正则拟阵类$\mathcal{M}_3$的新禁用子式
New excluded minors for the class $\mathcal{M}_3$ of regular matroids
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中文总结 AI 辅助
研究正则拟阵类$\mathcal{M}_3$的禁用子式，通过展示五个新禁用子式，丰富了其禁用子式列表，证明过程经线性代数计算和交叉检查，还记录了$\ell = 5$时余秩为 8 切片的相关数据。
中文摘要 AI 辅助
恩格尔、德盖·福特曼和施赖德为每个素数$\ell$赋予一个正则拟阵的禁用子式封闭类$\mathcal{M}_\ell$，其禁用子式决定了非常一般的主极化阿贝尔簇上曲线类的积分霍奇猜想的不成立情况。对于$\ell = 2$，该类是余图拟阵，根据 Tutte 定理，禁用子式为$M(K_5)$和$M(K_{3,3})$；对于$\ell = 3$，目前明确确定的唯一禁用子式是$M(K_{3,5})$，一般特征是[EGFS，问题 8.8]。我们展示了$\mathcal{M}_3$的五个新禁用子式，秩分别为 8、9、9、9 和 10；它们都不包含$M(K_{3,5})$或其他四个中的任何一个作为子式，所以问题 8.8 的禁用子式列表至少有六个成员，秩从 7 到 10 都有禁用子式。结合同一作者证明的$M(K_7) \notin \mathcal{M}_3$以及$\mathcal{M}_3$的禁用子式封闭性，$M(K_7)$的某个秩在 4 到 6 之间的子式是另一个禁用子式，所以列表至少有七个成员，其中六个已明确确定。该列表在结构上多样：两个具有自同构群$S_4$的二分顶点构造，以及三个没有这种顶点结构的非二分、非平面秩 为 9 的图。每个判定都通过有限的$\mathbb{F}_3$线性代数计算并带有明确的机器可检查见证进行认证，并由独立实现交叉检查；证书还表明这五个拟阵不在更大的类$\widetilde{\mathcal{M}}_3$中。我们还记录了据我们所知关于这个余秩为 8 的切片的第一个$\ell = 5$的数据：通过明确的证书表明$M(K_{3,5})$以及秩为 8 和 10 的子式在$\mathcal{M}_5$中，并且精确确定了根距离$d(M(K_{3,5})) = 6$。
英文摘要
Engel, de Gaay Fortman, and Schreieder attach to each prime $\ell$ a minor-closed class $\mathcal{M}_\ell$ of regular matroids, whose excluded minors govern the failure of the integral Hodge conjecture for curve classes on very general principally polarized abelian varieties. For $\ell=2$ the class is the cographic matroids, with excluded minors $M(K_5)$ and $M(K_{3,3})$ by Tutte's theorem; for $\ell=3$ the only excluded minor explicitly identified so far is $M(K_{3,5})$, and the general characterization is [EGFS, Problem 8.8]. We exhibit five new excluded minors for $\mathcal{M}_3$, of ranks 8, 9, 9, 9, and 10; none contains $M(K_{3,5})$ or any of the other four as a minor, so the excluded-minor list of Problem 8.8 has at least six members, with excluded minors at every rank from 7 through 10. Combined with $M(K_7) \notin \mathcal{M}_3$, established by the same authors, and the minor-closedness of $\mathcal{M}_3$, some minor of $M(K_7)$ of rank between 4 and 6 is a further excluded minor, so the list has at least seven members, six of them explicitly identified. The list is structurally diverse: two bipartite apex constructions with automorphism group $S_4$, and three non-bipartite, non-planar rank-9 graphs with no such apex structure. Every verdict is certified by a finite $\mathbb{F}_3$-linear-algebra computation with explicit machine-checkable witnesses, cross-checked by independent implementations; the certificates further show that the five matroids lie outside the larger class $\widetilde{\mathcal{M}}_3$. We also record what are, to our knowledge, the first $\ell=5$ data on this corank-8 slice: $M(K_{3,5})$ and the rank-8 and rank-10 minors lie in $\mathcal{M}_5$ by explicit certificates, and the radical distance $d(M(K_{3,5}))=6$ is determined exactly.
