Constructive non-commutative rank computation is in deterministic polynomial time

  • Gábor Ivanyos
  • Youming Qiao
  • K. V. Subrahmanyam
Article
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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.

Keywords

Edmonds’ problem symbolic determinant identity test semi-invariants of quivers non-commutative rank 

Subject classification

13A50 68W30 

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Copyright information

© Springer International Publishing AG, part of Springer Nature 2018

Authors and Affiliations

  • Gábor Ivanyos
    • 1
  • Youming Qiao
    • 2
  • K. V. Subrahmanyam
    • 3
  1. 1.Institute for Computer Science and ControlHungarian Academy of SciencesBudapestHungary
  2. 2.Centre for Quantum Software and InformationUniversity of Technology SydneySydneyAustralia
  3. 3.Chennai Mathematical InstituteChennaiIndia

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