  We extend our techniques developed in our earlier paper appeared in
Computational Complexity, 2017 (preprint: arXiv:1508.00690) to obtain a
deterministic polynomial time algorithm for computing the non-commutative rank
together with certificates of linear spaces of matrices over sufficiently large
base fields.
  The key new idea is a reduction procedure that keeps the blow-up parameter
small, and there are two methods to implement this idea: the first one is a
greedy argument that removes certain rows and columns, and the second one is an
efficient algorithmic version of a result of Derksen and Makam. Both methods
rely crucially on the regularity lemma in our aforementioned paper, and in this
manuscript we also improve that lemma by removing a coprime condition there.

Computational Complexity;Data Structures and Algorithms;Commutative Algebra;Representation Theory