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 $(k,\mathit{\ell })$-uniform, it is shown that this is equivalent to the condition that every rank-$(r(M)-k)$ flat of M has nullity less than ℓ. 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 $(k,\ell )$-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 $Z_5﹨t,AG(4,2),AG{(4,2)}^{⁎}$ 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 $k+\ell \le 4$.

拟阵是一致的当且仅当它没有次要同构于 ${你}_{1,1}\oplus {你}_{0,1}$ 并且正在铺路当且仅当它没有次要同构到 ${你}_{2,2}\oplus {你}_{0,1}$. 本文更一般地考虑，当拟阵M没有${你}_{克,克}\oplus {你}_{0,\ell }$-minor 用于固定的正整数对 $(克,\ell )$. 调用这样的拟阵$(克,\mathit{\ell })$-uniform，说明这等价于每一个等级的条件-$(r(米)-克)$M 的平坦度小于ℓ。概括 Rajpal 的结果，我们证明对于任何对$(克,\ell )$正整数和素数q，只有有限多个简单的余简单$GF(q)$-可表示的拟阵是 $(克,\ell )$-制服。因此，如果 Rota's Conjecture 成立，那么对于每个素数q，都存在一对$({克}_q,{\ell }_q)$ 正整数，使得每个排除的次要 $GF(q)$-代表性是 $({克}_q,{\ell }_q)$-制服。我们还确定所有二进制$(2,2)$-uniform matroids 并显示最多 3 个连接的成员为 $Z_5﹨吨,\mathrm{一种}G(4,2),\mathrm{一种}G{(4,2)}^{⁎}$ 和一个特定的自对偶拟阵 ${磷}_{10}$. 结合Acketa和Rajpal的结果，这完成了二进制列表$(克,\ell )$-均匀拟阵，其中 $克+\ell \le 4$.