arXiv 2102.114222021-02-22math.CO

一致拟阵的推广

A generalisation of uniform matroids

George Drummond

备注：: 10 pages, 8 figures

摘要：

一个拟阵是一致的当且仅当它没有子同构于$U_{1,1}\oplus U_{0,1}$，并且是铺路的当且仅当它没有子同构于$U_{2,2}\oplus U_{0,1}$。本文更一般地考虑了拟阵$M$对固定的正整数对$(k,\ell)$没有$U_{k,k}\oplus U_{0,\ell}$ -子式的情形.称这样的拟阵为$(k,\ell)$ -均匀拟阵，证明了这等价于$M$的每个秩- $(r(M)-k)$平坦的零度小于$\ell$的条件.推广了Rajpal的一个结果，证明了对于任意正整数对$(k,\ell)$和素数幂$q$，只有1000个单余单$GF(q)$ -可表示拟阵是\kl -一致的.因此，如果Rota猜想成立，则对于每个素数幂$q$，存在一对正整数$(k_q,\ell_q)$，使得$GF(q)$ -可表示性的每个排除子式是$(k_q,\ell_q)$ -一致的。我们还确定了所有的二元$(2,2)$ -一致拟阵，并证明了最大$3$ -连通成员是$Z_5\backslash t, AG(4,2), AG(4,2)^*$和一个特殊的自对偶拟阵$P_{10}$。结合Acketa和Rajpal的结果，这完成了二元$(k,\ell)$ -一致拟阵的列表，其中$k+\ell\leq 4$。

英文摘要：

A matroid is uniform if and only if it has no minor isomorphic to $U_{1,1}\oplus U_{0,1}$ and is paving if and only if it has no minor isomorphic to $U_{2,2}\oplus U_{0,1}$. This paper considers, more generally, when a matroid $M$ has no $U_{k,k}\oplus U_{0,\ell}$-minor for a fixed pair of positive integers $(k,\ell)$. Calling such a matroid $(k,\ell)$-uniform, it is shown that this is equivalent to the condition that every rank-$(r(M)-k)$ flat of $M$ has nullity less than $\ell$. Generalising a result of Rajpal, we prove that for any pair $(k,\ell)$ of positive integers and prime power $q$, only finitely many simple cosimple $GF(q)$-representable matroids are \kl-uniform. Consequently, if Rota's Conjecture holds, then for every prime power $q$, there exists a pair $(k_q,\ell_q)$ of positive integers such that every excluded minor of $GF(q)$-representability is $(k_q,\ell_q)$-uniform. We also determine all binary $(2,2)$-uniform matroids and show the maximally $3$-connected members to be $Z_5\backslash t, AG(4,2), AG(4,2)^*$ and a particular self-dual matroid $P_{10}$. Combined with results of Acketa and Rajpal, this completes the list of binary $(k,\ell)$-uniform matroids for which $k+\ell\leq 4$.

机构、总结、中文摘要均由AI分析而来，可能存在错误，请以论文正文为准。

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英文摘要：

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