Non-commutative Edmonds’ problem and matrix semi-invariants

TLDR: This paper considers the non-commutative version of Edmonds’ problem: compute the rank of T over the free skew field by using an algorithm of Gurvits, and assuming the above bound of sigma for R(n, m) over Q, deciding whether or not T has non-Commutative rank < n over Q can be done deterministically in time polynomial in the input size and $$sigma}$$σ.

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 Semin 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 (Hrubes 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: