发表机构
School of Mathematical Sciences, South China Normal University; Jiangsu Provincial Scientific Research Center of Applied Mathematics; Department of Applied Mathematics, The Hong Kong Polytechnic University（华南师范大学数学学院; 江苏省应用数学研究中心; 香港理工大学应用数学系）
机构由 AI 辅助整理，请以论文原文为准。AI 中文总结
本文证明递归线Zarankiewicz数在(15,6)处等于60，与带符号版本一致，消除了引入后者的动机，并给出部分精确值。
AI 中文摘要
Löfberg 和 Qi 引入了二阶 Zarankiewicz 数 \\(z_2\\)、递归线 Zarankiewicz 数 \\(z_{RL}\\) 以及带符号 Zarankiewicz 数 \\(z_{SL}\\)，用于双重简单双二次型。他们证明了对于所有 \\(m\\) 和 \\(n\\)，有 \\(z_2(m,n)\ge z_{SL}(m,n)\ge z_{RL}(m,n)\\)。然而，没有证据表明存在特定的 \\(m\\) 和 \\(n\\) 使得 \\(z_{SL}(m,n)>z_{RL}(m,n)\\)。引入 \\(z_{SL}\\) 的动机如下：在研究例外情形 \\(m=15\\)、\\(n=6\\) 时，Löfberg 和 Qi 证明了 \\(z_2(15,6)=z_{SL}(15,6)=60\\)，但当时 \\(z_{RL}(15,6)\\) 的精确值未知。在本文中，我们证明 \\(z_{RL}(15,6)=60\\)。这消除了引入 \\(z_{SL}\\) 的动机。一般情况下 \\(z_{SL}(m,n)=z_{RL}(m,n)\\) 是否成立仍然是一个开放问题。最近，Lebedev 提出了一个显式构造，在 \\(m=n=1893\\) 处将增广 Zarankiewicz 数 \\(z_A\\) 与受限增广 Zarankiewicz 数 \\(z_L\\) 区分开来。我们希望 \\(z_{SL}\\) 和 \\(z_{RL}\\) 的分离问题也能得到解决。我们还给出了 \\(6\le m\le 16\\) 时 \\(z_{RL}(m,6)\\) 的精确值。
英文摘要
Löfberg and Qi introduced the second order Zarankiewicz number \(z_2\), the recursive-line Zarankiewicz number \(z_{RL}\), and the signed Zarankiewicz number \(z_{SL}\) for doubly simple biquadratic forms. It was shown that \[ z_2(m,n)\ge z_{SL}(m,n)\ge z_{RL}(m,n) \] for all \(m\) and \(n\). However, there was no evidence that there exist particular \(m\) and \(n\) such that \(z_{SL}(m,n)>z_{RL}(m,n)\). The motivation for introducing \(z_{SL}\) was as follows: during the study of the exceptional case \(m=15\), \(n=6\), Löfberg and Qi showed that \[ z_2(15,6)=z_{SL}(15,6)=60, \] but the exact value of \(z_{RL}(15,6)\) was unknown then. In this paper we show that \[ z_{RL}(15,6)=60. \] This eliminates the motivation for introducing \(z_{SL}\). Whether \(z_{SL}(m,n)=z_{RL}(m,n)\) in general remains an open problem. Recently, Lebedev presented an explicit construction separating the augmented Zarankiewicz number \(z_A\) from the limited augmented Zarankiewicz number \(z_L\) at \(m=n=1893\). We hope that the separation problem for \(z_{SL}\) and \(z_{RL}\) can also be solved. We also present the exact values of \(z_{RL}(m,6)\) for \(6\le m\le 16\).
