computational complexity

, Volume 26, Issue 3, pp 717–763 | Cite as

Non-commutative Edmonds’ problem and matrix semi-invariants

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

Abstract

In 1967, J. Edmonds introduced the problem of computing the rank over the rational function field of an \({n \times n}\) matrix T with integral homogeneous linear polynomials. In this paper, we consider the non-commutative version of Edmonds’ problem: compute the rank of T over the free skew field. This problem has been proposed, sometimes in disguise, from several different perspectives in the study of, for example, the free skew field itself (Cohn in J Symbol Log 38(2):309–314, 1973), matrix spaces of low rank (Fortin-Reutenauer in Sémin Lothar Comb 52:B52f 2004), Edmonds’ original problem (Gurvits in J Comput Syst Sci 69(3):448–484, 2004), and more recently, non-commutative arithmetic circuits with divisions (Hrubeš and Wigderson in Theory Comput 11:357-393, 2015. doi:10.4086/toc.2015.v011a014).

It is known that this problem relates to the following invariant ring, which we call the \({\mathbb{F}}\)-algebra of matrix semi-invariants, denoted as R(n, m). For a field \({\mathbb{F}}\), it is the ring of invariant polynomials for the action of \({{\rm SL}(n, \mathbb{F}) \times {\rm SL}(n, \mathbb{F})}\) on tuples of matrices—\({(A, C)\in {\rm SL}(n, \mathbb{F}) \times {\rm SL}(n, \mathbb{F})}\) sends \({(B_{1}, \ldots, B_m)\in M(n, \mathbb{F})^{\oplus m}}\) to \({(AB_1 {C}^{{\rm T}}, \ldots, AB_m {C}^{\rm T})}\). Then those T with non-commutative rank <  n correspond to those points in the nullcone of R(n, m). In particular, if the nullcone of R(n, m) is defined by elements of degree \({\leq \sigma}\), then there follows a \({{\rm poly}(n,\sigma)}\)-time randomized algorithm to decide whether the non-commutative rank of T is full. To our knowledge, previously the best bound for \({\sigma}\) was \({O(n^{2}\cdot 4^{n^2})}\) over algebraically closed fields of characteristic 0 (Derksen in Proc Am Math Soc 129(4):955–964, 2001).

We now state the main contributions of this paper:
  • We observe that by using an algorithm of Gurvits, and assuming the above bound \({\sigma}\) for R(n, m) over \({\mathbb{Q}}\), deciding whether or not T has non-commutative rank < n over \({\mathbb{Q}}\) can be done deterministically in time polynomial in the input size and \({\sigma}\).

  • When \({\mathbb{F}}\) is large enough, we devise an algorithm for the non-commutative Edmonds problem which runs in time polynomial in (n + 1)!. Furthermore, due to the structure of this algorithm, we also have the following results.
    • If the commutative rank and the non-commutative rank of T differ by a constant there exists a randomized efficient algorithm to compute the non-commutative rank of T. This improves upon a result of Fortin and Reutenauer, who gave a randomized efficient algorithm to decide whether the commutative and non-commutative ranks are equal.

    • We show that \({\sigma\leq (n+1)!}\). This not only improves the bound obtained from Derksen’s work over algebraically closed field of characteristic 0 but, more importantly, also provides for the first time an explicit bound on \({\sigma}\) for matrix semi-invariants over fields of positive characteristics. Furthermore, this does not require \({\mathbb{F}}\) to be algebraically closed.

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 2016

Authors and Affiliations

  • Gábor Ivanyos
    • 1
  • Youming Qiao
    • 2
  • K. V. Subrahmanyam
    • 3
  1. 1.Institute for Computer Science and Control, Hungarian Academy of SciencesBudapestHungary
  2. 2.Centre for Quantum Computation and Intelligent SystemsUniversity of Technology SydneySydneyAustralia
  3. 3.Chennai Mathematical InstituteChennaiIndia

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