发表机构
Southeast University; Taiyuan Normal University（东南大学; 太原师范学院）
机构由 AI 辅助整理，请以论文原文为准。AI 中文总结
针对二阶Zarankiewicz数，通过有限穷举与结构链方法，证明其若干精确值，并分类极值构型，同时给出5×5情形上界。
AI 中文摘要
双二次形式的SOS秩问题与无C4二部图的Zarankiewicz极值问题之间存在自然联系。经典Zarankiewicz数z(m,n)控制与单项式平方相关的二部骨架。允许两个单元形成一个双线性平方，则得到增广Zarankiewicz构型与二阶Zarankiewicz数z2(m,n)。与通过递归充分条件定义的zRL和zSL不同，z2在不施加(RW 3+)条件的情况下，对所有不可约展示的SOS分解取最大值。因此，要证明z2(m,n)≤R，必须证明每个具有更多展示平方的简单受限构型都是可约的；充分条件的失败不能作为反证。我们证明了z2(4,4)=10，z2(7,4)=19，z2(8,4)=21，z2(5,5)=17，并在所有这些情形中得到z2=zSL=zRL。具有16个展示平方的极值不可约6×4构型在行和列重标号下形成单一同构类，而具有19个展示平方的极值不可约7×4构型恰好形成三个同构类。四列结果构成一个结构链：先对低阶极值构型进行分类，然后利用完全平方删除下的遗传不可约性来约束下一阶。有限穷举步骤使用候选剪枝、必要的相容性图、团枚举和轨道约简，并对每个剩余轨道给出可验证的可约性或不可约性证明。对于5×5情形，存在两个普通极值骨架；有限排除仅留下两个高度对称的18平方候选。它们定义相同的十平方多项式，该多项式具有显式的九平方表示，从而给出z2(5,5)的上界。
英文摘要
There is a natural connection between the SOS rank problem for bi-quadratic forms and the Zarankiewicz extremal problem for C4-free bipartite graphs. The classical Zarankiewicz number z(m,n) controls the bipartite skeleton associated with monomial squares. Allowing two cells to form a single bilinear square leads to augmented Zarankiewicz configurations and the second-order Zarankiewicz number z2(m,n). Unlike zRL and zSL, defined through recursive sufficient conditions, z2 maximizes over all irreducible displayed SOS decompositions without imposing (RW 3+). Hence, to prove z2(m,n)<=R, one must prove that every simple limited configuration with more displayed squares is reducible; failure of a sufficient condition cannot serve as a counterargument. We prove z2(4,4)=10, z2(7,4)=19, z2(8,4)=21, z2(5,5)=17, and obtain z2=zSL=zRL in all these cases. The extremal irreducible 6x4 configurations with 16 displayed squares form a single isomorphism class under row and column relabeling, whereas the extremal irreducible 7x4 configurations with 19 displayed squares form exactly three isomorphism classes. The four-column results form a structural chain: classify lower-order extremal configurations first, then use hereditary irreducibility under deletion of complete squares to constrain the next order. Finite exhaustive steps use candidate pruning, a necessary compatibility graph, clique enumeration, and orbit reduction, with a verifiable reducibility or irreducibility proof for each remaining orbit. For 5x5, there are two ordinary extremal skeletons; finite exclusion leaves only two highly symmetric 18-square candidates. They define the same ten-square polynomial, which admits an explicit nine-square representation, yielding the upper bound for z2(5,5).
