z_2(4,4)=z_{RL}(4,4)=10\) is the exact value of Xu and Yan.
