发表机构
School of Mathematics and Information Science, Guangzhou University（广州大学数学与信息科学学院）
机构由 AI 辅助整理，请以论文原文为准。AI 中文总结
该研究针对非空简单无余环二元拟阵，证明了基数量与回路数量的秩界不等式，证实了Oxley1983年的猜想，还推导了删除元素后的基数量关系及连续系列扩张下的基增长下界。
AI 中文摘要
设\b(M)\b、\b(M)\b和\b(M)\b分别表示拟阵\b(M)\b的基的数量、回路的数量和秩。我们证明，每个非空、简单、无余环的二元拟阵都满足\b2b(M)≥(r(M)+1)d(M)\b，当且仅当\b(M)\b同构于Fano拟阵时等号成立，这证实了Oxley在1983年提出的猜想。对于同一类拟阵，我们证明删除任意元素后，剩余拟阵的基的数量至少等于原拟阵的回路数量，即对每个\be∈E(M)\b，有\bb(M\backslash e)≥d(M)\b。我们确定了所有等号成立的情况，并由此推导出连续系列扩张下基增长的严格线性下界。主要计数步骤是对三种最大可能的回路大小进行联合估计，该估计从收缩归一化的基本回路计数和精确的折叠立方体边对应关系中得到。
英文摘要
Let \(b(M)\), \(d(M)\), and \(r(M)\) denote the number of bases, the number of circuits, and the rank of a matroid \(M\), respectively. We prove that every nonempty simple binary matroid with no coloops satisfies \[ 2b(M)\ge(r(M)+1)d(M), \] with equality if and only if \(M\) is isomorphic to the Fano matroid. This confirms a conjecture recorded by Oxley in 1983. For the same class, we prove that deleting any element leaves at least as many bases as there are circuits in the original matroid: \(b(M\backslash e)\ge d(M)\) for every \(e\in E(M)\). We determine all equality cases and deduce a sharp linear lower bound for basis growth under successive series extensions. The main counting step is a joint estimate for the three largest possible circuit sizes, obtained from contraction-normalized fundamental-circuit counts and an exact folded-cube edge correspondence.
Comments19 pages
