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Constructive non-commutative rank computation is in deterministic polynomial time

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Abstract

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.

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Authors and Affiliations

  1. Institute for Computer Science and Control, Hungarian Academy of Sciences, Budapest, Hungary

    Gábor Ivanyos

  2. Centre for Quantum Software and Information, University of Technology Sydney, Sydney, Australia

    Youming Qiao

  3. Chennai Mathematical Institute, Chennai, India

    K. V. Subrahmanyam

Authors
  1. Gábor Ivanyos
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  2. Youming Qiao
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  3. K. V. Subrahmanyam
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Correspondence to Youming Qiao.

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Ivanyos, G., Qiao, Y. & Subrahmanyam, K.V. Constructive non-commutative rank computation is in deterministic polynomial time. comput. complex. 27, 561–593 (2018). https://doi.org/10.1007/s00037-018-0165-7

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  • DOI: https://doi.org/10.1007/s00037-018-0165-7

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