2015/12/11 by Ivanyos, Gábor, Qiao, Youming, Subrahmanyam, K. V.
#Commutative Algebra (math.AC) #Computational Complexity (cs.CC) #Data Structures and Algorithms (cs.DS) #FOS: Computer and information sciences #FOS: Mathematics #Representation Theory (math.RT)
paper · doi:10.48550/arxiv.1512.03531
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.