Constructive non-commutative rank computation is in deterministic polynomial time

We extend the techniques developed in Ivanyos et al . (Comput Complex 26(3):717–763, 2017 ) to obtain a deterministic polynomial-time algorithm for computing the non-commutative rank of linear spaces of matrices over any field. The key new idea that causes a reduction in the time complexity of the algorithm in Ivanyos et al . ( 2017 ) from exponential time to polynomial time 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 & Makam (Adv Math 310:44–63, 2017b ), who were the first to observe that the blow-up parameter can be controlled. Both methods rely crucially on the regularity lemma from Ivanyos et al . ( 2017 ). In this note, we improve that lemma by removing a coprime condition there.