Non-commutative Edmonds’ problem and matrix semi-invariants

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.