We extend the techniques developed in [IQS17] 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 [IQS17] 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 and Makam [DM17b], who were the first to observe that the blow-up parameter can be controlled. Both methods rely crucially on the regularity lemma from [IQS17]. In this note we improve that lemma by removing a coprime condition there.

In 1967, J. Edmonds introduced the problem of computing the rank over the rational function field of an n × 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 e.g. the free skew field itself (Cohn 1973), matrix spaces of low rank (Fortin-Reutenauer, 2004), Edmonds' original problem (Gurvits, 2004), and more recently, non-commutative arithmetic circuits with divisions (Hrubeš and Wigderson, 2014).It is known that this problem relates to the following invariant ring, which we call the F-algebra of matrix semi-invariants, denoted as R(n, m). For a field F, it is the ring of invariant polynomials for the action of SL(n,. 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 ≤ σ, then there follows a poly(n, σ)-time randomized algorithm to decide whether the noncommutative rank of T is full. To our knowledge, previously the best bound for σ was O(n 2 · 4 n 2 ) over algebraically closed fields of characteristic 0 (Derksen, 2001). We now state the main contributions of this paper:• We observe that by using an algorithm of Gurvits, and assuming the above bound σ 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 σ.• When F is large enough, we devise an algorithm for the non-commutative Edmonds problem 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, then there exists a randomized efficient algorithm to compute the non-commutative rank of T . This improves 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 σ ≤ (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 σ for matrix semi-invariants over fields of positive characteristics. Furthermore, this does not require F to be algebraically closed.

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