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.

Highlights

  • 1.1 From the bipartite perfect matching problem to the commutative and non-commutative rank problemsGiven a bipartite graph G = (L ∪ R, E) where |L| = |R|, the celebrated Hall’s marriage theorem states that G has a perfect matching if and only if G has no shrunk subsets: S ⊆ L is called a shrunk subset, if |N (S)| < |S| where N (S) denotes the set of neighbours of S.Consider the following linear algebraic analogue of the bipartite perfect matching problem

  • Each edge may be viewed as a “partial function” from the left vertex set to the right vertex set

  • In the linear algebraic setting we shall think of one linear map from U to V as one edge

Read more

Summary

Introduction

1.1 From the bipartite perfect matching problem to the commutative and non-commutative rank problems. While perfect matchings and shrunk subsets are two sides of the same coin for bipartite graphs, non-singular matrices and shrunk subspaces are not in the linear algebraic setting. This gives rise to two natural algorithmic problems. While these two problems are different in general, as suggested by the 3×3 skew-symmetric matrix space, they do coincide in some special cases For this let us recall how the bipartite perfect matching problem and the linear matroid intersection problem can be cast as special instances of both problems. Using Edmonds’ matroid intersection theorem, Lovász showed that in this case the commutative rank and the non-commutative rank of B coincide, and both are equal to the matroid intersection number

Backgrounds to the commutative and non-commutative rank problems
Certifying matrix spaces without shrunk subspaces
Deterministic efficient algorithms for the non-commutative rank problem
Over small finite fields
A technical improvement of the regularity lemma
Proof of the main theorem
The algorithm for the main theorem
Proof of Corollary 8: the case of small finite fields
The full constructive regularity lemma
A Constructivizing the result of Derksen and Makam
B Efficient construction of cyclic field extensions of arbitrary degrees
Full Text
Paper version not known

Talk to us

Join us for a 30 min session where you can share your feedback and ask us any queries you have

Schedule a call

Disclaimer: All third-party content on this website/platform is and will remain the property of their respective owners and is provided on "as is" basis without any warranties, express or implied. Use of third-party content does not indicate any affiliation, sponsorship with or endorsement by them. Any references to third-party content is to identify the corresponding services and shall be considered fair use under The CopyrightLaw.