Non-commutative Edmonds' problem and matrix semi-invariants

In 1967, 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. It is known that this problem relates to the ring of matrix semi-invariants. In particular, if the nullcone of matrix semi-invariants is defined by elements of degree $\leq \sigma$, then there follows a $\mathrm{poly}(n, \sigma)$-time randomized algorithm to decide whether the non-commutative rank of $T$ is $<n$. 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, 2001). In this article we prove the following results: (1) We observe that by using an algorithm of Gurvits, and assuming the above bound $\sigma$ for $R(n, m)$ over $\mathbb{Q}$, deciding whether $T$ has non-commutative rank $<n$ over $\mathbb{Q}$ can be done deterministically in time polynomial in the input size and $\sigma$. (2) When $\mathbb{F}$ is large enough, we devise a deterministic algorithm for non-commutative Edmonds' problem in time polynomial in $(n+1)!$, with the following consequences. (2.a) If the commutative rank and the non-commutative rank of $T$ differ by a constant, then there exists a randomized efficient algorithm that computes the non-commutative rank of $T$. (2.b) We prove 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.