Given two matroids on the same ground set, the matroid intersection problem asks to find a common independent set of maximum size. In case of linear matroids, the problem had a randomized parallel algorithm but no deterministic one. We give an almost complete derandomization of this algorithm, which implies that the linear matroid intersection problem is in quasi-NC. That is, it has uniform circuits of quasi-polynomial size \(n^{O(\log n)}\) and O(polylog(n)) depth. Moreover, the depth of the circuit can be reduced to O(log² n) in case of zero characteristic fields. This generalizes a similar result for the bipartite perfect matching problem. Our main technical contribution is to derandomize the Isolation lemma for the family of common bases of two matroids. We use our isolation result to give a quasi-polynomial time blackbox algorithm for a special case of Edmonds' problem, i.e., singularity testing of a symbolic matrix, when the given matrix is of the form \(A_{0} + A_{1 }x_{1} + \cdots + A_{m} x_{m}\), for an arbitrary matrix A₀ and rank-1 matrices \(A_{1}, A_{2}, \dots, A_{m}\). This can also be viewed as a blackbox polynomial identity testing algorithm for the corresponding determinant polynomial. Another consequence of this result is a deterministic solution to the maximum rank matrix completion problem. Finally, we use our result to find a deterministic representation for the union of linear matroids in quasi-NC.

给定相同地面上的两个拟阵，拟阵相交问题要求找到一个最大大小的共同独立组。对于线性拟阵，该问题具有随机并行算法，但没有确定性算法。我们对该算法进行了几乎完全的随机化，这意味着线性拟阵相交问题在准NC中。即，它具有准多项式大小\（n ^ {O（\ log n）} \）和O（polylog（n））深度的均匀电路。而且，电路的深度可以减小到O（log ² n），如果特征字段为零。对于二分式完美匹配问题，这得出了相似的结果。我们的主要技术贡献是使两个拟阵的公共碱基族的隔离引理非随机化。当给定矩阵的形式为\（A_ {0} + A_ {1时， 我们使用隔离结果给出一种针对爱德蒙兹问题特殊情况的准多项式时间黑盒算法，即符号矩阵的奇异性测试} x_ {1} + \ cdots + A_ {m} x_ {m} \），对于任意矩阵A ₀和1级矩阵\（A_ {1}，A_ {2}，\ dots，A_ {m} \）。这也可以视为对应行列式多项式的黑盒多项式身份测试算法。该结果的另一个结果是对最大秩矩阵完成问题的确定性解决方案。最后，我们使用我们的结果找到准NC中线性拟阵的并集的确定性表示。